Con tents
Pr eface
11
Introduction 1-1 Bas ic Defin itions 13 1-2 Con version of El ectric Energy by the Tr an sforme r 15 El ectromechanical Energy Con versio n by an 1-3 Electrical Machin e 18 1-4 Functiona l Classifi cation of Elect romagnetic En er gy Conve r t ing Devices 24
13
1 Chapter
1
1-1 1-2 Chapter
2
2-1 2-2
2-3 2-4 2-5
2-6 2-7 2-8
Chapter
3 3-1 3-2 3-3
Transformers
An Outline of 'I'ransiormers P urpose, Applications, Ratings 27 Const ru ction of a Transformer 31 Elec tromagne tic Processes in the Transformer a t No-Load Th e No-Load Condition 43 Voltage Equations 45 Vari ations in EMF with Time. An EMF Eq uation 46 Th e Magnetization Curve of the Transforme r Th e No-Load Cur ren t W aveform 49 T ra nsformer Eq ua tio ns a t No-Lo ad in Compl ex Form .50 No-Loa d Losses 52 The Effect of the Core Loss on t he T ransformer 's Pe rform ance at No-Loa d 53
27
43
47
El ectromagnetic Pro cesses in the Transformer on Load 56 The Magnetic Field in a T ran sform er on Load. Th e MMF Equati on. Th e Leakage Inductance of th e Windings 56 Voltage E qua ti ons of the Tr an sform er Windings 60 Transferring t he Secondar y Quan tities to t he Primary Sid e 6? . . .
6
Contents 3-4 3-5 3-6 3-7 3-8
Chapter
4 4-1 4-2 4-3 4-4
The Phasor Diagram of a Transformer 65 The Equivalent Circuit of the Transformer 68 The Per-Unit Notation 69 The Effect of Load Variations on the Transformer 72 Energy Conversion in a Loaded Transformer 75 Transformation of Three-Phase Currents and Voltages 79 Methods of Three-Phase Transformation. Winding Connections 79 A Three-Phase Transformer on a Balanced Load 83 Phase Displacement Reference Numbers 84 The Behaviour of a Three-Phase Transformer During Magnetic Field Formation 89
Chapter
5 5-1 5-2
Measurement of Transformer Quantities The Open-Circuit (No-Load) Test 99 The Short-Circuit Test 102
Chapter
6 6-1
Transformer Performance on Load 106 Simplified Transformer Equations and Equivalent Circuit for 11» 1 0 106 Transformer Voltage Regulation 107 Variations in Transformer Efficiency on Load 111
6-2 6-3 Chapter
7
7-1
7-2
Chapter
8 8-1 8-2
Chapter
9 9-1 9-2
99
Tap Changing Off-Load Tap Changing 113 On-Load Tap Changing 114
113
Calculation of Transformer Parameters No-Load (Open-Circuit) Current and Mutual Impedance 117 Short-Circuit Impedance 119
117
Relationship Between Transformer Quantities and Dimensions Variations in the Voltage, Current, Power and Mass of a Transformer with Size 121 Transformer Losses and Parameters as Functions of Size 123
121
125
Chapter 10 10-1 10-2
Multiwinding Transformers. Autotransformers Multiwinding Transformers 125 Autotransformers 133
Chapter 11 11-1 11-2
Transformers in Parallel 138 Use of Transformers in Parallel 138 Procedure for Bringing Transformers in for Parallel Operation 139 Circulating Currents due to a Difference in Transformation Ratio 141 Load Sharing Between Transformers in Parallel 14:/
11-3
tH
7
Contents Chapter 12 12-1 12-2 12-3 12-4 12-5 12-6 12-7
Three-Phase Transformers Under Unbalanced Load 145 Causes of Load Unbalance 145 Transformation of Unbalanced Currents 146 Magnetic Fluxes and EMFs under Unbalanced Load Conditions 151 Dissymmetry of the Primary Phase Voltages under Unbalanced Load 154 Dissymmetry of the Secondary Voltages under Unbalanced Load 156 Measurement of the ZPS Secondary Impedance 160 Single- and Two-Phase Unbalanced Loads 161
Chapter 13 13-1 13-2
Transients in Transformers Transients at Switch-On 164 Transients on a Short-Circuit Across the Secondary Terminals 167
164
Chapter 14 14-1 14-2
Overvoltage Transients in Transformers Causes of Overvoltages 171 The Differential Equation for the Initial Voltage Distribution in the Transformer Winding 172 Voltage Distribution over the Winding and Its Equalization 175
171
Special-Purpose Transformers General 177 Three-Phase Transformation with Two Transformers 177 Frequency-Conversion Transformers 178 Variable-Voltage Transformers 179 Arc Welding Transformers 180 Insulation Testing Transformers 181 Peaking Transformers 182 Instrument Transformers 182
177
Heating and Cooling of Transformers Temperature Limits for Transformer Parts under Steady-State and Transient Conditions 184 Transformer Cooling Systems 186
184
14-3 Chapter 15 15-1 15-2 15-3 15-4 15-5 15-6 15-7 15-8 Chapter 16 16-1 16-2 Chapter 17 17-1 17-2 17-3
2 Chapter
18
189 Transformers of Soviet Manufacture USSR State Standards Covering Transformers 189 Type Designations of Soviet-made Transformers 190 Some of Transformer Applications 191 A general theory of electromechanical energy conversion by electrical machines
E1eclromechanical ~a.chjn!lS
Processes
in
Electrical
Contents
8 18-1 18-2 Chapter 19 19-1 19-2 19-3 19-4 19-5 Chapter 20 20-1 20-2 20-3 20-4
Chapter
21 21-1 21-2
Chapter 22 22-1 22-2 22-3 22-4 22-5 22-6 22-7 22-8 Chapter 23 23-1 23-2 23-3 23-4
Classification of Electrical Machines 192 Mathematical Description of Electromechanical Energy Conversion by Electrical Machines 195 Production of a Periodically Varying Magnetic Field in Electrical Machines 201 A Necessary Condition for Electromechanical Energy Conversion 201 The Cylindrical (Drum) Heteropolar Winding 202 The Toroidal Heteropolar Winding 206 The Ring Winding and a Claw-Shaped Core 206 The Homopolar Ring Winding and a Toothed Core 206 Basic Machine Designs Modifications in Design 207 Machines with One Winding on the Stator and One Winding on the Rotor 211 Machines with One Winding on the Stator and Toothed Rotor and Stator Cores (Reluctance Machines) 214 Machines with Two Windings on the Stator and Toothed Cores for the Stator and Rotor (Inductor Machines) 218 Conditions for Unidirectional Energy Conversion by Electrical Machines The Single-Winding Machine 227 Two-Winding Machines 230
207
227
Windings for A. C. Machines 235 Introductory Notes 235 The Structure of a Polyphase Two-Layer Winding 235 Connection of Coils in a Lap Winding. The Number of Paths and Turns per Phase 240 Coil Connection in the Wave Winding 244 The Selection of a Winding Type and Winding Characteristics 246 A Two-Pole Model of a Winding. Electrical Angles between Winding Elements 247 Two-Layer, Fractional-Slot Windings 250 Field Windings 255 Calculation
257
Contents Chapter 24 24-1 24-2 24-3 24-4 24-5 24-6 Chapter 25 25-1 25-2 25-3 25-4 25-5 25-6 Chapter 26 26-1 26-2 26-3 Chapter 27 27-1 27-2 27-3 27-4 27-5 27-6
Cha pter 28 28-1 28-2 28-3 28-4
The Mutual Magnetic Field of a Phase W inding and Its Elements 267 The Magneti c Field and MMF due to a Basic Set of Currents 267 The Effe ct of Core Sali ency. The Carter Coefficien t 270 The MMF due to a Basic Coil Set 272 Expansion of th e Periodic MMF du e to a Basic Coil Set into a Fourier Series. Th e Pitch Factor 276 The Phase MMF. The Distribution Fa ctor 280 Pulsating Harmonics of the Phase MMF 287 The Mutual W~~
Magnetic Field of a Polyphase ~
Presentation of th e Pu lsating Harmonics of the Phase MMF as the Sum of Rotating MMFs 288 Presentation of Pha se MMF H armonics as Complex Time-Space Functions 291 Time and Spa ce-Time' Complex Quantities and Functions of the Quantities Involv ed in Opera tion of a Polyphase Machine 294 Th e MMF of a Pol yph ase W inding , Its R otating Harmonics 297 The Fundamental Component of the Magn etic Flux Density in a Polyphase Winding (the Rotating Field) 303 Magn etic Flux Density Harmonics in th e Rotat ing Magneti c Fi eld of a Pol yphase Winding 306 The Magnetic Field of a Ro lating Field Winding 316 The Magneti c Field of a Concentrated Fie ld Winding 316 Th e Magnetic Field of a Distributed Fie ld Winding 319 The Rotatin g Harmonics of th e Excitation Fie ld 321
Flux Linkages of an d EMFs Induced by Rotati ng F ields 323 Introductory Not es 323 Th e Flux Linkage and EMF of a Coil 323 Th e Flux Linkage and EMF of a Coil Group 328 The Flux Linkage and EMF of a Phase 330 Th e Flux Linkages and EMFs of a Pol yphase Winding . A Space-Time Diagram of Flux Lin ka ges an d EMFs 333 Th e Flux Linkages and ' EMFs du e to th e Harmonics of a Nonsinusoidal Rotating Magnetic Fielrj 3/15 The In du ctances of Polyphase Win dings 341 Th e Useful Fi eld and the Leaka ge Field 341 The Main Self-Induc tan ce of a Phase 342 The Main Mutual Inductance Between the Phases 343 The Main Mutual Inductance Between a Stator Ph ase and a Ro tor Phase 341
10
Contents 28-5 28-6 28-7
Chapter 29 29-1 29-2 29-3
Th e Main Self-Inductance of the Complet e Winding 345 Th e Main Mutual Induct ance betwe en a Primar y Ph ase and th e Second ar y Wi n ding 347 Th e Leakage Inductance of th e Complete W inding 348 Thc Elcctromagnctic Torque The Torque E xpressed in Ter ms of Vari ation s in th e Energy of th e Magnetic Field 351 Th e El ectromagneti c Torque Expressed in Terms of El ectroma gnetic Forces 358 Electromagnetic Force Distribution in a Wound Slot 367
351
Chapter 30 30-1 30-2
Energy Conversion by a Hota ti ng Magnetic Field 372 El ectromagnetic, El ect ri c and Magnetic Power 372 Energy Conversi on in an El ectrical Machine and Its Model 376
Chap ter 31 31-1 31-2 31-3 31-4
Energy Conversion Losses and Efficiency Introductory Notes 379 Electrica l Losses 380 Magnetic Losses 387 Mechanical Los ses 396
379
Bi bliog raphy
397
Index
399
Preface
The subject matter in the text is presented in the sequence traditionally followed in the Soviet Union. It starts with transformers, passes on to induction and synchronous machines, d.c. machines, and concludes with a.c, commutator machines. Separate chapters are devoted to a general theory of electrical machines, machine design and engineering, and transients in electrical machinery. The electromagnetic processes that take place in electrical machines are examined from the view-point of electromechanical and mechanoelectrical energy conversion . With such an approach, it has been possible to extend the mathematics used to both conventional and any other conceivable types of electrical machines. In addition to electromagnetic processes, consideration is given to the thermal, aerodynamic, hydraulic and mechanical processes associated with electromechanical and mechanoelectric energy conversion. In view of the importance attached to the above accompanying processes, the text discusses general aspects of machine design and engineering. The chapters on transients are based on the theory of a generalized machine. The material includes the derivation of differential equations for induction and synchronous machines in terms of the d, q, 0 and the el, B, 0 axes, and their transformation to a form convenient for computerassisted analysis and design. The chapters dealing with specific types of machine (induction, synchronous, d.c.) are largely concerned with the conventional design. In each case, however, there is a short discourse on the operating principle and arrangement of the most commonly used special-purpose modifications. The electromagnetic processes occurring in conventional a.c. machines are described in terms of the resultant complex functions of electric-circuit parameters or their projections on the axes of a complex plane. As far as pr acticable, il unified or generalized approach has been taken to deve-
---~---------12
Preface
loping equations for and describing the physical processes in the two basic types of machine-the induction machine and the synchronous machine . This concerns electromagnetic torque, electromagnetic active and reactive power , saturable magnetic circuits, machine inductances, etc. More space is given to thyristor-controlled machines gaining an ever-wider ground, than to a .c. collector machines used on a limited scale . In the ligh t of new findings, the effect of core saliency on the harmonics of the airgap flux density has been treated in a more rigorous form . A nove l approach has been taken towards the equations for mmfs, emfs , electromagnetic forces, electromagnetic torque, and machine characteristics . Among other things, the equations for the synchronous salient-pole machines are developed in terms of the d, q, 0 axes, the analysis of transients includes the short-circuit condition in the synchronous generator, the starting of the induction motor, and events in the single-phase motor . The material marked with an asterisk (*) may be omitted on first read ing, without disrupting the integrity of the exposition. A. V. Ivanov-Smolensky
Introd uction
1.1
Basic Definitions
The utilization of natural resources inevitably involves the conversion of energy from one form to another. Quite aptly, devices doing this job by performing some mechanical motion may be called energy converting machines. For example, heat engines convert the heat supplied by the combustion of a fuel into mechanical energy. In fact, the same name goes for devices converting energy in one form into energy of the same form but differing in some parameters. An example is a hydraulic machine which converts the mechanical energy of a reciprocating fluid flow into mechanical energy further transmitted by a rotating shaft. A sizeable proportion of the energy stored by nature in chemical compounds, the atoms and nuclei of substances, the flow of rivers, the tides of seas, the wind, and solar radiation is now being converted to electric energy. This form of conversion is attractive because electricity can in many cases be transmitted over long distances, distributed among consumers and converted back to mechanical, thermal, or chemical energy with minimal losses. However, at present thermal, chemical or nuclear energy is converted directly to electricity on a very limited scale, because this still involves heavy capital investments and is wasteful of power. Rather, any form of energy is first converted to mechanical by heat or water machines and then to electricity. The final step in this sequence-conversion of mechanical energy to electricity or back-is done by electrical machines. From other electromechanical energy converting devices, electrical machines differ in that, with a few exceptions, they convert energy in one direction only and continuously. An electrical machine converting mechanical energy to electricity is called a generator. An electrical machine performing the reverse conversion is called a motor. In fact, a generator can be made to operate as a motor, and a motor as a generator-they are reversible. If we apply mechanical energy to the movable member of an electrical
c
14
Introduction
machine, it will operate as a generator; if we apply electricity, the movable member of the machine will perform mechanical work. Basically, an electrical machine is an electromagnetic system consisting of a magnetic circuit and an electric circuit coupled with each other. The magnetic circuit is made up of a stationary and a rotating magnetic member and a nonmagnetic air gap to separate the two members. The electric circuit can be in the form of one or several windings which are arragned to move relative to each other together with the magnetic members carrying them. For their operation, electrical machines depend on electromagnetic induction and utilize the electromotive forces (emfs) that are induced by periodic variations in the magnetic field as the windings or magnetic members are rotated. For this reason, electrical machines may be called electromagnetic. This also applies to devices that convert electric energy at one value of current, voltage and/or frequency to electric energy at some other value of current, voltage and/or frequency. The simplest and most commonly used electromagnetic energy conversion device which converts alternating current at one voltage to alternating current at some other voltage is the transformer. Its coils and core remain stationary relative to each other, and periodic variations in the magnetic field essential for an emf to be induced in the coils are produced electrically rather than mechanically. Electromagnetic energy converting devices with moving or, rather, rotating parts are more customarily called rotary converters. They do not differ from electrical machines in either design or the principle of operation. In fact, rotary converters can sometimes double as electric-to-mechanical (or mechanical-to-electric) energy converting machines. Therefore, we may extend the term "machine" to transformers and rotary converters as special kinds of electrical machine. Apart from electromagnetic electrical machines, some special applications involve the use of electrostatic machines in which the electromechanical conversion of energy is based on electrostatic induction and utilizes periodic variations in the electric field of a capacitor in which the plates are free to move relative to one another. However, electrostatic machines are no match for electromagnetic machines in terms of size, weight and cost, and are not used in commercial ' or industrial applications.
15
1-2 Conversion of Energy by Tra nsformer
As energy converters, electric al m achines are import an t elements in any power-generating, power-consuming, or in dustrial installation . They are widely used as generators" mot ors, or rotary converters at electric power stations, factories, farms, railways, automobiles , and aircraft. They are finding an ever increasing use in automatic control systems. Electrical machines are clas sed into alternating-current (a.c.) and direct-current (d .c.) machines, according as they operat e into or from an a.c. or a d.c. supply line. 1-2
Conversion of Electric Energy by the Transformer
In sket ch form , the arrangeme nt of a simple single-phase two-winding transforme r is shown in Fig . 1-'1. As is seen , it consists of two windings , land 2, wi th turns WI and W 2 ,
V2
F ig. I-I Electric and magnetic cir cuits of a tra nsformer
which are wound on a magnetic core . For better coupling between t he coils, the core is assembled from laminat ions punched in electric-sheet steel having a high relative permeability, !-tT' with no air gap left around the magnetic circuit. The laminations or punchings are made th in in order to reduce the effect of edd y currents on the ma gnetic field which altern ates at an angular frequency co , Let us open , say, coil 2 and connect coill t o a source of a sinusoidal alternating current of frequency f = Ul/2n and of voltage VI = V 2 VI cos Ult, where VI is the rms value of voltage. This will give rise to an alternating current, i1 = io, in the coil , which can be found from the voltage equation for the '
16
Introduction
circuit where it is flowing: VI
where R 1
=
- 81
+ R io
(I-i)
1
= resistance of winding 1
8 1 = -d'P'11/dt = emf of self-induction 1V11 = w 1c!) = flux linkage cD = BA c = magnetic flux B = magnetic induction (magnetic flux density) A c = cross-sectional area of the core. On setting flr constant and applying Ampere's circuital law to the magnetic circuit
~ HI
dl
=
~
(B/flrflo) dl = cD/AIJ. =
iOW1
(1-2)
where Aj.1 = ~lrfloAc/lc is the permeance of the core and lc is the mean core length, it is an easy matter to find the inductance of winding 1 L ll = w1 CP/i o = w:Aj.1 " nd t he mutual inductance L 12
=
wlIJ/i o
=
WIW2A~~
and to express in their terms the flux linkage and the emf 81 =
-L 11 dio/dt
Using Eq. (I-i) and neglecting R 1 i o, we obtain the magnetizing current io =
V2 locos (rot '-- n/2)
which produces an alternating magnetic flux cD = i Ow 1 A j.1 Variations in the flux cD linking coil 2 induce in the latter a sinusoidal emf of mutual induction 82
= -d1V 21/dt = -L 12 dio/dt
Thus, coil 2 can be used as a source of an alternating current of the same frequency t. but at another voltage, V 2 = 8 2, As is seen, the ratio of the instantaneous and the rms emfs across windings 1 and 2 and of the respective rms vol-
17
1-2 Conversion of Energy by Transformer
tages , is equal to the turns, or transformation, ratio: el/e 2 = E 1/E 2 = V 1/V2 = WI/W 2 (1-3) If VI is specified in advance, we may use Eq. (1-3) to find the turns numbers WI and W 2 such that V 2 will always have the desired value. Winding 2 can be used as an a .c. source by connecting it across a load resistance, R L . Then the emf e 2 will induce in it a sinusoidal alternating current i 2 = e2/(R L + R 2 ) which can be found from the voltage equation for the circuit thus formed * (1-4) where V 2 = R L i 2 • The secondary current i 2 will bring about a proportionate change in t he primary current i l . The relationship between i l and i 2 can be established by again using Ampere's circuital law written by analogy with Eq. (1-2) and recalling that in a transformer under load both windings contribute to the magnetic flux
~ Hz dl = (J)/A)J, =
i 1wI
+ izw z
(1-5)
Also, in writing the voltage equation for the circuit containing coil 1 (1-6) and neglecting R1i l as in Eq. (I-i), we find that under load the emf e l remains about the same as when coil 2 is opencircuited. This implies that el is induced by variations in the same flux (J) and in the same magnetizing current i o in coil 1, as exi st when coil 2 is open-circuited. If so, we may equa te the right-hand sides of Eqs. (J-f ) and (1-5) and argue that the sum of the magnetomotive forces in coils 1 and 2 is equal to the mmf due to the magnetizing current i o in coil 1 (1-7) In an adequately loaded transformer with a closed (noairgap) core, iOw l is negligible
I iOwl I ~ I i1w1 I ~ I i 2w2 I * Here and in Eq. (1-6), the emfs induced by leakage fluxes are not included. 2- 0 16 9
18
Introduction
So, without introducing an appreciable error, we may set iOWI
= 0
On this assumption, the directions of currents in the windings are such that their mmfs balance each other: (1-8) It follows from Eq. (1-8) that the ratio of the absolute values, [ r ], and of the rms values, I, of the currents in coils land 2 are inversely proportional to their turns ratio (1-9) Using Eqs. (1-3), (1-4), (1-6), and (1-8) and neglecting the losses associated with the cyclic magnetization of the core and with variations in the energy of the magnetic field, let us consider the balance of the instantaneous powers in the transformer. The power delivered to coil 1 by the supply line is PI = vIiI = -eli l iiRI
+
Some part of this power, iiRI, is dissipated as heat in coill, and the remainder, -eli l = ezi z, is transferred by the electromagnetic field into coil 2. The power supplied to coil 2
is partly dissipated as heat (i~R2)' whereas the remainder, vzi z, is delivered to the load. 1-3
Electromechanical Energy Conversion by an Electrical Machine
In sketch form, the arrangement of a simple rotating electrical machine is shown in Fig. 1-2. As is seen, it consists of a stationary member called the stator, and a rotating member called the rotor. The stator core, 4, is made fast to a base-plate, whereas the rotor core, 3, is mounted on a shaft carried in bearings, so that it is free to rotate, remaining aligned with the axis of the stator. On its cylindrical surface, the rotor core 3 has slots which receive a single-coil rotor winding, L, with turns WI' The stator core has similar slots which receive a single-coil stator winding with turns W 2.
19
1-3 Energy Convers ion by Electri cal Machine
The stator and rotor cores are assemb led from ringshaped laminat ions .py nched in electrical-s.heet ste~l hav ing a high parmeahil it y for better magnetic coupling between the windings. For the same purpose, t he coils are sunk in slots rather than put on the outer surface of the
rem Fig. 1-2 Electric and magnetic! circuits! of a simple electrical rria~ chin e Irr[the generating mode (il > 0, T em < 0) . . .
cores . With this arrangement, the air ga p between the stator and rotor may be made very small and the magnetic circuit presents a very low reluctance . The shaft carrying the rotor coup les 'it to another machine with which it exchanges mechanical energy (deliveri ng it in motoring, and receiving it in operation as a generator). The stator and rotor windings are connected to lines with voltages V 2 and VI' respectively. In motoring the li nes (or one of them) deliver electric energy to the machine. In operation as a gene rator, the machine delivers electric energy to the li nes (or one of them ). Electromechanica l energy conversion by an electr ica l machine utilizes the emfs t hat are induced in the windin gs as a result of variations in their relative position in space. To begi n with, su ppose t hat wind ing 2 is energized with i 2 = constant, and winding 1 is open-circuite d, so t hat i l = O. In t he circumstances, a stationary magnetic fiel d is set up , with its north pole, N , l ocat ed in the bottom part , an d the south pole, S, in the top part of the st ator core, 2*
20
Introduction
Assuming that the permeability of the stator and rotor cores, !-La,c is infinitely large in comparison with that of t he air gap, !-Lo (!-La,c ~ !-Lo) , we may neglect the magnetic potential difference across the core. Then, on writing Ampere's circuital law for any loop enclosing the current i 1w 2 in coil 2 (for example, the loop shown by the dashed line in Fig . 1-2)I l
we find the magn etic induction in the air gap due to coil 2 to be B 2 = !-LOi2W2/28
(I-tO)
where «5 is the radial air gap length. The flux linkage 1¥12 of this field with winding 1 varie s with the angle y that it m akes with winding 2. When y = 0, the flux linkage h as a m aximum positive value lf12. m
=
B 21:lw 1
(1-11)
where l is the core length in the axial direction and 1: = scR is the pole pi tch. As the ro tor turns through an angle y anywhere from zero to 180°, the flux linkage varies linearly as a function of the angle y lf 12 = lf 12 , m (1 - 2y/n) (1-12) Wh en y = n/2 , t he flux linkage is zero, lf12 = O. When y = n, it has a m aximum nega tive value , lf 12 = -lf 12. m • As the rotor keeps rotating, the flux linkage builds up linearly as a function of the angle y lf 12
=
-lf12, m (3 -2y/n)
(1-13)
and completes a period of variations when y = 2n. The mutual inductance between the win dings, L 12 = lf12/ i 2, varies in a similar way:
L 12 L 12
= L 1 2 , m (1 - 2y/n) =
-L 1 2 ,m (3 -
2y/n)
for 0 for n
< Y< < y<
rr 2n
(1-14)
where L 12 • m = !-LOW1W2h/2«5 is the maximum mutual inductance between the windings.
1-3 Ene rgy Conversion by Electrical Machine
21
If the rotor is turning with an angular frequency Q, the angle I' = Qt increases linearly, so that the emf induced in winding 1 is given by e1 = -d'¥12/dt = - i 2 dL l 2/dt = - i 2Q dL l2/dl' (1-15) It is called the emf of rotation or the motional emf. As is seen, the motional emf is proportional to the angular displacement, angular frequency and derivative of the mutual inductance with respect to the angular displacement of the rotor. From Eqs. (1-14) and (1-15) it follows that e1 = (2/:rr.) L12 ,mi2Q for 0 < I' < :rr. e1 = -(2/:rr.) L12 ,mi2Q for rr < I' < 2:rr. The sign applies when the emf is in the positive direction of the current in coil 1; the "-" sign applies when it is in the negative direction. The positive directions of currents in windings 1 and 2 are such that the magnetic fields are directed upwards as in Fig. 1-2, with I' = O. Thus, with i 2 held constant, a square emf waveform is induced in winding 1 of the elementary machine. The flux linkage, mutual inductance and emf vary with a period T = 2:rr./Q. Hence, these quantities vary with a frequency given by (I-16) f = Q/2:rr.
"+"
Using Eqs. (1-10) and (1-14), we can express t he motional emf defined by Eq. (1-15) in terms of the magnetic induction B 2 in the air gap e1 = 2B 2luw1 for 0 < I' <:rr. where U = rQ is the tangent velocity at the middle of the air gap. Therefore, the direction of e1 can be determined not only from Eq. (1-15). using Lenz 's rule, but also using the right-hand rule. Of course, both approaches give t he . same result (Fig. 1-2). If we, now , conn ect winding 1 having an internal resistance R 1 across' a load resistance R L , the circuit thus formed will carry an alternating current given by t, = e1/(R L R 1) (1-17) varying with the same frequency f as e1 does. The power generated in winding 1 will then be P,li 1 = - i ,i 2Q (dL I 2/dl') = (VI i 1R 1) i] (1-18)
+
+
:l2
Introduction
Some of this power, iiR I , will be dissipated as heat in winding 1; the remainder PI =
vIiI
= iiRL
will be delivered t o load. The voltage across winding 1, VI
=
ilR L
which is the same as the load voltage, will likewise vary with frequency f. On the assumption that i 2 is constant, winding 2 is energized from a source of d.c . voltage The power
it receives does not undergo electromechanical conversion and is comp letely dissipated as heat . The interaction of the magnetic fields set up by i 2 and i l produces an electromagnetic torque T em acting on the rotor. In determining T em' we may proceed from the fact that the work it performs as the rotor is turned through a sma ll ang le dl' is equal to the change in the energy of the ' magnetic fie ld, dW, caused by a change in the mutual inductance , dL 1 2 , assuming that both t, and t« remain constant, or mathematically T ern
dl' = dW = i Ii 2 dL I 2
H ence, Tern =
i Ii 2 dL 12/dl'
(1-19)
If the angular dis placement of the rotor, d'l" is in the direction of rotation, the torque in Eq . (1-19) acts likewise in the direction of rotation and is posit ive. If dl' is in the opposite direction, the torque in Eq . (1-19) is in the opposite direction, too, and negative. In the generator mode of operation , the torque is, as is shown in Fig . 1-2, negative, T ern < O. Using Eqs . (1-10) and (1-14), we can express the electromagnetic torque in terms of the magnetic induction in the air gap , B 2 , as well:
(I -20)
1-3 Energy Conversion by Electrical Machine
23
The direction of the tangential electromagnetic force, = 2B zli l w l , and of the torque in Eq. (1-20) can be ascertained, using the left-hand rule, as is done in Fig. 1-2 for the generator mode. Under steady-state conditions, when the rotor is spinning at a constant frequency Q, the electromagnetic torque Tern must be balanced by an external (mechanical or load) torque T m (1-21) T m = -T em = - i l i z (dLlz/dy) For this to happen, a mechanical power must be applied to the rotor via its shaft (1-22) T m Q = -ilizQ (dL l2/dl') which is converted to an equal electric power, eli l, given by Eq. (1-18). When y is anywhere between zero and 180°, both i z and i l are positive and dL 12/dy < O. In contrast, when y is anywhere between 180° and 360°, i z is positive, i l is negative, and dL 12/dy> O. Accordingly, the power in Eq. (1-22) is positive, T mQ > 0, not only when the rotor takes up the position shown in Fig. 1-2, but in any other angular position . This implies that an elementary electrical machine can perform electromechanical conversion of energy in one direction only (in our case, it can only operate as a generator). The same elementary machine can operate as a motor, thereby converting electricity to mechanical energy. To this end, winding 1 must be connected to an a.c. supply line of voltage VI and frequency [, so that i l is always in opposition to el (Fig. 1-3). On writing the voltage equation for the circuit thus formed F
VI
=
-e l
+ ilR I
and multiplying it by i. . we obtain the power delivered by the supply line to winding 1: VIiI = -eli l
+ i~Rl
Some of this power, iiRI' is dissipated as heat in winding I» and the remainder -eli l = ilizQ (dLlz/dt) is converted to mechanical power
T em Q = ilizQ (dLl2/dt)
Introduction
24
transmitted by the rotor to the shaft of the driven machine. Using the right- and left-hand rules, it is an easy matter to see that in motoring the torque is positive (T em > 0) and is in the direction of rotation.
Fig. 1-3 Electric and magnetic circuit s of a simple electrical ma chine in the motoring mode (il < 0, Tern> 0)
To sum up, the elementary electrical machine we have examined is reversible-it can operate as both a generator and a motor. This is in fact true of any electrical machine. 1-4
Functional Classification of Electromagnetic Energy Converting Devices
The analysis of simple electromagnetic energy converting devices set forth in Sees . 1-2 and 1-3 shows that transformers and elementary electrical machines can only operate from an a.c. supply line operating at frequency f. If a transformer or an electrical machine is to convert d.c. 'energy , the d.c. supply must first be converted to an a.c. form by a suitable device. This may be a semiconductor device, or a mechanical one as in electrical machines (in the form of a commutator whose bars are connected to the respective coils of the rotating winding, and fixed brushes riding the commutator).
25
1-4 Classification of Electrical Machines Tab le 1-1 Functional Classification of Electromagnetic Energy Conve rting Devices BIocI( d ia gram
Description
Transformer
V
T
D. c.-e-n. C .
Conversion of al terna ting curren tat one voltage to alternating current at another voltage
V
~_f~ ~
--.
A. C.-to-D.C. converter, D. C.-to-A. C. inverter
converter
Function(s) performed
Conversion of d.c., or back
--V,f-
-
f
T
VZ=
f
~
-
A. C. el ect ri cal machine
~ ---- ---
D. C. electrical machine (commutator- or rectifier-type)
~ ---- --
A. C. rotary converter (A. C.-D. C . electrical ma chine)
A. C.-to-D . C. rotary con vertel'
'V f rv
=
'=
2
CD f
o-;--D> ,Q
,Q
a.c. to
Conversion of d .c . at one voltage to d.c . at an other voltage
Conversion m echanical hack)
of a .c, to energy (or
Conversion mechanical hack)
of d ,c . to ener gy (or
Conversion of a.c, at 11 to a.c . at 12 =f= / 1 and to mechanical ene rgy (or in an y other diroc tion)
Conversion of a.c , at II to d.c. or mechanical energy (or in an y other d irec ti on)
26
Introduction Table 1-1 (cont inued)
Descr i p t io n
B l ock d i agr am
D. C. rotary converter o--I--e:t:> 52
F un ction(s ) p erfor m ed
Conversion of d.c, at V~ to d.c . at V2 =FV l and to mechanical energy (or in any other direction)
If we consider an electromagnetic energy converting devi ce in combination with a rectifier as an entity performing a specif ic funct ion, we shall obtain a func tional classification as given in Table 1-1. II:
Transformers
1
A n O utline of Tr a nsformers 1-1
Purpose, Appl ications, Ratings
A transformer is an electromagnetic energy converting device which has no moving parts and two (or more) windings fixed relative to each ot her, int end ed to tra nsfer electric energy between circuits or systems by virtue of electromagnetic induction . El ect ric energy in the form of an alternating current taken from a supply line with m l phases at a phase voltage V l and frequency t, is impressed on the input , or pr imary, winding whence it is transferred by a magnetic field into the output, or secondary , winding with m 2 phases at voltage V 2 and frequency f 2. In most cases , transformers only change voltages, VI =1= V 2 ' or currents, II =1= 1 2 , without affecting freq uency or numb er of phases. As a ru le , there is no conductive connection between the primary and secondary windings, and energy transfer between t hem is only by ind uc t ion ("transformer action"). A transformer having two single- or polyphase windings with no conductive connection between them is termed a two-winding transfo rmer (Figs. l -la and 1-2, respectively). A t ransformer having three or more win ding s (Fig. l -lb ) with no conductive connection between them is called a threeioinding transformer or a mul.tuoinding transformer (see Sec . 10-1) . Standing apart from other transformers is the autotransformer in which some of the energy delivered by a supply li ne is transferre d Lo t he secondary winding conducti vely (see Sec. 10-2) owing to a connection between the primary and secondary sides .
28
Part One. Transformers
v, , l~ 1
2
f \
\
\ r\ ---,
-
.....- r-:-
2
..s-- ' /
4
\1
3
-
it
v Vi
(a
V3
(6)
Fig. 1-1 Single-phase transform ers: (a) two-winding and (b) threewinding: I - primar y winding; 2-secondary winding; 3-secondary winding ; 4-m agneLic circuit (core)
Fig . 1-2 Three-phase, two-winding transformer : I-star-connected three-phase primary coil s; 2-star-connected threephase secondary coils; 3-magneti c circuit (core)
~
II ,II
db. 1 An Outline of Transformers
As already noted, energy supplied by a line is impressed on the primary wi1!-ding which m.ay be sin.gle-.or polyphas~. If energy converSIOn proceeds in the direction shown in Figs. 1-1 and 1-2, the primary windings are those which are labelled by the numeral "1". The secondary windings deliver power to a load line; in Fig. 1-1, windings 2 and 3 are the secondary windings, and in Fig. 1-2 it is winding 2. As is seen, a multiwin~in~ transformer may have several prima~y and secondary windings, For example, the transformer in Fig. 1-1b has two secondary windings, 2 and 3. polyphase windings are formed by star- or delta-connecting the phase windings of which there are as many as are phases in the supply line. Each phase winding is a multiturn coil mounted on a separate limb (or leg) of the transformer core. In terms of phases, there may be single-phase transformers (Fig. 1-1a and b), three-phase transformers (Fig. 1-2), and polyphase transformers. As electric energy converters, transformers have found many uses. Among other things, they are involved in the transmission of power from electric stations to consumers. As often as not, this calls for the voltage to be stepped down or up more than once. Therefore, the overall installed capacity of transformers in present-day electric systems is five to seven times the installed capacity of generators. Apart from transformers and autotransformers used in power transmission and distribution syst ems and referred to as power transformers, wide use is made of transformers intended to transform the number of phases and frequency. Also, special-purpose transformers are used in various industrial installations, communications, radio, television, aut omat ic control, and measurements. Commercially available transformers are made with power ratings from fractions of a volt-ampere to several hundred megavolt-amperes, for voltages from fractions of a volt to several hundred kilovolts, for currents up to tens of kiloamperes, and for frequencies up to several thousand hertz. Among special-purpose transformers are pulse transformers, variable-voltage transformers, stabilized-voltage transformers, etc. (see Chap. 15). Transformers are manufactured to relevant specifications or standards and are designed to perform specific functions. Accordingly , they are rated in terms of frequency, current,
30
Part One. Transformers
vollage, power, or some other values, all of which are called ratings or rated values. They are given on a nameplate attached to each transformer . In this text we shall denote them by the subscript "R" . The voltage rating, or rated voltage, is the line (or lineto -line) voltage as measured across the line terminals of a particular winding, and is designated (in Soviet practice) as Vt, R, line or V 2. R, line. The power rating, or rated power, of a transformer is its total power, which is
8i, R
= V 1, R
l i,R
for a single-phase transformer, and 8 i, R
I
I
=
V3 Vi, R. line]i, R,
line
= 3V i , Rlt, R
for a three-phase transformer* . In a two -winding transformer, the power rating, or rated power, of the primary winding, 8 i , R, is the same as that of the secondary winding, 8 2 , R, and equal to the power rating of the transformer, 81, R = 8 2 , R = 8 . The rated frequency, JR' of a harmonically varying quantity (current or voltage) for general-purpose transformers is 50 Hz in the USSR and 60 Hz in the USA and some other countries . Rated currents are found from the power rating and the rated voltage of the respective winding: I i• R =
8 RIVI , R
for single-phase transformers,
Ii. B , line = 8 R I V3 Vi. B , line for three-phase transformers (line current) and
I~ :
Ii,R
=
8 R /3V i , R
for three-phase transformers (phase current). The nameplate data are not to be understood as a prescription to operate the transformer only at its rated capability . Actually, its secondary current is allowed to vary
* Here and elsewhere in the text, the line quantities have the subscript "line", whereas the phase quantities have no subscript. For example, VI, line is the primary line voltage and VI is the primary phase voltage .
Ch.1 An Outline of Transformers
31
from zero to 1 Z, R , with short-duration overcurrents [13J. Also, app licable standards permit sli ght variations in voltage and frequency . It is to be noted t hat if we hold the primary vol tage constant, the secondary voltage will vary with the magnitude and nature of load and may differ from its value at no-load (open-circuit voltage) , when the secondary current is zero. It would seem that the rated secondary voltage should be taken equal to t hat at the rated power SR' Unfortunately, this voltage depends on the phase relation between the secondary current and voltage . Therefor e, to avoid ambiguity, the rated secondary voltage , Vz,R, is taken to be equal to the no-load (open-circuit) voltage, when the secondary current is zero. Arbitrarily , the rated secondary current is t aken to be equal to that computed from the rated power at the rated secondary voltage : 1 2, R = S RIV2 ,R for single-phase transformers,
1 2• R , line = SRI V3V 2 , R . line for the line cur rent of a three-phase transformer, and 1 2 ,R = S R /3V 2,R for the phase current of a thr ee-phase transformer. A transform er can step up or down the applied voltage. In a step-up transformer, the primary wind ing is on the lowvoltage (LV) side, an d the secondary winding is on the highvoltage (HV) side . In a step-down transformer, they are arranged the other way around. For ex ample, the transformer in Fig. 1-2 will be a step-up one if Vl ,R is lower than V 2,R, or a step-down one if V l, R is higher than V 2,R (the arrows in the figure show the direction of powe r tra nsfer ). 1-2
Construction of a Transformer
(i) The Core and Coils
The actual energy conversion in a t ra nsformer t akes place in its core and coils . For better energy conversion , the coils are placed on, or enclosed in , a magnetic circuit fabricated from a ferromagnetic materia l having a high permeability , [La' which
32
HVL
Part One. Transformers
I,
I
II
'-
Fig. 1-3 Transform er windings: (a) coaxia l and (b) interleaved
sandwich
Fig. 1-4 Two-l ayer cylindrical strip-conductor winding
is hundreds of times that of
~ free space, ~~o (see Figs. 1-1
and 1-2). To have a high permeability, the magnetic circuit ought not to be excessively saturated , and its magnetic induction (magnetic flux density) at a maximum magnetic flux ought not to exceed '1.4 to '1.6 T* . The required reactive power can be reduced by minimizing the leakage !luxes each of which links with only the primary or only the secondary winding. One way to reduce leakage fluxes is t o reduce the gap between the primary and secondary windings. To this end, the primary and secondary coils of a phase are put on the same leg or limb (see Figs. 1-1 and 1-2). The windings may be in the form of cylindrical coils taking up the whole length of, and arranged coaxially on, the limb (Fig . 'i-3a) or as a series of pancake or disc coils with the primary and secondary sections alternating in an interleaved or sandwich arrangement (Fig. 1~3b). Of a larg.e number of various coaxial windings, the cylindric al winding is the simplest (Fig . 1~4). An important aspect in improving the efficiency of energy
* T stands for the tesla, tho unit of magnetic flux density in the International Sys tem (51).Tran slator' s note,
33
Ch. 1 An Outline of Transformers
onversion is to reduce the amount of power lost as heat.
~o this en d, t he win di ngs are m a.d e of a m a terial \~i th a 10:,"
resistance an d a large cross-sect ional ar ea , an d with a n11nimum acce p table turn length. The . m agn etic circ uit is designed so as to kee p eddycurrent and hyst eresis losses to a minimum . Th is is usually done by using magnetically soft , ele ctrical-sheet steels wh ich
(a)
(b )
Fig. 1-5 Si ngle-phase transfo rme rs : (a) core typ e and (b) core-andshell (five-leg cor e) typ e: i -limb (leg); 2-yoke; 3-outer li mb (leg)
have a low hyst eresis l oss and hi gh res istivity, an d assem bling the core fr om indivi du ally insul ated l am inati ons with a thicknes s ch osen such tha t eddy currents would not affect the mai n magne tic fie ld an d woul d not lead to increased eddy-current loss. The lam ination thick ness d depends on the frequency t of t he magnetizing current (see Sec . 31-3) , and is t aken as 0.35 mm 0 1' 0.5 mm for 50 Hz . With a core fabricated as outlined above , t he iron (or core) loss can be kep t at a level comparable with the copper loss, an d the dem agnetizing effect of edd y cur rents can be reduced t o a mi ni mum . Transformer cores mos tly come in anyone of two designs, the core type an d t he shell type. In a core-type single phase transformer, the core cons ists of two vertical li mbs around wh ich the prefo rmed circular windings are pl aced. The win dings cons ist each of two coils which m ay be connected in serie s or par allel an d are pl aced on different limbs . The .top an d bott om memb ers , called the yokes , join the two li mbs in to a closed magnet ic circuit (see Fi g . 1-5a). 3- 0169
34
P art One. Transformers
In a core-type three -phase transformer, a primary and a secondary winding of one phase are wound on each lim b (see Fig. 1-2). The three equal limbs are join ed by the two yokes into a closed magnetic path. In a single-phase shell-type transformer , t he core is divided so that parallel magnetic paths encircle the single group or coils on two sides as if by a shell (see Fig. 1-1). As is seen, the yokes ar e built up to a cross-sectional area half as large as that of the limb carrying the coils . To reduce height and to facilitate transit by rail, highp ower transformers have fiv e-leg core-type circuits calle d
Fig. 1-6 Three-ph ase core-and-shell -t ype transform er : l-limb; 2-yoke; 3- oul er limb
the core-and-shell type in Soviet usage (Figs. 1-5b and 1-6). A core-and-shell transformer is lower in height because the yokes have to carry half as large a flux and may therefore have a lower height, too. As an illustration, Fig. 1-5 shows single-phase transformers of the core and the core-and-shell type of construction having the same power rating. Th e height can be reduced by about the same amount in a threephase core-and-shell transformer (Fig . 1-6) where the yok es have to carry a flux which is 1/rS times that in the limbs. In core-type transformers, the yok es carry the same flux as the limbs do . At t he corners of a core , t he yokes and limbs may be . joined in any one of two manners. One gives butt jo ints; and the other, interleaved (or imbricated) joints. With butt joints, the limbs and the yokes ar e stacked up sep arately, the coils ar e put on the lim bs , and the top yoke
Ch. 1 An Outline of Transformers
35
is then placed on (joined with) the limbs to form a closed magnetic circuit. The butt joints are filled by insulating spacers to avoid eddy currents at those places . The spacers form a virtual air gap which absorbs reactive power over and above that required by the iron itself. Because of this, butt joints are seldom used, although they simplify assembly and disassembly . Interleaved (or imbricatedi joints are used on a wider scale. In this case, the successive layers of laminations in the yokes and limbs are interleaved so as to give an overlap at the corners to reduce the joint reluctance (Fig . 1-7). Even so, the flux has to cross the insulation between the laminations at the overlapped portions, but the virtual air gap thus formed absorbs less reactive power than a core of the buttjoint type. A disadvantage of the interleaved type is that a core already assembled has to be disassembled (unbladed) at the top yoke so as to let the coils be put on the limbs. After that the top yoke is assembled (rebladed) again. Modern electrical-sheet steels disp lay directional (anisotropic) properties produced by cold rolling so that in the direction of rolling they have a reduced specific loss and an increased permeability [131. However, there is an increase in loss and a reduction in magnetic intensity at the joints between the limbs and yokes, where the magnetic lines of force turn through 900 from the direction of rolling . This drawback can to a marked degree be minimized by using mitred joints or overlaps as shown in Fig. 1-8. In low-power, low-voltage transformers, the coils may be wound on rectangular formers and the limbs may be given a rectangular cross-section. In high-power transformers, the coils are wound on a cylindrical mandrel, and the limbs are given multistep cruciform cross section approaching the area of the circumscribing circle so that the area within the coils has a more efficient iron-to-air ratio (Fig. 1-9). The yokes usually have a rectangular or a cruciform section with a limited number of steps. Clamping and packing arrangements for transformers vary from size to size. In power transformers rated at under 1 MVA (per limb), this is done with wooden or plastic battens and bars which fill the space between the limb and the insulating sleeve carrying the LV winding which is placed next to the limb iron (Fig .1-9a). 3*
36
Part One. Transformers
~ _
lami~1~~
1--,--
1--...---
I-..L-..L--j '---'-
-'---l'
~ lamination~
~ (a)
2''' ~ (6)
Fig. 1-7 Imbricat ed (interl eav ed) joints in a magnetic core : (a) sin gle -phase core-type transform er ; (b) three-phase core-type tran s-
forme r
II
Fig . 1-8 Mitred joints for a three-phase core-t ype tr ansformer using cold-rolle d gra in-orie nte d steel sheet lamin at ions
Fig. 1-9 Yok e clamping: (a) by wooden bat tens; (b) by steel s tuds (i-steel st ud; 2-insula ting tube; 3-pressboard wash er; 4-steel washer; 5-pressboard washer)
Ch. '1 An Outline of Transformers
37
In high-power transformers, the limbs were at one t ime clamped by steel studs insulated from the iron by syntheticresin-bonded paper cylinders (Fig. 1-9b), whereas the yokes were clamped with similar studs extending through wooden or steel yoke clamps (Fig. 1-tO) . The more recent practice is to clamp together the laminations in transformer limbs and, often, yokes with circumie-
Fig. 1- 10 Transformer frame
rential bands usually made of glass fibre bonded with thermoset ti ng epoxy compounds . (Such bands can be seen on the limbs in Fig. 1-tO, and on the limbs and yokes in Fig. 13.) With epoxy-resin-bonded bands , one nee d not use clamping studs or punch holes in the core laminations (such holes reduce the reluctance of the core and add to no-load losses) . The core and the yoke clamps along with the other parts serving to h ol d the core and coils in place make up the frame of a transformer (Fig . 1-10) . Microtransformers rated for units to tens of volt-amperes use far simpler core designs. As often as not, their cores are assemb led with one-p iece punch ings as shown in Fig . 1-11a , or two-piece laminations (one piece being E-shaped, and the other l-shaped) as shown in Fig. 1-11b.
38
Pa rt One. Tra nsformers
In the lamination shown in Fig. 'i-Ha, the middle li mb is cut through, so it can be bent away during assembly,
[lJ (a)
- -
== =r---- -
(0)
Fig . 1-11 Core assembly for microtransformers: (a) a-shaped laminations; (b) E- and I-sh aped lami nations
and coils can be put on it and inside the laminat ion. The next lami nat ion is inserted from the opposite end of the
Fig . 1-12 Strip-wound transformer : I-primary; 2-secondary; 3-core
coil. After assemb ly, the core is clam ped t ight by pressure end plates and studs. Another popular core design is that using long strips or ribbons of transformer steel wound on a r ing -sha ped former, and the coils in turn wound on the core by a suitable machine (Fig . 'i-12) . (ii)
Structural Parts of a Transformer
The fu nction of the structural parts in any transformer is to provide electrical insulation between the windings, to hold the core and coils in place, to cool the transformer, to provide connection between the transformer windings and the associated electric. li nes, and the like. Actually, the yoke clamps .and the other clamping and packing-on t parts may also be classed as structural parts.
Ch. 1 An Outline of Transformers
39
Let us take a closer look at !.he structural parts, choosing all oil-immersed three-phase power tr ansformer as an example . Its general arrangement is shown in Fig. 1-13.
Fig . 1-13 Three-ph ase, t wo-winding, 40-MVA, 110-kV transformer with split LV windings and on-lo ad t ap-cha ngin g on the BV side: I-HO-kV bu shing; 2- 1O-kV LV bushing; 3-lifting lu g; 4- tank; 5-tubular cooler ; 6-therm al siphon filter; 7- jacking lu g; 8- oil dr ain cock; 9-blow er; IO- cas tors; ll-yokeb ands ; 12-frame tie-rod; 13-yok e cla mp ; 14 -HV ta p-ch anger ; 15-limb band s; 16- core-andcoil li ftin g lu g; 17 -conserva tor; 18-oil gauge ; 19 - expl osion stack
Wind ing insulation. The turns of a transformer winding must reliably be insulated from orie another, from ' the turns of other windings, and from the transformer frame . In oil -immersed transformers for '10 kV and high er , this purpose is served by oil-paper barrier insulation . It is obtained by impregnating cab le pa per or electric-grade pressbo ard with transformer oil whi ch is also used to fill
P art One. Transformers
40
Fig. 1-14 Windings of th e transformer in F ig. t -t3 (dimensions in mm) : 1 - steel pressure ri ng; 2-fine vo 1tage-control winding; 3-coarse voltage-control winding; 4- H V windi ng; 5- LV win ding; 6-corner washer; 7-intercoil spacers; 8-insula t ing cyli nder ; 9-pressb oard support rings ; 10 - pr essb oard yoke ins ula tion; ll-edge bl ock ; 12wooden pac king strip; 13-wooden fillin g b ars; 14 , 15-press-b oar d cleat s
Yoke line
-'1
Y"o~l' un~J~~~~~!~~t
¢55lf
• 52
1.1
LV
Radi l1L section of !f0ke
2.7
Ch, 1 An Outl in e of Transformers
41
the space bet ween t he coils an d the frame. Apa r t from pr oviding electrical insulation, the transformer oil fill ing the transformer tank also doub les as a coolant. Int ertu rn in sulation is prov i7 ded by the oil-im pregnated in8 sulation on t he coil cond uctors (which may be round wire or str ip con ductors) . The arrangement of the ma jor insulation separating the windings fro m each other, from the 4tank , and from the frame is 5 shown in Fi g. 1-'14. 6 9 Leads and terminal bushings. ~.-rl..,-llIl vv<.l<"~10 The L.V . an d H . V . win dings of a transformer are connected to external circ uits by means of leads (insulated con ductors mounted ins ide the transformer tank) and terminal bush ings (dev ices 12 cons isting of a porcela in cylinder, a central current-carrying / conductor, and a m ounting fl ange). Fig . 1-15 Out door-service, 35 1,V, 250 A bushing wi th The conductor of a t erminal central conductor connected bushing must reliably be insul atto a lead : ed from the grounded top cover l-copp er termin al ; 2-brass on either ("oil" and "a ir" ) side nut; 3-bruss cap ; 4-steel st ud; 5-nut; 6-washer; (Fig. 1-15). The si ze and com7- rubber gro mme t ; 8- por- plexity of terminal bushings grow celain insulator; 9-steel flan ge; lO-lug; ll-rubbel' with the voltage rating of transformers . For '110 kV an d higher, seal ; 12-central conductor inside insu la tin g tu be oil-filled terminal bushings are used. Tan k accessori es an d fit tin gs . If the tank of an oil -immer sed transformer were filled full with oil and completely sealed off, it woul d inevitably burst un der the action of oil pressure bu ilding up in the tank with rising temperature. One way to prevent bursting is to keep the oil level in the tan k some distance below th e t op cover an d let the tank's insid es communicate with the atmosphere . In such a case, however 1 the oil is exposed to air over the entire area UlHIE;lr
42
Part One. Transformers
the cover-a feature which speeds up oil ageing through oxidation and moisture pick-up, so the oil loses its valuable properties too soon. Another course of action is to fit a conservator (or expansion tank) to the tank-a cylindrical
Fig. 1-16 Accessories of a transformer tank: 1-oil gauge; 2-filler cap ; 3-breather; 4-sJudgc sump; 5-COllscrvator shut-off cock; 6-Buchholz relay; 7-relief stack
vessel communicating with the tank and limiting oil expo sure to air (Fig . t-:12). In a transformer with a conservator, the oil needs to be dried, purified and regenerated or changed "far less "often . The conservator is usually fitted "with an oil level gauge (see Fig . 1-16), and a sump to collect sludge and moisture. The space at the top of the conservator communicates with the atmosphere via a breather tube brought out to terminate under the conservator (so as to keep drops of moisture from finding their way into the conservator). Any transformer generates a large amount of heat in operation, and this calls for a proper cooling arrangement. On l arge transformers, this is done by tubular radiators (see Fig. 1-13) which are attached to ports welded into the tank. The ports are fitted with cocks so that the radiators can be shut off and detached while keeping the tank filled. The temperature of oil is indicated by a thermometer mounted in the top part of the tank. On small and medium-size
Ch. 2 Processes in Transformer at No-Load
43
transformers, mercury thermometers are used, whereas on large units a better choice is filled-system thermometers or remote-reading resistance thermometers with their indicators mounted at an instrument board. Any fault which occurs inside a transformer in operation (insulation puncture, shorted turns, poor contact or sparking due to poor grounding) is generally accompanied by the evolution of gas as a result of the decomposition of oil or solid insulation. The gas bubbles rise to the surface and finally find their way to the conservator. On its way there, the gas is collected in what is known as the Buchholz or gas-formation relay (see Fig. 1-16) installed on a stub pipe between the tank and conservator. The Buchholz relay has an upper and a lower float. As gas collects in the relay housing, it displaces oil out of it. The top float drops, and its mercury switch completes an alarm circuit . In the case of a more serious fault, such as an inter turn short (or shorts) and the like, gas is usually liberated in an explosive fashion, and a large amount of oil is forced from the tank into the conservator. This causes the lower float to rise and close its mercury switch, thereby activating a tripping circuit which disconnects the transformer from the supply line and averts a major breakdown. To avoid irrepairable damage to the tank in the case of a heavy gas evolution, a device known as the relief or explosion stack is installed on transformers (see Figs. 1-13 and 1-'16). It is a long steel pipe communicating with the tank at one end and closed by a disc of thin glass at the other. When the pressure inside the tank rises dangerously, the disc bursts, so that excess oil and gas are expelled into the atmosphere 'before the tank has time to be deformed.
2
Electromagnetic Processes in the Transformer at No-Load
2-1
The No-Load Condition
On the primary side, transformers are excit ed by a harmonically varying voltage (2-1)
44
Part One. Transformers
As the load varies , the peak-value
F 1,m
and the frequency
I of the primary voltage ch ange but little , so it is usually assumed that they are constant and equal to their rated values F1,m = V l ,mn = constant I = In = cons ta nt This also goes for t he angular frequency W = 2'Jt1 = Wn = cons tant The secondary current is inversely proportional to the impedance of the line to which it is connec ted
I Z I = V H~+ X2 At a certain defin it e value of thi s impedan ce, Z the second ary winding car ries its r at ed current 12 = 12 ,H At Z
<
=
Zn ,
Zn, the secondary current exceeds its rated value
12
>
1 2 ,R
and th e transformer is somewhat overloaded . At I Z I> I Zn l
1 2 < 1 2 ,n and the transformer is underloaded . When I Z I is infinity, which occur s when the transformer is disconnected from the io V, e, ~
\
1, ~o
-
@
r-r--r-«
hI
iz=O
I
~
F ig . 2-1 Single-p hase tw o-windin g transformer on no-l oad
receiving line on the secon dary side (the secondary is opencircuited), the secondary current falls to zero . In th e circumstances , the transformer supplies no-load current, which is why this state is calle d the no-load (open-circuit) condition,
Ch. 2 Proc esses in Transformer at No-Load
45
The electromagnetic pro cesses occurring in a transformer at no load ar e far simpler than they are und er load, with 10 > 0, so their study can best be begun with the no-load c~ndition.
Consider the electrom agnet ic proce sses at no-load in the single-phase two-winding t r ansf ormer shown in sketch form in Fig . 2-1. This is a core-t ype transformer whose primary and secondary windings are shown for conv eni ence located on different limbs. (The ac tual arrangement of the windings on a core-type transformer has been described in Sec . '1-3, see Fig. 'I-Sa.) 2.2
Voltage Equations
The supply voltage VI im pressed on the primary winding gives rise in it to an alternating current i o , called the noload curr ent . This current produces two fluxes , namely the mu tual (usefu l) magnetic f lux which has its path wholly within the core of a very high permeabili ty, ~tr ~ '1, and links all the turns WI an d W 2 of the primary and secondary windings, and als o the leakage flux which links onl y the primary turns . If we find the mutual magnetic flux cD at an y section of the dosed magnetic circuit , we sha ll be able to find the mutual flux linkage with the primary winding 1f on = w1 cD and with the secondary winding "If 02 1 = w2 cI) The leakage flux has its path completed through nonmagnetic materials (air gaps, insulation) with a permeability equal to that of free space , ~to, and substant ially smaller than that of t he magnetic core . Therefore, the leakage flux linkage with the primary winding at no-load , 1f a O, is a small fraction of the mutual flux linkage with the primary, "If on (Fig . 2-'1 ). Th e periodicall y varying mutual and leakage fluxes induce electromotive forces in the windings with which they link. For t he posi tive directions of currents, voltages, emfs and magnetic lines of force shown in Fig . 2-'1, the primary emf oj mutual induction is e1 = - W I dCV/d t = - d"lf on /dt (2-2) whereas the secondary emj oj mutual induction is ez = - w z dcD/dt = - d"lfOZl/dt (2-3)
46
Part One. Transformers
and the leakage primary emf is e ao = -elcDao/elt
~ e1
(2-4)
Interpreting VI as an emf impressed on the winding from the supply line, we may write Kirchhoff's voltage equation VI
+ e + e ao =
R 1i o
l
(2-5)
where R 1 is the resistance of the primary winding. The no-load voltage across the secondary is the same as the emf induced in it V 2 = e2 2-3
l I
Variations in EMF with Time. An EMF Equation
1
For all power transformers (and for most microtransformers) ,II we may neglect in Eq. (2-5) both the voltage drop across R 1 . and the leakage emf e cu I
I u.i, I ~ I el I I e ao I -e; I ell I'l---l----+--'.--'lf-----,I-z-f{
and deem, with sufficient accuracy , that the primary emf of mutual induction is in antiphase with the primary voltage (Fig. 2-2):
el
=
-VI =
=
-EI,~
V1 ,m cos rot cos rot (2-6) -
Fig. 2-2 Time variations in voltages, emfs and magnetic flux of a transformer
It follows from Eq. (2-6) that the emf of mutual induction . varies with time harmonically, and its peak (rms) value does not differ from the peak (rms) value of the voltage
E 1 ,m = VI,m,
(E I = VI)
(2-7)
From a comparison of Eqs. (2-2) and (2-3), we may conclude that the ratio of e2 and l is time-invariant. This ratio is called the transformation, or turns, ratio
e
e21el
= E 2 , mlEl,m = E 2/E1 =
W 2/Wl
=
1221
(2-8)
47
Ch. 2 P rocesses in Transformer at No-Loa d
On the basis of E qs . (2-6) an d (2-8), we may arg ue that e2 varies likewise harmonically and is in phase with el . We may express the magnet ic flux (D in terms of el by integrating t he differential equation (2-2) subject to Eq . (2-6): t
t
. ~ E (1)= - - J e 1 d t= \' cos or dz 1
WI
WI
U
"
0
(2-9)
- Q)msin ro t where
(D m = E1 ,m/Wlro (2-'10) is the peak va lue of the magnetic fl ux . Using Eq . (2-10), we can der ive an equati on giv ing the rms va lue of e1 from the given peak magnetic flux or flux linkage E 1 = E 1 •n/ j/2 = row1cDm/ 1I 2 = ro'P'o11 ,m/ V 2 or E 1 = (2n /
11 2) j W1(I)m
. (2-11)
Accordingly, the rms va lu e of e2 is E 2 = rowlI)m/
11 2 = rolfo:d ,n,l 11 2
or (2-12)
Referring to the plot of Fig. 2-2, the magnetic fl ux l ags behind VI by 90° (it is said to be in qu adrature lagging with the primary voltage) , and leads e1 and e 2 by 90° (it is said to be in qua drature leading with the two emfs). 2-4
The Magnetization Curve of the Transformer
The thick ness and m aterial of the lami nations for a transformer core are always chosen accord ing to t he frequency of the magnetizing current , so as to keep eddy currents t o a minimum . The instantaneous magnetic flu x may th en be determined from t he inst ant aneous pr imary mm f, iOw 1, at no-l oa d. The r esultant r el at ionship between t he in st ant aneous va lues of the two quantities, cD = j (i o), is i dent ical
Part One. Transform ers
48
to that obtained with d .c., when eddy currents are nonexi stent. Graphically , the nonlinear re lationship betwee n t he flux cD in the core an d the direct curren t i o in the primary winding is dep icted by what is called the cl.c. magnetization curve (or characteristic) of a tran sformer. It can be construc ted on the basis of Ampere's circuital la w in in t egral form . On aligning the loop enc losing t he current in all the primar y turns, iOw 1, with a line of for ce of t he mutual magnetic flux in the core, Ampere's circuital law ma y be written
I
iowt =
I
~ Hzdl
The procedure yielding the cir culation of the H vector is as follows: (1) Assign a desired va lue to the magnet ic flux in th e core . (2) Bre ak up the core into n portions of length I" each , such that within each portion the act ive iron cross-sectional are a A" and the permeabili ty r em ai n constan t for the specifi ed m agnetic flux . (3) Calculate the magnetic induction within each port ion, B" =
iowt = :~ HI dl
=~ "= 1
n
H,Jh
=:2; (Bh /~la") I" "= 1
n
= cP :2;
I ,J~la"Ah =cD/Afl
(2-13)
" =1 n
where AJ.l
= 1/
~ Ih /~n hAh is the permeance of the core.
11= 1
On solving Eq . (2-13) for several values of cD and finding each time i o = cDlw1A fl' we can then plot the magnetiza-
Ch. 2 Processes in Tranalor mer at No-Load
49
tion curve (J) = t (io) for the transform er. An approximate sha pe of the m agnetization cu rve is shown in Fig. 2-3 which also gives the A~t = t (i o) curve . As is seen, A~ is a maximum near the knee on the magnetization curve . As i o and (I) keep increasing, the permeance decreases . 2-5
The No-Load Current Waveform
o
As already noted, if we neglect the effect of eddy curren ts and the core Fig . 2-3 Magnetizalosses, the rel ationship between the t ion .cur ve, QJ = f (io), instantaneous flux and no-load curof a transform er rent is the same as with direct current. Therefore, since the magnetic flux has been found to vary with time sinusoidally, as defined in Eq . (2-9), we can use the d .c , magnetization curve shown on the left of Fig. 2-4 in order to see how the no-load current varies
"it
Fig . 2-4 No-lo ad cur rent wav eform (ignoring core Joss)
with time, i o = t (t) or t (wt) . To do this , we should plot var iat ions in the magnetic flux with time, el) = ())m sin wt (on the right of Fig. 2-4) an d determine the instantaneous values of no-load curren t for some selected values of the magnetic flux . The relevant gr aphical procedure is indicated in the figure by arrows . T aking the value of (J) at point 1 and moving through point s 2, 3, and 4, we find the corresponding no-load curre nt i o at the intersection, 5, of the horizontal extending 1, -0 1 GU
50
Part One. Transformers
from point 4 and the vertical extending from point 1 . With a sinusoidal magnetic flux, the no-load current as depicted by the i o = f (roz) curve turns out to be nousinusoirlal. The no-load current grows increasingly more nonsinusoidal as the peak magnetic flux cD m raises the level of saturation in the core or, which is the same, as cD m exceeds increasingly more the fluxes corresponding to the linear portion of the magnetization curve where (I) is proportional to i o. 2-6
Transformer Equations at No-load in Complex form
Complex notation is applicable to equations that connect sinusoidal currents, voltages and emfs. Therefore, before we may write the transformer equations at no-load in complex form, we must replace the nonsi nusoidal no-Ioad current i o by a sinusoidal current i or = V2 1 0 1' )( sin wt (Fig. 2-5) equivalent in terms of the reactive power consumed. (Since the copper loss in the primary is low, the iron loss is likewise low, so we may take it that the active power and the art resistive current are small in c;;z:....--4------=~-com parison with the reactive power and the reactive current.) For this change not to affect the reactive power consumed, the rrns value of the equivalent sinusoidal current, Ion must be equal to the rms value of the nonsinusoidal no-load current, that is Fig. 2-5 Replacement of no-load current i o by an equivalent sinusoidal current i Or
I 01' =
V
1 /
1
f Iii·o d t
T J
(2-14)
o
In our further discussion, we call this current the reactive (magnetizing) component 0/ the no-load current . As is seen from Fig. 2-5 , the current i or must be in quadrature lagging with VI ' (A transformer at no-load and free from iron loss
51
Gil. 2 Pro cesses in T ra ns former at No-Load
Illay be rega rded with respect to the supply line as all inductor having a negligible ohmic res ista uco.) As will be shown later, even if we in clude losses, the active component of t he no-load current is very sm all in comparison with the reactive component , I Oa ~ l or . Therefore, it is legitimate to deem the rms value of t he no-load curre nt, 1o, equal to t he rm s value of the re active current
10 =
-V I ij a + n; ~ 1
0 ,.
It i ~ important to note that the no-load primary current usually ranges anywhere from 0.1 to 0.005 of the rated primary current . For each rms value of the primary voltage , VI ' we find loT' and compute the equivalent prim ar y inductance related to the main or mutual magnetic flux or between the primary and secondary. This will be referred to as the main or mutual primary inductance . It is defined as the ratio of the peak flux linkage to the peak reactive component of the no-load current: (2-15)
If we express the magnetic flux in terms of the equivalent perm ean ce A 1 2 of the t r ansformer
=
A 1 2 (w1 i or)
this permeance an d the mutual primar y inductance may be connecte d by a relation of the form (2-16) From this inductance, we can com pute the mutual ituiucti oe reactance of the primary winding
X 12
=
UlL~ 2 =
Ulw~ A 1 2
(2-17)
Now that we have introduced the necessary definitions, we may write complex relations connecting the emf and the sinusoidal reactive component of the no-load current . As follows from Eqs . (2-2), (2-15), an d (2-16), (2-18) 4*
52
Part One. Transformers
N ow we shall express e1 and i or as tli e real parts of the resp ec tive complex nnipl itud cs ruul tipli ed by ex p (j(f)t)*
[V 2" E1 exp (j(f)t )J
(2-1 8a)
ior = Re [V 2 i; exp (j(f)t) ]
(2-18b)
e1 =
Re
On substituting the above expressions in Eq . (2-18) and differentiating, we obtain Eq. (2-18) re-wri tt en in complex notation : Re [V2E 1exp(j(f)t)] =
-L~2
= Re =
[V'2jorexp(j(f)t)])
L~2 ~JV 2" j Or exp (j(f) t)]}
Re (-- j(f)L~2
.
or
{-
ddt fRe
rv'2 i., exp (j(f)t )l) .
.
E 1 = -j (f)L~ 2Ior = -jX 12Ior (2-19) Graphically, Eq . (2-19) and also Eqs . (2-6), (2-9), (2-11), and (2-12), as wr itten in com plex notation
- V = E = - j(f)W d>n/ l! 2" V2 = E 2= - j(f)w2(b~/V2 1
1
1
(2-20) can be de picted by a phas or diagram for a transformer at no -load, such as shown in Fig . 2-6 . Fig . 2-6 Phasor diagram of a transformer on no load (ignoring losses and active current)
2-7
Mo-l oad losse s
Although the core of a transformer is assembled from thin insulated electrical-sheet steel laminations, t he iron (core) loss accounts for 0.1 % to 0.2 % of the transformer's power rating. For example, in a 100-MVA transformer, the core loss is up to 200 kW . In microtransformers rated from 0.1 to 10 3 W, the core loss rises to from 2 % t o 20 % of the power rating.
*
The symbols with a dot above refer to complex quantities.
I
CII. 2 Proc esses in Tr ansformer at No-Load
53
The core loss is"the sum of the hysteresis loss Ph which is propor tional to" the frequency 1 and the square of the magnetic induction , B~ , and t he edd y-current loss , P e which is proportional to the frequ ency squared , 12 , and the magnetic induct ion also squared , B~ . In practical calculations, it is customar y to find directly the t ot al core loss, Peore, in t he various core elem ent s with an active iron cross-sectional area A Ii , magnetic induction B li =
+ e, = );
PI.olsoB~ (1 /50( 3 Inli
(2-21 )
where Pl.O/ 50 (in watts per kg) is t he spe cific core loss at a frequency of 50 Hz and a magnetic induct ion of 1 T [13J. At no-load and the rated primary voltage , VI R, the core loss is about the same as at rated load. Th erefore (as will be shown later), the magnetic flux and induction in the core at VI are nearly in dependent of the lo ad condition . The copper loss in the primary at no-load, PCu ,o = RJ~ , may be neglected , because the no-lo ad current is small and t his loss is a fr action of that at the rated primary current P Cu,l R . RIII,R
If we neglect the copper loss, the no-load loss of a transformer, Po, may be deemed equal to its core loss :
Po ~ 2-8
r..:
The Effect ·of the Core Loss on the Transformer's Perfo rmance at No-Load
A transformer with the core loss P eore draws from the supply line an active power given by TTI / oa' Th e nns value of the sinu soidal active current is lo a
=
P eore/TT I
(2-22)
This current is in ph ase with the applied volt age, so it may altern at ive ly be expressed in te rms of an equivalen t resistance R l 2 lo a = F 11R 1'!. (2-23) From a com parison of Eqs. (2-22) and (2-23), the equivalent resistanc e R 12 m ay be expressed in te rm s of voltage
54
Pa rt One. Transformers
and core loss as follo ws*:
R IZ = Vt lP corc (2-24) The cur rent drawn by the primary winding at no-load is the sum of the active current J Oa and t he reactive current t.;
i, = i.; + i; = V11R 12 + TT1/j X 12 =
V1Y o
(2-25)
The admitt ance (2-26) is equivalent to R IZ and jX 1 2 connected in parallel. Equation (2-25) describing the events occurring in a transformer at no-load, with allowance for the core loss, corre sponds to the equiva lent circuit shown in Fig. 2-7a. In practical cal cul ations, howe ver, it is more convenient to re-draw
I
~
...i,
....0
I
10 ioa.
V1
I
E,l
Ro
R12
jX 12
Iv,
E,l
jxo
'If
(6)
(a )
Fig . 2-7 E quiva lent circuit of a transfor mer on no-load : (a) wit h R an d X connect ed in p ar allel; (b) with R and X connecte d in
series
the equivalent circuit as shown in Fig . 2-7b whi ch includes t he prim ar y imped ance at no-lo ad
Zo
I'
=
Ro
+ jX o
(2-27)
On expressing Zo in terms of Y o in Eq . (2-26)
Zo = R o + jX o = 1/Y o = 1/(l IR 12 + 1/j X 12) and equati ng the coefficient s of the im aginary an d real par ts, we fi nd t hat 2 Ro - .R ~1 2 X 12 I(R 2t e -~1- X21 2 ) (2-28) X o = X12R ;2/ (R~ 2 + X ~ 2 )
*
The loss across R 12 a t VI is equal to th e core loss.
55
Ch. 2 Pro cesses in Transformer at No-Load
becau se when I oa « l or> it is inevit able that R 12 ~ X 1 2 . Fina lly , we may write
n, = X ~ /R12 ' x, = X 1 2 , R o « R 12 (2-29) The quantity X 0 retains the name of the mutual inductive reactance of the primar y winding. R o is a fictitious res istance the loss across which at l ois equa l to th p 1',(>1'0. loss of the t ransformer P ear e = I~R o
.
As is seen from the equivalent circuit in Fig. 2-7b, the . products Rol o and jXol o are, res pectively, the active and reactive components of the primary voltage
T\ .
If 12
Fig. 2-8 Ph asor dia gram of a transform er on no-loa d
F ig . 2-9 Impedance of the equi " valent circuit and no-lo ad cur ren t 1 0 as functions of TTL
The relation hetween the pr imary voltage Tll and the no-load current 1 0 . . Tll = - E 1 = Z oI o (2-30) is illustrated b y the phasor diagram in Fig'. 2-8 which, with P ear e = 0, R 12 = 00, and lo a = 0, is the same as that shown in Fig. 2-6 . Because the magnetic circuit of a transformer is nonlinear , the no-loa d current 1 0 rises at a faster rate t ha n VI' so R o and X 0 depend substan tially on 111 (Fi g. 2-9):
.
X;
=
X 12
'"
Vl /l o
56
Part One. Transformers
and
R o ,.... (V)I o)2 In contrast, as the primary voltage is va ried, R 1 2 remains practically unchanged, because the core loss is pro portional to the square of the magnetic induction, Eq . (2-21), or the primary voltage, Eq . (2-20).
3
Electromagn etic Processes in the Transformer on Load
3-1
The Magnetic Field in a Transformer on load. The MMF Equation. The leakage Inductance of the Windings
When a transformer is operating on load, i ts secondary is traversed by a current
The load current gives rise to a change in the primary current. Proportionate changes also occur in the magnetic flux and the secondary voltage , aud thera is an increase
i,
if Ii.I ~f ~ p/ -
£1!
~f
~2
'--
=
!tZtt62 Z - Pz
Fig. 3-1 Single-phase, two-winding transformer
Oil
load
in the power lost. For a proper estimate of these changes in a transformer on load , it is essential above all to examine its magnetic field and to deve lop voltage equations for its pr imary and secondary windings.
Ch, 3 Processes in Transformer on Load
57
Figure 3-'1 shows a single-phase , two-winding transform er whose second ar y is conne cted acros s a load impedance Z. Assuming that all the relevant electric and magnetic quantities are var ying harmonically , we may write them in complex notation . In doing so, it is important to remember that the instantaneous value of a harmonic quantity is t o be construed as the re al par t of t he respective complex amplitud e multiplied by exp (jwt) i
= Re [V 2 j exp (jwt)]
[V 2Vexp (jwt)] e = Re [V2Eexp (jwt) J
v = Re
= Re [cP m exp (jwt)J lJf = Re [O/m exp (jw t)J
(1)
The adopted positive directions of the abo ve quantities are shown in Fig . 3-'1. Positive directions for II and 1 2 ar e chosen such that they set up a positive mutual magnetic flux . Posit ive directions for t he voltages and emfs across the win dings are the same as for the res pective curren ts . Positive diractions on load are chosen the sam e as for oper ation at no loa d. When a transform er is operating on lo ad, its magnetic flu x is established by the prim ar y current II traversing the primary winding and by the second ar y current 1 2 traversing the secondary winding . To simplify the matters , this magnet ic flux can be visualized as a superposit ion of two fluxes, namely the mutual (or m agn etizing) flux and the leakage flu x . Th e grea ter proportion of the flux linking the windings is the mutual flux whi ch h as all of its pa th within the core and com pletely encloses the wind ings fr om both sides. The mutual flux cD (Fig. 3-'1) is the same at an y section of the core ; it s linkage with t he prim ary is WlcPnll and with the secon dary , w 2 cD m . Und er Amp ere 's circuital law, the magnetic intensity du e to mu tual induction is t he sum of the prim ary and seconda ry mmfs
58
Part One. Transformers
Since t he mutual magnetic induction and the mutual flux are connected to the field int ensity in a well -defined manner (see Chap . 2), we may arg ue that the mutua l flux
is established by the sum of t he primary and secondary mmfs. This sum may be vis ua lized as the mmf due to some current i o traversing the primary winding (3-1) ilWl + i 2w2 = iOw l Therefore, the current given by i o = (ilWl
+ i 2w2)/ W l
may be called the mag netizing current, and Eq. (3-1), an mmj equation .
The non linear effects taking pl ace in the transformer core as it undergoes cycles of magnetization by the current i o may be accounted for as in t he case of no-load operation. The nonsinusoidal current i o may be rep laced by an equivalent sinusoidal magneti zing current the rms va lue of which is
=v
10 IBa + I ar and whose active component l oa is related to the core losses. Then we may write the mmf equation in complex notation as
.
I lwl ,;
"'!
I
+
.
.
I 2w 2
=
I owl
(3-2)
In our further discussion, the term "magnetizing current" will refer to the equivalent sinusoidal magnetizing current 1 0 , Now we are in a position to present the primary mmf i l w 1 as a sum of iOw l and (illVl - iolVl) = -i 2lV z which ba lances the secondary mmf i zlV 2 , and the magnetic flux in operation on load as a sum of t hree fluxes, namely: (a) the mutual magnetic flux and the leakage flux with flux li nk age 1Jf aO' set up by the primary mmf iolV l (Fig . 3-2a); (b) the leak age flux established by the mutually ba lancing mmfs, namely (illV l - iolVl) = - i zlV z on the primary side and i zlV 2 on the secondary side (Fig . 3-2b). Referring to Figure 3-2, it is seen that the lines of the leak age flux have their path comp leted through nonmagnetic (air, oil, etc .) gaps alb l and a zb2 comparable in leng th with the portions of the lines accommodated within the core (bla l and bza z) . These lines link either the primary turns (1Jf o i and 1Jf ao), or the secondary turns (1Jf az).
59
Ch. 3 Processes in Transformer on Load
The lines of the leakage flux in a transformer may be divided into two groups-those linking only the primary turn s and giving rise to the flux linkage 1Jf 00 due t o i o and / i,
a,
e,
iz=O
,---
iowrt
'I{,.
b, ~
iri o
J
(i,-i,)
w, t
a,
a2
'Pa,
'f;;z
b,
bz
1
iz=-(i,-ioJw,(W2
~
r-r-r-
'---
~ li,""w·l
~
(6)
(a)
Fig. 3-2 Magnetic flux on load as the sum of (a) mutual flux and (b) leakage flux
1Ya1 due to (i1 -
i o) , and those linking only the secondary turns and giving rise to the flux linkage 1Jf 02' To appraise the relationship between the flux linkages and the currents in the windings, we shall develop an equat ion by Ampere's circuital law for, say, a closed line of the leak age flux linking t he primary winding as shown in Fig . 3-2b:
~ H dl =
:r
b1
«i
«i
b1
Jr n, dI + ~I\ H eor e dl = (ii- i o)
Wi
Let us write the magnetic field in the nonmagnetic region, H 0 ' and the magnetic field in the core , H eore , in terms of the respe ctive induction and permeability: Ho
H eoro ~[. a. e ore
= = =
B oht o
B eore/~t a , eo re ~t r . eore ~t o
~r,eore ~ '1
Therefore , the leakage field ill the core is negligibly small lIeore
=
B eorehta ,eore =
0
60
Part One. Transformers
The total current is equal to the magnetic potential difference across the nonmagnetic gap bl
bl
:0 ) n, dI
) n, dl = al
(ii - io) Wi
=
0
It follows from the foregoing that 1.J.! o i is proportional to (i 1 - io). The same holds for 1.J.! co and 1.J.! cr2 and their respective currents i o and i 2 • Therefore, the leakage inductances of the windings L cr1 = Wcr1/ ( i 1 - io) t.., = 1.J.! cr2/i2 (3-3) L cro = 1.J.! cro/ i o are constant for a gi ven transformer and solely depend 011 the wid th of nonmagnetic gaps and the number of turns in the windings (see Sec . 8-2) . With a high degree of accuracy, the total leakage fl ux linkage with the primary winding may be written Wcrl = 1.J.! co
+ 1.J.! o o = i
Lcroi o
+ L cr1 (i 1-io) ~ L cr1i1
(3-4)
because in operation on load i 1 ~ i o, and we may neglect wh atever diff erence there may be between L cro and L cri and deem t hat L cr o ~ L cr1. By analogy with the mutual inductance [see Eq. (2-16)J, t he leakage inductances may be expressed in terms of the respective permeances, A cr i and A cr 2: L cr1
=
W~AUl'
L cr 2
= w;A U2
or in terms of perm eance coefficients L cr2
= f10 Wi Acrl = ~LoW~Acr2
AU!
= A u!hlo
L cr1
wher e
Acr2 =
3-2
A cr2/ f1 0
(3-5)
(3-6)
V o ltage Equ ations of the Transfor mer Windings
The emf induced in each of the transformer windings can conveniently be presented as the sum of the mutual emf E 1 (or E 2 ) and of t he leak age emf E U 1 (or E cr2).
61
Ch. 3 Processes in Trans form er on Load
The mutual flux shown in Fig . 3-2a does not differ from that in a transformer OIl no-load (see Fig . 2-1) . Th erefore, the mutual emf ma y he expressed in terms of the mutual flux in precisely the same manner as at no-load . Given a certain E I , the magnetizing current i o must be the same as at no-load, provided that E I and CD are the same in either case . Therefore, 1 0 and E I can he conn ected by an equation of the form (3-7)
where
Zo
=
Ro
+ jX o
Using the turns ratio, n 21 = w2 /w ll we can wri te the mutual emf on the secondary side as
.
=
.
=
.
(3-8) The primary and secondary leakage emfs, e crl and e cr2 , are induced by the leakage flux linkages "If o i and 'P" cr2, respectively, proportional to the primary and secondary currents : e crl = - d'P" crl/dt = -L crl dil/dt (3-9) e cr2 = -d'P" cr2/dt = - L cr 2 di 2 /dt - E2
- n 21E I
n 21Z0IO
Using complex notation and differentiating by analogy with Eq . (2-19), we get and, similarly (3-10)
Here, (3-1'1)
are called the leakage inductive reactance of the primary and second ary, respectively. As is seen from Eq. (3-'10), the leakage emfs ar e in quadrature lagging with the associated currents. Now that we have defined the primary and secondary emfs of a loaded transformer and recalling that all the quantities involved vary harmonically*, we ma y write Kirchhoff's
* The nonsiuusoidal magnetizing curren t is replaced b y an equivalent sinusoi dal cur rent .
II
62
Part One. Transformers
voltage equati ons for t he prim ar y and secon dar y windings ill complex form as
.
111
.. . + u, + E = RIll . . . . E 2 + E 0 2 = R 2l 2 + 11
(3-12)
01
2
where R, and R 2 are the resistances of the primar y and secondary windings, respe ctively, including add ition al losses due to altern at ing current (see Sec. 31-2). In writing Eqs. (3-12), positive directions were chosen as shown in Fig. 3-1. The volt age 111 is the supply emf impressed on the winding from an external source. The . . voltage 11 2 = Z l2 is the voltage drop across the l oad on the secondary side with an impedance of value Z = = R jX . Expressing the leakage emfs in (3-12) in terms of the respective leakage induct ive reactances and currents (3-10), we may re-write t he voltage equations as follows:
+
.
(3-13) . . 11 2 = E 2 - l2 Z2 where Zl = R, jX l and Z2 = R 2 jX 2 are the complex imped anc es of the primary and secondary windings, respectively .
+
3-3
+
Transferring the Secondary Quantities to the Primary Side
The performance analysis of a transformer can greatly be simplifi ed, if we transfer the qu an tities associated with the secondary to the primary winding. This technique consists in tha t the real t ransformer having in t he general case different numb ers of primary and secondary turn s, W I and W 2, is rep la ced by an equivalent transformer in which the secondary ha s the same number of turns as the primary, w~ = WI (see Fig . 3-3). The qu an tities associated with the equivalent secondar y ha ving WI turns are said to be transferred (or referred) to th e primary winding or side. They are expressed in terms of the original secondary quantities adjusted in value by a suita bl e factor so that transfer of secondary quantities to the primary side wi ll leave the magnetic fie ld, and the power fluxes PI' P 2, and Q2 unaltered . The procedure is as follows .
o..
63
Ch. 0 Processes in Trans former on Load
('1) To leave the ma gnetic flux (I) unaltered , we must retain the secondary mmf unchanged, that is
.
I~ Wl =
.
I zw z
whe nce
j~ = j ZWZ/Wl
(3-14)
Here and elsewhere , the prime on a secon dary quantity in di cates that it has been tr ansferre d to the prim ar y side . (2) With (I) ke pt constant, t he emf is proportion al t o the t urns number. Therefore, the emf acro ss the secondary 4>
iI
II
t?/!'f/!
~,
"it;z
'. ~
Pf -
-----
jE! It;,
z.~ tv;
- Pz
Fig. 3-3 Transformer of Fig . 3-1 with i ts secondary transferred to = WI the primary,
w;
winding transferre d t o the primary side will in crease W 1/W2 times: E~ = E Zw1/w Z (3-15) (3) To keep unchanged the values of P z an d Qz drawn by the load on t he secon dar y side, its R an d X mu st be repl aced by th ose t ra nsferred t o the primar y side: P 2 = RI~ = R'I~2 Q•... = XI 22 = X ' I'22 Using Eq. (3-14), we get R' = R (W 1/WZ)2 X ' = X (w1h v z)2 Therefo re, (3-16) We can see th at the secon dary impedance can be transferre d to the prim ary side, adjuste d in value by t he turn s ratio squared.
Part One. Transformers
64
The secondary voltage call likewise he transferred to th e primary side, adjusted in value by the turns r a tio
t
(3-17) V; = Z ' j~ = Z (w]/w z)z zWz/w] = VZw]/w z The secondary impedance Zz, its resistive component Hz and its inductive component X z can be transferred to the primary side in about the sam e manner: Z; = H; jX; = Zz (w]/wz) Z H; = Hz (w]/wz)Z (3-18) = (w]/w z)Z As a result, the secondary voltage equation takes the form E~ = Ez (w]/w z) = VZw]/w z Zz (w]/wz)Z jzw z/w j or
+
X;
X;
+
E2'
V'2 + z.i: 2 2
(3-19) Because the primary and secondary windings have the same number of turns, the transferred (or referred) secondary emf is the sam e as the primary emf : =
.
"
E; = E 2w]/w Z = E] The mmf equation for a transformer with its secondary parameters transferred to the pr imary side is ex tended to include the secondary mmf expressed in t erms of the secondary current referred to the primary wind ing
.
.
+
.
I]w] I~w] . lOw] Dividing the above equation through by equation of transformer currents
.
.
WI
give s th e
.
I ] + I ; = 10 (3-20) which has the same physical meaning as the mmf equation (2-30) . With a suffi cient ly heavy load, when the primary current markedly exceeds the magnetizing current, I] ~ 10' th e current equation can approximately be written as
.
.
I] = -I; = - I zwz/w] or
l II
I] /I z = wz/w] (3-21) As is seen, given a heavy load, the referred secon rlary cur rent, I;, does not differ from the primary current , I ].
Ch. 3 Processes in Transformer on Load
3.4
65
The Phasor Diagram of a Transformer
The voltage and current phasor diagram of a transformer is a graphical interpretation of the equations describing the performance of the transformer. These equations includ e - the winding voltage equations
.
111
.
.
+ ZIII
-E I
=
-E~ =
(3-22a)
-i1~
-EI =
-the load voltage equation
.
-11'2
= Z'
+ Z~ (-j~) .
(-1') 2
(3-22b)
(3-22c)
-the mu tual emf equation
- EI
-E~
=
=
z.),
(3-22d)
-the current equation
i,
=
i, -
i;
(3-22e)
Using a ph asor diagram constructed to a cer t ain definite scale, we can determine the voltages, emfs and currents of a transformer on load . The sequence in which a phasor diagram is constructed depends on which quantities are specified to define the operation of the t ransformer and which quantities are to be determined . Suppose that we know the secondary current 1 2 and the load impedance Z = R jX (for an inductive load , X > 0; and for a capacitive load, X < 0). We set out to find the . secondary voltage 112 , the primary emf E I , the magnetizing . . current 1 0 , the primary current II' and the primary voltage VI' Th e phasor diagram is usually constructed for the transformer with its secondary quantities referred to the primary side. Therefore , the first step is to determine the secondary quantities referred to the primary side (that is, adjusted in value by the turns ratio or the t urn s ratio squared). The referred secondary current is
+
I~ = ;;- OI6D
.
I2
(W~/W l)
Part One. Transformers
and t he referred imped ances ar e Z'
=
Z~ =
Z (WI/W2) 2 = R'
Z 2 (WI/W 2)2 = R~
+ jX'
+ jX~
The ph asor diagram is m ade more compact if the complex quantities referred to the primary side are t aken with a minus sign , -1~ an d The first to be pl otted (see
- r;
Fig. 3-4 Pha sor dia gram of a tra nsformer operating int o a resist iveinductive loa d (CP2> 0, X > 0)
Fig. 3:.. 4) should be -1 2 which ma y be drawn in an ar bitrary direction , say along t he positive axis of the complex t ime pl ane and on the sca le adopted for currents. Th en , using t he load voltage equation, we find the referred secondary . voltage: - V~. This voltage has an activ ~ component, R' ( - 1~), an d a re act ive component , j X' ( - 1~ ), wh ich are laid off t o the adopt ed sca le . The active comp-onent is la id off in the direction of -1~ , whereas the reac ti ve com ponent leads -1~ by 90 if the load is inductive and X > O. The actual second ary volta ge is found by Eq . (3-17): 0
,
V2
=
Vi (W2/Wl)
Ch. 3 Proc esses in Transformer on Load
Then we find graphically the mutual emf -E 1 and compute the magnetizing current 10 = E 1/ V Rfi
67
-E'
2
+ X~
and the ph ase angle CjJo =
arctan
(XolR o)
Now j 0 can be laid off on the phasor diagram . The mutual flux cD can be found from Eq . (3-7) and laid off on a scale of its own (the flux is in quadrature lagging with -E1) . The primary current II is deduced from t he current equation . The primary voltage VI is found graphically in a similar way. The construction thus obtained also gives the phase
-ii
R'(-ii)
Fig . 3-5 Ph asor diagra m of a tr ansformer operating into a resi stivecapaciti ve load (rr2 < 0, X < 0)
shift (P2 between t he secondary voltage and current, and the phase shift (PI between the corresponding primary quantities. With a resistive-inductive load , both the primary and the secondary currents lag beh ind the respective voltages in phase, so CjJl and CjJ2 are taken to be positive: CjJl > 0 and CjJ2> 0 (see Fig. 3-4). The ph asor diagram for a resistive-capacitive load is plotted in Fig. 3-5. As is seen, the secondary current leads the voltage by an angle (P2 (CjJ2 < 0). If the load is predominantly capacitive (see Fig . 3-5), the primary current like5*
68
Pa rt One. Transformers
wise leads the voltage by an angle (PI < O. If the capacitive component is less pronounced, the primary current may even lag behind the voltage . 3-5
The Equivalent Circuit of the Transformer
If we treat a single-phase, two-winding transformer as a two-port, the equivalent circuit stems from Eqs. (3-22a) through (3-22d), where the secondary quantities are trans-
ferred to the primary side . Given VI' the circuit equivalent to a given transformer must draw from the supply line the same primary current i, as the transformer itself. In order to identify the configuration of this equivalent circuit, we must express the primary voltage in terms of the primary current. To begin with, we shall express i, in terms of E1 and the circuit parameters
Hence, 11 l/Zo+l /(Z~+Z)
Substituting the above expression into the voltage equation gives
Vi =
j1 Z1 -E 1= i, [Z1 +-
1/Z
o
+ 1I ~Z~ +Z/)
] =
j1 Z eQ
(3-23)
It is seen from Eq. (3-23) that the transformer equivalent circuit drawing a primary current II must have an equivalent impedance given by ZeQ = Z1
+- 1/Zo +1I\Z~+Z/)
This impedance is presented by the circuit in Fig . 3-6 where ZI is shown connected in series with a parallel combination of Zo and (Z; Z'). A detailed analysis would show that the individual arms of the equivalent circuit carry the same currents as the
+-
Ch. 3 Proc esses in Transformer on Load
69
windings of the transformer in whi ch the secondary quantit ies are tra nsferred to the primary sid e . Also, the current s
Fig . 3-6 E quivalent circ uit of a t ransformer
ent ering the nod es of the circ uit and it s loop voltages satisfy the basic tra nsformer equat ions. 3-6
The Per-Unit Notation
E lectrical quantities (such as currents and voltages) and circuit paramet ers (reactances an d resistances) can be expressed each as a fraction of an arbitrarily ch osen base or reference quantity , thereby gi ving per-un it quantities. The per-uni t notation sim plifies the equat ions describing t r ansformer performance. It also simplifies a chec k on the design data an d result s , because the per-unit qu an titi es of different t r ansformers differ much less t ha n the same quantit ies expressed in absolut e units . The base quantities usually chosen for the primary side of t ransformers are : -the rated phase primary voltage, VI, R -the rated phase primary current, I I. H - the rated impedance presented by t he t r ansformer t o th e supply line, I Z1, n I = VI , nlII, n (3-24) -the power r ating of t he t r ansformer
8 1, H = VI, RII , R in the case of a single-phase transform er , and 81, H = 3V I , RII , R for a three-phase tr ansformer ,
70
Part One. Transformers
The base quantities usually chosen for the secondary side are: -the rated phase secondary voltage, V z, R = VI, R (WZIWl); -the rated phase secondary current, l z , R = 11, R (wl/w z); -the impedance presented to the line on the secondary side (at V z, Rand i ; R) I z; R I = V z , RlI z, R =
I z; R I (wZlwl)Z
(3-25)
-the base power on the primary side
S2, R = Sl, R To obtain a per-unit quantity on the primary side, its absolute value is divided by an appropriate base quantity taken in the same units
VI: l
=
Vl/V l , H
1,!:l = lIllI, R
I Z:I:O I = I z, III z; R I I Z*l I = I z, III z; R I P:I: l = Pl/S], R = Vi:l1,r.l cos
(3-26) (PI
where an asterisk stands for per unit. Sometimes, this index may be omitted, if the use of the per-unit notation is referred to in the text. The power equation in per-unit quantities is equally applicable to single- and three-phase transformers. The quantities associated with the secondary winding of a transformer can be expressed as per-unit quantities in anyone of two ways. For example, we may divide a given secondary quantity taken in absolute units by the corresponding secondary quantity taken as the base. Alternatively the secondary quantity may first be referred to the primary side by adjusting it in value by the turns ratio or the turns ratio squared, as the case may be, and the result may then be divided by the adopted base quantity associated with the primary side:
V 21V2 , H = V~fl1l, R 1,1:2 = 1 21I 2 , H = I~lIl, R (3-27) IZ:,:21 = I Z21/1 Z2,R I = IZ~ I/IZl,R I P*2 = P 2/S 2, H = P~/S l, R = V:J: 21,r. z cos crz Vr.z
=
For obvious reasons, the secondary quantities expressed on the per-unit basis carr y no referring index, .
71
Ch. 3 Processes in Transformer on Load
Anyone transformer equation may be written in per-unit notation . . To this end, it must be div ided through by the corresponding base quant it y . As an exam ple, let us do this for Eq. (3-13) which gives t he primar y voltage
or
.
.
V:I: I = -E:':I
. + Z:I:lI,1:l
(3-28)
For the current equation, we obtain
.
or
1*1
.
+ 1*2 =
.
(3-29)
1*0
As is seen , the per- unit equations are written in about the same way as those in absolute quantities, except that they have no indexes to show transferring to the primary side. Per-unit quantities are also helpful in expressing the parameters and quantities involved in equivalent circuits , and in constructing phasor diagrams . The per-u nit parameters and losses of a transformer va ry within a ralative ly narrow range of va lues and depend ma in ly on it s power rating . Let us establis h the relations between some of the perunit quantities . Among other things, we will find that the mutual inductive reactance varies inversely as the no-load current:
X:I: O = I Z:I:O I = I z, II I ZI, R I = (TTl, RII 0) (II, RIV I , R) = I I, RlI o
(3-30)
The resistance during magnetization can be expressed in terms of the no-load current and the core losses (the no-load or open-circuit losses) as
n., = =
Roll z; R I = P coreII, R/3I~Vl, (Pcore/3VI. RII, R) (II, RlI o)2
~ P:~, corel I~:Q
R
(3-31)
Part One. Transformers
Finally , the winding resistances are equal to the copper losses
R :I: 1 = RIll z; R I = 3RII~, R/3T1 1, RI 1 , R = PCu, /8 1, R = P:"cu, I R:1: 2 = R~/I Zl, =
R
(3-32)
I = 3R~I~, R/3VI , nIl , Ii
Pc«, 2/ 81, R = P:1: ClI ,
2
Using the above relations and data sheet values, the range of values for the basic per-unit quantities of three-phase power transformers rated from 25 to 500 000 kVA can readily be defined. Transformers with higher ratings have lower resistances and higher inductive reactances: I,~o =
+
P:':l, CU X*l
0.03 to 0.003
= P*o = 0.005 to 0.000 6
P:I', core P:1' 2, ClI
=
X:':2
=
P:I:, cu = 0.025 to 0.0025 0.03 to 0.07 (3-33) I Z:I:O I = X:I,o = 33 to 330 R:I: I = R u = 0.012 5 to 0.001 25 R,~o~= 5.5 to 65 As is seen from the above figures, as the power is changed by a factor of 20 000, the per-unit quantities change not more than ten-fold (in fact, X*l and X:I: 2 only change by a factor of 2). As can readily be checked , the same parameters expressed in absolute units will change by a factor of many hundred thousand . 3-7
=
The Effect of Load Variations on the Transformer
In a transformer, the primary and secondary windings are coupled by a mutual flux . Therefore, any change in load impedance (the impedance on the secondary side), with the primary voltage held constant, leads not only to a change in the secondary current, but also to a change in the magnetic flux, the magnetizing current, the primary current, and the secondary voltage. After the transients associated with a load change die out, the transformer settles down to a new steady state in which the electric and magnetic circuits are at equilibrium . In other words, the currents in the windings and the magnetic flux in the core take on va lues which again
Ch, 3 Proc ess es in Transformer on Load
73
satisfy the conditions of equilibrium for its electric circuits defined by the voltage equations, (3-13) or (3-19), and for its magnetic circuit define d by t he current equ ation (3-20) supplemented by the emf equations (3-7) and (3-8). A change in the secondary current immediately brings about a change in the peak magnetic flux cD m and the primary emf E l it induces. The state of equilibri um that exist ed on the primary side prior to that change and with which was associa ted a certa in definite pr imary cur rent is ups et, and a current is induced in the primary in accord with Eq. (3-13)
i,
=
n\ -
(-E\)] /Zl
The primary emf and the primary current keep varying un til t he magnetizing current (with the new value of 1 2 ) and the . . corresponding emf, -E l = ZoI o, build up enough for a st ead y-state current to appear in the primar y winding. Considering together the equations written earlier, the primar y current (Fig. 3-7) ma y be written
t. = l\. R/(Zo + Zl) . . =
10 • NL - I~Zo /(Z o
where
.
1 0 • NL
j~Zo/(Zo
+ Zl)
.
=
+ Zr)
VI, R/(Zo
(3-34)
+ Zl )
is the ma gnetizing current at no-load . Because Zl ~ Zo, with a sufficiently large load we have 1 0 • NL ~ 1
and
.
II
=
.
-I~
The magnetic flu x varies directly wi th E, which is in t urn a function of t he magnitude and phase of the primary current
..
-E l
or, in per-unit
=
.
VI. R - ZlI l
..
.
-E*l = V*l . R - Z*l!,l:l (3-35) At no-load , when I I = 1 0 • NL ~ 0, the emf and the flux are equal to the prim ar y voltage taken as unity
eon =
CD n = cD ~/cD ~ l R =
V'!'l, R
= 1
Part One. Transformers
74
At rated load (I*1 = 1*1, R = 1), the emf and fl ux change in significantly in comparison with their no-load' values. . . Ev en when t he phase of I I , R is such that ZlI1 is in the same or oppo sit e direction with VI , R, the emf is
where Z','1 = 0.03 to 0. 07 (see Eqs. 3-33). Thu s , even in t he wor st loading case, with the load rising fr om zero t o its full value (see Fi g. 3-7), t he emf an d flux change by as li t tle as Z,!:1 X tOO = 3 % 7% Given other ph ases for 1, an d 1 2 , the changes in the emf an d flux are still more insignificant . Referring to Figs. 3-4 and 3-5, t he emf decre ases in the case of a resistive-ind ucti ve load an d m ay in crease if t he lo ad is resistive-ca paci t ive ~====~~§~';I2~ and the ph ase sh ift is cl ose I<:. t o - n/2. The effect of lo a d 1 o var iations on the m agnetizin g current is likewise insignifiFig. 3-7 Flux , primar y emf, cant. It can be eva luate d bv magnet izin g current and priE q . (3-20) : . mary current as functions of
to
secon dary cur rent: soli d line , resistive-inductive load , CP2> 0; dashed line , resistive-capa citive load , CJl:i < 0; Io x=Io.NL
i o = - E1 /Z o
.
. .
= (VI - Z l I 1)/Z o
.
=1 0 ,
NL -
I 1Z 1 /Z o (3-3G)
In a linear approx ima t ion , this cur re nt varies in the same m an ner as t he pr imar y emf. If we in clu de t he nonlinear beh aviour of t he m agne ti c circuit whi ch causes Zo to vary as well , this cha nge bec omes more pr onounced . The effect of nonlin earity m ay be account ed for by using t he ma gnetizat ion curve, (J) = f (I o) , shown in Fig. 2-9. Plots of II , E 1 , (D , and 1 0 as fun ctions of 1 2 for inductive and capacitive load s ar e sh own in Fi g. 3-7,
75
Ch. 3 Pr ocesses in Transforme r on Load
3.8
Energy Conversion in a Loaded Transformer
The energy fed int o t he pr imary winding of a transformer from a supply li ne is customarily treated as t he sum of two parts. One pa rt is delivered to load and is partly lost in the transformer it self. The average time rate of this unidirectional flow of energy is called the active power drawn by the primary winding from t he supply line. For a single-phase transformer, it is given by PI
=
VIII cos
CPI =
VI lla
=
VIall
(3-37)
where I l a = I I cos CPI is the active current VI a = VI COS CPI is the active voltage The act ive power is taken as positi ve, PI > 0, if CPI li es anywhere between _90 0 and +90 0 (electrical) . The other part of in put energy is spent to establish magnet ic fie lds in t he transformer it self * and also electric an d magnetic fields in the load. The direction of this energy is changed twice every cycle , so the res pective power averaged over a cycle is zero . The transfer of energy between the supply line and a field (electric or magnetic) is described in terms of the peak inst ant aneous power , called the reactive power. The reactive power drawn by t he pri mary winding of a single-phase transformer from the sup ply li ne is given by QI
=
VIII sin
CPI =
VII l r
=
VlrI
I
(3-38)
where I l r = I I sin CPI is the rms value of reactive current Vir = VI sin CPI is t he rms va lue of react ive voltage The reactive power is assumed to be posit ive, QI > 0, if the reactive current is lagging beh ind the voltage, < CPI <:n, wh ich corresponds to a resistive-inductive load. The reactive power is taken to be negative, QI < 0, if the reactive current is lead ing t he voltage, > (PI> - :n , which cor resp onds to a resistive-capacitive load . Consider the conversion of active power in a transformer. Let us write the active component of the primary voltage, VIa = VI COS (PI, as the sum of projections of Eland the
°
°
* The energy associated with the electric field wi thin th e transformer is usually ne~lecte~,
Part One. Transformers
76
voltage drop R]I] (see the phasor diagram in Fig. 3-8a) VIa = V] cos cp] = E] cos 11J] R]I] and the active power p] supplied to the primary winding by a supply line (its direction is shown in Fig. 3-9 by an arrow) as the sum of two components p] = (11] cos (p]) I] = (E] cos 1h) I] (R]I]) I] (3-39) The term fiR 1 = P C ll , ] is the copper loss in the primary winding, that is, the power lost as heat dissipated in the primary turns (see the arrows in Fig. 3-9). Referring to Fig . 3-8b, the active component of the primary current, I] cos 1IJ], is shown as the sum of the active components of the magnetizing current locos CPo and of the secondary current I~ cos 1P2' Therefore, the term (E 1 cos 1P]) I] may likewise he written as the sum of two components: E]I] cos 1p] = E2I~ cos 'ljJ2 E]I O cos CPo
+
+
+
=
P e~
+ P eor e
(3-40)
The term P em = E]I~ cos 11J2 is called electromagnetic power. It is transferred inductively from the primary to the secondary winding. The flow of electromagnetic power crosses the channel between the two windings (Fig. 3.9). The term E]I o cos (Po = E]I oa = P eor e represents core loss in the transformer. Referring to Fig. 3-8c, the active component of the primary emf, E] cos 1P2' can be expressed in terms of the active component of the secondary voltage, V~ cos (P2' and resistive voltage drop, R~I~. Hence, we may write P em = (E] cos 1P2) I~ = (V~ cos (P2) I~ = P2 P CU, 2
+
+ (R~I~) I~
(3-41) Some of the electromagnetic power is expended to make up for the copper loss in the secondary winding, P CU,2
=
I~2R~
The remainder,
P 2 = V~I~ cos CP2 is transferred to the load conductively (see Fig. 3-9). The active power input to a transformer is p] = P CU, ] P rpm =
P CU,2
+ P eor e + P em + P~
(3-42)
77
Ch. 3 Processes in 'transforme r on Load
.
'r
- E,=-Ez
(a) (a)
Fig. 3-8 Phasor diagrams of a transforme r oper ating into a resist ive inductive load (see Fig . 3-4) ,
\
\ I I
If
V,
'\
7
P~qf ) -, /
( k-. .....
\
,11
\ >JlT", tl.ll ~
It. ~ "VI
r:rm
~
r-,
Pea;'
( )
(r ~ -:..c.
ow. I I I I
imr
..."
iTA
'I
~
[2 ~
I
Pco;e(Qo) Q6 f
l
6 2m
~
~r
'-'...l)"'\;
lJI
~I
IV
~I
~ fm
i
rQz
V2
)
1\
TV
~
I
Fig. 3-9 Flo ws of active and reac tive power in a loaded transformer
.... 78
Part One. Transformers
or where
2j Ploss = PCu ,1
+ P Cu, 2 + P eore
The conversion of reactive power in a transformer may be treated in a similar way, likewise referring to the phasor diagram in Fig. 3-8. The primary reactive power can be treated as the sum of the fo.llowing components:
Q1 = (VI sin (PI) II = (E 1 sin tPI) II
= (E 1 sin tPI) II
+ (XIII) II
+ QUI
(3-43)
where
E 1 (II sin tp1) = E 1 (I~ sin
= Qam Qem
=
~)2)
+ E 1 (1
sin CPo)
+ Qo
(E 1 sin tl)2) I~ = (V~ sin (P2) I~
= Q2
0
+ (X~I~) I~
+ QU2
The physical meaning' of the reactive components is as follows: Qem = E 1 (I~ sin tp2) is the reactive power transferred inductively from the primary to the secondary winding .Qo = E 1 (fo sin (Po) = ElIoT is the reactive power required to establish the magnetizing (mutual) flux QUI = X II; is the reactive power expended in setting up the leakage flux 011 the primary side QU2 = X ~I~2 is the reactive power expended in setting up the leakage flux on the secondary side Q2 = V~I~ sin CP2 is the reactive power drawn by the load The directions of these power components are shown in Fig. 3-9.
tho it Transformation of 3-phase Currents and Voltages
4
Transformation of Three-Phase Currents and Voltages
4-1
Methods of Three-Phase Transformation. Winding Connections
Three-phase currents and voltages may be transformed either by a bank of three single-phase two-winding transformers (Fig. 4-1) or by a three-phase , two-winding transformer whose windings are put on a common magnetic circuit of the core or the shell-and-core (five-leg core) construction (Fig. 4-2). The magnetic circuit of a core-type three-phase transformer can be formed from those of three single-phase transformers combined together. Arranging the single-phase transformers as shown in Fig. 4-3a and combining the limbs that do not carry any windings (Fig. 4-3b), it can be noted that with a symmetrical set of voltages (Fig. 4-1), the flux in the combined limb, equal to the sum of the phase fluxes, vanishes
.
CD A
.
.
+ cJ) + CDc B
=
0
Therefore, we are free to remove the combined limb altogether (Fig. 4-3b). The magnetic circuit thus derived is sometimes used in practice and is known as a spatial threephase core. In such a transformer, the instantaneous fluxes in limbs A and C have their paths completed through limb B, because . CD B = -cJ) A - CDc
.
.
Most frequently, however, the magnetic circuit of a threephase transformer is built as a flat (or planar) core-type structure (Fig. 4-3c), with the limbs arranged to lie in a common plane. It differs from a spatial core in that the phase B core has no yokes and the axes of all the phase legs and yokes lie in a common plane. A flat core shows a degree of asymmetry which results in an asymmetry of the magnetizing currents. However, this is of minor importance because these currents are small. The shell-and-core (five-leg core) form of the magnetic circuit is employed in high-power transformers (Fig . 4-2b) so as to reduce the yoke height. This is achieved owing
vAB
v.Be
rpA
A
cF
VA
X
:>-r-
Va
Pe
r-;:::;::::::8
C
Va
y
b ~
Vi,
't-
'-I
~
v,,+.
Vc
Z
c :>-r-
*=-
PC
'-F
Vi,e
F ig. 4-1 Three-phase transform at ion by a bank of si ng le-phase transformers
c
A
8
,L
I
T
.L,
.1
PA-
rPy e---:
,- v"b
X
Z
c
b
CL
I:r+-Vbe
Y
HV
HV
LV
LV
h Z
(CL )
Fig. 4-2 Three-phase t ra nsformers: (a) core- ty pe; (b) sh ell- and-core (five-leg ra re) t ype
_----.c
ij,=o
(b)
(0 )
Fig. 4-3 Three single-phase magn etic circuits transform ed in to (a) and (b) a "spatial" core and (c) a plana r core
ell. 4 Transf ormation of 3-Phase Currents and Voltages
81
to the formation of extra closed paths (through the si(~e limbs) for the magnetic fluxes . In a shell-and-core m agnetic circuit, the phase fluxes cD A' cD B ' and cD C may be visualized as composed of the individual loop fluxes cD u , c = CI)u _. cDc. As follows from the flux phasor dia gram in Fig . 4-2c, construct ed on the basis of design data for B
A
C
Fig. 4-4 Zigzag-star connection
the core , the fluxes in the yoke loops form a nearly symmetrical star (cD u is somewhat smaller than cD b = cDc), and the fluxes in yokes band care 1/ V if t imes the fluxes in the phase limbs. (It is to be recalled t hat in the core-form transformer of Fig. 4-2a the yoke fluxes do not differ from those in the phase limbs.) A three-phase transformer is far more economical , so banks of single-phase transformers are only used where a single three-phase transformer of t he same power r ating would have a prohibitively large weight or size . The phase windings may be connected in a star (Figs . 4-1 and 4-2a) , a delta (the LV winding in Fig . 4-2b) and , though seldom, a zigzag (Fig. 4-4) . In a star connection, V ll n e = V A B = V B C = V C A =
where .and JUn e
6-0169
=
Jph
V 3V p h
Part Dne. Transformers
In a delta connection , V ilhe
= Vp h =
Va
=
Vb
=
Vc
=
V ab
=
V bc = V ac
In a zigzag (or interconnected-star) connection, the vol tage and current relations are the same as in a star conne ction, but in order to obtain the same phase voltage the number of turns per phase must be increased 21Vf = 1.16 times. This leads to a higher cost which is to some degree offset by an improvement in performance (as regards t he waveform of phase emfs and fluxes) . The star connection is designated by a "y" sign or the let t er Y (sometimes spelled in full as "a wye connection") . The delta conn ection is designated by the Greek let t er f... or the Roman let t er D. The zigzag connection is designated by the let t er Z. A winding carrying the highest rated (line) voltage is referred to as a high-voltage (HV) winding. A winding carrying the lowest rated (line) voltage is referred to as a low-voltage (LV) winding . In the Soviet Union , the manner in which the windings of a two-winding transformer are connected is designated by a fraction , with t he form of connection of t he HV winding placed in the numerator and t ha t of the LV winding in the denomina t or . For exam ple, t he form of connection for the transform er in Fig. 4-2a will be designat ed Y /Y , and for that in Fig. 4-2, Y IJ],. Outside t he Sovie t Union , the same forms of connections may al ternatively be desig na t ed as Y-Y (or wye~wye) and Y-b. (or wye-delta). Under a relevant USSR standard, the start and finish of t he HV winding in a single-phase tr ansformer are marked as A and X, and those of the LV winding, as a and x . The st arts and finishes of the HV winding in a three-phase transformer will be designated as A, B , C and X, Y , Z , and those of the LV winding, as a, b, c and x , y, z. The neutral wire is designated as N , and the centre (or zero) point of a star connection is marked 0 on the HV side and 0 on the LV side.
th. 4 Transformation of 3-phase Currents and Voltages 4-2
A Three-Phase Transformer on a Balanced Load
The performance of a three-phase transformer on a balanced load may be described in terms of the t heory developed for single-phase transformers. In fact, all the relations derived for a single-phase transformer fully apply to any phase formed by a primary and a secondary on a common limb . Some adjustment needs only to be made for the magnetization of the core in a three-phase transformer (see Sec. 4-4) and the calculation of the magnetizing current (see Sec. 8-1). However, the magnetizing currents are negligible in comparison with the load currents, and the unbalance in these currents related to the dissymmetry of a flat (planar) three-phase magnetic circuit is of minor importance. Therefore, the actual calculations are based on an equivalent balanced set of averaged magnetizing currents to which corre spond the averaged mutual impedances (ZOA = Zo B = Zoc) accounting for the magnetic coup ling between the various phase windings. Because the leakage fluxes are concentrated in the space t aken up by the windings themselves (see Sec. 8-2), the leakage flux es of the individual phases may be considered independently of one another, whereas the leakage impedances of the phase windings equal in size may be deemed identical (X 1A = X 1 B = X 1 C , X 2 A = X 2 B = X 2 C ) . This also goes for the phase resistances (R 1 and R 2 ) . Therefore, with balanced primary line voltages and balanced load impedances, the phase currents and voltages are likewise balanced. In the circumstances, the line and phase quantities are connected by simple relations : [ph
= [line
V ph = vline/ -V 3
in the case of a st ar connection , and
V ph = V ll ne [Ph
= [line/ Y 3
in the case of a delta connection, and we may describe the ' performance of any of the phases, using the equations, equivalent circuit, and phasor diagram developed for a singleG*
84
'j I I
il
Part One. Transformers
phase transformer (see Chap. 3), extended to include the phase voltages, currents and impedances, and also the transformation ratio in terms of phase voltages or turns: 11,21
4-3
=
V 2 , R(ph/ V1,
H(ph)
=
W 2/W 1
Phase Displacement Reference ~umbers
For proper use of transformers in power systems, it is important to know the phase displacement between the emfs on the HV and LV sides, as measured across like terminals. For example, on the HV side the emf must be measured across terminals A and B, and on the LV side, across terminals a and b. In single-phase transformers, the phase displacement between the emfs on the HV and LV sides may be 0 0 or 1800 • The line emfs on the HV and LV sides in three-phase transformers can only be displaced in phase through an angle which is a multiple of 30°. Transformers having the same phase displacement between their HV and LV emfs fall in the same phase displacement (or reference phasor) group, each group being assigned a distinct reference number. Since an angle of 30° is exactly the angle between adjacent hour markings on a clock dial, a convention adopted internationally is to indicate phase displacement as a clock figure representing the hour read by a clock when the minute hand takes the place of the line emf ph asor on the primary side and is set at 12 o'clock, and the hour hand represents the line emf phasor on the secondary side. The "time" thus read is the reference number assigned. An example of this convention for phase displacement group 11 is shown in Fig. 4-5. The positive directions adopted are from A to B and from a to b. In the designation of a transformer, the reference number follows the symbol for the winding connection (for example, Y/Y-O or Y/ ~-11). If the phase windings on the HV and LV sides are wound in the same direction, the LV leads may be marked in any one of two ways, shown in Fig. 4-6. Because the windings link the same flux, the emfs labelled by the same letters will be in phase in case (a), and in anti-phase in case (b). (As the flux decreases, the HV and LV emfs will be directed
Ch, 4 Transformation of 3-Phase Cur rents and Volt ages
85
fr om X t o A and from x t o a in case (a), and the LV emf will be directed from a to z in case (b). ) As already n oted , sin gle-phase tran sform ers can onl y have zero or 180 0 ph ase displacem ent. Consequ ently , they m ay bear onl y reference number (fZ) o (12) or 6, respectivel y . For o brevit y, they are designated by t he symbols 111-0 (see Fig. 4-6 a) an d II I-6 (Fig. 4-6 b) . A cha nge from group 0 (12) to J 9 group 6 call s for n o connection cha nges in the transforme r it self; it will suffic e to rem ark lead a as x , and lead x S 6 as a. In the USSR , singleFig. 4-5 Clock-hour conve nt ion ph ase tra nsforme rs are ma nto design ate ph ase displ acemen t ufa ctured with a 111-0 windgrou ps ing connect ion . Extending the foregoing to the HV an d LV ph ase windings of a three-ph ase t ransfor mer an d referring to the pha sor diagr am, it can be seen t hat a Y/Y three-phase tra nsforme r with the leads m ark ed as shown in Fig. 1:-7a falls in ph ase
q; t
1/1-0 A
X LV
a
0
J
s: (a)
t
1/1-6 0 A
X oX
r x >t
a
(f(6 ) []J
Fig . 4-6 Lead markings and phase displacement gro up numbers single-phase transfo rmers
1"01'
disp lace ment group 0, so it s designation is Y/Y -O. (The phase emf ax is in the direction of the phase emf A X ; t he phase emf by is in the direction of the ph ase emf BY, cz -+ CZ, an d the line emf ab is in the direction of the line emf A B .) If we re-label the leads , going all the way r ound the circle , we can converL a group 0 trans former to a group 4 or grou p 8 t ran sform er withou t actually sh ift ing any connections in si de t he t r ansfo rme r. Wi th the leads m arked as
86
Part One. Transformers
shown in the parentheses, (a), (b) an d (c), the line emf (a) (b) is in the direction of the li ne emf B C (because these emfs are measured across th e windings put on the sam e limbs), and the t ran sfor mer is conver ted to one in phase displaceme nt group (4). W ith the lead s labelled as shown in ' the
W
y(y-o
[e]b~a] [Q] B
.
x
z
lLJ
(e)[b] (a)[e] (b)[a]
Y(Y-6
[b]a, (~ (c)
C
a [a~[e]
xm
.!
B
Z (z)[y] (x)[z] (y)[x] Y
y
b
A ~ --
(a.)b
c
z
abe (e)[b] (a)[c] (b)[a]
(z)[y] (x)[z] (U)[x] [8] [bJ
ABC
B
l?7(fJ
(e) arb]
(e)[b] (a)[c] (b) [a]
awe .
A
(&)
.b.
W
Y/A-11
ABC
b(a){a]e [e] . (b)
a
(MJ
x
y
z
x
y
z
lli]~X a (z)[y] (x)[z] (y)[x] .b Z
y
[a] (a)a --. (3) / A (b)
[7J l [bJ (e)
C
abc
(e)[b] (a)Ee] (b)[q]
Fig. 4-7 Lead markings and ph ase displacement group numbers for . th ree-phase t ransform ers
bracket s, the emf [ aJ [bI is in the dir ect ion of emf CA , aud the transformer is converted t o one fa ll ing in ph ase displace ment gr ou p [SI. If we wish to ob t ain a y / y-(j transforme r (Fi g. 4-7b), we must shift the neu tra l jumper that reverses the ph ase of all the emfs (the emf ab is in anti-phase with t he e mf AB) . If we re-la bel the lea ds all the way roun d t he circle , a group (i translonner will be conve rted to a group ('10) or a gr oup [21 t ra nsformer . (Th e resp ecti ve markings are given in Fig. 4-7b in paren theses an d brackets, respect-
Ch. 4 Transformation of 3-Phase Currents and Voltages
87
iv-ely.) This exhausts all the likely even reference numbers that can be derived for a Y/Y connection. . Odd phase displacement clock numbers are obtained for a Y /11 conn ection. With the leads marked without parentheses or brackets (a, b, c, x , y, and z in Fig. 4-7c), the line emf ab which is at the same time the phase emf yb is in the dir ect ion of the emf YB , and the transformer falls in phase displacement group H. If we re-Iahel the leads all the way round the circle as shown in the parentheses and brackets in Fig. 4-7c, we shall obtain group (3) and group [7I. (Each time we re-label the leads, a particular emf is turned through an angle 120° = 4 X 30° and the reference number is in cremented by 4.) If we in t erchange t he st ar ts and finishes of the phase win dings, a group 11 t ransforme r will become a group 5 tra nsformer (the respe ctive markings ar e given without parentheses in Fig. 4-7d). Finally , if we re-label the leads all the way round the circle as shown in Fig. 4-7d, we shall obtain group (9) and group [L]. . Of all the likely phase displacement groups , three-phase two-win ding trans form ers of Sovi et . manufacture are only ava ilable in group. 0 and gro up 11, wit h t he (neutral) lead of the star available for connection where necessar y (Y/Y n-O, Y/11-11, Y nll1-11). Additionally , some tra nsformers ma y have their HV windings connected in MY n-11. As is seen from Fig. 4-8, the delta connection in this case is obtained differently than in a Y /11-11 transformer . (A is connect ed to Z , whereas in a delta on the LV side a was connected t o y.) If t he delta on the HV side were connected in the same man ner as the delta on the LV si de in a Y /11-11 connection in Fig. 4-8c, the M Y conn ection would fall in ph ase displacement group 1, rather th an H. It is of interest to see how , in the general case, the phase displacement numb er will ch ange if we make the LV winding an HV one and t he HV winding an LV one , whil e retaining their connection and markings. Obviously , the pha se displacement between the HV and LV line emfs AB and ab will be the same as before, being 30° X N (Fig. 4-9) . Ho wever , the emf ab in t he diagram shown by dash ed lines will now lead the emf A IJ b y the same angle 30° X N by whi ch it lags behind in the dia gram shown by solid lines. Therefore , if we coun t the phase displacement from emf AB to emf ab always clockwise, the
Part One. Tra nsfor mers
88
ang le 30° X N ' in the second case will complement the original 30° X N angle to 360° 30° X N'
+ 30°
X
N
=
360°
Th us, after the above manipulation, the phase displacement reference number N ' can [Q] be found as
B~: Maz~ IW.
awe.
N ' = 12 - N
where N is the original clock For N = 11, N ' figure . b~C = 12 - 11 = 1. The /1 /Y n-11 connect ion x y z a. (N ' = 11) can be derived from t he Y nl /1-1 connection (N Fig . 4-8 i1/Y-1J Lra nsfor mer = 12 - N' = 1) whi ch is in o o turn derived from the Ynl/1-11 Ii connection by modifying the J delta connection (see below). J Which ph ase displ acement (fZ-N) JOo bA~ group a transforme r will fall O N'XJO in depen ds not only on the sequence of marking the starts a A A8a and finishes of its LV winding, NIIJO but also on how the phase windings are connected in a Fig . 4-9 Using the clock figur e delta . Under a relevant USSR and lead markings of th e LV (BV) winding to deri ve th e standard, the delta on the LV clock figure and lead markings side must be formed by confor th e HV (LV) winding necting lead a to lead y, lead b to lead z, and lead c to lead x, as shown in Fig. 4-7 or 4-10 by solid li nes. If, instead, we form a delta by connecting term inal a to te rminal z, terminal b t o term inal x, and terminal c to terminal y (as shown in Fig. 4-10 by dashed lines) , t he LV emf, say ab, will be turned t hrough 180 - 120 = 2 X 30° clockwise , and the clock figure will be in cremented by 2. (With the leads marked as shown in Fig. 4-10, group 3 will change to group 3 + 2 = 5.) Wi th t he conn ect ion shown by soli d lines, t he li ne emf ab, which is at the same t ime the phase emf yb , is in t he direction of t he emf ZC. With the connection shown by dashed li nes,
'\];1
8tt
I II
c!j
Ch, 4 Transform ation of 3-Phase Currents and Voltages
t he line emf ab which is now the phase emf ax is in the direct ion of the emf BY . That is , it is turned from its original direct ion through 2 X 30°. This rule extends to any other odd phase displacement groups . So , wh en the delta is formed in any other way t ha n recommended, reference number N will becom e reference number N = N 2. More specifica lly , inst ead of group 11
+
I
a a.
.0
cL~~0JV ~ o b
c
ab
'm'
I
,I ,I r
L
z
I
s
x
....J
C
+
'V
B~C
~~ . b' L~J \
r-,
Fig . 4-10 Phase disp la cement group resulting from th e manner of delta connection
there will be group 1; in stead of group 3 th ere will be group 5; instead of group 7 there will be group 9; instead of group 1 there will be group 3; in stead of group 5 t here will be gr oup 7; and in stead of group 9 there will be group 11. A relevant USSR standard recommends that the zigzag connect ion should solely be used on the LV sid e, and prescribes only one group , namel y Y/Z n-11 , t hat is , one wi th the neutral li ne at the zigzag available for connect ion . 4-4
Yhe Behaviour of a Three-Phase Transformer During Magnetic Field Formation
In discussing single-phase tran sformer in Sec. 2-5, we have seen that when the magnetic flux is sinusoida l , cD = cD m sin wt, the magnetizing current i o is nonsinusoidal. In addit ion to the fund amental component, I ol • m sin wt , varying with an angular frequency oi , the magnetizing
90
current with an integers
Part One. Transformers i o contains odd harmonics", 1011, In sin kest, varying
angular frequency kes, where k stands for the 3, 5, 7, 11, '13 , and so on, i o = lot 'ln sin
wi + ~ 11
101l .1n
sin kwt
Distortion in the waveform of i o increases (the odd harmonics grow in amplitude) as the magnetization characteristic CD = f (to) becomes progressively more nonlinear. In three-phase transformers, the nonlinearity of the magnetization charncteristic lead s 10 far more complex effects,
r
Fi g. 4-11 H armonic compone nts of a symmetrical set of three-phase cur rents
and the manner in which they manifest themselves depends on the type of winding connection and the core design. Three-phase transformation may be accompanied by distortion in the sinusoidal waveshape not onl y of the magnetizing currents, but also of the magnetic fluxes and phase voltages. Before going any further, it appears worth while reca lling some features of the harmonic components in symmetrical three-phase systems of emfs, voltages, currents and fluxes . " Here and else wh ere, only th e re active components of th e no-lo ad current are considered.
91
Ch. 4 Transformation of 3-Phase Currents and Voltages
A sy mmetric al t hree-phase syst em h as t hree sets of nonsinuso idal phase quantities (currents, voltages, an d fluxes) t h at, at an y ins tant, are equal in magnitude , waveform and fundamental frequency, but are separ at ed in time-phase by one-third of a period T 1 = 2'Jt/ oi . A symmetrical three-phase system of nonsinusoidal currents i A , i B, and i c is shown in Fig . 4-11 . The fundamental terms of the phase quantities (say, currents) are likewise separated in t ime-phase by a third of a cycle and form a symmetrical syst em t hat has a p ositive phase sequence, PPS (Fig. 4-11)
= 1/ 2" 1 A 1 sin wt
iA I
'V21 sin (wt - 2'Jt/3) =V2 1A I sin (wt + 2'Jt/3)
iB I = iCI
A 1
(
The sum of phase quantities (say, currents) in t he case of the positive ph ase sequence is always zero
I
(4-1)
I
This can readily be proved if we write t he ph ase qu anti ties (say , cur rents ) as a symmetrical set of complex qu an tl ties
.
1 AI'
1 B1
=
.
1 A 1 ex p (-j2'Jt/3),
~/C 1 = j A l exp (j2'Jt/3) Similar symmetrical systems with a positive or negative phase sequence are formed by all harmonics whose order k is not a multiple of three (that is , other than triplett harmonics*) (4-2) k = 6c + 1 where c Thus,
=
0, 1, 2, 3,
=V 2" 1,lh sin kiot i B h = V 2I.Ah sin k (wt- 2'Jt/3) =V2I 'l h sin (kw t + iA h
i C h =V 2 I A h sin k (wt
+ 2'Jt/3) = V 2l
Ah
sin (kwt
2'Jt/3)
+ 2'Jt/3)
* Triplen harmonics refer to all harmon ics whic h are mu lt iples of three.
92
Part One. Transformers
+
When k = Gc 1, in whi ch case the upper signs in the arguments of the sines apply, a positive ph ase sequence of quantities (say, curre nts) is formed. When k = 6c - 1, in which case the lower signs in the arguments of the sines apply, a negative phase sequence (NPS) of quantities (say, currents) is formed. The sum of the kth harmonics of the phase quantities is likewise equal to zero
+
°
+
iBh i Ch = (4-3) iAh The harmonics of phase quantities , whos e ord er is an integral multiple of three (triplen harmoni es), k
=
Gc
+3
(4-4)
where c = 0, 1, 2, 3 , .. ., form a zero phase sequence system .(ZPS) . The t riplon-harm onic terms are all in phase iBh = V2I Ah sin k(wt+2n/3) (CI ,)
= V ZI Ah sin (kwt) = i Ah
=
iB h
=
i,lh
(4-5)
i Ch
For the third-harmonic t erms , t his rule is illustrated in Fig . 4-11. Now we sha ll see wh at restraints are imposed on nOIJsinusoidal curr ents hy the v arious winding connect ion types. All harmonic terms other t ha n trip1en , that is , the Ist, 5th , 7th, 1'11h, 13th, etc. harmonics, form a positive or negative phase sequence (PPS or NPS) system and exi st in the phase windings connected in any manner. In the neutral wire, these harmonics do not exist , he cause their sum is always zero . The line currents of these ·harmonic terms , with th e windings connecte d in a delta , are V 3 times the phase cur rents, for example
I A B 1 = I BC l = I C A l = V 31.'1.1
.
.
.
(4-G)
where l ABI = I Al - I B1' All triplen harmonics , that is , those harmonics whose order is an integer multiple of three , that is 3, 9, 15, etc ., cannot exist in a wye -connection without a neutral wire (Fig . 4-12). In a three-phase star- connected system with the neutral wire , Y n- availahle for connection, the neutral
Ch. 4 Transformation of 3-Phase Currents and Voltages
93
wire carries a current equal to three times the phase current . For example . (4-7)
In a three-phase delta-connected system, the third-harmon ic phase currents circulate within the closed path formed by the delta, and are not present in the line wires .
A
B
C
id N
3i 3
A
B
C
,W
A
i.f X
I iJ ~O I
Fig . 4- 12 Third-harmonic currents in various forms of winding connections
A similar situation exists for nonsinusoidal fluxes in the various core designs . * In a three-phase bank of single-phase transformers, such as shown in Fig. 4-13a, the third-harmonic phase fluxes cI> AS = cI> BS = cI>cs = cI>s have their path completed within the core in the same manner as the fundamental terms . The depe ndence of cI> ~ (l) l cI>s on i o is re presentable by the magnetization curve cI> = f (io) (see Fig. 4-13a). In a shell -and-core (five-leg core-type) transformer, the outer limbs play the same part as the neutral wire in a starconnected winding. They form . a sp lit "neutral" core which provides a closed path for the third-harmonic fluxes. In each outer limb, the third-harmonic flux is 3cI>s/2. The outer limbs als o provide a closed path for the fundamental fluxes cD AI' (l) B l' and cI> 01 ' Therefore, the depencD 2 on i o may, dence of the nonsinusoidal fl ux cD ~ (l)l to a first approximation, be deemed similar to the magnetization characteristic ' of a she ll -and-core transformer with a sinusoidal fl ux (see Fig. 4-13a).
+
+
* In our further discussion, we shall only be concerned with the fundamental and third-harmonic terms.
Part One. 'I'ra nsformerc
In a throe-phase" core-type '. ,transformer-. .which has no "neutr al " core in th e form of outer limbs , th e th ird-h armonic phase fluxes (Fig . 4-13b) have the ir path comple te d thr ough the tank walls and run in to the appreciable opp osition pr esen ted by nonmagnetic gaps. Because of t h is, the reluctance seen by the third-h armonic flux es is t ens of t im es (r:) ep
rp
=o=oU m J
,.- --....---
J
J
I
/
' 'PJ
_
-
cf>3 -:
-
rf>;y-
/
/ / --- -
/
\
\
\
~
I'-
\--
"""'~- -- -~--
/.
Tank
(6)
rp
(a)
(b)
Fig. !i-13 Third -h armoni c flux es in various core des igns
higher than that seen by the fun dament al fluxes traver sin g a closed path within the core. In determining the fund amental an d third-harmoni c te rms, we have to invoke different magnetization characteristics . For the third-harmonic flux, this is the lin eal' magnetizing characteristic, cDs = is (is). For the fund amental flux, th is is the nonlinear magnetization cha racterist ic, cD 1 = II (i o), derived for the sinusoidal flu x upon replacing i o with (i o - is) which gives rise t o the mmf associat ed with the fundamental flux (Fig. 4.13b)* .
* This is t r ue, if we consi der the fun da me nta l and t h ird-ha rmo nic terms on ly .
- ,
'
.
,
I
.
. '
,
.
.
,
Gil, 4 Transfo rm ation of 3-Pllase Curr ents and Voltages
Now we shall examine the waveformsof magnetizing -currents, flux es and voltages asso ci ated with the va ri ouswin ding connections an d core designs, assuming t hat at no -l oad the transformer is energized from the HV side . 1. A three-pha se hank of single-phase tr an sf ormer s . M Y connection. With the supply voltage impressed on the delta-connected HV side , the phase voltage is the same as t he sinusoidal line voltage . Therefore, all t he single-phase transformers in the bank are connected t o carry a sinusoidal voltage , an d t hey are magnetized in the same manner as an individual single-phase t r ansforme r is magnetiz ed with a sinusoidal volt age (see Sec. 2-5) . In other words, the flux varies sinusoidally and the magnetizing phase current, nonsinusoidally. The m agnetiz in g current has t he waveshape shown in Fig . 2-4. The line conductors carry harmonic currents whose ord er is no t a multiple of three (esp ecially , t he fu n damental term i OI ,lI n e) ' Their rms values are V:3 tim es the rms values of t he phase quantities l
o1,llne =
VS l o1
[see Eq . (4-6)1. The t riplen h armonics (especi ally i o3 ) t ra verse a closed path within the delta, and ar e no t present in the line con duc to rs (see Fig. 4-12). Bec aus e t he ph ase fluxes con t ain sol ely the fun damental t erms (<3) AI '
96
Part One. Transformers
fundamental flux and the third-harmonic flux, <:t> ~ <:t>1 -+ <:t>3 (if we reca ll that they will not induce the 5th and 7th harmonic phase voltages) . The waveform of the magnetic flux is determined graphically , using the magnetization curve <1) = t (i o) for a sinusoidal magnetizing current, i o ~ i 01 ' (Actually , this current
wt
Fig . 4-t4 Harmonic compon ent s of curren t, flux and ph ase emf in a hank of Y/Y single-phas e t ransformers
contains the 5th and 7th harmonics, because these terms are not present in the flux.) The phase flux in Fig . 4-'1 4 is determined to within the t hird harmonic. As is seen , t he flux waveform is heavily fl at tened. This leads to a distortion in the sinusoidal waveform of the phase emfs and voltages . With a fla t tened flux waveform , the phase emf has a well-defined peak (see Fig. 4-14) which may exceed the fundamenta l peak by as much as 60 % to 90 %. Accordingly, the t ransformer insulation must be designed for this peak, and this leads to a more expensive transformer. This is the reason why the Y/Y connection is not used for banks of t ransformers or where the magnetic circuit is of t he she ll -and-core form.
97
Ch. 4 Transformation of 3-Phase Currents and Voltages
Fro m this point of view, it is pr eferable t o use th e M Y or Y I ~ connection. If such a transformer is energized on the star-connected HV side, the departure of the fluxes and phase emfs from the sinusoidal waveshape will be negligible . With t his form of connection and with an y core design, the third-harmonic fluxes are reduced by the third-harmonic currents for which the path is closed within the delta-connected LV winding . The third-harmonic fluxes (Fig . 4-15) :
~\
30
,/>;,
-- -- --~.rA
eJi
(/1"
P"L1 .'
8"1l
4>"
P.r.l1
-0;.
lSi!>
fr
3<-
It"
Fig. 4-15 Damping of third-harmonic currents by the currents circul at ing around a closed delta
induce in the phases of the delta-connected winding the third-harm onic emfs e3/'. which give rise to third-harmonic currents i 3~ . Because the delta presents a low (practically inductive) impedance , the currents lag behin d the emfs by an angle close to nl2 and set up third-harmonic flu xes
98
Part One. Transformers
Assigning some value t o cD I and using the
+
The current i o sets up
5 '
4'3
2
wi
(Pt Fig. 4.-16 H armonic components of curre nt and flu x in a Y/Y threephase core-typ e transform er
As is ~seen from the plot of
+
th. 5 Measurement of Transformer Quantities type transformers are pr eferably connect ed Y /Y n , Y /I1, or Ynl 11. Then t he third-harm onic flux es are reduced still more by t he third-harmonic currents traversing a closed path around the delta or in the neutral wir e, Y n- The wi nding losses due to these currents are smaller than those due to the third-harmonic fluxes in the structural parts and tank walls. In a delta-connected winding with its neutral wire available for connection, the third-harmonic currents flow in the line wires and have their path completed through the transmission-li ne capacitances and the neutral wire. They interfere with the operation of nearby communication lines and produce ext ra losses in the cable sheath because, as follows from Ampere's circuital law, their ex ternal magnetic field is non-zero. For this reason, the YIY n-O connection is only used in small transformers supplying local loads. In all other cases, a relevant standard recommends to use the Y111-11 . or Ynl11-11 connection. The YIY connection is not considered in this standard at all.
5
Measurement of Transformer Quantities
5-1
The Open-Circuit (No-Load) Test
The transformer quantities, including losses, can conveniently be measured by an open-circuit (no-load) test and a shortcircuit test. The performance of a transformer at no-load has already been examined in Chap. 2. The equations for a transformer at no-load, wi th all owance for t he primary impedance ZI, can be derived from the gener al equations (3-8), (3-13), (3-19), and (3-20), if we set t he load impedance, Z, t ending to infinity and the secondary current equal to zero
.
.
VI = -E 1
.. + ZII = II (ZI + Zo)
-V~ = -E~ = II
=
10
l
-E
1
=
z.i,
(5-1)
tOO
Part One. Transformers
On open circuit, thei load impedance in the equivalent circuit of Fig. 3-6 must be set to infinity, Z' = 00. Recalling [see Eq. (3-33)] that ZI ~ Zo, it is legitimate to take it that
= 0 and VI = -E1 = z.i», The open-circuit test does not call for expensive equipment which would be necessary if the transformer quantities were measured by tests under load. As its name implies, the open-circuit test is carried out with the secondary opencircuited, and with the test gear arranged as shown in the
Zl
o ( b) '
(IL )
Fig. 5-1 Measuring the parameters of a single-phase transformer by (a) an open-circuit test and (b) short-circuit test
test set-up in Fig. 5-1a. The power rating of the variablevoltage source which energizes the primary may be as low as a few per cent of that of the transformer under test. During a test, VI is gradually raised from zero to 10 % above its rated value. Holding the frequency at its rated value, too, the experimenter measures II = 1 0 (doing this for each phase of a three-phase transformer) and the power Po drawn by the transformer under test. Using the data thus obtained, he plots the no-load phase current 1 0 , the power Po, and the power factor cos rp = P o/V1I o as functions of the phase voltage VI' In the case of a three-phase transformer, the plots are constructed for the average phase current
10
=
(lOA
+
lOB
+1
00)/3
and the average phase voltage VI
=
(VA
+ VB + V c )/3
* We assumed that ZI = 0 already in Chap. 2, because ZI is no more than one-thousandth of ZOo
Ch. 5 Measurement of Transformer Quantities
101
Using the values of 1 0 and VI thus found, the experimenter finds the power factor at no-load, cos ero. The following transformer quantities are found by an open-circuit test at the rated voltage. . 1. The transformation ratio defined as the ratio of the secondary to the primary voltage at no-load n 2 1 = w 2/wI = E 2,RIE I,R ~ V 2,RIVI,R
2. The no-load current, found either on a per-unit basis (as a fraction) or as a percentage
10
=
(Io,oeIII,R) X 100%
The no-load current must lie within the limits given in Eq. (3-33). 3. The mutual imperlance, defined for ZI ~ 0 from Eq. (5-1),
its resistive component being
n,
=
P o,oe/31 5,oe =
I z, I cos ero
and its reactive component being
x, 1/ Zfi -Rfi
~
I z, I sin era ~ I z, I
4. The no-load loss. At VI = VI,R it does not practically differ from the no-load core loss, P eore , De, because the primary copper loss under these conditions, P eu, I, De = 315,oe R I , is a small fraction of the core loss, 1 0 , De being very small. As has been shown in Chap. 3, the magnetic flux at rated load remains about the same as it is at no-load (provided VI is held unchanged). Therefore, given the rated applied voltage, the core loss at rated load, P eore , is approximately equal to the core loss at no-load, P cOl,e , DC' and the total noload loss, Po, De P qore
= P eore, ge = Po
(5-2)
102
5-2
Part One. Transformers
The Short-Circuit Test
In this test, the secon dary is short-circuited, so that the load impedance is Zo = 0, an d the secondary voltage, V 2 , is likewise zero . (In a three-phase transformer , it is presumed that the secondary leads are all commoned so as to give
0 - - - --+--------' ta )
Fig. 1)-2 Equivalent cir cuit of a transformer on a short circuit : comple te; (b) simplified
(a)
a balanced short-circuit.) The t r ansforme r equations for a short-circuit test can be derived from the general equ at ions (3-8), (3-13), (3-19), and (3-20)
~EI
VI =
-E
I
il
+ ZJI
= - i ; = -ZJ; = ZJo = jo - i;
(5-3)
Using the above equations or t he equivalent circuit in Fig. 5-2a drawn for the short-circuit t est, we can find the primary and secondary currents, t he m agnetizing current, an d mutual emf on short-circuit •
V
L, = Z +Z~Zo/(Z~+Zo)
-t: = i, = =
~e! =
•
•
= Vi/Z sc ~ Vii (Zi + Z~)
jiZOI (Z; + Zo) ~ ji jiZ;I(Z;
+ Zo) ~ (Tr1/Z o) (Z~/Zsc)
• Io , o cZ~/Zs c
zqio ~ zqio. ocZ~/Zsc =
ViZ;IZ~
(5-4)
Gh. 5 Measurem ent of Transform er Quantities
where
Zse
= Zl + Z~Zo/(Z~ = R se + jX se
"103
+ Zo) ~ Zl + Z~ (5-5)
+ R; x ; ~ x, + X; R se ~ R 1
is the short-circuit impedance, resistance and reactan ce of the transformer (that is, the impedan ce of a transfo rm er wi th its secondary short-circuited , as seen by the supply line) . The approximate expressions for 11 , I :, 1 0 and E 1 have been derived for 1 0 ~ 1 1 , I z, I ~ I ZO I, and • •f I Z ~ I -e; I Zo I [see Eq . (3-33)1, 1,=-12 and are fairly accurat e. Th e corresponding equivale nt circui t is shown in Fig. 5-2b, and Fig . 5-3 Phasor dia gr am of a the corresponding ph asor dia tra nsformer with it s secondary short-circuit ed (10 ;:::: 0) gram in Fig. 5-3. As is seen from the ph asor diagram , the . short-circuit voltage V 1 = Zse 11 is the hypothenuse of a triangle whose legs are the active vol tage R sel 1 an d the reactive voltage jX se 1 1 • Th e right-angled voltage (or impedance) tri ang le drawn for t he sh or t-circuit condition is referred t o as the shortcircuit triangle, and the angle
.
we will obtain still simpler expressions for the magnetizing current and emf
~·1 = -:j~ = V1/~se 10
,-E. 1
=
V 1/ 2Z o
=
I o,oc/ 2
= zoi 0 , oe/2 = 111./2
(5-7)
104
Part One. Transform ers
H ence it follo ws t hat the referred secon dary current on short circui t does no t diff er from the primary current, which is also t rue of the rated-lo ad condition; the magnetizing cur ren t is ab out h alf as lar ge as the no-load (open-circuit) cur rent with the same prim ar y v olt age appli ed; the mutual emf is ab out h alf as large as t he open-circuit primary voltage or emf . If the secondary is sh ort-circ uited wi th t he r ated primary voltage applie d, VI = VI. TI, t he n t he tra nsient s (to be dis cussed in Sec. 13-2) will giv e rise to short-circui t primary and secondary current s [see Eqs . (5-6) an d (5-7)1 dangerous to the transformer . On a per-unit basis [see Eq. (3-33)1 these currents are 1,~ 2
=
= V*l. TIl I Z*. sc I = 7 to 16
(5.8) that is, they are 7 tuIf t imes as he av y as the ra te d currents in the win dings . If such currents were all owed to exist for a lon g t ime, the resultan t t emperature rise would impair the electrical and mechani cal stre ngth of the insulation . For this rea son, a short-circuit test is con ducted at a reduced primar y voltage wh ose va lue is chosen suc h that t he currents in the windings coul d not exceed their ra ted va lues . On a per-uni t ba sis , this voltage should not exceed 1,1:1
< v'n ,sc = I Z'!:sc 11,':1, R = I Z*, sc I = 0.06 to 0.14 A likely set -u p for a short-circuit test is sh own in Fig. 5-1b. As in the open- cir cui t t est , it does not require any bulky load resistors or a large te st voltage source. (Wi th t he shortcircuit current ad jus ted t o the rated value, the power r ating of the test source is not over 0.06 t o 0.14 of t he power rating of the tr ansformer under t est.) The voltage is gradually raised from zero t o anywhere from 0.06 to 0.14 of the rated primary voltage. While holding the frequency at its rated value, I = I r a t erl' r eadings are taken of the sam e qu antities as in the open-circuit te st, namely the prim ar y current II and the power drawn by t he transformer, P scUsing t hese readings, I I' P sc an d cos cvsc ar e pl ot t ed as functions of the phase voltage VI' and t he plo ts thus obtained are us ed to det ermine graphically VI • SC , P £c and cos
~
Ch. 5 Measu rement of Transform er Quantities
105
fac tor is found from the average values of 11 and VI as cos
P se/3TTl Il
crse =
W ith the cur rent m ai nt ained at its r ated value, the shortcir cuit te st yields the following transformer quantities. 1. The short-circuit impedance from Eq. (5-7) as
I Z 5e I = VI , selIl, R its resistive component
I z., I cos
R se = P se/3IT. R
epse
and its reactive component X se
= V I Z~e I -
R~e =
I z., I si n
CPse
The resistive component is the sum of the winding resistan ces, R se = R 1 R ;. During a short-circuit test, it is important to note th e winding temperature e at which R se is measured. Th e me asured value of R ~e is then adjusted t o a temper ature of 75°C:
+
en
R se. 7 5 = tt.; [1 + 0.004 (75 The reactive component of the short-circuit impedance is the sum of the leakage inductances, X se = Xl X~, whi ch , as has been explained in Chap. 3, are independent of the current s tra versing t he respective windings. For the sa me reason , X r e is in dep en dent of the current at which it is measured . The short-circuit impedance and power factor are likewise adjusted to a temperature of 75°C:
+
I Z se, 7 5 I =
11 R ;e, + X~e 75
cos crse , 75 = R se , 75 / 1 z.;
75
I
2. The short-circuit loss Pse . At II = 11, R, it does not practically differ from the copper losses in the primary and second ar y windings carry ing r ated currents
Peu. R = =
P e ll . r , R
2 1
sn 1
.)
10 R
+ Peu, + 3R ' I '2
because the copper loss loss, P r;gre . W .
2. R
= 3R1I~, R
2 2' R =
i~
3R sc 1 1.2
+ 3R
2I:. R
R
Illany times the short-circ uit yQn:l
I·
106
Part One. Transformers
With VI, se equal to 0.06 to 0:14 of VI, R, the short-circuit primary emf is E I ; Be = VI , se/2 = 0.03V I, R to 0.07V I, H The short-circuit flux and induction in the core, which are proportional to E I , sc- amount to anywhere between 0.03 and 0.07 of their open-circuit values . The short-circuit core loss, which is proportional to the magnetic flux density squared, ranges between 0.9 X 10-3 and 5 X 10-3 of the core loss under rated con ditions, and between 2 X '10-4 and 12 X 10-4 of the copper loss also under rated conditions. 3. The impedance voltage. It is defined as the voltage that must be applied to one of the wind ings, with the other short-circuited, so as to circulate rated current at a tem perature of 75°C, with the windings connected as for rated voltage operat ion. If this voltage is ap plied to the pr imary winding, the impedance voltage expressed in abso lute units is
VI, se = I z.; 75 I II , R Usually, the impedance voltage is expressed in per unit or per cent of the rated voltage of the winding in which the voltage is measured V
se = V*l, sc = VI;se/ VI; R = Z",, se
(5-9) (VI, se/ VI, R) X 100% Similarly , the resistive component of the impedance voltage is given by Va = R se , 7511, R/VI• R = R*, se = V se COS CPsc and it s reactive component is given by u; = XseI I, R/V I , R = X 'I', se = V se sin CPse (5-10) (see the ph asor diagr am in Fig. 5-3). or
V se
=
6
Tra nsformer Performa nce on Load
6-1
Simplified Transformer Equations and Equivalent Circuit for 1 1 »1 0
In service, the load on a transformer is varying all the time. As a result of variations in the load impedance Z, the secondary current may vary from zero to its rated va lue, and it~ phase relative to the secondary yolta~e also varies ,
107
CII. 6 Tr an s form er Perform an ce on Load
As h as been expla ine d in Sec . 3-7, variati ons in the secondar y curren t are accom panied by nearly proportionate variat ions in the primary cur rent , and th is le ads to slight variations in the.magneti c flux . This ch apter will deal with the effects that variations in the secondary current m ay ha ve on t he secon dary voltage an d efficiency of a t ransformer . Z'2 ", -VI Analysis will be carried out /J for the most frequently encountered load cond itions, namely VI, R = constant and II » 1 0 , Fig . 6-1 Simpli fie d equivalen t When 11 » 10, we may, as circuit of a transformer with II ~ 1 0 in the case of a short-circuit , set 1 0 =0 and IZo 1= CXJ . On this assumption , the primary current [see Eq . (3-20)] does not differ from the secondary current referred to the . primary 11 = - I;, and the voltage equations for the prim ary and secondary wi ndin gs , Eqs. (3-13) and (3-19), may be combined in to a single equation
.
.
.
VI
+
=
.
+ Zscll R sc + jXsc is -
V~
(6-1)
where Zsc = ZI Z~ = the short-circuit impedance of t he transform er . Therefor e, the equivalent circuit in Fig. 3-6 m ay be simplified by removing the arm carrying the magnetizing current, and the sum of im pedances ZI Z~ m ay be replaced by Zsc. The simplified equiva lent circui t an swering Eq. (6-1) appears in Fig. 6-1.
+
6-2
Transformer Voltage Regulation
Graphically, th e depen dence of V 2 on 1 2 , with the power factor cos CPl\ an d VI, R held constant, can be shown by an external characteristic. Plotted in arbitrary units, it takes the form shown in Fig. 6-2. The manner in whi ch V 2 var ies with 1 2 depen ds on the char act er of load . If the load is resis tive-inductive ((P2 > 0), V 2 decreases as 1 2 increases. If the load is predominantly capa citive ((P2 ~ - n /2), t he on-load secondary voltage may exceed its rated no-lo ad va lue , .
108
Part One. Transformers
A measure of how much changes in 1 2 will cause V 2 to change is given by voltage change defined as L1V2 = V 2 • R
a
Iz
-
V2
with VI = VI, R held constant. I t is usually expressed . in per-unit and known as voltage regulation
l '
Fig. 6-2 External characteristics, V 2 = f (1 2 ) , of a transformer with VI held constant: solid line, resistive-inductive load, CJl2 = const > 0; dashed line, resistive-capacitive load, CJl2 = const < 0
L1v= L1 V 21V2 • R =
(V~,
=
(VI, R -
R -
= L1 V' lVI, R V~)IVI, R
V~)IVI, H
or L1V
=
V,~1,
R -
= 1 - V'I'2
V*2
(6-2)
If the per-unit voltage regulation is known, the per-unit secondary voltage can be found from
Fig. 6-3 Simp lined phasor diagran, of transformer voltage Te~ulation~ with I~ ~ I Q
The voltage regulation equation can be derived from the voltage regulation diagram in Fig. 6-3. (For convenience, the voltage drop phasors R:I:.scI I and X* .scII are shown on an exaggerated scale.) Construction of the voltage regulation diagram begins at point 2 which is the tip of the - V~ ph asor . The - V~ ph asor is drawn through this point in an arbit ary direction, and I I = I ~ s then drawn to make an ngle ep~·.with the
109
Ch. 6 Transformer Performance on Load
- V~ phasor. Now point 2 is used as the origin fo: the phasor representing the resistive voltage drop R scl1 and the reactive voltage drop jX s Point 1 occurs at the tip of the VI phasor. The value of - V~ is found at the intersection of the - V~ phasor and the circle with point 1 as centre a~d with VI. R as radius. At the same time, the direction of VI . and the angle a between VI and - V~ are found. _ The diagram in Fig. 6-3 has been plotted in per-unit and its components are given [see Eq. (5-10)1 by the following equations: \, .
ci1'
..
V*l. R = VI, R/V1, R = 1 1*1 = II/II, R = 1*2 = 1 2/12, R = I~/I1, R = ~ V~2
=
V~/V1. R
R*. SCI *l = R sc (II. R/V1, R) 1*1 = Va~ X*, scI*l = Xsc (II. R/V1, R) 1*1 = vr~ Let us write the voltage regulation as the difference between V*l, Rand V*2: "' /),v = V*l,R - V*2 = 1 - V*2 Referring to the voltage regulation diagram
V*2 = V~2 = V'H, R cos a - c = cos a - c In turn, because the angle a is negligibly small cos a= V 1- sin 2 a ~ 1- (sin 2 a )/2 = 1-d2/2 On expressing the segments c and d as the sum and difference of projections of R*, SC!,H and X'i<, SC!,i
+ X*. sc sin CP2) 1'1<1
c = (R'I<, SC cos CP2 = (Va cos CP2 u; d = (X'I<,SC cos CP2 = (vr cos CP2 - Va
+
sin CP2) ~ R*. sc sin CP2) sin CP2) ~
!,H
we obtain an expression for the voltage regulation /),v = c d2/2 = (va cos CP2 o; sin CP2) ~
+
+
+ (vr cos qJ2 -~a sin qJ2)2 ~2
(6-3)
110
Part One. Transformers
As is seen, the secondary voltage regulation depends substantially on the load phase angle (P2' A plot of !'!..v as a function of cr2 with ~ = 1 for a transformer with Vse = 0.1. Va = 0.04 and u; = 0.0918 is shown ill Fig. 6-4. The dashed line gives the same dependence on neglecting the second
Llv
5~Z
0,00
tf I
'fz
Fig. 6-4 Voltage regulation, S», as a function of (jl2, with 1 2 = 1 2 , R ana ~ = 1: solid line, by Eq. (6-3); and dash ed line , with the second term in the equation ignored
term in the equation. Because the inclusion of the second term affects the final result but little, an approximate equation, convenient for analysis,
+
(va cos cr2 u; sin cr2) ~ ~vse cos ((Pse - cr2) (6-4) is used in many cases (especially where Vse is low). As follows from Eq. (6-4) the voltage regulation is a maximum, !'!..v = Vse, when cr2 = crse, because cos ((Pse - cr2) = 1. Conversely, the voltage regulation is a mini0 mum, !'!..v = 0, when (Pse - (P2, 0 = 90 and cr2, 0 0 = _(90 crse), because then cos (crse - (P2, 0) = 0 (see Fig. 6-4). With some other load phase angles, the shortcircuit triangle takes up the characteristic positions shown in Fig. 6-4, such tha\ . !'!..v = Va for CP2 = 0 0 !'!..v = +vr ~ = +90 !'!..v
=
=
HI
Ch, 6 Transformer Performance on toad
Becau se the second term in Eq. (6-3) is small, the dependence of voltage regulation on the relative secondary cur rent ~ , with qJ2 held cons tan 1., is practi cally linear. 6-3
Variations in Transformer Efficiency on Load
Electric energy should preferably be transformed with as low relative losses as practicable or, which is the same, with as high an efficiency as can be achieved. Here, t he efficiency of a transformer is defined as the ratio between the active power delivered to the line on the secondary side and the active power drawn from the supply line on the primary side 11 = P 21Pl = m2V212 cos qJ21mlVlIl cos
qJl
(6-5)
The primary active power may be written PI = P 2
+ P eor e + P eu. + Pc« 1
2
We shall limit ourselves to the operation of a loaded transformer with the rated primary volt age, VI . R' , held constant. We shall make the same assumptions as in Sec. 6-2. That is, we shall deem II ~ 1 0 , 1 2 = I~, and I Zo I = 00. We will also neglect the difference in core loss between operation on load and at no-load and assume that P eor e
=
P eore, oe
=
Po
where Po is the no-load loss with the rated primary voltage appli ed. Then the copper losses may be expressed in terms of the short-circuit loss P se at the rated primary current: Pee ,
1
+ P eu.
2
= I~Rl
=
+ I~2R ~
= IiR se
I~ .RR se (lIllI, R)2 = Pse~2
The secondary active power is given by* P 2 = m2T{2Iz cos (P2 = mlVl. RI~ cos qJ2 _ _ _ _ ml RI I• R (I~lIl, R) cos qJ2 = SR~ cos eP2
V"'\
.
* In setting V~ = ~R' we negl ect the effect of voltage regulation on the secondary active "power .
112
Part One. Transformers
Substituting the above ex pressions in E q. (6-5) gives us the dependence of the effi ciency on ~: 1']-
-
P l-(P COl'e -/-PeU .I -/-P cU. 2) PI
=1-
PO+B 2PsC BSR cos CP2+ PO+ B2P sC
(6 6) )-
The effect of the secondary current on t he secondary voltage may be accounted for as follows : V~ =
VI, R (1 - L1v) and its effect on the iron loss thus: Pc ore = Po (EIIV I, R) 2 = Po (1 - L1v) where, with sufficient accuracy , E I = VI, R (1 - L1vI2) Accordingly, the equation for the secondary active power may be re-written as P 2 = m 2 V 2I 2 cos CJl2 = mi V~I~ cos CJl2 = SR~ (1 - L1v) cos CJl2 Then, the effi ciency equation ma y be refined as 11=
1
-
Po (1- ~ v) +B 2Ps c
r3(1- ~v)SRCOS (P2+Po(1 -~v) +B2Psc
(6 ~) -{
Equation (6-7) holds for the entire range of changes in secondary currents. On both open- and short-circuit
P2 =
~ (1 -
L1v) S R cos CJl2 = 0
and the efficiency reduces t o zero . This can be pr oved form ally from Eq. (6-7) , recalling th at at no-lo ad ~ = 0, whereas on short-circuit, L1v = 1 - V 2* = 1, because V 2 :1: = O. Although approximate, Eq. (6-6) derived for ' L1v = 0 is sufficiently accurate for the relat i ve secon dary current varying from ~ = 0 t o ~ ~ 1. Let us fi nd the value of ~ at whi ch the effi ciency is a maximum . E quating the deri vative df]/d~ t o zero and simplifying t he equat ion, we obt ain
Po or
=
~ ~axPsc = I
~ ll1ax =
Peu. f -
-
I
-I- Peu.
1- PolP sc
2
(6-8)
cu.
r r:
113
7 Tap Changing
This implies that the efficiency of a transformer is a maximum when the load is such that the no-load core loss at rated primary voltage, Po, 10 is equal to the copper loss, cos
r
+
~ rn a x =
V 2.45/12 .2
=
0.45
In the range from O .4~max to 2.5~rnax, the efficiency falls off insignifi cantly. Such variations in efficiency are typical of all power transformers.
7
Tap Changing
7-1
Off-Load Tap Changing
As follows from the analysis given in Sec . 6-2, in the worst case (when the load phase angle CP2 is equal to the phase angle on a short circuit, CPsc), the per-unit voltage regulation may be anywhere between 0.06 and 0:14 . Th is is far more than is permitted by relevant service codes. To maintain the secondary voltage constant against such variations, tappings on the coils are brought out to terminals so that the number of turns can be changed. This tap-changing can be effected on either the primary or the secondary side. In transformers operating at a fixed pri mary voltage, this is done by changing the number of turns on the secondary side, while holding the primary turns unchanged . With this arrangement, t he magnetic flux, the core l oss, and the magnetizing current (whi ch is a function of the ratio V1/wl) remain practically constant. 8- 0160
Part One. .Transformers
in transformers operated at constant load (or, which is the same, at a constant secondary current) and a varying primary voltage, VI' it is preferable to change taps on the primary side so as to maintain the ratio VI/WI nearly constant. All power transformers have tappings on the primary or secondary coils which permit voltage adjustment within +5 %. Low- and mediumpower transformers (Fig. 7-1) usually have three taps per phase (+5 %,0, and -5 % variations in the turns ratio), whereas transformers of higher power ratings have five taps
(+5%, +2.5%, 0, -2.5%, and -5 % variations in the turns ratio). Tap stepping operations are performed by contact switches, usually called tap changers. Tap changers can be made simple and inexpensive if taps are stepped with the transformer out of circuit and the taps are made at the neutral point of a three-phase star-connected winding (see Fig. 7-1). This avoids a short circuit between adjacent taps or breaking a live circuit during a transition. The operating handle of the tap-changer is passed outside through the tank side. Fig. 7-1 Tap-changing by a switch: 1-transformer winding; 2-tapchanging switch
7-2
On-Load Tap Changing
Voltage adjustment can be made far more accurate and automated if tap changing is done with the transformer left on load, without breaking the circuit. This, of course, calls for a more sophisticated tap-changing arrangement, notably one incorporating what is known as a transition impedance. Impedance in the form of either resistors or iron-core inductors is introduced to limit the circulating current between the two tappings. Most frequently, resistor transition is used ~o r on-load g" Ior tap change". The arrangement of such a tap
CIln
115
eh. 7 Tap Chan gin g
one ph ase is sh own in Fi g. 7-2. This is seen to be a combination of a fa st-acting di verl er switch D S , an ev en t ap selector TS l' an d an odd tap selec to r T S 2 ' The di ver ter switch and the transition resistors, R 1 and R 2 , are usually in stalled in a sep ar ate oil-filled t ank. The di verter switch is designed t o carry t he current usually developed when the two taps ar e bridged . The tap selectors may be moved from tap to tap onl y when their circuits are de-en ergized . Figure 7-2 shows t he diverter swi t ch an d t he even tap sel ector in the position when the T 2 tap is brought in circuit. To move to the next, T 3 ' tap, Fig. 7-2 On-load ta p changer t he odd t ap select or should with cur rent-li miting resistors first be moved t o that tap, and the diverter switch ma y then be rotated clockwise. The ensuing sequence of events is as follows: con tacts . 1 an d 2 break , contacts 1 and 3 make, contac ts 1 and 3 br eak, and contacts 3 and 4 make. When fully automated , a tap stepping operation is completed in a m atter of a spli t secon d.
Fig . 7-3 On-l oad t ap- changer with a transition i nducto r
Changer~r
The arr angement of a t ap one phase) whi ch uses the t ra nsit ion impedance in the Iorrn of an iron-cored in ductor is shown in Fig. 7-3 . In addi; ion to a transition cent re-t ap inductor (or rea ctor), L , whi ch is wound in two hal ves , 1 an d 2, put on a comm on no-gap core, the arrangement includes two t ap sel ectors , TS 1 and TS 2 ' whi ch can 8*
116
Pa rt One. Transformers
move fro m tap to tap after their circuits have been de-energized, and two on-off switches , 8 1 and 8 2 , to de-energize the respective tap-selector circuits. The tap selectors and the centre-tap inductor are located in the transformer tank, and the on-off switches are enclosed in a separate tank mounted on the transformer. In Fig. 7-3a, the load current is shown passing from tap T4. through the halves of the inductor in opposition, and hence noninductiv ely (wi t hout magnetizing the inductor core). Therefore, t he inductor presents to the load current onl y a small resistance, while it s reactance may be igno red . Transition from tap T 4. 10 , say, tap T 3 may be visualized as consisting of a sequence of seven steps li st ed in Table 7-1. The most significant steps Table 7.1 Tap -to -Tap Transition Steps Position Step T81
1 2 3
,
'I
4 5 6 7
T4
T4 Ts Ts Ts Ts Ts
I
T8 2
T4 T4 T4
T4 T4 Ts Ts
I
81
ON OFF OFF ON IJN ON ON
I
Figur e No . 82
ON ON ON ON OFF OFF ON
7-3a 7-3b 7-3b (dashed) 7-3c
are illustrated in Fig . 7-3. In Fig. 7-3b, one of the two tap select ors ha s opened , and the load current is carried through one half of the inductor induct ively (setting up a magnetic flux ar ound it). However, the induct or is designed so that the instantaneous reactive vol tage drop during this step ha s but an insignificant effect on the secondary terminal voltage of the transformer. In Fig. t:3~ the inductor is sh own bridging the two adjacent tappings,"T 4. and T 3. The load current is shared equally between the~two tappings and passes noninductively in opposition through the ha lves of the inductor. The tap step voltage is applied to the whole of the inductor wind ing and the circulating current, I e' is limited by the total impedance of the inductor whose fie ld is now directed aiding to the mmf due to the circulating current (shown by the dashed lines in the fig ure ).
117
Ch. 8 Calculation of Transformer Parameters
8
Calculation of Tra nsformer Pa rameters No-load (Op en-Circuit) Current and M utual Impedance
8-1
In Sec. 2-6 it has been shown that the reactive component of the no-load current, Ion can be deduced from the parameters of the magnetic circuit. It is, however, simpler and more convenient to determine it from the reactive power required to magnetize the transformer. The reactive power may be expressed in terms of either the mutual emf and the reactive component of Acore the no-load current Qo = s.r; (for a single-phase transformer) lsap or the core flux. Let us do this with reference to Fig. 8-1. In terms of the peak flux, E l is given by
Wf
Fig. 8-1 To calculation of reactive currents required to magnetize the core, [or core, and the gaps, [or, ga p
Ei
= 2'Jf,fw/Pm/ V2
where
In accord with Eqs. (2-13) and (2-14), the reactive component of no-load current can be written as
+
lor = lor, core lor, gap The first term on the right-hand side sustains the magnetic potential drop in the core lor, core = H mlcore/ 112 Wi ~ and the second term sustains the. magnetic potential drop in the air gap lor, gap = B ml gap /;l/ 2 Wi~to The total reactive power is the sum of the reactive powers required to set up the flux in the core and the gap Qo = Ell or = Ell or, core
=
mcoreqcore
+
'+ Acoreqgap
Ell Or,
gap
118
Part One. Transformers
Ell or• core/mcore = niBmH n/r is the specific magnetizing power of the core qg ap = nIBinlgap/f.lo is the specific magnetizing power of the air gap m cor c = mass of the core A core = cross -sectional area of the core r = density of the core material The values of the specific magnetizing power as a function of induction (magnetic flux density) for imbricated-joint cores are given in [131. The reactive power required to magnetize a core of any design is given by
where
Qo
qco r e =
=
qlegmleg
+ qyo] lemyoke + n ga p , l egqgap, legAl eg
+ n g a p , y ok eq gap , yokeAyoke
(8-1)
where ml eg and myo]w are the mass of the legs and yokes, A 1e g and A y ok e are the cross-sectional areas of the leg and yoke, and ql e g' qYol,e' q ga p, 1 e g and q gap, yol!C are the .specific magnetizing powers of the legs, yokes, leg gaps, and yoke gaps . In a single-phase core-type magnetic circuit, n g a p , leg = 2 and n ga p, y oke = 2. In a three-phase core-type magnetic circuit, n g a p , leg = 3, and n g a p, y oke = 4. The active power, equal to the core or no-load loss, is deduced from the specific loss for legs, PI e g' and yokes, P yoke' which are given in [13J (8-2)
This power is ordinarily calculated for the rated primary voltage, VI, R = E I , R, only. Once it is found, it is an easy matter to determine the no-load current components (see Sec. 2-8):
lor
=
QO/mVI,R,
loa =Po ~
!
the no-load current 10 = 11IBn
+ 18;'
and the components of the mutual impedance
Ch. 8 Calculation of Transformer Parameters
"* 8-2
119
Short-Circuit Impedance
On a short circuit (see Sec. 5-2), the primary mmf, i1w1, and the secondary mmf i 2 w 2 , balance each other almost completely. Therefore, without running into a serious errol', wemaydeem that i1w1= - i 2w2, and that, on a short circuit, only the leakage flux exists, whereas the mutual flux is non-existent, because dV iOw1 = i1Wl i 2w2 = 0 The flux pattern applicable to this case, with the windings H arranged coaxially, is shown in Fig. 8-2. The magnetic field intensity H within and between the windings acts / along the leg axis. With sufficient accuracy, the magnetic field may be taken as being symmetrical about the leg axis. Therefore, the value of H remains practically the same within a distance D/2 of the leg axis and along the coil height h. The magnetic intensity distribution along the radial coordinate x reckoned from the inside surface of the coil area is shown in the same figure. By Ampere's circuital law, the magnetic field intensity is a maximum between the windings (a 1 < x < a 1 a1 2) , where the magnetic lines of force link all of the]JrIiilary Fig. 8-2 Leakage flux in a transformer on a short-circuit current: ( (i1lV 1 = - i 2 lV 2 ) H = H m ~ i1w1/h
+
+
+
In a first approximation, the magnetic potential drop in a core material with m-, core = CXJ and H core = 0 may be
120
Part One. Tra nsform ers
neglected. W ithin the wind ings, t he magnetic fie ld intensity varies linearly from zero to H m- For ex ample, when x is anywhere from zero t o aI' t he magnet ic lines of force link with a current i1wrx/a r, and the magnetic fie ld int ensit y is
H = i1wrx/ha 1 = H mX/al The energy associated with t he leak age flux established by two magnetically coupled windings may be expressed in terms of the inductances of those windings W Recalling that
=
Lli~/2
+ L~i~2/2 + iri~L~2
and we obtain W
= (L ru
+ L;u) i~/2
=
Lsci~/2
(8-3)
where L sc is the short-circuit inductance. The same energy may be expressed in terms of the sp ecific energy of the magnetic fie ld W
= HB/2 = flOH2/2
In determining the energy, the integral may only be taken over the volume V = nD mean (a1 a r2 a 2) 12, occupied by the windings, which encloses the bu lk of the le akage energy
+
W = .\ w dV
+
= (flo/2) .\ H2 dV
v
v
Because H remains the same all the wa y round the circle with the diameter D = Dr x and along the height 12, the elementary volume is .
+
dV
= nDh dx = n (Dr
+ x )h dx
With an accuracy sufficient for engineering purposes, the diameter D = Dr :r may be replaced with a mean diameter
+
Ch. 9 Transformer Quantities vs. Dimensions
121
After this simplification, "1+"12+"2
W = (l1-oh:rtDmean/2)
+
.\ o
HZ dx = (11-0/2) nDmeanhauH;" .: (8-4)
+
(a1 a z)/3. where au = a l 2 Equating (8-3) and (8-4) yields an expression which ' connects the short-circuit inductance to the size and winding data of a transformer (8-5 L S0 = nl1-oDmeanwiaukRlh where k R = 1 - (al .+ al2 az)/nh is the Rogovsky coefficient (after its originator). It minimizes the error in calculations due to the assumptions made. The short-circuit inductive reactance is given by X se = (2n zfl1-o D meanauw~) k F l h ( 8 - 6 ) The resistive component of the short-circuit impedance is calculated as the sum of the referred resistances of the windings (8-7) n.; = n, R~ = n, tt, (w l/wz)2 where R l = P7.'inDmeanWlks/Awl is the resistance of the primary R z = P75nDmeanwzkslAwz is the resistance of the secondary AWl' A wz = cross-sectional area of the primary or secondary turns, respectively P75 = resistivity of the wire at 75°C after [13] k, = 1.05 to 1.15 is the series-loss coefficient.
+
+
+
9
Relationship Between Transformer Quantities and Dimensions
9-1
Variations in the Voltage, Current, Power and Mass of a Transformer with Size
Suppose we have a range of geometrically similar transformers. Two transformers out of this hypothetical range are shown in Fig. 9-1. Any dimension of any transformer
122
Part One. Transformers
in this range differs by a factor of k from the like dimension of any other transformer in the range . Taking any linear dimension, say, the height, l, of the core as the base or reference dimension, we may deem that all the other dim eilsions of the tr ansformers in a given range are proportional to it. For example, the mean turn diameter, D mean , is about equal i to l. The cross-sectional area of any element of a transformer is proportional t o the square of the base dimension , A ~ P . By the same token, liZ the v olume of any element is proportional to the base dimension cubed, V ~. P . Now let us see how t he rated electromagnetic quantiFig. 9-1 Geometrically similar ties of a transformer are reI asingle-phase transformers ted t o it s size. Suppose that all the transformers in a given range are fabricated of the same materials and that the magnetic flux density B in the core, the current density J in the wind ings, and frequency remain always constant . 1. Neglecting the difference between terminal voltage and generated emf, we get
-
VI. R ~ E 1 ,
R
= (2n/ V 2) !wiBA Jeg ~ w:rAJeg """ w'l.ZZ
(9-1)
That is, the primary voltage is proportional to the number of turns in the primary winding and the square of the base dimension . 2. Assuming that as l is varied , the total cross-sectional area of conductors in a wind ing r em ains proportional to l2, we get (9-2)
That is , t he primary current of the base dimension . (Here, area of the conductors in the 3. The total (or apparent) S
=
SR
= VI, nIl, R
is proportional to the square Al is the total cross-sectional
winding.) power of a transformer ~
l2WI (l2/ w1)
= l4
(9-3)
123
Gh. 9 Transformer Quantities vs. Dimensions
is proportional to the base dimension raised to the fourth power . Importantly, on the assumptions made, the power of the transformer is independent of the number of turns. It. The mass of the transformers in a range fabricate d of the same materials
m = ~ I'll "-' [ 3 is proportional to the base dimension cubed. The mass per unit power mJS "-' [3/ [4 "-' ill ("",1/S1/4 )
is inversely proportional to the base dimension. (The mass per unit power decreases as power r ating goes up.) 9-2
Transformer Losses and Parameter s as Functions of Size
1. The total loss of power in a transfor mer is t he sum of the core loss and t he copper loss. The core loss is proportional to the mass of the core elements, mcore, because the specific core loss in similar elements remains unchanged as the physica l dimensions are va ried . The copper loss may be expressed in te rms of the winding volume and 112 = nDmeanA 2 and als o the current density J and the resistiv ity p
Pc« = Peu. 1
+ Pc«.
+
pJ211I pJ2V 2 If the materials remain the same and the current density is he ld constant, the copper loss is prop or t ional to the base dimension cubed
P eu = pJ2 (VI , Thus, t he transform er loss P col'e
2
=
+ V 2) "-' [3
+ r-; "-' za
is proportional to the b ase li near dimension cub ed . :
(9-5)
/I
124
Part One. Transformers
The spe cific loss (the t ot al loss divided by to tal power) (P eu
+
Peore)/S '" l31l 4
'"
ill '" 1/S1 / 4
(9-6)
is inversely proportional to the base linear dimension or the fourth-power root of the total power. (In high-power transformers, the spe cific loss is lower.) The loss per unit of cooling ar ea, A e oo1
+
(P eu P eore)/A eool '" ZSll2 = l is proportional to the ba se linear dim ension and increases with increasi ng power rating. This is t he reason why highpower transformers must be pro vi ded with a well-developed cooling are a in the form of ducts in t he core and windings. 2. The short-circuit inductive react an ce (see Sec. 8-2) X se '" w~Dmeanaa/h '" w~ l
(9-7)
is proportional to the number of turns squared and the base linear dimensions. The short-circuit resistanc e R se = Pe ulI;. R
'"
l3/(l2/W1)2 '" w~/l
(9-8)
is proportion al t o t he number of t urns squared and inversely proportional t o the base linear dimension. The reactive and resistive components of the impedance (short-circuit) voltage are given by
=
u;
R*ae =
Va
X*se
=
Xsel l • R/V1 , R '" (Wil/Wll2) (l2/WI) '" l
= Raell• R/V1 • R '" (wi/l) (l2/WI) (1/w ll 2) '" 1fl
The short-circuit tangent is tan lVse
= X *aeIR*ae = vJ»; '" 1fl2
(9-9)
In other words, as a t ransformer gr ows in size , its v; rises and its va falls. This checks well with practice. 3. The reactive and active components of t he no-load current (see Sec. 2-6) lor '"
~ Hz. dl/wi '" llur,
loa = Peore/Vj.
R '"
Z3/w jl2 ,....,. 1/wj
(9-10)
are proportional to the base linear dimension and inversely proportional t o the number of turns.
Ch. 10 Multiwinding Transformers. Autotransformers
125
The relative no-load current (or the relative magnetizing power)
Qo/S
=
1 0 V I , Rll l , RV I , R '" (l/w l ) (w I/Z2 ) = ill
= lolli, n
(9-11)
is inversely proportional to the base linear dimension. In commercially available transformers, geometric similarity is never complete, nor can Band J be held constant. Nevertheless, the relationships set forth above are true, at least qualitatively. As follows from the foregoing, it is advantageous to use transformers with higher power ratings, because they take less materials per unit power, need lower reactive power, and dissipate less heat.
10
Multiwinding Transformers. Autotransformers
10-1
Multiwinding Transformers
(i) Three-Wlndlng Transformers In a multiwinding transformer, the core carries.more than two electrically isolated windings. Power systems mostly use three-winding transformers to couple electric systems or networks operating at three different voltages, VI' V 2 , and V 3 • Three-winding transformers may be single-phase (Fig. 10-1) and three-phase, with their windings connected Y n/Y nl~-0-11 (Fig. 10-2) and Y nlM ~-11-11. A three-winding transformer may have either one primary (1) and two secondaries (2 and 3), or two primaries (1 and 2) and one secondary (3). Our discussion will be limited to transformers having one primary and two secondaries. A three-winding transformer does the job of two twowinding transformers one of which connects network (or system) 1 to network 2, and the other, network 1 to network 3. Economically, a three-winding transformer is more attractive than two two-winding units. Among other things,
/
12B
Part One . Tra nsformers
it is less ex pensive to make and takes up less space at a substat ion It can transf er power not onl y from the primary 2
3
If
Fig. 10-1 Sin gle-phase, three-winding transformer: i-primary winding ; 2-secondary win ding ; 3- tertiary win ding; 4she ll-and-c ore-type m agnetic circuit
network (1) to any of the secondary networks (2 or 3), but also directly (by a single transformation step) from one of
Fig . 10-2 Y n/Y nl ~ -O- 11 t hree-phase , t hree-win ding t ransformer : i -LV three-ph ase win din g; 2-MV t hree- phase win din g; 3- HV t hree-phase win ding; 4-core-type magne tic circuit
the secondary networks to t he ot her (say , from network 2 t o network 3). With two t wo-win ding t ransformers, such power tra nsfer necessitat es two transfor mation st epsfirst from network 2 t o network 1, then fro m network 1
eh. 10 Multiwinding Transformers. Autotransformers
127
to network 3. Accordingly, the losses are about twice as heavy. On the demerit side, a three-winding transformer is less reliable. Should any of its windings be damaged, the entire unit must be removed from service. With two two-winding transformers, damage to any of them leaves the other unaffected . The electromagnetic processes in a three-winding transformer may be described by analogy with a two-winding unit (see Chapters 2 and 3) . As a preliminary step, however, its secondary and tertiary quantities must be referred to the primary side, with their values multiplied by the respective turns ratio or its square: I~ =
Iz
X
(WZ/w I)
I~ = Is X (Ws/wI )
= V z X (wi/W Z) = V s X (W I/W 3) I = I z, I X (wI/WZ) Z I = I z, I X (WI/W S)2 V~ V~
I Z~ I Z~
The mutual flux is set up by a magnetizing current 1 0 which is given by the current equation
.
II
.
.
+ I~ + I;
.
= 10
(10-1)
and the mutual emf is given by
-E
where The trical) The
+
I
=
-E~
=
-Ii;
=
z.i,
(10-2)
Zo = R o jX o is t he mutu al impedan ce. leakage flux is established by a balanced (or symmeset of currents, j~ , j~, and I~, where i; = i; - i; leakage emf in each winding is
E. UI =
-jXI~~ -jXI~
. .
E~2 =
-jXl~
E~3
-j X ;I~
.
=
where X l' X~, and X; are the equivalent leakage reactances of the windings, found wi th allowance for the effect of currents in th e other ' windings.
128
Part One. Transformer s
Formally , the vol t age equations for the t hree wind in gs are written as for a two-wind in g t ran sform er [see Eq s. (3-13) and (3-19)1:
.
.
=
.
. + RIll
.
. + -E~ = - ir~ + E~2 - RJ~ = - V~ + Z~ (-j~) -E~ = -i'; + E~3 - R/; = - V; + i ; (- j ;) VI
-E I - E cn
where Zj
=
Z~ = Z~ =
=
-E 1
Zlll
(10-3)
+ jX R ~ + jX~ R~ + jX~ RI
I
Equations (to-1) through (10-3) apply t o t he equ i valent circuit in Fig. 10-3. The mutual impedance Z~ is foun d by calculation or experiment in exactly t he sam e way as for a two-winding tra nsformer (see Sees. 5-1 and 8-1). The impedances Zl' Z~ and ~._---=:=-....,
z;
Fig. 10-3 Equivalent circuit of a three-winding transform er
Z;
are exp ressed in te rms of the sh or t-circuit impedances an d ZSC23 ' as determined by a short-circuit t est, using t he tes t set-u p shown) in Fi g. 10-4. ZSCl 2 = R SCl 2 jX s c l 2 is found with the te rti ary winding open-circuited. Z SCl3 = R SCl 3 jX s c l 3 is found with th e second ary winding open- circuited. Finally Z SC23 = R SC 2 3 jX s c 2 3 is . found wi th t he secon da ry energized a nd t he prim ar y opencir cuited , and is referre d t o the prim ar y by t he equation Z SCI 2 ' Z SCl3
+
+
+
Z~C 2 3
It is to be noted that
=
Z SC2 3
(WI/W2)2
12:1
Ch. 10 Multiwinding Trans forme rs. Autotransformers
Solving the above equations for ZI ' ZI Z~ Z~
Z~ ,
Z~,
and
we get
= (ZSCI2 + ZSC13 - Z~ C 23) /2 = (ZSC12 + Z~ C23 - ZSCI3) /2 = (ZSCI3 + Z~ C2 3 - ZscI2)/2
(10-4)
The resisti ve components of th e ab ove imped an ces are the resi stances R I , R ~ and R ; of th e respecti ve windings (see [2
2
2
V,
VI 3
3
ZSC,1Z
ZSc,1$
Zsc,23 "
Fig. 10-4 Short-cir cuit tes t on a single-phas e, three-winding transformer
Sec. 8-2), whereas their reactive components have the meaning of the equivalent leakage inductive re actances of the windings: x, = (X SCI2 + ~XS CI3~- X~C23)/2 X~ = (X SCI2 + X~C23 - X SCI3)/2 X; = (X SCI3 + X~C23 - X SCI2)/2 The secondary and tertiary voltages of a loaded threewinding transformer may be found analytically, using Eqs. (10-1) and (10-3), or graphically, using the phasor diagram shown in Fig. 10-5. On assuming 1 0 « II and deeming VI, R, i 2 an d i 3 known in ad vance, we can find EI , V~ and V~, and determine the per-unit voltage regulation for the secondary and tertiary windings: D.v 2 = (V~ - VI, R)/V1, R D.V 3 = (V~ - VI, R)/V I • R From the equivalent circuit or the phasor diagram , it is seen that when Zl =1= 0, the referred secondary voltage de9-0169
130
Part One. Transformers
pends not only on the referred secondary current, but also on the referred tertiary current (by the same token, V~ depends not only on I~ but also on I~)-a feature undesirable from the consumer's point of view. This effect can be minimized by reducing Zl at the expense of its reactive component, X l' Practically, this can be done by placing the primary winding between the secondary and tertiary windings, as shown in Fig. 10-1. If we express the short-circuit inductive reactances in terms of the winding dimensions (see Sec. 8-2):
+ al\)/3 + a12 X SC13 '"'" (a l + a 3 )/3 + a1 3 X~C23 '"'" (all + a )/3 + a 23 where a 23 = a l + a13 + a1 2' XSCll\ '"'" (a l
3
~~=~~~~~I7"t~
Fig. 10-5, Phasor diagram of a three-winding t.ransformer
+ +
we can see that with this arrangement Xl becomes negative (as in the phasor diagram of Fig. 10-5) and very small in absolute value
(X SC12 X sc13 X~C23)/2 '"'" 2al/3 alii a13 a 23 = ,-a l/3 < 0 Three-winding transformers are built with their windings differing in power ratings. A relevant Soviet standard stipulates the following ratios (as fractions of the primary power): Xl
,
St,R/St,R
,
+
SZ,R/St,R
S3,R/ St,R
1 1 1 1 1 2/3 1 2/3 2/3
The power ratios must be the same as the ratios between the respective referred currents. It is also required that i, = -j~ - i; (see Fig.' 10-5). However, the sum of the secondary and tertiary currents may exceed the primary current
Ch. 10 Multiwinding Transformers. Autotransformers
131
and the sum of the secondar y and tertiary powers mayexceed the primary power
+ VII ; ;;;;:: VII I 8 2 + S3;;;;:: 8
VII~
or
1
The same Soviet standard requires also that for the first power ratio for the second, and for the third
+
82 8 3 ~ 1 1/ 3 8 1 With any power ratio, however, a three-winding transformer must satisfy the active and reactive power balances
P1=P2+ P3+ 2J P Q1 = Q2 + Q3 + 2J Q where 2:.P and 2:.Q are the active and reactive power losses in the transformer itself (see Sec. 3-8). (i i) Split-Primar y (Spli t-S econda r y) Two-Windi n g
Transfor mers A split primary (or split secondary) consists of two electrically isolated parts , so, in effect, such a transformer is a three-winding t ransformer. Fro m a t hree-winding unit proper, it differs in that energy need not be transferred between the halves of t he split winding. The arrangement of a transformer having one primary (1) and a split secondary (2 and 3) is shown in Fig. 10-6. The magnetic circuit is of the core-and-shell (five-leg core) form, as shown in Fig. 1.5b. The halves of the split secondary are on the low-voltage side and are wound on different legs. The primary winding, which is on the HV side, has two parallel paths likewise wound on different legs. With this arrangement, magnetic coupling between the halves of the split secondary is very loose, and transfer of energy from network 2 to network 3 by virtue of a magnetic field is negligible. Because of this, such a transformer 9*
132
Part One. Transformers
may be looked upon as a combination of two separate transformers , one coupling network 1 to network 2, and the other coupling network 1 to network 3. If only one half of the LV secondary, say, LV2, is loaded, on the HV side only one of the parallel paths, wound on the same -leg LV2 HVt will likewise be loaded. Of Vi HVf course, such a transformer LV.; can transfer energy in the reverse direction as well. Then it will have two primaries, LV2 and LV3, each supplied from a separate source, and one secondary, HV1. Fig. 10-6J Single-phase, twoThe values of Vs and V 2 winding transformer with a split may be the same or different. LV winding The values of <1>2 and <1>3 and ofi,the refferred currents l~ and I~ depend on the relative magnitudes of VI and the referred secondary and tertiary voltages, V~ and V~. If
If. V~
then
=
V~
i; = i;
and eD 2 = cDs and, as a consequence, the fluxes in the outer (unwound) limbs of a five-leg core-type magnetic circuit are nonexistent. In the general case , when the voltages in networks 2 and 3 are such that V2 =1= V;, the referred currents and fluxes are likewise different
.
I~
.
. .
.
=1= I; and
.
and the difference flux, c})2 - <1>s, has its path completed via the outer legs. If the transformer had a two-limb core, then, with V~ =1= 1 ; , the difference flux between the upper and lower yokes would run outside the magnetic circuit, and appreciable eddy currents would be produced within the sides of the oil tank and other substantial structural parts, leading to increased eddy-current losses. This
i
133
Ch. 10 Multiwinding Transformers. Autotransformers
is the reason why it is preferable to use the core form of magnetic circuit for split-winding transformers . As compared with conventional two -winding transformers having one HV primary winding and one LV winding, split-winding units offer an unfailing ad vantage in that should a short-circuit occur across the secondary terminals, it will draw half as heavy a current from the supply line . This is because in a conventional two -winding transformer windings 2 and 3 are connected in parallel and its shortcircuit impedance ZSCi23 is half the short-circuit impedance of windings 1 and 2 (or 1 and 3) in a split-winding transformer , ZSC12 = ZSC13' Understandingly, split-winding transformers have gained marked popularity. 10-2
Autotransformers
In an autotransformer, the primary and secondary windings are coupled bothind uctively and conductively. In fact, it
if,
z x
v'l
z
i'=iz
Fig. 10-7 Autotransformer connection
has a single tapped winding which serves both primary and secondary functions. The auto-connection used to transfer energy from an input networlc[at voltage V to an output network at voltage V' > V is shown in Fig. 10-7. The figure shows two windings, 1 and 2, wound on the same core and enclosing each other (see Fig. 1-1a). The primary is on the LV side, V, and the secondary is connected between terminal a (X) on the input network and terminal x on the output network in such a way that its voltage V 2 is added to V to give V'.
134
Part One. Transformers
The secondary of an auto-transformer must be designed for V or V', whichever is the higher (in the circuit of Fig . 10-7, this is V'), rather than for V 2 , as in an ordinary transformer. The transformation ratio ri of an autotransformer is the ratio VIV' at no-load (1' = 0). For the circuit in Fig. 10-7, n
=
+E
VIV ' = El/(E l
2)
= 1/(1
+n
21 )
where n 2l = E 21E j = w 2/wl • Electromagnetic processes in an autotransformer can be described, using the usual transformer equations
.
V2
jj
+i
.
=
E2
=
i
.
(10-5)
- Z 2 /2
o 2 n 2l p..
•
E, = E 21n 2 1 = -Zo/o
To them are added equations describing the circuit itself, with the positive directions assumed as shown in Fig. 10-7:
'.
.
V' = V
.
V
=
.
+. V
2
-VI
(10-6)
A phasor diagram for an autotransformer is shown in Fig. 10-8. For insight into" the basic energy processes in an autotransformer, we shall neglect loan d the voltage drops in the windings which enter Eqs. (10-5) and (10-6) by assuming 1 0 = 0, I z, I = 0, and I Z2 I = O. Then, 1111 2 = V 21Vl = n 2 l V = VI
V' = V + V 2 = VI (1 + n 21 ) = Vln 1 = 11 + 1 2 = 1 2 (1 + n 21) = 1'1n
{10-7)
135
Ch. 10 Multiwinding Tr ansformers . Autotransformers
With the above simplifications and the active and reactive power losses neg lec ted. t the t ot al power ofj an autotrans-
v=-v,
Fig. 10-8 Ph asor di agr am of an autotransformer in th e case of a resistive-inductive load; cpr > 1, 1! 21 = 0.5, I! = 1/(1 0.5) = 2/3
-+
former ma y be wr itten as the sum of two components
S
=
VI = VIII
-+ V II
= ST(a) -+ S c = (VI -+ V 2) 1 2
=
2
=
V 2I 2
VIl' = S'
-+ V II
2
(10-8).
where S T (a)
= VIII = V 2 I 2
is the power transferred power), and
inductively (the tr ansformed
Sc =
VI I 2
is t he power t ransferred from the primary to the secondary network conductively (the condu cte d power). This is why an autotransformer needs t o be designed to withstand only the ST( a) term which accounts for only a fraction of t he to tal power, S , called t he auto fr acti on ST(a/S = T1 2I 2/V'I 2 = (V' - V)/V' = 1 - 12 (10-9) where 12 < 1. ..
136
Part One. Transformers
In an autotransformer as in a conventional transformer, the transformer size is sole ly determined by the transform ed pow er S T . As has been shown in Sec. 9-1, the transformer size is proportional t o S 1/ 4. For an ordinary transformer, ST = S . For an autotransformer, S.;' = S T ( a) = (1 n) S. Accordingly , given t he same power rating, an autotransformer will be smaller in size and less exp ensiv e to make. An autotransform er becom es more attractiv e as its transformation ratio n differs progressively less from unity . For example, when n = 0.9, the transformed power decre ases t en-fold, where as when n = 0.1 , it is about the same as that of an ordinary transformer . This is the reason why it. is common pra ctice t o m ake au totransformers with n ranging between 0.5 and 1. In su ch cases , the relatively mor e expensive insulation on the secondary is more than offset by the reduction in the weight of, and the losses in, the autotransformer. Autotransformers ar e widely used to power domestic appliances and control-system units and come in size s from 10 to under 1 000 VA . In the Soviet Union , they are common in hi gh-voltage power t ransmission lines where they are used t o t ie networks operating at closely spaced voltages, namely '110 and 220 kV, 220 and 500 kV, and 330 and 750 kV. The overall capacity of such autotransformers runs into hundreds of megavolt-amperes. Autotransformers may be used t o both step up and step down the applied voltage . For example , the autotransformer in Fig . 10-7 will step down the applied voltage, if the load is connected to receive V , and power input comes from a network at V' . Apart from single-phase, two-winding autotransformers (Fig. 10-9a), power syst ems oft en employ three-phase, twowinding autotransformers (Fig. 10-9c), and also singlephase (Fig. 10-9b) and t hree-pha se (Fig. 10-9d) three-winding au to transform ers . The st andard win ding conn ections and phase displ acement gro up s used in t he Soviet Union for autotransformers are li sted in Fig. 10-9. Auto-connect ed single-phase windings are deno ted by I an l a , whereas starconnected t hree-phase au to windings, with t he neutral av ailable for conne ct ion , are den ote d by Y n. au to ' Autotransform ers may constitute an electr ic hazard , especially when 1/n ~ 1, because of direct connect ion between the HV ne twork at V' and the LV network at
137
Ch. 10 Multiwinding Tra nsformers. Autotranslormers
v ~ V' . In t he abse nce of grounding, t he voltage between the LV con duc tors and ground is V' /2, wh ich appea rs owing to cap acitive coupling between the HV wires and ground. IIV ond MV
HVond LV A Q X
~
~ UJ-t
x
I I Auto
LV
~o /1-0-0
ta ;
HVand LV OAaBbCc
I (6)
B
ffi
A
Yn , aulo HVand M V
n
~r (C)
LV
A Am B 8 m C Cm
Fi g. 10-9 Winding connec tions and phase displace ment groups for autotran sform ers
For example, if an autotransform er were used t o ste p down from 3 kV t o 220 V, the voltage bet ween the 220-V wires and ground would be 3/ 2 = '1.5 kV. This is why applicable safety codes guard aga ins t usi ng au to tr ansformers with ~/n ;> 2, .
Part One. Transformers
138
The use of autotransformers with n ~ 1 runs into certain difficulties because fairly heavy short-circuit currents are likely to develop. If an autotransformer is energized with V' on the HV side (Fig. 10-7) and a short-circuit occurs on the LV side, V would reduce to zero, and winding 1 of the autotransformer would be short-circuited. At the same time, the voltage across winding 2 would rise from V 2 to V', and this would lead to a further increase in the shortcircuit currents. On setting V = -VI = 0 in Eqs. (10-5) . and (10-6), let us determine the steady-state short-circuit current in winding 2:
I SC2 = V'IZsC2I= (V2IZsC21) (T1'IV2) = I sc2. T/(1-n) (10-10) where ZSC21 = short-circuit impedance with winding 1 short-circuited and with the supply voltage applied to winding 2 I sc2. T = V21ZsC21 is the short-circuit current in winding 2 with V 2 impressed on winding 2. It is seen from Eq. (10-10) that the short-circuit current in winding 2 of an autotransformer is V' IV 2 = 1/(1 - n) times the short-circuit current of a conventional transformer used to transfer energy from a network at V 2 into a network at VI' The closer is n to unity, the larger the short-circuit currents, I sc2 and I SC1 = n21Isc2' and the more dangerous are their consequences.
II
Transformers in Parallel
11-1 ·
Use of Transformers in Parallel
Parallel connection of several transformers is widely used in electrical systems. In many cases, it is the only way to convey large blocks of power over large distances. Several transformers operating in parallel at a major substation cannot be replaced by a single unit of the same total power rating, because it would be prohibitively large and unwieldy both to manufacture and move it to its permanent location. Even at not so large substations, the use of several transformers operating in parallel offers a more convenient way to tackle the problems of reliability and plant expansion.
139
Ch , 11 Tra nsf orme rs in Parallel
Shoul d any unit fa il, the remaining ones will still be operabl e and take up the load previously carried by the faulty transformer. In the meantime, the fa il ing transformer can be replaced by a standby unit whose cost will undoub tedly be sm all in comparison with that of all the installed transformers. Also, if a substation has a sufficiently large number of t ransformers, it is always possible to combine in paralle l as many of them as may be necessary for optimal load sharing and energy conversion at a min imal loss (see Sec . 6-3). The choice of a number of transformers to be operated in parallel is both an engineering and an economic problem in optimization . In this problem, the variables to be opti mi zed are the t otal cost of manufacture and operation of the installed transformers . An important po int to bea r in mind is that the cost of the energy lost and the cost of manufacture decrease] with the increase in per unit rating , wherea s the redundancy cost increases. 11-'-
Procedure for Bringing Transformers
in for Parallel Operation
To avoid li kely errors, the transformers to be operated in parallel must be interconnected at ident ically marke d term ina ls. An example of two transformers connected for parallel A(% if
t.;
E"A x r-
~
a ~
i flJ 11/3
.J
r--
/3
Ef,B
X
~t2~
'--
Xcr; Sf
IX
Ii;
12a:
'-----
f
ap
ZA
.
1/2
tE,d
12p
t. E
St
aa:
Vi Z
T
x
Fig . 11-1 Par all el operation of t wo l/i -O singl e-phase, two-winding transformers
operat ion is shown in Fig. '1'1-1. As is seen, the ident ically ma rked te rminals of transformers a an d ~ (A a and A 13' X a and X (3' aa and a (3 ' :ra. and XII) ar e respecti vely connecte d t o the same bus. . .
140
Part One. Transformers
Let us formul ate the rules for paralleling two transformers, with their ·load Z disconnected (that is, with switch 8 2 open). Obviously, the primary terminals of the two transformers, namely A a , A 13' X a , and X 13' may be connected in the above way to the input network without having to meet any additional requirements. After the primaries are connected for VI = VIa = V 2j3, the voltages existing between the disconnected secondary terminals aaxa and aj3xj3 will be as follows: V 2a = E 2a = V Ia/nl2a = EIa/nI2 a
and V 2j3 = E 2j3 = V I j3/nI2j3 = E I j3/n l2j3 Terminals X a and xj3 may be commoned without running any risk. However, commoning terminals aa and aj3 may give rise to an emf across switch 8 1
.
.
.
E I>. = E 2a - E 2j3 (11-1) Commoning terminals aa and aj3 will not give rise to any circulating currents in the windings only if o
0
•
EI>.=E 2a - E2j3=O or when the secondary emfs are the same
.
.
E 2a = E 2j3 For this to happen, the transformers to be brought in for parallel operation must meet the following requirements: 1. Have the same transformation ratio. If n l2a = nI2j3 and VIa = VI 13' the secondary emfs J will be~ the same, E 2a = E 2j3. 2. Fall in the same phase displacement (clock figure) group. If so, N a = N 13 = N, the secondary emfs, E 2a and E 213, will be turned through the same phase angle relative to the identical primary emfs, E l a = E l 13 = -VI' and will therefore be in phase o
o
E2a
0
0
0
(E l a/n l2a ) exp (jON) = (EI 13 /nl2j3 ) exp (jON) = E2j3 =
The above requirements are also applicable to three-phase transformers. When they are brought in for parallel operation, connections must be made between the identically
141
Ch. 11 Transformers in Parallel
marked line and neutral terminals (A a and A
13'
°
B a and B 13'
Ga and G 13' a a and a B' b« and b B, C a and cB ' O a and 13 ) , If this condition is met , the secondary line emfs will be iden-
tical in both m agnitude and phase . 11-3
Circulating Currents Due to a Difference in Transformation Ratio
Consider two V1-0 single-phase transformers ex and p. If their transformation ra tios ar e not the same, n I 2a =1= n 12B ' and E t>. =1= 0, the circulating currents l la , lIB' 1 2a , and 1 2 13 which will appear in the windings upon closure of switch 8 1 , can be estimated on neglecting the magnetizing . . currents (Io a = 1 0 /3 ) an d writing the equations for transformers ex and p (see Sec . 3-3) for the positive directions shown in Fig. 11-1:
.
VIa
VI B where
. . + IlaZla, .. -E I B + I I BZI B'
= -E l a
.
.
1 2a =
..
V 2a
= E 2a
-
1
V 2B
=
-
1 2 [3Z 2 [3
- I l a n 12a ,
. . 2a
E 2 [3
.
1 2 [3 =
Z 2a
(11-2)
. - I I [3 n 12 B
Also, we must consider what happens when t he two transformers are brought in parallel while the load is disconnected (switch 8 2 is open and 1 2 = 0):
. .
. .
. .
=
V I 13
=
II =
I la
+ 11 [3 '
VI
. .
V Ia, V 2
1 2a
=
.
V 2a
.
+ 1 2 [3
(11-3)
.
=12 =0
Solving Eqs . (11-2) and (11-3) for the secondary circulating . . current 1 2a = - 1 2 [3 gives (11-4)
+
+
where Z~c[3 = Z2[3 Z I BlnT2 13 and Z~ca = Z2a Z I a 1n T2a are the short-circuit impedances of the two t ransfor mers, w hen the functions of the windings are reversed .
142
Part One. Transformers
Using E qs. (11-4) and (11-2), we can readily establish t?e relationship between the voltages V2 = V2a = V2/3 and VI that exist aft er switch SI is closed •
v 1 -where n 21/3
"Z 'sea +z'se
V• 2
-
. Zse
/3
(11-5)
an21/3 + Z~ e/3n 21/3
= 1/nI2/3' n21a = 1!n 12a , and for
.
.
+
Z~ea
=
Z~el3
VI = - V 21n 21 (11-6) n 21/3)/2 is the mean transformation
where n 21 = (n 2la ratio. If the difference in t ra nsformation rat io between the two transformers is small (n12:1.lnI2/3 ~ 1), where
.
.
i.; .
=
- i 2/3
~ E!>,I(Z~e/3
.
+ Z~ea) .
E!>, = E 2a - E 2/3 = E 1 (1!n I2a - 1InI 2/3) ~ V 2 (~n) (11-7) is the difference between the secondary emfs given by Eq. (11-1), ~n is the per-unit difference in t ransformat ion ratio, and n 12 is the mean t ra nsform ation rat io. The circulating currents defined by Eq , (11-4) and appearing at the ra ted primary voltage.] VI = V I, R , when V 2 ~ V 2 , R, can conveniently be ex pressed in per-unit, t aking the rated current of, say, transformer a as the base quantity: l*ea ~ 1 2al1 2a,R = I l al l l a , R (11-8) ~ ~nl(VBe,a v Se/3S a, RISI3 ,R) where Vsea = 1 2a, RZ~e a1V2a , R v se/3 = 1 213 , RZ~e I3 IV2 I3 , R
+
are the impeda nce voltages of t he two transformers, and Sa ,R S I3 . R
= 1 2a ,RV2a , R = 1 213 , R V 213 , R
are the power ratings (rated powers) of the two transformers . It follows from Eq . ('1'1-8) th at eve n with a small difference in the t ransformat ion ratio, the circula t ing cur rents may be comparable in m agni tude with t he rated currents of
143
Ch. 11 Transformers in Parallel
the paralleled I,transformers. For example, when 'Sa , RIS f\,jR = 1, Vsca = vscf\ = 0.05, and /),n = 0.05, the circulating current will be 1*. c = 1 21I2 R = 0.05/(2 X 0.05) = 0.5 per-unit To avoid hazardous circulating currents, the transformers to be paralleled may differ in their transformation ratios by not more than 0.005. 11-4
Load Sharing Between Transformers in Parallel
If paralleled transformers meet all the requirements, no circulating currents will be flowing in their windings when the load is disconnected. Let us load the paralleled transformers by closing switch S2 (see Fig. 11-1) and see how the load current will be
--if z
-v;
Fig. 11-2 Equivalent circuit of two transformers in parallel operation
shared. This can be done by reference to the equivalent circuit in Fig. 11-2. In fact, it is a combination of the equivalent circuits for transformers CG and ~, as given in Fig. 6-1. The currents in the parallel branches formed by the short-circuit impedances of the transformers Zsca
=
Zscf\ =
+ n;2 Z2a = Zsca exp (jqJsca) Zlf\ + n;2 Z2f\ = Zscf\ exp (jqJscf\) Zla
are inversely proportional to the impedances
jlalI~f\ =
Zscf\/Z sca
=
(ZsCf\/Z sc a)
exp
(j/),qJsc)
(11-9)
144
Part One. Transforme rs
The sum of the currents gives the load current
..
+
.
I = I 1a I 1fl If the paralleled tra nsformers are fully identical and their short-circuit impedances are the same in magnitude, Z sc a = Zscll' and in phase angle, (Psc a = qJsc fl ' then each transformer will carry half the total current I 1a = Il~ = 1 1/2 If the short-circuit impedances are t he same in magnitude, but differ in phas e angle , qJsca < qJscfl ' then
.
.
I 1a = II fl exp (j L1qJs c) and the current phasor diagram looks like one shown in Fig. 11-2. Ea ch transformer carries a current given by
I 1a = I 1fl = 11/2 cos (L1qJsc/2) which is 1/cos (L1qJsc/2) times the current I i/2 existing when the load is shared equa lly. Fortunately, even with the largest possible value of qJsc fl (about 90°) and the least possible value of qJsc a (about 60°), their difference is about 30°, and the resultant overload does not exceed 1/cos (L1qJsc/2) :::;;; 1/cos (30°12) = 1.03 Therefore, the overload due to a difference in qJsc may be neglected, and consideration should only be given to the difference in magnitude between t he short-circuit impedances. It follows from Eq. (11-9) that the current ma gni tudes are inversely proportional t o the "magnitu des of the shortcircuit impedances
I 1aJI 1fl . Zscfl/Zsca Simple manipulations give (Iia/lill) (Vl ,RIV 1 ,R) Zsc flI lfl,R V1 ,R
Not ing that
and
I1a,RV1,R I11l ,RV1, R
Ch, '12 3-Phase Transformers un der Unbalanced Load
'1 40
we can read ily find t hat per-u nit load s on t he paralleled transformers, S 'l,a and S * ill are inversely proportional to th e impedance voltages, Vsc a and vs cfj : S* aIS*fl = vscfjlv sca (11-10) If vsc fj = Vsc a , the per-unit load is abo ut the same on either of the paralle led transformers, and eac h is being utilized to full advantage . If one carries its rated per-u ni t load, S'I:a = 1, the ot her, too , will carry its rated per-unit load , S*fj=1. If, say, vsc fj > Vsc a , transformer ~ will be under-loaded, although transformer ex is carrying its rat ed load
S* fj = (vscalvscfj) S 'I:a < 1 Conversely, if transformer ~ is carrying its rated load, transformer ex. will be overloa ded
s.; =
(vSCfjlv sca ) S'I: fj > 1 This is the reason why the transformers to be paralleled must have identical relative impedance voltages . (I n pr ac.t ice, t he maximum difference is allowed to be as high as 10% .)
12
Three-Phase Transforme rs Under Unbalanced Load
12- 1
Causes of Load Unbalance
In the preceding sections, we discussed three-pha se t r ansform ers operated in networks with symmetrica l voltages and ba lanced l oads. Unfo rtunately, an ideally ba lanced load is practically nonexistent in power systems, and there is always some degree of unbalance present . This unbal ance increases with increasing power rating of single-phase loads drawing their power from three-phase systems, an d is especially pronounced under abnormal conditions, suc h as tw oand sing le-phase fau lts to ground, fa ilure of one of t he phases , and the like . To form a reliable estimate of the unbalance that m ay be to lerated in an operating system, we need a mathematical description of what happens in a transformer in t he case of an un ba lanced load. 10-0169
. Part One. Transformers
.
.
. In the most general case, a transformer may be not only
~ '~
carrying unbalanced secondary line currents I c .Itne- I b.\Ine \a~nd je. llne, but also operating from a network with unba., Iance di .Iine voltages 11A D , 11B C , and V CAo To obtain a of the events taking place in such a case, .complet e picture ... . . . 'we must determine the phase secondary currents I a' I b' and } e (if the :secondary is delta-connected), phase and
.
.
. ..
.
.
.
line primary currents I A, I B, I C and I A.llne, I B.llne, I c.une (the latter only if the primary is delta-connected), primary .. . phase voltages 11 A' VB' and V c (the primary is star-connect. . . cd), :secondary phase and line voltages Va' Vb' V e and Va b , Vbe' Vea (the latter only if the secondary is star-connected). ;), Most commonly, these quantities are found by the method _Qf ) ymm etr ical (phase-sequence) 'com ponents. By this meth.9d;' an unsymmetrical (unbalanced) set of phase voltages, currents ·0 1' fluxes is resolved into symmetrical systems equal in number to the number of phases and formed by the respective components in the positive, negative and zero phase sequences. An important point to bear in mind is that the phase sequence in the supply network has no bearing on what happens in a transformer under balanced load contitions. This implies that its winding impedances for the negative-phasesequence (NPS) currents do not differ from those for the ..positive-phase-sequence (PPS) currents, Zl' Z2 and Zo (see 'Chap . 8). Special treatment is only needed for zero:phase-sequence (ZPS) currents. ; 12-2
Transformation of Unbalanced Currents
. (i) Star-Connected Secondary With the secondary star-connected, the specified unbalanced
~ Iine. currents j a, line' jb. .\.t ime, phase currents
l lne i
and
Ie, l Ine
Ia=Ia.line, Ib=Ib .line,
are, at the same
Ie=Ie.line
Ch. 12 3-Phase Transformers und er Unbalanced Load
147
The phase secondary currents may be represented as sums of symmetrical current compon ents , namely, PPS currents i.. = (t a 2i b ai e)/3
+
+
(12-1) NPS currents
+
+
ja2 = (i a a2jb ai e)/3 . . I b2 = I a2a i.; = j a 1a 2 where a = exp (j2n/3), an d ZPS currents
a.
to
(12-2)
+
= i. , = jeD = +ib i e)/3 (12-3) How a set of phase currents is resolved into symmetrical components is illustrated in Fig. 12-1.
i
ut
iao!H ico ibO
Fig. 12-1 Resolution of an unba la nced system of currents into sy mmetrical components -
The neutral wire of a Yn-connected winding carries a current . . . . (12-4) In = r , i, t , = 3I ao
.
+
+
As is seen, this current is three times t he ZPS current. Assuming t hat t he system (network) is linear and neglecting the magnetizing currents in comparison with the load currents, we may deal with the transformation of each of the symmetrical systems individually. 10*
Part One. Transformers
The relationship between t he PPS primary and secondary currents has been established in Sec. 3-7. It has been shown that whate ver the conn ection of the second ary and primary windings,
.
II
=
"
-I~
+1
~ -I~
0
This equation may be written for any of the three phases, using the notation adopted for an unbalanced load: (12-5)
where j ~I ' hI' and j~I ar e the secondary currents referred to the primary side. * Because the phase sequence in a transformer is of no importance, the relationship between the NPS secondary and primary currents will be the same
This relation may be extended to ZPS currents in cases where they can flow in t he primar y winding, that is, when t he primary is star-conn ect ed, with the neutral brought out, or del ta-conn ected:
. lAO
=
.
-I~o
.
= I BO
-i/,o = I
-I~o
co
(12-7)
Thus, when the primary is connected in a Yn or /'0", its phase .current s are equal to the corresponding secondary phase .current s:
.
. . + I A 2 + lAO
.
IA
=
I
iB
=
-j;,
A1
-I~
(12-8)
t,
= -j~ When the primary is connect ed in a Y n its line currents do not differ from its phase currents i
j A,lIne =
iA ,
j B,lIne =
j
B'
i C ,lIne
=
Ie
* Equation (12-4) and all th e other eq ua tions in this section ar e written for th e winding connec tio ns where the id en ti cally marked phase wind ings (A and a, Band b, C and c) are wound on the sam e-leg.
149
Ch. 12.3-Phase Transformers under Unbalanced Load
andthe current in the neutral wire
jN = si AO= - sjao -
-
', In
is equal to the referred current in the neutral wire on t he secondary side , In a delta-connected pr imary, t he li ne currents do not contain ZPS components
.
.
.
+I
- (I Bl + I BZ
=
a.. + jAZ) -
=
-(tl
+ h2)
BO)
+ j B2) + (hi + h 2) (jBl
(12-9)
The ZPS current l AOhas its path completely around the delta and does not appear in the line wires (Fig. 12-2).
.,.
-(i~-if,)
-Ia.
A
"
; -Ib
-j:
(i~-iD
a
B (i:-i~)
,I;
C
c
(jk-i~ o)
Itt i, i,
u-u." )
A 'tu
B
6
(i~-j~o)
c I' =3Il1.o , n
C
i n=ajl1.o
n
,.---r-~
A
B
~
C
,
n -i~
ia a
t,
A
-i;
-t;ir :
6
i,
B
-I:
-i;
c
jn=ajl1.O
C
---
'
-16
n
ai~o=O
N
ia. ,....-_IL ::J
"
h,llne=It,-Ie
..
Fig. 12-2 Transformation of unbalanced currents by various winding connections
In a star-connected primary, there is no neutral wire that might carry ZPS currents. Therefore, no ZPS currents into the primary .'are . . induced -
150
Part One. Transformers
and the ph ases of this win ding onl y carry PPS and NPS currents
t, = =
j
B
= =
t.:
=
i., + i A 2
= -j~l -
h2 =
-(i~ - i~o)
i, B) . j Bl + j B 2 = -ib1 - Ib2 -ii; = -(j;, - h o)
-lea) (12-10)
As follows from t he foregoing, wh ate ver the form of conn ect ion , the PPS and NPS curren ts are transformed identically . Therefor e, it appears reas onable t o treat separately only the ZPS phas e curren t s, and t o lump t ogether the PPS and NPS currents
.
. .
.
.
I a = I( a) + l ao
.
+
where I(a ) = I al I a 2 is the PPS and NPS currents in phase a, shown by the da shed line in Fig. 12-1.
(ii) Delta-Connected Secondary . When t he secondary is connected in a delta, t he specified line currents always sum t o zero
.
+
.
+
.
I a ,line I b,lIne I c,line = 0 Because the line currents are t he differences of phase currents
.
...
...
..
la, line =Ia-I b, I b,lIn e =Ib-I c, I C ,line=Ic - Ia (12-11) and the phase cur re nt s do not contain ZPS components and sum to zero . . . Ia Ib t ; = 3I ao = 0 (12.-12) we may wri te the phase currents in terms of t he line currents ~
+
+
i a = ( t,llne - j c,llne)/3 i, = (i, , llne - i, .llne)/3 i. = (j c,lIne - i b•lI ne)/3
(12-13)
151
Gh. 12 3-Phasc Transform ers under Unbala nced Load
Graphically, the phase currents are defined by the centroid of a line-current triangle, lying at t he intersection of its medians. As will be recalled from school ma thematics, the
-r .
a ~----',------=;;P}c
..........
\
"
". ,
\
Ie,line I I
\
<,
I
rIa, line
\ <,
,
.~\
" ":
.
.
1
\
<,
, -'
\
1
\ 1 ''J
Fjg, 12-3 Determining th e ph ase currents in a delta- connected windnig .
-
, . j ").
intersection of the medians in a triangle li es two -thirds ..qf the way from its apexes (see Fig. 12-3). Because the secondary phase currents do not containany ZPS currents , they are fully transformed into theprim.ary, no matter how it is connected (see Fig. 12-2): . . s>
-h,
j B = - j b' i C = -j~ When the primary is star-connected with its neu trrl;!l brought out,
jA =
- : " :~ I
t N = 3j AO = -3ho = 0 12-3
•
.i .
". : , . ' 01
.r
i :: '~.
", ' 'HC'<: Magnetic Fluxes and EM Fs under Unbalan ced Load Conditions
Under unbalanced load conditions, the total magnetic flux may, in a linear approximation, be visua lized as ~the superposition of the fluxes set up by PPS , NPS :" and ; ZPS currents (Fig. 12-4). » : <: } " J J r
The balance between the PPS primary cui~erit~:aJ~ ~, I Bl' I CI) and the PPS secondary currents (j aI', is never complete. The unbalanced fractions of the .BPS primary currents, which are the magnetizing curr~nts:'1B
Al;J 'cD
:
'.
..~ .~~ 1
Part One. Transformer:'!
f52
+
+
+
hI' JBI hI' JCI i.. give rise to a balanced set of PPS fluxes
,pe2 Wz
Fig. 12-4 Balanced components of magnetic fluxes and emfs und er an unbalanced load
transformers under a balanced load (see Sec. 4-1). What is especially important is that these fluxes sum to zero.
cD A l
+ (DEl + cD
c I
= 0
So, they are free to traverse a closed path in any form of magnetic circuit. The same goes for the systems of NPS currents in the primary winding (1A2 ' 1 B2' 1 C2) and in the secondary winding (I a2' 1 b2, 1 e2)' They, too, are not completely balanced and form a symmetrical set of NPS fluxes (see Fig. 12-4)
.
cP A
.
2
.
+ cP + cP B2
C2
=
0
In contrast, the ZPS fluxes established by the ZPS currents and their paths substantiallydependjon how the windings are connected and the form of the magnetic circuit. As has already been explained in Sec. 7-3, the ZPS fluxes have their paths completed within the magnetic circuit only in the core-and-shell (five-leg core) type of transformer and also in a three-phase bank of single-phase transformers. In a core-type transformer (see Fig. 12-4), the in-phase
153
Ch, -12 3-Phase Transformers under Un ba lanced Load
ZPS flu xes
.
·
.
.
.
cD B o
=
have their paths outside the magnetic circuit and within the nonm agnet ic ga ps, tank sides, an d ferro magnetic structural par t s. Because the ga ps offer a high opp osi tion, the ZPS flux es in a core-type transformer are m arkedly sm alle r than t hey are in a five -leg core-type transformer or in a bank of transformers (With the same mmf, I a ow 2 ) . The ZP S fluxes are especiall y strong when the ZPS currents flowing in a star-connected secondary with its neutral brought out are not balanced by the currents in the primary, which usua lly happens when the latter is star-connected with no neutral wire available (see Fig. 12-4). As with PPS and NPS currents, it appears reasonabl e t o lu mp together the PPS and NPS flu xes and t o treat sepa rate ly only t he ZPS fluxes :
.
.
where .
.
.
cD(A)
=
cD c I
.
.
.
CV( C)
.
+ cD"
cD( B ) = CV BI + cD B2 ' and
= .cD Al
+
.
.
+
CV c =
(1) A2 '
fluxes. Sinusoidal PPS and NPS fluxes induce in the primary phases the mutual emfs of positive and negative phase sequence s [see Eq . (3-7)]: E( A ) =
E A I
. " E (B) = E B I
+E = -
j
(wll!2) w lcD(A) =
+E
j
(wll! 2) wj(1) (D) = E(b)
A2
B2 =
-
_.
Eia)
(12-14)
where Eia) = E(a)wI /w2 is the PPS and NP S mutual emf of phase a referred to the pr imary sid e. Sinusoidal ZPS fluxes induce in the primary phases mutual emfs of zero phase sequence (see Fig . 12-4):
E.40
=
-
j
(w/ 11 2) wjcDo= E~o
(12-15)
where E' aO = Ea ow I /w 2 is the ZP S mutual secondary emf referre d to the pr imary side . The ZPS mutua l emf may be expressed in t erms of the ZPS currents set ting u p the ZPS flu x cD o (the ZPS flu x
t:
Part One. Transformers
154
needs to be treated separately only when the primary is star-connected and it carries no ZPS current, as in Fig. 12-4) (12-16) where
Zoo
+
Roo jX oo is the mutual impedance to ZPS currents X oo = Ulw;Aoo is the ZPS mutual reactance proportional to the permeauce for the ZPS flux Roo = resistive component of the mutual impedance, due the hysteresis and eddy-current losses in the ferromagnetic structural parts, associated with the sinusoidal ZPS fluxes. Because t aO is the magnetizing current for the ZPS fluxes, Eq. (12-16) is written by analogy with Eq. (3-7) defining the relation between 1 0 and E 1 • =
.
12-4
.
Dissymmetry of the Primary Phase Voltages under Unbalanced Load
The equations defining the primary phase voltages under unbalanced load are written by analogy with those for the balanced load conditions, Eqs. (3-13). The mutual emf E 1 is replaced by the mutual phase emf which is the sum of . . the PPS and NPS emfs, and the ZPS emf, E AO = E BO = E c o: · . . . VA = -E(A) E AO ZII A
· VB ·
=
Vc =
+ . -E(B) E A o + ZII B . . . -E(c) - E A O + ZlI c .. . .
.
(12-17)
The primary line voltages V AB, V BC and V CA, which are in the general case unbalanced, are specified in advance. When the primary is delta-connected, the primary phase voltages are the same as the specified line voltages and need not be determined. Also, the ZPS current lAo around the delta balances the secondary ZPS currents, and EAO in Eq. (12-17) vanishes. When the line voltages are symmetrical, the phase voltages in a delta-connected primary are likewise symmetrical.
155
Ch. 12 3-Ph ase Tra nsf orme rs under Unbalan ced Load
When the primary is s tar-connected with its neutral wire isolated, the specified line voltages are the differences of the respective primary phase voltages
.
.
(12-18)
.
V BC= V C-V B
Also, adding tog ether the right- and left-hand sid es of Eqs . (12-17) and recalling that t he emfs an d currents containing no ZPS currents sum to zero
+ E BI + Ecl ) + (E A2 + EB2 + EC2) = 0 . . . . . . I iA) + I(B) + I (C) = (I AI + I BI + I cl) . . . + (I A2 + I B2 + I C2) = 0
E( A)
+ E (B) + E (c ) =
(E AI
we obtain an important equat ion
VA + V B + VC
= - 3E AD = 3h oZoo
(12-19)
Sub tracting t he secon d line in Eqs. (12-18) from t he firs t and re calling Eq . (12-19), we get
.
VAB -
.
...
.
+ V B + V c) + 3V B 3 (E AD + VB)
V BC = -(VA
= VB =
CV A B- VBc )/3 - EAo =
V (B) -
BAO
and by analogy,
V
C
= CrT BC -
.
VcA )/3 - EAD = .
V ec ) -
EAO'
et c. (12-20)
Here, V( B ) and V( c) are t he phase voltages with no ZPS current flowing in the secondary, t hat is , when I c o = 0 and
.
.
E AO = -Zoo/Ao
=
0
As. is seen from Fig. 12-5, the phase voltages V( A), V(B), and V(C) are directed away from t he centroid of the line':' voltage triangle, N, towards its apexes [see also Eq. (12-13) and Fig. 12-3 for phase currents].
'I
Part One. Transformers
_ When the line voltages form a symmetrical set, V AB .VB C = VCA' and there is no ZPS current flowing in the secondary, i ; = 0, the primary phase voltages are likewise symmetrical VeAl
=
V( B)
=
V( c)
=
VA
=
VB
=
VC
The appearance of ZPS currents ii a O =1= 0) causes the centroid of the line-voltage triangle to shift by a distance Eo (from point N to point N 0) and upsets the balance of the , .J11 phase voltages. Now, even if " laO the primary line voltages are balanced the phase voltages -tAO will be unbalanced, V.4. =1= VB=I=V C ' In core-type transformers, the primary phase voltages c are distorted considerablv t: &t.~-:~~~---less, because the reluctanc~ to the ZPS fluxes is many times that existing in a fiveFig. 12-5 Phasor diagram for primary voltages under an unleg core (shell-and-core) type balanced load or in banks of single-phase transformers. It follows from Eq. (12-20) that the unbalance of the phase voltages may arise from the dissymmetry of the line voltages even though there are no ZPS currents flowing. As regards the symmetry of phase voltages, it is preferable to connect the primary in a delta, because, given symmetrical line voltages, the phase voltages will not be distorted even when the secondary carries a ZPS current.
j
12-S
Dissymmetry of the Secondary Voltages under Unbalanced Load
The equations defining the secondary voltages may be written by analogy with Eq . (3-19) applicable to balanced load. .. .. The referred mutual emf E~ = E 1 is replaced by E(a) = E C A)
Ch. 12 3-phase Transformers under Unbalanced Load
or E( Bl, and E (Cl, and also
-11~ =
-E( A l -
E~ o =
EA O +
157
.
E A O:
z~h
-v;, = -E(B) - EA O + ZJ;"
etc.
(12-21)
Eliminating the emfs between the above equations by invoking Eq. ('12-17), we can express the secondary phase voltages directly in terms of the primary phase voltages:
-V~
=
T1A
- 11;'
=
VB -
-
ZJA + ZJ~ ZJ + z~h, B
(12-22) etc.
It · may be added that Eqs. (12-22) are applicable to any form of primary and secondary winding connection. If the primary is delta-connected and the secondary is star-connected with its neutral brought out, the ZPS se
condary current I aD is balanced, from a magnetic point of view, by the primary ZPS current lAo flowing around the delta, there is no ZPS flux, and the primary currents are equal to the respective secon dary currents referred to the primary side:
i;
= jAl + j A2 + jA o = -j~l -
j
B
= -i;,
j
C
= -j~
hz - ho
= -I~
Now the referred secondary phase voltage differs from the primary phase voltage by a relatively small voltage drop across the short-circuit impedance (as under balanced-load conditions)
-11;' =
+
VB + zsci;,
(12-23)
where Z sc = Zl Z~ is the short-circuit impedance . When the specified primary line voltages are symmetrical , the primary phase voltages are, as already noted, likewise symmetrical, and the dissymmetry in the secondary phase voltages due to a dissymmetry in the currents is relatively small.
Part One. Transformers
158
Equations (12-23) may be used to determine the secondary voltages also when the primary is star-connected, and the secondary is delta-connected (a Y/!1 transformer), because then the secondary and primary currents contain no ZPS currents (I AD = I en = 0). In the circumstances, the primary and secondary currents balance themselves as well
-hI -h2 =
i A =jAI +jA2 =
.
.
-j~
I B = -Ii,
.
I
C
.
-I~
=
whereas the ZPS flux and the ZPS emf reduce to zero. Therefore, [see Eq . (12-20)1 the primary phase voltages are determined by the position of the centroid of the linevoltage triangle
.
VA
=
.
.
V(A),
VB
=
.
.
V(B),
V
c =
.
V(C)
and, given symmetrical line voltages, are themselves symmetrical, and Eq. (12-22) reduces to Eq. (12-23). . The ZPS currents I aD may cause a more noticeable dissymmetry in the primary and secondary phase voltages when the primary is star-connected and the secondary is starconnected with its neutral brought out, and there is no ZPS current flowing in the primary [see Eq. (12-10)1:
i,
= i(A) = i AI
i; = hI + j~2
+ i A2 =
+ j~o
-hI -
= j,a)
h
2
= -j,a)
+ j~o
In view of Eqs. (12-10) and {12-20), we may re-write Eqs. (12-22) as - V~ = (V(A)
+ h1Zsc) + j~2ZSC + j~oz~ - Vi, = (V( + h1Zsc) + ib2Zsc + hoz~, etc. where Z~ = Z~o + Z~ is the ZPS impedance of the
(12-24)
B)
secondary referred to the primary side. When the primary line voltages are symmetrical, V(A)' V( B) and V( C) are likewise symmetrical, and the dissymmetry is related to the voltage drops due to the ZPS currents (j a2Zsc, ji,2Zsc, j~2ZSC) and the ZPS currents, I~oZ~.
Ch. 12 3-Phase T ran sformers und er Unb alanc ed Load
159
As is seen from Fig. 12-6, the voltage drops due to the PPS currents (j~ tZsc , hnzsc, j~lZSC) do not lead t o a voltage unbalance. It is to be noted, however , that even in a core-type transformer in which t he ZPS impedan ce is rela tivel y small (Z;~n = 0.3 to 1.0) and onl y sev eral times the per-unit shortcircuit impedance (Z*sc = 0.05 to 0.13), the unbalance o~ B
Fig. 12-6 ~Phasor diagram of a YIY 11 transformer und er an unbalanced load (the primary line volt ages are balanced)
phase voltages is more noticeable due to ZPS currents than to NPS currents of the same magnitude. The effect of ZPS currents is especially troublesome in shell-and-core (five-leg core) transformers and in banks of Y/ Y11 tra nsformers. This is because the ZPS flux es h ave their paths wi thin the magnetic circuit in the same manner as the PPS fluxes. In such transformers, Z~n = Z,j;O= -10 t o 100, so even small ZPS currents gi ve rise t o a prohibitive dissymmetry of phase voltages. This is why the Y/ Y 11 connection ought not to be used in shell-and-core (five-leg core) transformers and in banks of t ransformers. In core- type transformers using the Y/ Y 11 connection, it is important to limit t he ZPS currents. Subtracting the second line in Eqs. (12-24) from the first, we will find t hat t he diss ymmetry in the second ary line
,. ~ .
160
Part One. Transformers
voltages is solely related to the NPS currents
V ab
.
=
=
Vb 11(A ) -
etc.
.
Fa
ir( + (hI B)
-
hD Zsc + (j~2 - i b2) Zsr ,
The diss ymmetry of phase voltages due to ZP S currents in shell-and-core transformers, banks of transformers, and large core-type transformers using the Y/ Yn connection can be minimized by providing an additional com pensating delta-connected winding. The ZPS currents ind uced in the delta will damp out ZP S fluxes, thereby substantia lly reducing t he unbalance of phase voltages in the main windings. Somet imes, t his delta-connected winding is use d as a tertiary winding and connected to the network. In suc h a case, the delta-connected winding is designe d not only t o ba lance out ZPS currents, but also to transform some power into the network to whi ch it is connecte d.
* 12-6
M easurement of the ZPS Secondary Imped ance
The secondary im pedance to ZPS currents is measured by producing ZPS currents , l ao = I , in t he secondary phases. The simplest way to do this is by series-connecting the secondary windings into an pen delta (Fig . 12-7a).
(a, )
~
i ao' 1.a,
Zz
1.1 Wa
~~
l ao .~ oo
Zn
s
(6)
Fig. 12-7 Measurement of ZPS Impedance: (a) test set-up; (b) equ iva lent circuit for ZPS current
Once the voltage, current and active power are measured by t he instruments connected as shown in the diagram, we can readil y find the phase im pedance to ZPS currents Z n = F/31,
R; = P/312.,
Xn=
V Z~
-
R~
Ch. 12 3-Phase Transform ers und er Unbalanced Load
161
The rea cti ve component, X n , is t he sum of the ZPS mutual inductive reactance, X 00' and the leakage inductive reactance, X 2' of t he secon dary :
Xn
= X oo + X 2 =
where A oo and A (J 2 are th e flu x and th e leakage flux , The resisti ve com ponen t R n • is the sum of the ZPS secondary resistan ce R 2 :
R;
=
wAoow~
+ WA (J2W ~
perm ean ces to the ZPS mutual resp ec ti vel y. of th e ZPS mutu al imped ance, mutu al resistance Roo and the .
Roo
+R
2
If, in addition t o Y/ Yn-connect ed windin gs , a transform-
er has one more winding delta-conn ected, then , relative to the ZPS emf , it may be cons idere d short-circuite d. The ZPS current appearing in the della, I fl , m arkedly reduces the ZPS fluxes an d impedan ce (see Fig. 12-7a). In such case, the ZPS impedance should be measured , with switch S closed (see Fig. 12-7b). When the delta is open, the ZPS impedance of the secondary winding (see th e equi valent cir cu it ) is
a
+ Zoo
Zn = Z2
When the delta is closed, it is su bstantiall y reduced to Z n6.
= Z2+
Z6 Z 00
,
Z 6. + Zoo
~ Z2
+ Z~ « Zn
because Z~ « Zoo.
* 12-7
Single- and Two-Phase Unbalanced Loads
Single-phase load, Y/Y o or Y/MY n connection (Fig. 12-8a). The quantities specified in advance are the primary line voltages V A B = V Be = V CA = V1 ,ll ne, and the load impedance Z. Phase a carries a curren t I a which has its path through the load impedanc e Z. The other phases carry no currents: The ZPS current
j aO 11-01 6 9
.
Ib=Ic=O to be
is~found
(ja + j
b
+ j c)/3 =
j a/3
Part One. Transformers
162
The .sum of the PPS and .NP S currents in .phase a is .
i: + i: = i; -
j(~) '
"
iao
=
2ia13
The currents in the primary winding [see Eq. (12-10)] are
" ;.
.iB=
ho
--:-i(b) = i;, -
= -j~/3 = j c
The voltage across the load ' impedance Z and the 'ph~se 'a Va
A
E I':
.
... . . ,Ii
-»-
in
I
Z
ia a
b
•
ic
c.
~
: wz
. c ,
G
.'
c+ i;: C 5 a
_ Va
'J_
(0)
d ' . . ..r
Ie
z
C
A
··
8
.-
c
t::t::: . ', . (6)
, i
.-
i a Va
ia,line ' ,
z
,.
Fig. 12-8 Single- and two-phase loads in various types of connection ·, 'f .:
[see Eq. (12-24)] is
-v~
=
iT A + (hi
+ j~2) Zsc+ hoz~ =-z'i~
where VA = Vll n e /3. .i. Recalling' the relations' between currents, ' the 'load current 'is ' found to be " . :, " . .' . • •
I~ = : ~ r.
,
.. ..
'
I
•
I
~ 3V AI (2Z sc
+ Z~ + 3Z')
-' -
.:
j'
' 'I~ = -V3'Vll~e/ I:2Z sc + Z~ +3i, ,.
(1:2~25)
On setting Z' = 0 in Eq. (12-25); we obtain an equation for a single-phase short-cirouit cllrrent. · . '. '.' . . ' '.. ' Single-phase load, MY n eonneetlon] (Fig. 12-8b). The quantities specified in .advance. ara .the primary phase vol-
163
Ch. 12 3-phase Transformers under Unbalanced Load
tages which are the same as the phase voltages, V A = V CA = VB = V A B = V C = V BO = V li ne , and the load impedance, I Z I. The primary currents are given by Eq. (12-10) :
t, = -i~, i B =
j
0
= 0
The load voltage is equal to the phase a voltage [see Eq, (12-23)1
-V~ =
VA + zscj~
=
-z'h
The load current is found to be
i d = VA/(ZSC + Z')
or
I~ ·
VI,line/ I Zsc
+ Z'
(12-26)
On setting Z' = 0 in Eq. (12-26), we obtain an equation for a single-phase short-circuit current (which, in our example, does not differ from the current flowing in the case of a balanced three-phase short-circuit). Two-phase load, Y/Y connection (Fig. 12-8c). The =:quantities specified in advance are the line voltages V A B = V BC = V OA = Vll ne , and the . load impedance Z. Phases a and b carry a load current I b = - I a' whereas phase c carries no current, Ie = O. The primary phase currents definj B = -i;" ed in Eqs. (12-10) are t, = -j~ = 1 0 = O. The load voltage is equal to the line voltage V a b [see Eq. (12-23)1:
.
hJ'
V~b =
V;' -
V~ =
V
A
-
V + Zsc (i~ - j;,) = zi; B
The load current is .given by
i;,
=
CV A- VB)/(2Zsc + Z')
or
I;'
=
=
-VA B/(2Zsc + Z')
VI, lIne/ I zz.,
+ z'
I
(12-27)
On putting Z' = 0 in Eq. (12-27), we obtain an equation for a two-phase short-circuit current. Two-phase load, MY connect ion. If the primary is deltaconnected and the load is arranged as shown in Fig. 12-8c,. 11*
II
164
Part One. Transformers
the load current may be found from Eq. ('12-27). Noting that in a delta-connected winding 11A
= VI,line and I VA - VB I = of load current is found to be
V 3VI,line,
the
=
VB
magnitude
Ib = V 311 1,lInel I zz., -I- z- I
(12-28)
On putting Z' = 0 in Eq. (12-27), we obtain an equation for a two-phase short-circuit current. Two-phase load, Y It-" connection (Fig. 12-8d). The line . . load current is I a, line = I c.Itne i whereas the line current in phase b is zero. According to Eq. (12-13), the secondary phase currents are
i a = (t,lIne - i c,lIne)/3 i b = -ia,Hne/3 i, = -t.s;»
=
zia,lIne/3
The load voltage is equal to the phase a voltage, Eq, (12-23), or the line voltage V ca:
-V~ = -V~a =
VA
-I-
zsch
= -i a ,lI neZ '
The load current is given by or
i a,lIne
= -
VAI(2Z sc/3 -+ Z')
I a.lIne = V 1,lInei (V "3 I 2Z sc/3 -I- Z' I)
(12-29)
On setting Z' = 0 in Eq. (12-29), we obtain an equation for a two-phase short-circuit current.
13
Transients in Transformers
13-1
Transients at Switch-On
Each time a change occurs in the load or the primary voltage, a transformer does not reach a new steady state until all transients die out. Sometimes, the currents accompanying the transients may exceed their steady-state values manyfold. The winding temperature and the emfs, all of which are current-dependent, rise substantially and may even
165
Ch. 13 Transients in Transformers
exceed the maximum safe values. Obviously, if the designer fails to take a proper account of the transients that are likely to occur in a transformer, he will not be able to choose the correct dimensions, service conditions, and the extent of protection needed. To begin with, let us consider the transients that occur when a transformer is just switched on. Suppose that the secondary winding is open-circuited (i 2 = 0). At time t = 0, the primary is switched into a supply network (or system) with a phase voltage V1
=
V 1, m cos (rot
+ 1p)
The transients in the primary circuit of a transformer can be described by a nonlinear differential voltage equation i oR 1 + W 1 d/dt = V1 (13-1) where i o is the transient no-load current , = f (io) is the mutual flux which is a nonlinear function of i o (see Fig. 2-3). Since i oR 1 ~ W1 d/dt, we may, without committing a serious error, write i o in terms of as i o = w1/L O where L o = const is the mean primary inductance d/dt = (f)R1/L o = V 1/W1
(13-2)
The solution of a linear differential equation with constant coefficients is the sum of two terms, a free (or transient) term and a forced (or steady-state) term. In our case, (f) =
o, + ss
The transient term t = C exp
(-~ot)
is the general solution of a homogeneous equation d(f)/dt + R 1/L o = 0 where -R 1/L o = ~o is a root of the characteristic equation. The steady-state term ss = m sin (wt + 11)), where m = 111 m / ro w ] is the mutual flux [see Eq. (2-9)1 which is estahlished in the transformer core at no-load and ['1 = 111m cos (wt + 1p). The constant C is determined from initial conditions, ,_~
166
Part One. Transformers
If we neglect the residual flux, eD res = 0, then at t = 0 the flux in the core is zero:
cp 1=0 = eDt Hence,
+ eD ss
= C
+ eD m sin 'ljJ =
0
and
c:D
=
- c:D m sin 'ljJ exp (-aot)
+ c:D
m
sin (rot
+ 1p)
(13-3)
The worst case at switch-on occurs when 'ljJ = +n/2, because at t = 0, VI = O. In the circumstances, the initial
I
I t I
(IT
I
14
II I'
i omar
1 0
f 2.3 lj
5
io i
<[>
wt/Jt
~.,rzIoR
Fig . 13-1 Variations in the flux and magnetizing current of a transformer at switch-on; 'IjJ = -n/2, cD core = cD exp (-aot)
value of the t ra nsient flux is equal t o t he peak value of the steady-state flu x , C = +eDm , and, as is seen from Fig . 13-1, a half-cycle after switch-on the flux in the core rises to a maximum va lue equal to twice the peak value, eD ma x ~ 2c:D m . (By this instant, the transient term subsides very little, exp (-aon/ oi) ~ 1.) The current in the winding following switch-on can be found graphically from Fig. 13-1, using the magnetization curve in Fig . 2-3. * The maximum switch-on current io, max observed a halfcycle after the onset of -t he transients may exceed the peak
* The replacement of the nonlinear magne tization curve by a linear one results in an enol' in the va lue of a o and the damping of the transient flux . . ,
Ch. 13 Transients ' in Transformers
value of . the -rat ed load current i o, max~ l!2 It.R~ l! 2 I o;w
This point mU1'1t be borne in .mind when adjusting the setting of the protective relay's and carrying out' an open-circuit test. .'. . . .. :. . "
'13-2
Transients on a Short-Circuit' Across . the Secondary Terminals
.
•
,~ l
.
~ .
We shall 'li~it ou~~elves to a baian~ed (three-phase) shortcircuit across the secondary terminals of a transformer. Suppose that prior to a short-circuit, that is , at t < 0, .the primary was energized with . . V1,m
l.!1
cos (wt
+ '¢)
."
and the secondary was open-circuited. If we ignore the magnetizing current and deem i l = -i~, the transients on a short-circuit may be solved, using an equivalent circuit for a short-circuited transformer. Referring to Fig. 5-2b, the equivalent circuit of a shortcircuited transformer contains a resistive ' component,
R se = R I
+ R~ ·
and an inductive component
I •
+ X~ = '£ 1' + ;L' ~
X se = X 1
such that
X'scl (0
~ La c
: :' Thetrarisfents in s~ch 'a ·circ~it . can - be 'desoribed 'by " ~ linear differential equation with constant coefficients; ·R se - ' 'const ant , and ·.Lse . constant: ' .' " :'
Rsei l
+ Ls~ dit/dt =
VI
'whose solution is The transient term it,l = C exp
(~aset)
is the solution of a homogeneous ' equation dil/dt
+ Rseil/Lee
= 0
. (13-4)
Part One. Transformers
168
where - R sel L se = ase is a root of the characteristic equation. The steady-state term ii, ss =
1/ 2 [se cos (cut +
fp - fPse)
is the particular solution of Eq. (13-4) at t = 00, or the steady-state short-circuit current. The amplitude and phase of this current can be ascertained from a short-circuit equivalent circuit (see Fig. 5-2b):
V2 [se =
V imlV R~e + (cuLs e),l = Vim! I Zse I fP se = arctan (cuLseIR se)
The constant C may be defined from initial conditions. Prior to the short-circuit, the transformer was running at no-load, so (neglecting the no-load current) we may deem that at t = 0, i..
1=0
= -
i~,
= C+
Hence, C=
-
1=0
= ii, t + ii, ss
V2 [se cos (11; - fPse) =
V2 [se cos N' -
0
fPse)
and the transient current is it = - i~ = - l!2 [se cos (11; - fPs e) exp (-as et)
+ l!2[secOS(cut+fP-fPse)
(13-5)
The transient time is in fact the time required for the transient current iI,t to die out. In timet = 1!a se after the onset of the transients, the transient current falls to lie of its original value. In time 3la se, it falls to lle 3 , or onetwentieth of its original value and is practically non-existent. The time required for the transient component to reduce to 1!e of its original value is called the transient time or decay modulus, r = l la. For power transformers (see Sec. 3-6), 'r se = Vase = Xse/cuR se = 0.01 · to 0.2 s that is, the decay modulus increases as the power rating of transformers goes up. An increase in the initial value of the free (transient) com ponent leads to an increase in the short-circuit current,
169
Ch, 13 Transients in Transformers
A short-circuit is most severe if it occur s when til = (Psc or CPsc + rr. In such cases, the initial value of the free component is equa l to the peak value of the forced compo nent - 11 2 I sc cos (1/-1 - (Ps c) = + V"2 Is«
tp =
The waveforms of short-circuit currents when til = (Psc + n are shown in Fig. 13-2. Th e currents in the windings attain
Jl'jw
Fig . 13-2 Var ia ti ons in transform er cur re nt du rin g transients followa sh or t-circuit on the secondary si de (lp = qJsc rt)
+
i n~
their maximum values a half-cycle after the onset of the transients ij, max = I i~ Imax =
112 I sc [1 +exp ( -
n /coLsc)!
Dividing t his current by the peak rated current and assuming that the primary voltage is at its rated value, we get
t-:
maxi
11 2 i ; R = (Is c/IR) [1 + exp (-
n/coLsc)]
= (tlv sc ) [1 + exp (- n/coLsc)]
(13-G)
where vsc is the per-unit short-cir cui t current. In power transform ers, the maximum short-circuit current ma y be as high as ij, maxi V 2IR = 25 t o 15 per un it
(13-7)
(The larger va lues app ly to transform ers of lower :power ratings .)
170
Part One. Transformers
A transformer must be designed so that a short-circuit would not put it out of service or cut down its service life. In choosing the winding design and clamping arrangement, preference should be given to those preventing"the damage that the electromagnetic forces might do to the windings during a short-circuit. The point is that the .prlmary and secondary windings of a transformer carry currents flowing in opposite directions. On a short-circuit, i l . max and i 2 • max are many times the rated values. The electromagnetic forces arising from the interaction of the oppositely directed currents in the primary and secondary tend to compress the turns in the inner winding and to expand the turns in the outer winding. The electromagnetic forces arising from the interaction of currents flowing in the same direction in the turns of each of the windings tend to compress the windings in height. During a short-circuit, the electromagnetic forces, which are proportional to the current product, t-: maxi2. max' or the square of each current, i; .max or max, increase 225 to 625 times. These forces are pulsating at a frequency 2/ = 2 X 50 = 100 Hz without reversing their direction. The windings will not be damaged if the accompanying mechanical stresses do not exceed 50 to 60 N mm -2. No less dangerous is the heat effect of short-circuit currents, because the copper losses (proportional to the current squared) increase many-fold, and the temperature of the windings abruptly rises. Since the free component of the short-circuit current decays in 3..sc = 0.03 to 0.6 s, the rate of temperature rise may be evaluated from the steadystate short-circuit current. This current causes the copper loss to grow 49 to 225 times. The current density in the windings builds up appreciably and may be as high as 20 to 40 A mm -2. If the windings are assumed to be heated adiabatically (that is, the heat liberated within the windings is not transferred to the surroundings), the temperature of the windings will rise at the rate given by J2/170 ~ 2.4 to 9.5 °C S-1
i;.
Prior to a short-circuit, the maximum safe temperature of the windings may be as high as 105°C (see Sec. 16-1). The short-time maximum safe temperature of the windings, at which their insulation still remains intact, is set at 250°C.
171
Ch. 14 Overvoltage Transients in Transformers
If we know the rate of temperature rise, we can readily find the time, t sc' during which the winding temperature will go up from 105°C to 250°C: t sc
~
2.5 (100vscIJR)2 = 5 to 25 s
As a rule, the protective relay(s) will disconnect a transformer from its supply much earlier, and the winding temperature will not rise to it s limit.
14
Overvoltage Transients in Transformers
14-1
Causes of Overvoltages
In service, transformers are often subjected to overvoltages. For example, an overvoltage may deve lop when an element (or elements) of an electric system is turned on or v
a (aJ
SP
i
(b)
?/////T/T/T,7 //} '/./
Fig. 14-1 Overvoltage waves
off (switching voltage surges), but this contingency is usually provided for in t ransformer design. More dangerous to transformer insulation are overvoltages produced by lightenings which strike line conductors and induce high-voltage waves in them (lightening voltage surges).
172
Part One . Transformers
An overvo ltage wave propagates both ways from the point of occur rence at a velocity very close to that of light. An overvoltage wave has the shape of an overdamped pulse with a steep leading edge (Fig. 14-1a) . The rise time of this wave is usually a split microsecond or millisecond; the pu lse duration equal to the wave length runs into tens of microseconds . Figure '14-1a shows a standard total overvoltage wave used in testing transformers for pulse strength. Its length, in terms of the fall or decay time to half its peak va lue, is 50 X 10- 6 s. To minimize overvoltages (see Fig. 14-1b) , it is usual to equip transformers with a spark-gap , SP, which will break down at V m - Ahead of a spark-gap, the overvoltage wave may have a very la rge peak value , V mo- Past the spark-gap, its peak value, V m, ought not to exceed the voltage used in testing the insulation for pulse strength (see T able 14-1)*. Ta ble 14-1 P ul se Test Voltages for Tranaior mer Insulation used in the USSR Windi ng voltage class, kV
3
G
10
Peak value of test voltage (total wave), kV 44 60 80
15
35 110 115 220 330 500
108 200 480 550 750 1050 1550
The transients that take place in a transformer when the incident wave has a fast rise time, t; (a few microseconds) are of a very complex character . 14-2
The Differential Equation for the Initial Voltage Distribution in the Transformer Winding
The transients occurring in a transformer as the ap plied voltage is raised from zero to V m during the rise time t; may be likened to what ha ppens when the same transformer is energized with an alternating voltage having the same peak va lue, V m' an d a peri od T t = 4t r (see t he dashed line
* The wave reflected f rom th e input ca pacitance of the transformer is combined with the incident wave, hut tho total wavo does not raise the voltage above V m at which the spark-gap breaks down .
173
eh. 14 Overvoltage Tr ansients in Transformers
in Fig. 14-1a). This alternating voltage has a fairly high frequency . For the sta ndard test wav e, this frequency is f t = (jJ t/ 2n = 'li T t = 'lI4tr = 1/(4 X 1.2 X 10-G) = 2.08 X 105 Hz At such a frequency, we may no longer igno re the capacitive coupling existin g between t he wind ing elements, Cd, and betwe en the winding elements and the grounded parts of the
. . . ----,r, -
I
- ....
L.J1f
A I
-_.-..rL,
rlo!f
LJO
1
LJ
~
I2l
CJ IZl
. X
cqr
r--""'-...,....---, I I
I
I
A
L
I
f
..L C' ..LC" ,0sit ,0sit
Cd.
I
L
L
I
"""""'-;..,..--. cit
c''I
=~c,f
-
....
f
C lid.
cit
II·
L/
~-
Cd
II
II
V
.J... e'" ,T Sh , L I
,;:c,f
c'q.
- L'f.k
x
r
II
=;=c~
,;:c~
~6) ,
F ig . 14-2 Overvoltage in a windi ng: (a) cross-section al view; (b) equivalent circuit : 1-winding conduct ors ; 2-grounded parts; 3-through 6-electrostatic shields around the coils next to th e winding start; Cd-capacitance between coils; Cq-capacitance of coil to ground; C~h , C ~;J, C~ l; - cap acitances between shi elds and winding
transformer, C~ (Fig. 14-2a). Nor may we use the usual equivalent circuit which only considers inductive coupling and is applicable to operation at the rated frequency (which is 50 Hz for power t ransformers in t he Soviet Union). Now we must use an equivalent circuit which includes both the inductances of the various winding elements, L ', and the capacitances between t hem , Cd, and their capacitances to the grounded parts, C~. Such an equivalent circuit, wi th the te rminal X grounded. is shown in Fig. 14-2b.
174
Part One. Transformers
If we are only interested in the initial voltage distribution, when the voltage at the winding start is V m' then, because f t is very high, we may deem that no currents can flow in the turns because their inductive reactance co tL' is very high. Rather, they flow in the series capacitive reactances, 1!roCd, and the shunt capacitive reactances, 1/roC~. Therefore, we put ro tL' equal to infinity and assume that the voltage distribution is solely dependent on the capacitances inclu'ded in the equivalent circuit in Fig. 14-2b (for the time being, we ignore the part of the circuit shown by dashed lines). O'21----\+-----"'d--I----"\k--l Let us replace the lumpedcapacitance circuit in Fig . .0 x 14-2b, composed of n elements, 1.0 0.8 ' 0.6 O. lf 0.2 0 :": " d./J'x ' ' by a distributed-capacitance circuit shown in Fig. 14-3. . '~R: "" If-Q.x~d.{lx A A)------9X Then the total series capaciI tance will be Vm Kdx ..( dQ.x '"
'" .
t
CqdX,
d.x
'
x
' ,I
Fig. 14-3 Voltage distribution in the -capacitive circuit of a winding -with its finish grounded: ' . , . .. ., . . . . i-initial distribution at a . = 10; 2':"""initial distribution at a = 5; 3-initial distribution in a transformer with electrostatic shields; 4-initial distribution at a = 0 and final distribution at any value of a
c, =
1/LJ (1/Cd) = Cdln
and the total shunt capacitance will be
"', c, =.LJ Cq = »c;,
'
Taking the winding length as unity, we may, for a winding element of length dx, find its shunt capacitance C q dx and the shunt differential parameter K dx, where K = 1/C dWith V m at the start of the winding held constant, the voltage v'" within a distance x of the grounded terminal X can be found by solving a set of differential equations for the shunt charge on the element K dx, equal to
Q", = dv",/K dx
(14-1)
and the voltage across the capacitance C q dz, equal to
Ch.14 Overvoltage Transients in Transformers
175
On finding the derivative dQx/dx from Eq. (14-1) and substituting it in Eq. (14-2), we obtain a linear differential equation with constant coefficients for V x : d 2vx/dx 2 - (C q/Cd) V x = 0 (14-3) 14-3
Voltage Distribution over the Winding and Its Equalization
The solution of Eq. (14-3) has the form V x = D 1 exp (ax) D 2 exp (-ax)
+
where a = 11 C q/Cd and -a are the roots of the characteristic equation. Applying the boundary conditions that exist when the winding terminal is grounded ,'(i ) V x = D 1 exp (a) D 2 exp (-a) = V m for x = 1 ~ (2) V x -:. D 1 D z . 0 for x = 0 we can find the constants D 1 and D 2' and the initial voltage distribution will be V x = V m sinh ax/sinh a (1~) As is seen from Fig. 14-3, for the values of a most frequently found in practice, a = 5 to 10 (curves 1 and 2), the initial voltage distribution is rather nonuniform, and becomes the more so as a increases (that is, with an increase in the shunt capacitance or a decrease in the series capacitance). When the initial voltage distribution is ideally uniform (curve 4) corresponding to a ~ 0, and , ' V x = V m sinh ax/sinh a ~ V max/a = V mX, the -voltage existing across the element I1x nearest to the start of the winding is I1v = V m I1x , In a real winding (a ~ 3), the voltage existing across the winding element I1x nearest to the start [see Eq. (14-4)] will be I1v = (dvx/dx)=l I1x = (V ma coth a) I1x . Vma Ax which is a times the voltage in the case of a uniform distribution. . '
+
+
176
Pa rt One. Tran sformer s
.The fur th er propaga tio n of the over voltage wa ve along the transformer winding can conveniently be examined, if we assum e that the wave is rect angular in sha pe , as shown by the dash ed line a t. the bottom of Fig. 14-1. In this case, V m at the winding start, appearing a t. t; ~ 0, remains unchanged, and soon all poin ts on the winding come by a steady-stale voltage. This is th e final voltage distribution. With the win ding t erm inal ground ed, this will be a linear distribution V x = V mx' (curve 4 in Fig. 14-3). The propagation of an overvoltage wave along the winding is in effect a transition from the initial voltage distribution at t = t; ~ 0 to the final voltage dis tribution at t = oo , Because the equivalent circuit of the winding is composed of capacitances and inductances which form , between them, a cascade of resonant circuits, the transition from the initial to the final voltage distribution at each point is oscillatory. Owing to the losses in the ' resistanc es, these oscillations gradually decay. The swing of oscillations and the associated overvoltage inc rease wi th increasing diffe rence between the initial and final voltage distributions. To minimize the hazards associated with such oscillations, the value of a mu st be kept as small as practicable. Also, a decrease in a leads to a decrease in the initial voltages existing across the elements close to the start of the winding. Unfortunately, an ample spacing between the winding and the grounded par ts cannot be obtained without a marked increase in the size and cost of the transformer. The best way to equali ze the initial voltage distribution and to mak e it comparable with the final distribution is to use electrostatic shields in the form of open metal rings (labelled 3 through 6 and shown dashed in Fig. 14-2a). When such shields are connected to the start of the winding, the capacitive coupling of the first coils to the winding start (via capacitances C~h' C~h, and C~;/ in Fig. 14-2b) is sub stantially inc reased, and the initial voltage distribution becomes more uniform and close to the final one (curve 3 in Fig. 14-3). In Soviet-made transformers, the use of shields around the windings gu arantees the required pulse strength of insul ation in transformers for 110 kV an d higher (see Table 14-1).
Ch, 15 Special-Purpose Transformers
* 15
Special-Purpose Transformers
15-1
General
177
This chapter discusses special-pur pose transformers which either transform or convert some parameter(s) of electric energy (frequency, number of phases, or voltage waveform) or serve some special purposes (continuous voltage adjustmen t, supply of high voltages, isolation of the secondary current from the load impedance, supply of secondary current or voltage proportional to the primary one , et c.). . 15-2
Three-Phase Transformation with Two Transformers
There are ways to transform three-phase with only two transformers. One of the most commonly used conne ctions for this purpose, known as C B A the Scott transformer or the Vca T-connection, is shown in Fig. 15-1. In the T-connection, one transformer, b, has its primary connected directly acros s two lines; t his is t he "main" transWz Wz former ha ving W I turns and a a primary voltage equal to the line voltage V C B = VI, line' The voltages V c o and V o B are equal, becaus e the emfs in the two hal ves of t he Be winding are induced by t he same magnetic flux (Db' The other transformer, a, is called the teaser. Its primary has ~~~~i~: -l Three/two-phase con- V 3' wl /2 t urns and is connec te d between t erm inals A and 0 of the main transformer, so t hat its primary voltage is V3V I ,li n e / 2 . The second ary voltages Va and Vb form a balanced two-phase sys tem, becau se they are equal in 12-0 16 9
i18
PArt One. '1'rnnsrormerA
magnitude 11b= 11 C B (10 2 /10 1 ) = VI, I it le (10 2/W 1 ) Va = llAO (210:/ V 3 WI) = Vi. line (10 2110 1) and are shifted in phase by the same angle as 11AO and V C B, that is, 'Jt/2. 15-3
Frequency-Conversion Transformers
(a) Frequency trebling. This purpose can be served by a bank of three single-phase transformers whose primaries are star-connected and energized from a three-phase supply at freq 8ncy t.. As has been shown in Sec. 7-3, the fluxes of + 10
f3
C& "'2
Wz
fz=2f{
Ftg, 15-2 Frequency doubling
such a transformer contain a sizeable proportion of the third harmonics for which f3 = 3fl' The third-harmonic emfs, E 3 , induced into the secondary are in phase, so when the output winding is a combination of three single-phase secondaries connected in series aiding, the output will only contain 3E 3 at 3f1' but no emf at the fundamental frequency, because in a balanced system the fundamental emfs sum to zero. (b) Frequency doubling. This purpose can be served by a transformer having two independent magnetic circuits (at o: and ~ in Fig. 15-2). The primary energized from a supply at t, encloses both magnetic circuits, so the emf induced in it is due to the sum of two fluxes, (I)a and
eh. 10 Speelal-Purpose Transformers
170
The secondary in whi ch the emf is induced at twice the fundamen tal fr equ enc y , I'!. = 2/1' has lo'!. t urns contributed by two halves whi ch are wound on diff eren t cores and are conne ct ed in opposition. Wi th this arrangement, the flux linkage of the secondary is proportional to the difference flux, cD a - cD 13 ' As is seen from the plot in Fig . '15-2, as the sum flux cI) a cD 13 al ternates a t frequency 11' t he difference fl ux o, -
+
15-4
Variable-Voltag e Transformers
Stepwise voltage adjust men t by tap cha nging has been examined in Chap . 6. Because of an added comp lexity in transformer design and t ap-changers, only several steps of ad justment can be provi ded, a ?+~ and t he overall ra nge of adj ustment does not exceed +5 %. U U A more continuous voltage f ~ adjustmen t in low-volt age a Vic transformers and low-power ~2 f::b.. v~l au t otransformers can be obtained with brushes or sliding y; .~ v.I z contacts that can be mo ved acr oss a skinned portion of pt he transformer winding. This =~ 2 ~~T v,p Iform of adjustment changes P 1 f= the transformer voltage in t: 0'-steps equal to the voltage small 11 11 acr oss one t urn ; the range of 0 a adjustmen t can be ex tended Fig . 15-3 Vari abl e transformer considerably. This arrangement with a d.c.-biased core is utilized , for example, in dim mer-control transformers for auditoriums and theatre stages. In a 250 kVA unit, the output voltage can be varied from zero to 220/380 V. In high-power or high-voltage t r ansformers, cont inuous voltage adjustment can be obtained by biasing the core with direct current. A li kel y arrangem ent utilizing d.c. bias is shown in Fig. '15-3. It is a combination of two single-phase transformers differing in the transformation ratio, n'!.la '---.,
~~
12*
b
1.g0
Part One. transformers
=1=
nZlll' Each transformer has a split core biased with d.c, ill the same manner as ill the case of frequency doubling. (When the core halves are biased in opposite senses, the biasing current is nearly sinusoidal.) . . The primaries with voltages VIa anrl Vl ll are series-con. . . nected for a supply voltage VI = VIa Vll~ ' The secondaries with voltages -Vza = nZlaVla and -V ZIl = nZlllVl ll are likewise series-connected and loaded into an impedance the voltage across wh ich is
+
.
Vz = V za
.
+ V ZIl
When the cores of transformers a an d p are biased separate ly , it is possible to vary the ratio between the resistances of the primary windings traversed by a common current II, and the voltage ratio
1;
= VlllIV l a
For example, if we increase the bias on transformer p, the voltage ratio will decrease. A change in 1; brings about a proportionate change in t he output _yoltage _
~r
__
2 -
V2a _ V2/3 ---= V1 n21a'1+£,n21/3 + £,
When 1; = 0, - V z = Vln Zl a; when 1; = 00, - V z = VlnZlll. In practice, the output voltage can be varied within narrower, but sufficiently broad limits. 15-5
Arc Welding Transformers
Arc welding transformers ha ve to operate intermittently, with frequent transitions from no-load to an arc often accompanied by instantaneous short-circuits. It is usually required that the short-circuit current of a welding transformer be not more than two or three times its rated voltage. Another requirement is that variations in the circuit (load) impedance ought not to produce marked variations in output voltage. To meet these requirements, the short-circuit impedance of a welding transformer must be many times that existing in ordinary transformers. As a rule, the short-circuit impedance of welding transformers is raised at the expense of the inductive reactance . To this end,
Ch. 15 Special-Purpose Transformers
181
the windings are placed on different sections of the core and are series-connected . A further increase in short-circuit inductance can be obtained by pl acing adjustable reactors in the secondary circuit. 15-6
Insulation Testing Transformers
Insulation tes ting uses voltages from 1 MV upwards. Such voltages can only be supplied by a cascade of series-connected transformers (Fig . 15-4). The to tal output voltage V is the
Fig. 15-4 Three-stage casca ded l aborat ory transform er: (a) circuit diagram; (b) external appearance of a 1.5-MV, '1.5 MVA cascaded transformer (in contrast to the arrangement in (a), the trans-
former tank in the firs t st age is isolat ed fro m g roun d)
sum of the second ar y voltages, V 2 , supplied by each st age in a cascade. In a three-stage cascade, it is V = 3V 2 • Each transformer in the cascade is installed in a separate tank and has three windings, namely winding 1 energized from the previous stage, and windings 2 and 3 which are autoconnected (the last unit in the cascade has only Winding 1 and 2). The tanks of the second and third stages stand on pedest al insul ato rs and are at a voltage of V 2 and 2V 2 relative to the ground, respectively. The tank of the first
Part One. Transformers
182
stage is grounded . Accordingly, the winding insulation of the first and second transformers] is designed for V 2 V 3' and that of the third, for V 2 •
+
15-7
Peaking Transformers
Peaking transformers are employed in electronics as sources of recurring peaked voltage pulses which have a short duration in comparison with the pulse period. Such a voltage can be produced across the secondary of a transformer with a heavily saturated core , if the primary is connected to a source of sinusoidal voltage, VI = VIm sin tat, via a highvalue resistance R or a linear inductive reactance. Then the primary current i l will be sinusoidal, the flux waveform will be flattened, and the emf will be peaked. A similar effect, although less pronounced, can be obtained with a star-star three-phase transformer (see Sec. 4-4 and Fig. 4-14). 15-8
Instrument Transformers
Most power systems operate at voltages and currents too high to be measured by ordinary instruments directly. Instead, instruments must be connected to an H.V . network via instrument (or measuring) transformers which may be designed for voltage or current measurements. Instrument transformers are also used to energize control and safety relays and other automatic control devices. Instrument load on the secondary of a measuring transformer is called its burden and is expressed in volt-amperes at a certain power factor. Voltage transformers. Instrument transformers in this group are designed to step down the primary voltage to around 100 V. The burden should be no Iess' than some specified value, ZR, and the transformer must be designed so that its referred secondary voltage changes little as the load is varied from zero to its full (rated) value. In Sec. 5-1 it has been shown that
v: ~ - T\Z' /(Zsc + Z') Therefore, if Z' ~ Z sc, then V~ ~ -VI. When this condition is met, the primary to secondary voltage ratio will always
Ch,-15 Special-P urpose Transformers
183
be the same, and
V2
=
-
VI W 2 / W 1
-
V1n 2 1
For all measures taken , Zsc will alw ays be greater than zero , and therefore a voltage tra nsformer in troduces two kinds of error in the measuremen ts being made: the ratio error and t he phase-angle error. The ratio error is given by -Ii, , t v = V 2 lV / W 2 - VI X 100%
-vi ..
j
(a)
..
-i~
VI
an d the phase-angle error 6v refers to the phase angle between . . Fig. 15-5 Ra tio 'and pha se-angV j and - V~(see Fig. 15-5a). le errors of voltage and curre nt Th ese errors increase with instrument transformers increasing Z and ought not to exceed cer ta in limits specified in appropriat e standards. The limits of error define the "accuracy class" of a transformer and are state d for the rated burden Z = ZR and t he rated primary voltage. The error limits for three accuracy classes adopted in the USSR are given below. (6)
Accuracy cla ss 1: t v
= +0.5 %, 6 v = +20' ,
Accuracy class 2: t v = +1.0% , Accuracy class 3:
6 v = +40'
tv = + ~.0 %, ·6v
, no .liin~t
Current transformers. Instrument transformers, in this group ar e intend ed to change curren ts in power networks ~Q values acceptable :.to' in~ters , usually' down t o 5 A. ' ,' -::',', As 'already noted ;' the' secondary of a currentfransforfner is connected t o, 'an ammeter, a wattmeter, or an automatic': cont rol device. if several in struments ar e powered 'by: the Same current transformer, they 'are series-conn ected. . ' For propel' operation, a current transfor iner ' must: he heldin astate close to a short-circuit (Fig. 15-5b)'. Its 'burden Z ',ought not to exceed a certain rated value, ZR. As follows from "t he basic equations and equivalent circuits' of transformers (see Sec. 3-5) , the secondary current is related to th'E{primary ' current by 'the following equation: ' , " , I:
.;
j~ = j2W2/WI =-i1zO/(Zo
+ Z~ + Z')
"
' "
,
.
-r
~
I:'~ ~
' :': !
Part One. Tr ansformers
184
It is an easy matter to see that the ratio error and the phaseZ' angle error will progressi vely decr ease as t he sum Z~ decreases in com paris on with ZOo This is why t he designer makes every effort to t urn out a curre nt t ransformer having the highest possible value of Zo, the lowest possible value of Z~ , and with Z < ZR' At the rated burden, the curre nt ra t io errors '
+
t, = 12W2/~ 1-Il 1
X
100%
.
an d t he ph ase-angle error 6; (see Fig. 15-5b) should not exceed t he limits stated in applicable standards. In the Sovi et Un ion , t he following error limits and accuracy classes are adopted for current transformers. Accuracy cl ass 0.2: tv = 0.2 %, 8 i = 10' . Accuracy class 0.5: tv = 0.5 %, 8 i = 40' . ' Accuracy cla ss 1: tv = 1.0%, 8i = 80' Accuracy class 3: tv = 3. 0% , 8 i = no limit Accuracy cla ss 10: tv = 10 %, 6 i = no limit
16
Heating and Cooling of Transformers
16-1
Temperature Limits for Transformer Parts under Steady-State and Transient Conditions
Energy conversion by transformers involves a loss of power. The magnitude of power loss varies with the conditions under which a t ransformer is operating (see Sec. 6-3). The bulk of t he power lost is dissipated as heat in the core and coils. The core loss ma y wi th sufficient accuracy be deemed proportional to the primary voltage squared, and the copper loss to the primary and secondary currents squared. A change in load mostly affects the copp er loss, whereas the core loss, given a constant primary voltage, remains nearly ,unaffect ed. . Some of the power loss is dissipated also in the structural parts (the t ank , clamping arrangement, etc.) lying within the magnetic fi~ld of the transformer.
Ch, 16 Heating and Cooling of Tran sformers
185
The heat dissipates in the transformer parts, and they rise in temperature above the surroundings. As the t ransformer keeps rising in t emp erature, a progressiv ely la rger amount of heat is transferred to the surroundings , because the hea t flux is proportional to the temperature rise (the degrees above the ambient temperature). Given a sufficiently long time (theoretically , an infinitely long t ime), t he temperature will cease rising, because all of the heat dissip ated will be transferred to the surroundings . (In more detail, heating and cooling is discussed in Sec. 35-3.) The steady-state t emperature of the t ra nsformer parts depends on the cooling arrangement used. A transformer an d its cooli ng system must be~designed so that the temperature rise does not exceed the limit specified in each particular case. The limits given in relevant standards most apply t o the parts coming in contact with the insulation, oil or any other dielectric liquid that ma y be used. The reason for this is th at elevated temperatures lead to an accel erated ageing' of insulating materials with the resultant loss of electrical and mechanical strength. E xperiments have shown that an increase of 8 degrees C in t emperatur e will halve the service life of an oil -immersed t ransformer . A transformer will serve reliably for 15 t o 20 years, provid ed the temperature rise of its par ts does not exceed the limits stated below (the figures are taken from an applicable Soviet standard). Oil-I mrnersed Tran s] 0 rmers
Wi ndi ngs 65 dog. r. Exterior sur faces of core and stru ct ura l work 75 deg. C Top layer of oil: in totally encl osed uni ts 60 deg. C in oth er types of enc losu r e 55 d eg. C Dr y Tra nsj ormers
Wind in gs 'a nd core surfaces in cont act with insulation accor ding to in sul a t ion class: Clas s Y 50 deg. Clas s A 65 deg. Cla ss E 80 deg. Cla ss B 90 dog. Class F 115 deg. Class H 140 'dsg.
C
C C C C C
~
I
186
Part One. Tra nsformers
The limits of temperature rise stated above are fixed, assuming that the ambient temperature is 40°C. In the case of water-cooled transformers, the inlet temperature of cooling water is assumed to be 25°C, and the respective limits of temperature rise may be raised by 15 deg. C. The design temperatures of transformer parts are assumed to ensure a service life of 15 to 20 years, in view of the observed daily and yearly variations in ambient temperature and transformer load 'under actual service conditions. Most of the time, the load is less than rated and the amb ient temperature is lower than 40°C, so the transformer insulation reaches its design temperature but seldom, and this extends the service life of transformers . The limits established for the winding temperature under steady-state short-circuit conditions are as follo ws. Oil -Immersed Transf ormers
Copper wi nd ings Aluminium windings Dry Trans formers
Copper windings and insu lation of the classes 1isted below: iSO°C Class A 250°C Class E 350°C Classes B, F, H Aluminium windings and insula Lion of the classes listed below: Class A Classes E , B, F, H
The short-circuit duration (see Sec. 13-2) must be li mit ed so that the temperature limits stated above could not be exceeded . In Soviet practice, it is under 5 s. 16-2
Transformer Coo ling Systems
Small transformers are air-cooled and insul ated, which is why they are usually referred to as dry transformers. For units of larger rating and higher voltage, oil cooling is more economical. Oil cooling may be natural or forced. In the former case , transformers are referred to as oil-immersed air-cooled. The core and coil assembly of such a transforme r is encl osed in a tank filled with transformer oil. H eat dissipated by the coils and core is transferred t o the filling oil. The hot oil
Ch. 16 Heating and Cooling of Transformers
187
is lighter than the cold oil next to the tank sides, and this difference gives rise to a natural circulation of oil in the tank. On picking up heat from the hot parts, it rises; near the tank sides, it gives up its 1 heat and sinks. The heat transferred to the tank sides is then reje cted to the surrounding air (Fig . '16-1). Under steady-state cond it ions, t he temperature distribution in each h ori zontal layer is such (Fig . 16-2) that /fO 60 80 100 r . the temperature rise of the core and coils relative to the Fig . 16-1 Variations in temp eoil, on the one hand, and the rature wi th height of a transformer : temperature rise of the oil l-oil temperature; 2-tank side relative to the surrounding tem perature; 3-winding temair, on the other, is sufficient perature; 4-core temperature for all the heat dissipated by Core the core and coils to be transl V winding ferred by convection to the oil HY windin!l and from the oil to the tank Tank side sides, and by convection and radiation to the ambient air . As is seen from the figure, the temperature wi thin the core, coi ls and tank side changes 90 but little because they are fab ricated of metals having a high 80 thermal conductivity . The 70 temperature change is more marked in the coil insulation 60 and also when heat is transferred from the core and the outer 50 surfaces of coil insul at ion to /fo L.. oil and from oil to the tank Fig. 16-2 Horizontal temperasides . The temperature graditure distribution in an oil -iment is especially pronounced mersed, natural air-cooled transbetween the outer surface of former the tank and the ambient air. In transformers of hi gh power ratings, the withdrawal ·of heat fr om t he tank sides is a problem calling for · special ~_
188
Part One. Transformers
treatment. The point is that the heat dissipated in a transformer per unit area increases in proportion to the linear dimension . In simpler words, an increased cooling surface is necessary. This extra surface may be obtained by making ducts in the core and coils, providing fins and corrugations on the tank sides or, which is the most common method, using a tubular radiator (Fig. 16-3). In oil-immersed, aircooled transformers, oil circulates through the radiator (s) naturally, by convection. In air-insulated, natural-aircooled transformers, the core and coil assembly is in direct contact with the ambient air and heat is abstracted by Fig. 16-3 Radiator convection; some heat is withdrawn by radiation. Large transformers use natural oil circulation and ail' blast. In them, by directing an air blast onto an ordinary
Fig. 16-4 Radiator blowers
tubular tank or onto separate radiators , the rate of heat dissipation can be increased several-fold (Fig. 16-4). Better heat withdrawal is obtained by a combination of forced oil circulation and air blast. A still better arrange-
Ch. 17 Transform ers of Soviet Manufacture
189
ment, especially for very large transformers, is to combine oil immersion and water cooling. In fact, two arrangements are possible in this case, namely natur al oil circulat ion and water, and force d oil circulation and water . In the former case, an intern al cooler is employed, whereas in the latter, the oil /wa ter heat exchangers are external to the transformers. Oil-immersed , wa ter-cooled tr ansformers require large amounts of running water, so t hey are mostly installed at hydraulic power st ations ,
17
Transformers of Soviet Manufacture
17-1
USSR State Standards Covering Transformers
In his studies or work, the reader ma y run into transformers of Soviet manufacture. If so, it will be useful, as we believe, for him t o know whi ch USSR state st andards, GOSTs, are applicable to various transformers. GOST 16110-70. Power t ra nsformers. Terms and definit ions (see also CMEA* Standard 1103-78) . GOST 11677-75. Power t ransformers. Gen eral specifications (see also CMEA Standard 1102-78). GOST 721-77, GOST 21128-75. Rated ph ase-to-phase voltages . GOST 18619-73. Power tra nsformers, three-phase, natural air cooled, general-purpose, '10 to 160 kVA, up t o 660 V. GOST 14074-76. Power transformers, dry, protected , general-purpose, 160 to 1.6 MVA, 6 to 15.75 kV inclusive. GOST 12022-76. Pow er t ransformers, three-phase, oilimmersed , general-purpose , 25 to 630 kVA , 35 kV inclusive . GOST 12965-74. Power t ransformers, three-phase , oilimmersed, general-purpose, '110 kV . GOST 17546-72. Transformers (and autotransform ers), three-phase, power , oil-immersed , general-purpose, '150 kV . GOST 15957-70. Transformers (and autotransformers), power, oil -immersed, gener al-purpose, 220 kV . GOST 17545-72. Tr ansformers (and autotransformers), power , oil-immersed, general-pur pose, 330 kV.
* CM EA stands for th e Coun cil of Mu tual Economic Assistance of which th e USSR is a memb er.-Trunslator 's not e.
1M
PlIrt
One. Trnnsformers
GOST 17544-72. Transformers (and auto transform ers) , power, oil-immersed, general-purpose, 500 kV . GOST 3484-77. Pow er tr ansform ers. Test pro cedu res. GOST '15'16 :1-76. A.C. electrical equipment for 3 to 500 kV. Insul a tion streng th requirements . GOST '15'16 .2-76. A.C . electrica l equipment and installation for 3 kV and higher . General proc edures for insulation t esting. GOST '1 4209-69. Tr ansfo rmers (and autotransform ers) , power , oil-immersed . Load capac ity . 17-2
Type Designat ions of Soviet-made Transformers
The and A T
typ e design ation of a t ra nsformer consists of letters numerals. The letters are used as follows . stands for an autotrans form er . stands for three-phase . A second T, for three-winding. o s tands for single-phase . P stands for a split LV wind ing (see Sec. '10-'1). H st ands for on-l oad t ap changing. (If there is no H in the t ype designation, t he transf orm er is designed for offload tap changing or has no t ap chang er at all. ) The numerals in the numerator, following the let ter(s), give the power rating in kVA, an d the numerals in the denominator, its kV class on the HV sid e. The designations used for the various cooling arrangements are listed in the t able t hat follows. Tablc 17-1 Des tgna tion of Cooling Arrangem en ts Dry trausformers Natural air cool ed, open Same, pr ot ected Same , seale d Air-blas t cool ed Oil- Immersed 'I'ransform crs Oil natural Oil -na tural , air-blast For ced-oil , ai r-bl as t Oil-natural, wate r Forc ed-oi I, wa t el'
Dcsignation C
C3
cr
CJI: M
JI:
Jl.L~
MB
u
eh.
11 transformers of Soviet. Mnnufncture
17·3
HH
Some of Transformer Applications
Dry transformers are mainly intended for installation in dry indoor locations with a relative humidity of not over 80 % and in the absence of corrosive substances and currentconducting dust. TIleY are fire -safe and are gaining popularity in residential buildings, in laboratories, etc. A dry transformer may be built into an enclosure so as to kee p fore ign objects from finding their way into the core and coil assembly, but to give free access for cooling ail'. Low-power transformers (under 4 kVA for single-phase uni ts and under 5 kVA for three-phase units) find use in radio, electronics, automatic control, communications, in dustrial drive, domestic appliances, and to energize handheld power t ools. All transformers are designed for moderate climates, for tropical climates (tropicalized), and for cold climates.
A General T heory of Electromechanical Energy Conversion by Electrical Machines
18
Electromechanical Processes in Electrical Machines*
18-1
Classification of Electrical Machines
An electrical machine operating by electromagnetic in duction consists essentia lly of a station ary member and a movable memb er (Figs . 18-1-18-5) . The st ationary part is m ad e up of a suitably shaped core, one or more windings, and structural parts intended to hold the stator in its designated position . The movable part consists of a core , one or more windings, and structural parts ena bling t he m ovabl e part to move relative to the stationary part and to tran smit mechanical energy to or from the m achi ne . The movable and stationary windings may be connected to external lines directly or thr ough a suitable device. Conn ection to the movable windings is by sliding contacts. As a rule, the mo vable part of an electrical ma chine has one degree of fre edom (motion in an y other directions is prevented by bearings or supports whi ch may be of one of several designs). In most electrical machines, t he mo vable member rot ates relative to t he stationary member. Quite aptly, they are called rotating machines, and their movable member is called the rotor, and t he stationary member t he stator.
* Th e auth or refers primarily to th e moto r mo de of operation. By th e reversibility pr in ciple, however , th e rea der m ay readily extend th e reasoni ng to th e generati ng mode where necessar y.-Translater 's no te .
Fig. 18-1 Ro tatin g cyli ndrica l ma chine: I - stator wind ings; 2- rotor windings; 3-st a tor core; 4-rotor core; 5- stat or st r uctura l parts ; 6-rotor sha ft ; 7-axia l-radia l bearin gs (supports)
Fig. 18-2 Rotating diso-t ypej machine (the notation is the same as:in Fi g. IS-l)
.z ~
6
u 7
~
~
J
'f-
2 5'
.
f~
Fig . 18-3 Flat lin ear machine: I -stator windings; 2-movable-ll1ell1ber windings; 3-stator core; 4-movable-mell1ber core; 5-stator structural parts; 6-movablemember connec ting-rod ; 7-supports I 3- · Ut li 9 ·
Part Two. En ergy Conversion by Elect rical Machines
Most fr equently, the roto r is a cylinder rot ating ins ide the stato r which is likewise a cylinder bu t a holl ow one (see Fig. 18-1). Somet imes, to incre ase the moment of inertia of the rot at ing parts , the rot or is made in t he form of a ring enclosing the stator. As an alt ernative, a rotating machi ne can be built so that the stator and the rotor are discs facing each oti er (Fig . 18-2). A less frequent variety of electrica l machi nes is one in which the movable part recipr ocates rel ative to the stator x
U
Fig . 18-4 Tubular linear machi ne (the notation as in Fig. '18.3)
in a linear fashion. Quite aptly, such machines ar e calle d linear . They ma y be flat and t ubu l ar . In a fl at linear m achine, the movable an d t he stationary cores are each the shape of a parallelepiped , with their broad sides facing each other (Fig. 18-3). In a t u bu lar linear machine, the movable cylindrical core is free t o move axiall y inside the stationar y annular core (for exa mple, a plunger moving inside a solenoid), as in Fig. 18-4. Flat linear machines can serve as, say, drives for electric-powered rail-riding vehicles, especially where high speeds (over 200 or 300 kmph) are in vol ved or desired. Tubular linear machines can be used to actuate the reciprocating parts of various mechanisms. Both rotating and linear machines can be built for restricted to- and-fro motion . Restricted rotary motion may be utilized t o operate, for example, the balance wheel of an electr ic clock , and restricted linear motion ma y ser ve t o actuate an electric pick.
195
tho 18 Proces ses in Electrical Machines
Sometimes, it may be necessary to link an electrical machine to a source (or sink) of mechanical energy so as to transform some parameter(s) of the mechanical energy being converted . Thi.s i.s done by what may be called mechanical
7
8
5
'I
3
1 Z Fig. 18-5 Geared rotating electrical machine: I-frame; 2- st at or core; 3-stator winding; 4-rotor; 5-rotor shaft ; 6- ball bearing; 7-gear train; 8-gear-train shaft
converters. A mechanical converter is often made integral with the associated electrical machine. The most commonly used form of mechanical converter is a step-up or a stepdown gear box (Fig. 18-5). · Rotating motion can be transformed to reciprocating motion by gears, a worm and gear combination, or friction transmission. Osolllatory motion can be transformed into rotating or translational motion by a variety of ratchets and pawls. Most frequently, however, electrical machines are built without any mechanical converters . 18-2
Mathematical Description of Electromechanical Energy Conversion by Electrical Machines
Let us consider a rotating electrical machine in which the windings have an arbitrary number s of parallel paths (or circuits) embedded in slots or on the outer surface of the stator and rotor. Each path may consist of many coils connec ted in some particular manner. The cores, too, may be 13*
196
Part Two. Energy Conversion by Electrical Machines
of any configuration. As an example, Fig. 18-6 shows a rotating electrical machine with s = 5 parallel paths of which two (labelled "1" and "2") are located on the stator, and three (labelled "3", "4", and "5"), on the rotor. The electromagnetic processes taking place in an electrical machine can be described by Kirchhoff's mesh (or loop)
Fig. 18-6 Multiwinding rotating electri cal ma chine
equations and the equations of motion for the rotor. In a linear approximation (that is , assuming that the core material has an infinitely large permeability) , the flux linkage of, say, the kth path (where k may take on any value from 1 to s) can be expressed in terms of the winding currents im the self-inductance of the kth winding, L k 1tl and the mutual inductances between the !dh winding and all the other windings, L k n , where n can take on any value from n = 1 to n = s, except n = k s
\f" =
>;
n=l
s
\f kn = ):
inL"n
n= l
In many cases, the mutual and self inductances of the windings are markedly affected by whether or not the cores have saliencies. With saliencies, rotation of the rotor causes variations not only in the mutual inductance between the paths on the stator and rotor, but also in the mutual inductances between the paths on only the stator and on only the rotor, and in the self-inductance of each path.
Ch, 18 P rocesses in Electrical Machines
197
In the general case, all t he self-induct an ces L k h an d all t he mutual induc tances L im are functions of the coil and core size and of the angul ar position of the rot or , L k n = f (1'). Using Kirchhoff's voltage law , we may write a set of s volt age equations, each describing one of the parallel pa ths. For t he kth pa th, such an equation takes t he form Vh = R hi h + (f'P'h/ dt =
wher e L im din/d t
R hi h +
s
'>:
n=1
(L hn di n/dt + inQ dLhn/d 'V)
(18-1)
t he transformer emf rela t ed t o var iat ions in the cur rent in the nth path in dL lm/ dt = the rotational emf relate d to vari ations in t he mu tual inductance with the nth pa th (when n ==1= k) or to variations in the self-inductance of t he ktb path (when n = k) Thus, as follows from Eq. (18-1), the emfs induced in the kth loop are the sum of transformer emjs related to vari ations in coil currents when the mu tual or self inductances remain unchanged, =
s
- nI:=1
Lim din/dt
and rotational emjs relat ed to variations in the mu tual or self-inductan ces, with t he currents held constant s
-Q
2J in dLkn/d'l' n= 1
The term "transformer emf" refers to the fact that a similar emf is induced in transformers where the primary and secondary are stationary relative to each other. The term "rotational emf" refers to the fact that it can only be generated when the rotor is moving at some angular velocit y Q = dl'/ dt For t he loop s connected t o an extern al circuit, Vh in Eq . (18-1) can be in t erpreted as t he emf of t he circuit . For shortcircuited loops , Vh = O. The mechanical power derived by an elec trical machine from electrical energy can be expressed in t erms of the associated circuit parameters , proceeding fr om the law of conservat ion of ener~y. To begin with, let us determine thy
Par t Two. Energy Conversion by Electrical Machin es
198
instantaneous electric power that the kth loop draws from the associated supply line
=
Ph
+.
. R' 2 V h~ h = h~h
s
~h
~ L din LJ hn crt
s
+r\~ ' ~'h
n=l
~. dLh1l LJ ~n ----cry-11= 1
The total instantaneous electric power drawn by all the loops can be found by adding together the powers of all the loops:
2j
Ph =
1<= 1
2j R/,i.~ + 2j h= 1
ih
h= 1
2j
L hn
~itn
n=1 .
L
n
el L h1l ely
(18-2)
--
s
The t erm ~ R hi'f, is the power dissipated as heat in the h=l
loop resistances R h and gives the power los t on conversion. Th e rem ainder of the input power goes to sustain variations in t he field energy owing to variat ions in t he loop currents and in duc t ances . Because t he magnetic field energy is
+2j s
s
i h 2j inL hn (18-3) h=l n=l its t ot al change over a t ime dt during which i/o in , and L h1l change by di h, d in, an d dL /m , is given by
H!
=
dfiV = (oW/iJih) di h s
=~
h= 1
+ (o W lOi n) di n + (8W/aLhn) dL hn
S
ih
~
L/m di n
+
+
S
~
h=l
n =l
S
i h• ~ i n dL hn n =l
Th erefore , the power spent t o sustain variations in t he magnetic field energy is s
Pw =
~ = L; h=l
s
ih
~ n =l
u;
(~;'
s
s
+ (Q/2) ~ h=l
ih
L;
in
el~~"
n=!
(18-4) It corre spo nds to t he second and half the third te rm in E q. (18-2). In other words, t he power sp ent to sustain variations in the energy of t he magnet ic field is all of the sum of t he powers defined as the pro du cts of t he loo p currents by the
Ch. 18 Pro cesses in Ele ctrical Machines
199
transformer emf, and half the sum of the powers defined as the products of the loop currents by the rotational emf. The remainder is the mechanical power transmitted by the shaft t o the driven machine (in motoring) or from a prime mov er (in gener ating) S
Pme ch
=
.'\
"" LJ Ph- Pe- Pw
S
~~ "" . LJ "" In . ~ dLk n =""2 LJ lk
k= 1
h= 1
(18-5)
n=!
As follows from Eqs . (18-1), (18-2) and (18-5), the mechanical power is equ al t o half t he sum of the powers defined as the products of loop currents and the rotational emf. Hence, we ma y conclude that electromechanical energy conversion involves only the rotational emf , whereas the transformer emf does not contribute to this conversion. It is to be noted that the power spent t o sustain variations in the magn etic field energy is not wasted irrevocably , bu t sums on the avera ge t o zero. This is because ie a ro tating electrical machine all quantities (currents, self-inductances, mutual inductances, etc.) vary periodically. At the end of a cycle of alternation , all quantities, including the magnetic field energy, t ake on the same value they had at the beginning of t he cycl e, i.e. , W et ) = W(t +T ) ' This implies that variations in the energy of the magnetic field over a cycle, or period, sum to zero , that is , t+T
.\ dW =
W ( t+T ) -
H l (t)
=0
t
During t hat part of a cycl e when t he ma gnetic field energy builds up (dW> 0), the power Pw given by Eq. (18-4) is positive (Pw > 0), and the energy required to set up the magne ti c field is t aken by the loops from t he line. During the rema ining par t of a period , Pw < 0, and t he energy st ored by t he m agnetic field is again returned t o the line. This exchange of energy betw een the machine and the line goes on in such a manner that the energy drawn from the line averages over a period t o zero. A measure of t his exchange is wh at is called the reacti ve (or magnetizing) power. In the case of a single-phase supply an d sinusoidal variations, this is the maximum instantaneous power drawn from the line to set up the magnetic field in the machin e;
Q=
I dW/dt
Im¥! 1\
Part Two. Ene rgy Conversion by Electrical Machines
200
Recalling t h at the electromagnetic t or que, T em, acting on the rotor at a given instant can be expressed in terms of Pmech defined in Eq. (18-5), an d comparing the resultant expression with Eq . (18-3), we get s
T
= em
Pm ech Q
s
=~ " 2 LJ
i " i el Llm I< LJ n el y
1< =1
=
elW ely
(il< = cons ta nt)
n= 1
(18-6) Thus, in a machine with a linear magnetic circuit t he electromagnetic t orque is the partial derivativ e of the magnetic field energy W with respect to t he angular pos ition y of the ro tor , with t he loop currents held constant (i h = = cons tant , and in = constant). If this derivative is positive, t he torque act s in t he direction of ro tation (or in t he direction of increasing y), and elec tric energy is converted to mechanical. If the derivative is negati ve, reverse conversion takes place . Equation (18-6) may be extended t o .machin es with non linear magnetic circuits, if variations in the magnetic fi eld energy, dliV, as the rotor turns through an angle dy can be foun d not onl y for i k = constant , but als o for f!aj = constant . In each jth elemen t of t he ma gn etic circuit f! aj must be found for i h = constant and the angular posit ion '\' of the rotor. If the te rminal coil voltages VI<, the angular velocity Q of the rotor , and the rela tion Lim = f (y) are known or specified in advan ce, the currents ca n be found from Eqs. (18-1). J,hen t he electromagnetic torque can be found by Eq . (18-6) where y = Qt . If the angula r velocity is not known , hu t the external torque T ext is specified in advanc e, then Eqs . (18-1) and (18-6) must be solved simultaneously with the equ ations of motion (18-7) T em- T ext
= J c!Q/dt
Q =Q lnit +
I (dQ/ d t) dt o
t
l' =
'\'Inlt -/-
I 1J
Q elt
(18-7)
Gil. t9 Production of Periodic Magnetic Field
20t
The mathematical description derived above for a multiloop (multipath) rotating machine can be extended to a linear machine whose movable member reciprocates relative to the stator. The equations for a linear machine differ from the above equations only in that the angular displacement y is replaced by a linear displacement x , the electromagnetic torque Tern is replaced by an electromagnetic force N acting in the direction of displacement, the angular velocity Q by a linear velocity u, the angular acceleration dQ/dt by a linear acceleration du /dt, the external torque T ext by an external force Next ' and the moment of inertia of the rotor J by the mass of the movable member, m:
19
Production of a Periodically Varying Magnetic Field in Electrical Machines
19-1
A Necessary Condition for Electromechanical Energy Conversion
From inspection of Eq. (18-6) , we may conclude that a necessary condition for an electrical machine to perform electromechanical energy conversion is a change in the self or mutual inductances of the coils as the rotor turns through an angle. An electrical machine will perform its function if the derivative of at least one quantity with respect to the angular position of the rotor is non-zero dLkn/dy =1= 0 because it is only then that Tern =1= 0 and P mech =1= O. This is a necessary, but not a sufficient condition for a continuous, unidirectional electromechanical (or mechanoelectrical) conversion of energy. It is also required that Lhe currents in coils 11; and n should vary in such a manner that not only the instantaneous, but also the average values of T em and P m ech be sufficiently large. Because in technically feasible designs the magnetic fields, flux linkages,iself and mutual inductances cannot be monotonically rising functions of currents and the angular position pi the rotor" the only possible case is when these quantities
202
Pa rt Two. Ene rgy Conversion b y El ectrical Machines
var y periodically as functions of 1', when the derivative dL/m/d'V likewise varies periodically. For L ll n to be a periodic function of 1', it is essential that t he current tr aversing coil n sets up a magnetic field periodically varying in sp ace (t an gentially t o the air gap). Some of t he coil an d core desi gns capable of producing a periodicall y varying magnetic field are discussed in the sections t hat follow. 19-2
The Cylindrical (Drum) Heferopolar W inding
The conductors of a drum winding are laid in slots on the side surface of the core which ma y be in the form of a toothed (or salient-pole) cyli n der or toroid*. As is seen from Fig. 19-1a, the curre nt in the conductors on the core surface facing the air gap alternates in direction periodically. This gives rise to a magnetic field which varies periodically in space - the core is magnetized heteropolarlyin going r oun d the circumference, an N pole is followed by an 8 pol e, and an 8 pole is foll owed by an N pole. The sp acing between zones A an d X occupied by conductors carrying currents which alternate in the direction of flow varies from design to design. Accordingly, a drum windin g can set up a magnetic field wi th a varying number of periods , cycles of altern at ions per r evolution , or , as mo re commonly stat ed, a vary ing number of pole pa irs, (Fig. 19-2). This spacing is me asured along t he periphery of the airga p with a mean radius R an d is call ed the pole pitch. If we designate the pole pitch as .. (see Fig. 19-'1a and Fig. '19-2), t hen the numb er of pol e pair s (or cycles of alternation per revolution) will be give n by p
= nD /2.. = nR h:
(19-'1)
The simplest of all drum windings is the two-pole winding for which p = '1 , and the magnetic field comple tes one cycle of alternat ion per rev olution . Those with p > 1, are called multipole windings.
. * In mach ines wi th smooth cores (those havi ng no slo ts), the coil conductors are bound ed t o t\Ie oute r surface of the core :
Ch. ill Production of Periodic Magnetic Field
203
In a drum winding, the conductors lying on the surface of the air gap may be interconnected in anyone of several
..__Ll~ -:....
~..., ,,,,
.
•
N
lol ~
I
(a)
Fig. 19 -1 Production of a periodically varying magnetic field in rotating electrical machines: (a) cylindrical (drum) heteropolar winding; (b) toroidal heteropolar winding; (c) ring winding and claw-shaped core; (d) ring homopolar winding and toothed core
ways. Whatever the form of connection , however, the coilends will never encircle the yoke of the core . Each coil may be wound with one or two turns . Each slot may contain one side (Figs, 1~-1a and 19-2a) 0):' tW9 sides
A
Fig . 19-2 Heteropolar cyli ndrical (drum) windings : (a) sin gle-layer, single-phase concentrated winding; (b) two -layer, si ngle-phase concent rated wind in g; (e) two-row, single-phase concent ra ted winding; (d) single-layer, single-phase dist ributed winding (q = 3); (e) si ngle-layer, two-phase distributed winding (q = 3) _,
Ch. 19 Production of Periodic Magnetic Field
205
(Fig . 19-2b and c) of a coil. In t he former case, we have a single-layer winding, and in t he lat ter, a double-layer wi nding if a given coil has one of its sides at the bottom of a slot and the other side at the top of the same slot (Fig. 19-2b). If the sides occupying the same slot li e in the same plane, we have a double-row wi ndi ng (Fig . '1 9-2c). Frequently, it is convenient to pl ace con duc t ors carrying currents flowing in t he same direction in several, say three, slots (Fig. 19-2d) rather than in one. The number of slots occupied by a phase belt (that is , by a belt of phase conductors carrying currents flowing in t he same direction) in a single-layer winding is called the num ber of slots per pole, denoted by q. When q = '1, the winding is called concentrated. When «> 1, we have a distri buted winding. The slots of t he same core may carry sever al identical heteropolar windings producing between them fields with the same number of cycles of change, p, and energized from (or supplying power to) a polyphase line. This structure is called a polyphase winding. As is seen from Fig. 19-2e which shows a two -phase winding, each phase is a distributed heteropolar winding with q = 3 (see Fig. 19-2d). Phase A consists of belts with conductors carrying the current flowing in the forward direction (A) an d belts with conductors carrying the current flowing in t he reverse direction (X), and the spacing between the adj acen t bel ts is equal to the pole pitch 'to Phase B consists of belts with conductors carrying the current in the forward direction (B) and belts with conductors carrying t he current in t he reverse dir ect ion (Y) . The bel ts of phase B are laid bet ween the bel ts of phase A and are displaced from the ph ase A belts through a quarter of a cycle, or a half pole pitch, 't/2. A similar arrangement is applicable toa polyphase winding with m phases. In such a case, the number of slo ts per pole per ph ase, q, is given by q
=
Z I2pm
('19-2)
where Z is the total number of slot s on the core. The adjacent bel ts in a given phase are displaced from one another by a pole pitch 't, and the belts in the adj acent phases by a distance equal to -clm:
208 19-3
Part Two. Energy Conversion by Electri cal Machin es
The Toroidal Heteropolar Wi nding
A toroidal winding (see Fig. 19-1b) differs from a cyli ndrical in that the connections between its conductors carrying currents in the same direction, that is, the coil end s or overhangs, are wound around the toroidal core . If the Conductors on the surface facing the air gap are arranged in the same manner as in a cylindrical coil, a toroidal coil does not differ from the latter as regards the production of a magneti c field periodically varying in space . In fac t, it comes in the same modifications as t he cylindrical winding. It offers some advantages in t he manufacture of small elec trical machines. 19-4
The Ring Winding and a Claw-shaped Core
80 far we have dealt with forms of winding in which a periodic heteropolar magnetic field was produced by an alternation in the direction of current flow in the conductors. In a ringshaped winding (see Fig. 19-1c), a periodic field is obtained due to an alternation in the direction in which the clawshaped tee th of the core enclose the energized ring-shaped winding. As regards the production of a periodic field, this arrangement is equally efficient as the previous designs. A limitation of this design is an increased magnetic leakage between t he claw-shaped polepieces. An advantage is simplicity in manufacture. Its application is mainly in small machines and also in special-purpose medium-power units.
19-5
The Homopolar Ring Wi nding and a Toothed Core
A ring winding whose coils enclose the shaft of an electrical machine produces a homopolar field in the air gap. For the direction of current flow shown in; Fig. 19-1d, the surface of the inner core is in N polarity, and that of the out er , in 8 polarity. Periodic variations. in the magnetic flux density within the air gap occur owing to the saliencies made on the core surface facing the air gap. If the surface of the other core is smooth or has a slight salience, then within the low areas (slots) the specific permeance will be smaller than it is within the saliences (teeth). Accordingly, the magnetic
Gil. 20 Basic Machine Designs
207
flux density within a salience (tooth) will be higher than it is within a low area (slot). The magnetic flux density in the air gap will vary with a space period equal to the tooth pitch, or spacing between adjacent teeth, t z . The number of pole pairs will be p = nD/t z = Z ':. ('19-3) where Z is the number of teeth in the core. An advantage of this design is that the resultant magnetic field undergoes a larger number of alternations per revolution than with any other design because the coil conductors need not be laid in slots (the ring winding is external to the core), and there is no limit to slot size-in fact, they may be however small.
20
Basic Machine Designs
20-1
Modifications in Design
In the previous chapter, we discussed the ways and means of producing a periodic magnetic field in an electrical machine. Now we shall see how an electrical machine must be arranged for the self and mutual inductances of its windings to be functions of the angular position of the rotor and to vary periodically as the rotor rotates. This effect can be obtained in anyone of three basic machine designs, namely: (1) in a machine with one winding on the stator and one winding on the rotor; (2)- in a machine with one winding on the stator and a toothed rotor; (3) in a machine with two windings on the stator and a toothed rotor. Each design may come in several modifications. As is explained in Chap. 19, the magnetic field in the air gap of an electrical machine may be either heteropolar or homopolar. Respectively, one uses two varieties of windings, heteropolar and homopolar. Heteropolar windings may be single-phase and polyphase. Homopolar windings may only be single-phase, and they may operate on a.c. or d.c. Instead of a single-phase heteropolar winding, use is sometimes made of a ring winding in a claw-shaped core.
208
Part Two. En ergy Conversion by Electrical Machines Table 20-1. Conceivable Wind ing 1 on
Hetero Winding 2
Rotor core A
1
H et eropolar the rotor
I
To ot hed s tator co re
Toothed rotor core
on 2 Smoo th rotor core
3 T oothed r otor core H eteropol a r th e stator
on
4
Sm ooth rotor core
5
Toothed cor e
rotor
Uncapahle of Sim ilar to C3
Homopolar a ll th e stator 6 Smooth core
rotor
Uncapab le of
I
Ch. 20 Basic Machine Designs
209
Designs of El ec trical Mach i nes the s t ator Ho rnopo l ar
polar
I I B
Sm ooth st a t or cor e
c
I Too th ed s t a tor co r e I D 1Smo o t h sta to r co re
Sim ilar to C3 (bu t needs exter nal l eads)
Uncapab l e of energy conversion ill ei ther direction Sim ilar to D3 (but needs ex te rnal leads)
energy con versi on ill eit he r d i rec tion Sim ila r to D:l
energy conversion in eit her d ir ec tion
Uncapable of ele c trom echani cal or mochanoel ec t r ical conversion
2.10
Part Two . En ergy Conver sion by Eiectrical Machin es
Winding 1 on Hetero W inding 2
Rotor core A
7
Tooth ed core
rotor
8
Smoo th core
rotor
I
Toothed st a t or core
None
Not e. H et er opolar win ding 1 may be
Uncapabl o of single- or pol ypha se. Ramap o
The core s urfa ce faci ng the air ga p may be smooth or to othed . Accord ing ly, there may be smo oth cores and t oothed cores. In toothed cores, the opening and, sometimes, the shape of slots and teeth have a direct bearing on the permeance of the air gap . In fact, the teeth can be suitably shaped to control the permeance of the air gap . In smo oth cores , the slots have a limited opening (as compared with the air gap) , and the air gap between the face of a t ooth and the mating core remains constant as the rotor rotates. In such cores, the opening of slots has a negligible effect on the perm eance of the air ga p . Toothed cores are used in electrical machines in which energy conve rs ion is based on peri odic variations in the permeance of the air gap . In some cases, the teeth of such cores are.dimensioned so' as to obtain a desired shape for the field in the air gap (this is true of the salient-pole roto rs of synchronous machines and the salient-pole stators of d.c, machines) . Also , cores in which the slots are made open for ease of coil placement behave like toothed cores. The use of open inst ead of semi-closed slots leads to in creased pu lsational losses and is justified only inasmuch as the manufact ure is simplified . Wherever one may use slots with a sma ll openi ng , a roun d core will be preferable, as it will keep the additional losses to a minimum.
21'1
Gh. 20 Basic Machin e Designs
Table 20-1 (con ti nue d) the st ator
polar
I I B
Hornop ola r Smo oth s t a tor core
c ,
'I'oo tl icd st ator COr e
I I D
Smooth st ator core
Uncapabl e of energy conver sion in either direction
ene rgy conversion i n either d i r ect ion ar wind ing' 1 can only be single -phase .
I
Some of the conce ivable machine designs are list ed in Tab le 20-1. It gives combinations of heteropolar and homopolar windings and toothed and smooth cores for the st ator and ro tor. Combinations using ring windings and claw-shaped cores are not included because such machines are identical to those using a:single-phase heteropolar winding. For the same reason , there has been no need to include cylindrical and toroidal heteropolar windings. The most important of the modifications li sted in Table 20-'1 are examined in the pages t hat follow. 20-2
Machines with One Winding on the Stator and One Winding on the Rotor
In a machine carrying one winding on the stator and one winding on the rotor, electromechanical energy conversion occurs mainly owing to variations in the relative position of, and in the mu tual in duct ance between , the windings as the rotor rotates . Variations in the self and mutua l inductance of the windings due to the saliency of the cores are of secondary imp or t ance. In the arrangement considered, only heteropolar windings are used on the stator and rotor. The rotor core may be 14*
212
Part Two. En ergy Conve rsion by Electrical Machines
t oot hed (notably , with sali ent pol es) a nd smoot h (rou nd or cylindrical) . The stator core may likewise be toothed or smooth. This gives a total of four combinations labelled as A1, A2, B1, and B2 in .T able 20-1. Figure 20-1 shows a fourpole machine with single-phase heteropolar wi ndings on the stator and rotor, and smooth stator and rotor cores (modification B2 in Table 20-1). The winding curr ents i, or i 2 set up a four-pole (p = 2) magnetic field (the figure onl y shows t.he magnetic lines of
Fig. 20-1 Machine wi th one stat or winding (1) and one roto r winding (2) (PI
=
P2
=
2)
force due to i 2 ) . A plot of L 1 2 an d £11 (£22) as functions of the an gle I' between the ax es of the t wo windings is shown in t he same figure . As is seen , the mutual inductance, proportion al to the flux linkage of the magnetic fie ld due to i 2 with the turns of winding I is a maximum when y = 0, that is, when the axes of t he coil s run in the same direction . Wh en t he axis of coil 2 m ake s wi th the axis of coil I an angle y = n /4, which corresponds to a linear displacement along the periphery of the air gap through 1:/2 , or a quarter-cycle of change in the field , t he Il ux linkage with coil 1 and t he mu tu al
Ch. 20 Basic Machine Designs
213
inductance will be zero . A cycle of change in the mutual inductance is completed as the rotor moves through 2-r or through a pole pitch an gle ,,?p = rr. In the general case , when the windings set up a p-cycle field , the mutual inductance undergoes a complete cycle of change as the rotor moves through 21: or the pole pitch angle given by ,,?p = (2rrJ2:n;R) 2.. = 2:n;/p
(20-1)
If the rotor is ro ta ting a t angular velocity Q, the mutual in ductance will alternate with a period given by
T = ,,?p/Q = 2n /pQ
(20-2)
Accordingly, the frequency of change in the mutual inductance, f, and the angular frequency of change in the mu tual inductance, ro, are given by
f
1/ T = pQ/2:n; w = 2:n;f = pQ =
(20-3) (20-4)
The sha pe of the plots for £12' £11' and £22 is typi cal of round (cylindrical) cores with q = 1: a half-cycle of change in £12 is triangular in shape, £ 11 and £22 are nearly constant; i t is onl y wh en the slots in the stator and rotor are aligned tha t the self-inductances show slight variat ions, bu t these may safely be ignored . As the number of slots per pole per phase, q, increases, the pattern of change in £12 takes on a shape close to sinusoidal, which ha s a wholesome effect on the performance of the machine . Thus, by increasing the number of slots on a round rotor core carrying a singlephase winding the £12 pattern can be made nearly sinusoidal in the round-core synchronou s machine shown as an example of modifi cation A2 (see Table 20-1) . A practically sinusoidal pattern of change in £12 can be obtained with q = 1 as well , if the rotor cor e is so shaped that the air gap at the too th axis is two-thirds to one-half of the gap at its tips (or edges). This ty pe of rotor (a sa li ent-pole rot or) with a si nglephase winding is used in modific ations A1 an d BL Most frequ ently , electrical machines are built with singleor polyphase heteropolar windings (see Sec. 19-2) having the same num ber of pole pairs. This is true of induction ma chines an d conve nt iona l synchronous machines (see Parts 4, 5 and 6 of this text) .
214
20- 3
Part Two. Energy Conversion by Electrical Machines
Machines with One Wind ing on the stator and Toothed Rotor and Stator Cores [Reluctance Machines)
In a machine with one winding on the stat or and toothed rotor and stato r cores , electromechanica l energy conversion is bas ed on the variations caused in t he self-inductance of the winding by t he tee th on the cores. Figure 20-2 shows such a machine with a single-phase concentra ted het erop olar 'l'z
d.L/1
/dY
Ii I I I I
r-4I
I
I
I--+--I-o-I~ o I I Y I 7C/Z ,(
I
~-'
I
I L.J
Fig. 20-2 Machine with concentrated heteropolur st at or winding (1) (PI = 2, ql = 1) and toothed stator (3) and rotor (4) cores with an equa l number of teeth (Za = Z4 = 8)
windi ng on t he stator, and the rotor and stator cores having the same number of t eet h, Zs = Z 4 = Z (modification A7 in Table 20-'1). The tooth pitch angle of the rotor, '\'Z 4 = 2n/Z 4 , is the same as that of the stator, '\'za = 2n/Za . As is seen from the curves in Fig. 20-2, when Z:j = Z4' variations in L l l are suffi ciently large for an effective energy conversion t o t ake place. To avoid some undesirable effec ts in operation, it will be well-ad vised to choose t he number of teeth on the stator and rotor such tha t Z 4 - Z~ = +2Pl (20-5) The rationale of such a choi ce will he expla ined la ter. An example of a machine satisfying the condition in Eq. (20-5) is shown in Fig. 20-3. The rotor has Z4 t eeth,
Gh. 20 Basic Mach ine Designs
215
where as the salient-pole stator .has 2Pl = 4 poles. The coils of t he concentrated single-phase stator winding are wound around the pole-pieces and laid out in major slots between them. On the surface of the poles are made minor stator teeth displaced from one~fianother by ·:a tooth angle I' Z3 = 2n/ Z;, where Z~ is the number of tooth angles that can
r ·
Fi g. 20-3 Machi ne with concentrated heteropolar stator winding (1) (PI = 2, th. = 1) and toothed st ator (3) and roto r (4) cores with different numbers of teeth (Za = 12, Z; = 16, Z4 = 20)
be accommo dated rou n d the core . The number Z; must satisfy the condition defined by Eq. (20-5) and be, of course, a multiple of 2PI' that is Z~ = 2Pl (an in t eger) . This in turn requires t hat the rotor should have a number of t eeth which is a m ultiple of t he number of poles, t ha t is , Z4 = 2Pl (an integer). In our case, Z4 = 2 X 2 X 5 Z~
= 20 - 2
=
X 2
20
= 16
For the machine t o oper ate normally, it is essent ia l that each pole sh ould spa n 2/3 t o 3/4 of a pole pitch , t he rema inder being taken up by the major slots betwe en them. Accordingly , each pol e must carry an odd number, N 3' of mino r te eth . This numb er must lie within the limits given above
N3
=
two-thirds to three-fourths . of Z~/2p
(20-6)
216
Part Two. En ergy Conversion by Electrical Machine s
where Z~/2p is the number of minor tooth angles per pole pitch. In our case, N 3 = (2/3 to 314) X 16/4 = 3 Th e self-inductance of the stator wind in g, L Il = 'l\/i 1 , varies with the rel ative position of the stator and rotor tee th. For the machine in Fig. 20-2 (with Z3 = Z4) ' it is a m aximum when t he stator and rotor t eeth are ali gned, for exam ple when '\' = 0, n /4, n /2, etc . It is a minimum when a slot is oppo sit e a too t h , which happens when '\' = = n18, 3n18, 5n18, etc. , and the permeance of the air gap and the flux linkage 1Jf 1 for a given i 1 are minimal. For the ma chi ne in Fig. 20-3, the self-in ductance of t he winding is a maximum when the rotor tee th are align ed with the stator tee th lying on the axes of the winding poles (1, 1', 1", 1"'). This happens wheu v = 0, n/10, n/5, etc. In this posi tio n , all minor teeth on t he pol es ar e approxim at ely opp osite the rot or te eth, and t he flux linkage of the winding for a giv en conductor curre nt is a maximum. Conversely , t he self-inductance of the wind ing is a mi nimum when the rotor slot s ar e aligned wi th t he st ator t eeth lyin g on the axes of the winding poles, which happ ens when , say , Y = n/20, 3n/ 20, etc . The self-inductance undergoes a complete cycle of chang e as the ro tor moves t hrough one t ooth pitch , or one t ooth angle Yz = Y Z 4' (The pol e pi t ch an gle of t he winding, 1'Pl = 2nlp l' and the tooth angle of the stator, Yz:i, ha ve no effect on the perio d of change in the self-inductance .) If the rotor is rotating at an angular velocity Q , the time period of change in self-induct ance, it s frequ ency an d angular fre quency are given by T = Y'Z 4/Q = 2nlZ4Q f = Z 4 Q/2n (0
= Z4Q
(20-7) (20-8)
From comparison of Eqs. (20-8) and (20-3), it is seen th at in syn chr onous reluctance machines, the frequ ency is Z 4/P times that of conventional two-winding machines with th e same rotational frequency . Th e rationale of cho osing the number of t eeth subject t o Eq . (20-5), may be explained as follows. For variat ions in the self-inductance of the winding to be subs t antial, the
217
Ch. 20 Basic Mach in e Design s
rotor and stator teeth must take up the sam e relative position at every pole of the winding (this is true, for example, of the case in Fig. 20-3 wher e the rotor and stator t eeth are shown aligned at all the poles) . Let one of the rotor teeth (say, too th No.1) be aligned with a stator too th at pole 1' . Then the next adjacent rotor tooth will be displaced from the nex t adjacen t stator tooth through an angle YZ3 - YZ 4; the rotor tooth following it will be displaced from the corresponding stator tooth by an angle 2 (YZ3 - Yz,;) , etc. As a result, the rotor tooth in alignment with the stator too th at pole 1" will be separated from the first by an ang le YPl/2 or YPl/2Y Z4 t ooth pitches of the rotor . Wi th the respect to the nex t adj acent stator tooth , this rotor tooth will be displaced through an angle YPI (Y Z3 - YZ 4)/2YZ4 whi ch must be equa l t o t he t ooth angle of t he stator, that is YPI (Y Z3 - YZ4)/2YZ 3 = +YZ3 Hence, on recalling tha t 1'111 =
2n lp 1
we obtain the cond ition defin ed in Eq . (20-5). If stator winding 1 (Fig. 20-4) is a distributed one, and the conduct ors carry ing cur ren ts in the same direction ar e laid at each pole among several (q) slots (in Fig. 20-4, q = 3), the stator core 3 need not be a t oothed one (modification B7 in Table 20-1). In the modification using a distributed win ding, variations in the self-in ductan ce of t he winding can be produced by the teeth on the unwound core, 4. Th e slots in core 3 carrying the winding may have a limited opening. In the cir cumstances , core 3 may be treated as a smooth one. Variations in the self-induct ance will be a maximum when the rotor has the same number of t eet h , Z4' as there ar e poles on winding 1 (in Fig. 20-4, 2p 1 = 4 and Z 4 = 4). This design may be regarded as a specia l case of a machine with the numb er of te eth meeting the condition defined in Eq . (20-5) for a smooth st ator, when Z3 = 0, and Eq . (20-5) redu ces to Z4 = 2p1' In such a ma chine, the hi gh-perrneancc zones lie opp osite the rotor teeth, and the low-permeance zones li e oppo site the ro tor slots. If it has Z4 = 2Pl tee t h , such a rotor is called salient-pole . The time period , frequency and angular frequency of var iations in self-ind uctance are give n by Eq . (20-8). Apart from a heteropolar winding, the design in question may use a single-phase homopolar winding (modification
Part Two. Energy Conversion by Electrical Machines
218
C7 in Table 20-1). In this design , variations in the self-indu cta nce of the win ding are obt ained by using a toothed core for both the stator and rotor, the effect being a maximum when Z3 = Z4' In performance, such a machine is not unlike a single-phase heteropolar machine with Z3 = Z 4'
LUm
il
(1
Ln o
r 0
7(;
Fig. 20-4 Machine with distribu ted heteropolar stato r (1) windi ng = 2, ql = 3) and a toothed (salie nt-po le) ro tor (4) core (Z,I =
(PI
=
2Pl
=
3)
Th e t ime period, frequency and angular frequency of variations in the self-inductance of a homopolar winding are found by Eq. (20-8). The design examined in t his section is utilized in reluctance synchronous machines. It offers advantages of simple construction and freedom from sliding contacts in the electric circuit of the winding. 20-4
M achines with Two Winding on the Stator and Toothed Cores for the stator and Rotor (Inductor M achines)
In t his case, electromechanica l '(or mechanoelectric) energ y conversion occur s m ainly owin g to variations in the mutual inductance between the s tator windings as a toothed rotor core mov es relative to them .
2Hl
Ch. 20 Basic Machine Designs
The win di ngs m ay be heter opolar and have t he same nu mber of pol e pairs (PI = P2) or a diff erent number of pole pairs, or they ma y he homopolar . It is also possible to combine a het eropolar an d a homopol ar winding. The stato r m ay be built with either a t oothed or a smooth core. This leaves
a 2
I I I
-l
Fig . 20-5 Machine with twu hc teropolur stat or windings, ] and 2 [12 = 2) and toothed cores for the s ta tor (3) an d rot or (4), wit h th e same number of t eeth (Zs = Z4 = 8)
(PI =
us wi th five likely modificatio ns (A3 , B3 , C3, D3, and C5, see Table 20-'1 ). One cycl e of ch ang e in the mutual in duct ance is completed as the rotor (at 4 in Fig . 20-5) moves through a t ooth pitch t Z 4 or a too th angle, '\'Z 4 = ,\,z. If the ro to r is r otating at an angul ar veloci ty Q , the t ime period and an gular frequency of the mu tual inductance can be found by Eq. (20-8), assuming that the rotor core ha s a number Z = Z 4 of te et h . According to the m ann er in whi ch the mu tu al in du ct an ce between the win dings is m ade to var y, the machine can be built in one of four modification s. (i) The mutual inductance between the windi ngs varies owing to changes in the mean perm eance of the air ga p wi th the rot ation of t he to ot he d rotor core rel ati ve to the t oothed s tat or core , bot.h having the sam e numb er of t eeth (modifica tions A3 an d C5 in T able 20-'1) . In this arra ngemen t , the mutual induct ance has the same sig n in any po sition of the
220
Part Two. En erg y Conversion by Electrical Machines
rotor rel ative t o the stator . As the rot or rotate s, it oscillates about its mean value , being a maximum when t he stator teeth are aligned with the rotor tee th, an d a minimum when the stator teeth ar e aligned with the ro tor slots (an d, of course, when the s ta tor slots are aligned with the rotor teeth) . The numb er of t eeth
Z
= Z3 = Z4
(20-9)
is chosen t o give the requisite fr equ ency of cha nge in the mutual inductance, from Eq . (20-8) . Similar va riations in t he mutual induct ance wi th Z3 = Z4 can be obtained with modification C5 which uses two homo pol ar wi ndings. It is equ iv alent in performance to modification A3 wh en the latter uses single-pha se windings with PI = P2" (ii) The mutual inductanc e between the two het eropolar windings on the stat or is m ade t o va ry owi ng t o t he rotati on of a t oot hed rotor core (modification B3 in Table 20-1). In t his arrangement, t he sa liency of the sta tor is of minor signific an ce. For variations in the mutual inductance t o be as large as possible, t he numb er of pol es Pi and P2 and the numb er of teeth Z4 mu st be ch osen such t h a t
Z4
=
P2 + PI
(20-10)
For example, in the machine of Fig . 20-6 wi th a two-p ole st ator winding 1 (PI = 1) an d a four-pol e stat or wi ndi ng 2 (P2 = 2), the condition defined by Eq. (20-10) will be satisP2 = 3). fied if the ro tor h as t hree teeth (Z 4 = PI In su ch a machine , the permeance will be a maximum in zones B I , B 2 , and B 3 whi ch are ali gned wi th the ro to r teet h. It is an easy m atter to prove that t he mutual induct an ce between the stator windi ngs is a function of the posit ion that zones B I , B 2 , and B 3 take up relati ve to thes e windings. When the ro tor takes up t he posi tion sh own in t he figure (the angle i' between winding 1 and the rotor te eth is n /G) , the mu tual induc ta nce £1 2 is a posi ti ve maximum (the magnetic fie ld du e to the cur rent i 2 pro du ces a maximum positi ve flu x linkage with winding 1). If we rotate the rotor through n /3 , its tee t h will move in to the position previou sl y occupied by the slots an d, as can read ily be sh own , the flux linkage and the mutual inductanc e will change sign and take ea ch a m ax imum negative va lue equal to the positive
+
Ch. 20 Basi c Machine Design s
221
maximum va lu e in magnitude. With PI = 1 and P2 = 2, the av erage mutual inductance will be zero. It will likewise be zero if P2/Pl is an even number, that is , if t he windings are such tha t given smooth cores, the mu tual inductance Yz
Yz
Y
o
Fig . 20-G Mach ine with t wo hotoropolar sta tor windings , 1 and 2, with a diff eren t num bel' of pol e pairs (PI = '1 , P 2 = 2), a smoot h stator core (3), and a toothed rotor core (4). Z4 = PI P2 = 3
+
between them is zero . If P2/PI is an odd number, the average mutual inductance will be non-zero, and the mu tual inductance will be pulsating about its mean value . The number of rotor teeth Z 4 is uniquely fixed by the specified frequency and angular frequenc y . As a rule Z 4 is fairly large , and in order to satisfy Eq . (20-10) , winding 1 must be made with a moderate numb er .of pole pairs (PI = 1, 2, 3) and winding 2 with a large number of pole pairs, clos e to that of rotor te et h , P2 = Z 4 - Pi - This introduces some difficulties in t he m anufacture of winding 2. In fact , if Z 4 is very large, one has to use the modification described in (iii) below . Winding 2 is laid in the minor slots shaped so that t heir effect on the permea nce of t he air-gap may be neglect ed . Winding 1 is laid in the major slots whi ch can be formed by enlarging the cross-section of som e minor slots without increasing t heir total numb er (Fig. 20-6) , or t hey
222
i)urt Two. Energy Conversion
by Electrical Machin es
may rep lace a group of several minor slots and te eth by removing one or several coils from winding 2. When Z 4 = = 2PI' the two windings have the same number of pol e pairs, P2 = Z 4 - PI = 2PI - PI = PI If, however , this arrangement is to perm it variations in the mutual inductance , one of the windings mus t be distributed or, if both windings are left concent rat ed, they must be displaced from each other by a qua rter of a cycle, as shown in .t he cross-sectional view of modification B3. (iii) The mutual inductance between the two heteropolar windings on the stator is made to vary owing to changes in the position of a toothed stator core relative to a toothed rotor core, both having the same number of teeth . In this arrangement, the number of te eth on the stator and rot or must be chosen such tha t there are as many high-permeance zones as in a machine with a smoo th stator core in (ii ) (Fig . 20-6), that is, P2 + PI (when P = 1 and P2 = 2, the number of such zones will be 2 + 1 = 3 or 1). To obt ain this number, it is essential that the difference in the number of te eth between the stator and rotor be P2 + PI' Z 4 - Z3 = P2 + PI (20-1'1) To prove, at the centre of a high-permeance zone , say , B I in Fig . 20-7, a rotor tooth is opposite a stator tooth (or, whi ch is t he same , a rotor slot is opposite a stator slot) . An adj acent rotor tooth is displaced from an ad jacent stator too th by an angle I' Z3 - I' Z4; the nex t adjacent rotor tooth is displaced from the next ad jacent stator to oth by an ang le 2 ('\'z3 - YZ4) ' and so on . To arrive at the centre of the next high-permeance, say, B 2 the displacement must be 2n /(p2 + PI) or Z 4/(P2 + PI) roto r teeth. Then , because at the centre of zone B 2 the rotor too th mus t again be opposite the stator tooth, the displacement of this rotor tooth from the correspond ing stator tooth Z 4 (YZ3 - YZ4) /(P2 + PI)
mu st be equa l to the tooth angle of the stator, that is, Z4 (-\'Z 3 - '\'Z 4) /(PZ + PI)
=
'\'Z3
Hence, the number of tee th on the stator and rotor must sati sfy the condition defin ed by Eq . (20-11).
., .
"
' .
'
.
223
Gil. 20 Basic Machin e Designs
Also, for windings 1 and 2 to be able t o form balanced circuits, Z:i must be equal to 2p z (an integer) , if PiAs an example, Fig. 20-7 shows a machine having p z = 2 and PI = 1, as does the machine in Fig. 20-6. In view of the desired frequency ,
»->
Z3 = 2 X 2 X 5 = 20 Z 4 = 20 + 2 + 1 = 23 The pattern of changes in L 12 with t he angular posi tion of the rotor is qualitatively the sam e as for the machine in
y
a
77:
271.
23
23
37T: 23
Fig. 20-7 Machine wit h two heterop olar stator wind ings, 1 and 2, differing in the number of pole pair s (PI = 1, p z = 2). The stata l' and rot or are built with t oothe d cores differi ng in the num ber of teeth : Z4 = Zs
+ PI + pz =
20
+ 1+2=
23
Fig. 20-6. But the self-inductance varies nearly sinusoidally , and it completes a cycle of change in one tooth angle , I'Z4' It is to be noted that as the rotor moves through one tooth pitch or through I' Z4' the axis of a high-permeance zone moves through an angle 2'Jt/(Pz + PI) , so that zone B I takes up the place of B z , zone B z takes up the position of B 3 , and so on. As is seen , the high-permeance zones rotate at a higher spe ed
224
Part Two. En ergy Conversion by Electric al Machin es
tha n the rotor by a factor of
2n/Yz 4 (p z + PI)
=
Z4/(PZ + PI)
Th e t ime period and fre que ncy of vari ations in the selfinduct ance can be found from Eq . (20-8). (iv) The mu t ual induct ance bet ween a heteropolar an d a homopolar winding is m ade to var y by the rotation of
F ig. 20-8 Machine with a},heteropolar (2) and a homop olar (1) st at or windings, with a smoot h stator core](3) l and a 'ltoothe d rotor core (4). Z,j = PZ = 2
a to othed rotor core relative to a smo ot h st ato r core (modification D3 in Table 20-1) . As is seen from Fig. 20-8, the het eropolar winding is lai d out in t he stator slo t s (in the figur e, p z = 2). The homopolar .winding is wound as a ring around the rotor shaft. The figure shows the posi tive directi ons of the currents in t he windings and the magnetic fi eld set up by i 2 • If the stat or and ro tor cores were smooth, t he magne tic field established by win din g 2 would be periodic (as shown in Fig. 20-1). Its lines would close vi a the yokes and li nk the cur rent i 2 in the slots wi thout linking with winding 1 . To make variati ons in L 12 as large as pr acticable, the rotor core
Ch, 20 Basic Machine Designs
225
is made with teeth, and the number of teeth is taken equal to the number of pole pairs for winding 2, that is, Z 4 = P2 (20-12) As an example, the condition defined by Eq. (20-12) will be satisfied by the machine shown in Fig. 20-8 (P2 = 2) when Z4 = 2. When the rotor is in the position shown in the figure (the angle between the axis of the 8 pole on winding 2 and the axis of a rotor too th is 'Y = 'Yzj2 = n/2), the mutual induct ance L 12 betw een windings 1 and 2 has a ma ximum negative value. In this position, the high-permeance zones aligned with the rotor teeth lie opposite the N poles on winding 2. Conversely, the zones lying opposite the 8 poles have the lowest permeance, so the periodic magnetic field whose lines close around the cur rents in the slo ts, along the yokes, and across t he gaps is insignificant. In contrast, th e homopolar field whose lines close around the coil ends on the N poles of winding 2, across the gap zones having a maximum permeance, across the yokes, through the shaft, end-shields, an d frame (see Fig. 19-1d) is substantial (it is shown by dashed lines in Fig. 20-8). As is seen , the lines of t his homopolar fiel d link with homop olar winding 1, and the resultant flux linkage is neg ative. If we turn the rotor through 'YZ4/2= n/2, the axes of the rotor teeth will line up with the axes of the 8 2 poles on winding 2, the high-permeance zones will lie opposite the 8 2 poles on winding 2, and the result an t homopolar fiel d will produce a positive flux linkage with winding 1. It should be noted that in Fig. 20-8 we have chosen small values for Z4 and P2 only to simplify the illustration. In practical machines, t he relationship between f and Q is usually such that Z4 = P2 must be fairly high. As already noted , winding 2 with a large number of poles is difficult to make. In fact , if Z4 = P2 turns out to be too large, the modification being discussed' has to be replaced by that examined in(v)below. (v) The mutual inductance between a heteropolar and' a homopolar winding is made to vary by the rot ation of a too thed rotor relative t o a toothed st ator, having differen t numbers of tee th (modification C3 in Table 20-1). As follows from Fig . 20-9, this modification differs from that in (iv) only in that the stator core has teeth. To make variations in L 12 as large as practicable , the number of teeth on the stator (Z3) and the rotor (Z4) must be chosen such" 15-0 16 9
226
Part Two. En erg y Conversion by Electric al Machines
that the number of high-permeance zones form ed around the periphery of the air gap be equa l to Pz, as in the m achine shown in Fig. 20-8. As has been pro ved in (iii) , the numb er of high-perme ance zones for a t oothe d stator and a toothed L
r 2~
T
Fi g. 20-9 Machin e with a heteropolar (2) and 'a h omopol ar (1) wind , in g on th e stator an d toothed st at or (3) an d rotor (4) cores h aving a different number of t eeth (Z4 = Z 3 P2 = 12 2 = 14)
+
+
rotor is equa l to the diff erence in the number of teeth between the stator and rotor. Therefore , Za and Z4 must be chosen such that (20-13) As an example, for the machine in Fig. 20-9, which uses winding 2 with two pairs of poles (Pz = 2) and st ator 3 with Za = 12 t eeth, the condit ion defined in Eq. (20-13) will be sa t isfied when
Z4 = Za + pz = 12 + 2 = 14 When the roto r takes up the position show n in the figure (the angl e between t he axes of the st ator and rotor t eeth is I' = 31'Z4/2 = 3:rt/14), the mutual inductance between windings 1 and 2 has a maximum negative value. In this position, the high-permeance zones in the air gap, wher e the
Ch. 21 Unidir ection al Energy Conversion
221
stator teeth lie opposit e the rot or teeth , are aligned with the N2 pol es of winding 2. In contrast, t he zones lying opposite the S poles have the lowest permeance. If we turn the rotor through an angle "Yz 4/2 = :rt/14, the axes of the ro tor teeth will ali gn themselve s with those of t he stator tee t h on the axes of the S poles. The high-p erm ean ce zones will then lie opposite the S2 poles on winding 2 and produce a homopolar magnetic field which links wi th winding 1. The design with two windings on the stator is frequently used in special-purpose machines. Among its advantages are the relatively high frequency of variations in the self or mutual inductances at a relatively low rotational speed, and also freedom from sli ding cont act s in the elect ri c circuits of the windings (for which reason such machines are called brushless or contactless). In t he generator mode of operation, such m achines generate voltages at a high frequency, although the rotor is rotating at a medium velocity (in ductor generators) . In the motor mode of operation, their rotors rotate at a substantially lower speed than the m achines having windings on both the stator and rotor. Because in such motors the ro tor speed is reduced electromagnetically (without any gearing), they may be called electromagnetically down-geared motors.
2\
Conditions for Unidirectional Energy Conversion by Electrical Machines
21-1
The Single-Winding M achine
In this chapter, we shall discuss what cur rents the windings of a machine must carry for un directional energy conversion to take place. The discussion will be concerned with the same machine designs as are listed in Sec. 20-1. To begin with, we sh all turn to the equat ion of electromagnetic torque, Eq. (18-6), for a single- or a two-winding machine. In this equation, the self-inductance of one winding, L 11l or the mu tual inductance L 12 between two windings is a periodic function of t he angular position of the rotor or time. For unidirectional energy conversion, the currents in the windings must vary so that the mean electromagnetic torque is nonzero. 15 *
228
Part Two. Energy Conversion by Electrical Machine s
Because all the events involved recur periodically, it will suffice to determine the torque averaged over a period r 1
To=y
r Jo
T
Tdt
and to define the conditions for currents under which elect-r~r---/----II--....Y....---..;..t ric energy is converted to mechanical (To> 0) or back (To < 0). Let us consider a single-winding machine first (see Sec. 20-3). As follows from Eq. (18-6), when n = 1 and k = 1, the electromagnetic torque developed by a machine with t one winding 1 carrying a curI rent i 1 (see Figs, 20-2 through 20-4) is given by Lf/O
1 ' 2 dL ll T --2~1~
(21-2)
No matter where the winding is wound (on the stator or rotor) and how it is arranged, variations in L 11 will be t qualitatively the same, with o\ I ---7-an angular period (tooth angle) \ I To (generator) 'Y z or a time period T \ /\ = -2n/pzQ = 2n/ro [see Eq. \ ,I, rem (geflerator) (20-8)], about some mean selfFig. 21-1 Conditions for unidi- inductance, L 11 o • Expanding into a Fourier series and retainrectional energy conversion in a single-winding machine ing the zeroth and first terms, variations in the self-inductance with time may be described by an equation of the form (21-3) L 11m cos rot L 11 ~ L 110 - ,/
+
where ro = Qpz = 2n/T is the angular frequency of variations in the self-inductance (see Fig. 21-1).
Ch. 21 Unidi rection al Energy Conversion
229
The derivative of the self-inductance with respect to the an gular posi tion of the rotor is dL ll/dl' = (dLll/ dt) (dt/ dl') = - PZL ll m sin wt (21 -4) where I' = Qt , an d dt/dl' = '1/Q. As we have already learned , t he current in the onl y windin g of a m achine must be an alternating one . Using Eq. (21-1), it is an easy matter t o see t hat if the wind ing carri ed a const an t cur rent, the mean t orque T o would be zero. Let us li mi t ourselves t o the fund ament al component of current, responsible for the la rgest me an electromagnetic torque. Then, i i ~ 1 1m cos (WIt rp) (21-5)
+
N ow the qu estion is: What should WI and cp be for Toto be a maximum, with all other con dit ions being equal? Since the mean torque is given by T
To =
PZI~:;/l1m ~ cos- (Wit + rp) sin wt dt
-
o its ev aluation reduces to evaluating its in tegral. Upon trigonometric manipulations in t he integr and , we get T
~ cos'' (Wit + (p) sin wt dt o
T
= ~ ! .\
si n tot dt
T
+-} ~ cos (2Wit + 2cp) sin wt dt
o 0 The first t erm on t he right-h an d si de is equal to zero. The second term may be re-written as T
4- Jsin [(W+ 2W t+2 cp] dt i)
o
T
1 •
+4
.\ sin [(co - 2Wi) t - 2cp] dt o When the angul ar frequency of the current is WI = w/2 (21-6) the time period of the current, T I = 2T , is twice the time period of the self-inductance , and the period of the current
230
Part Two. Ene rgy Conversion by Electrical Machines
squ ared , T l /2 , is equal to the time period of the self-induct ance (see Fig. 21-1). Then t he mean electromagnet ic torque is To = (p zI\mLnm/8) sin 2cp (21-7) When (p = n /4, the mean torque is a maximum in the generator mod e of operation . When (p = - n/4, it is a ma ximum in the motor mode of operation . Also, the periodic component of the current squared is in phase with the selfinductance in the former case , and in anti-phase in the latter case. Th e respective plo ts of cur rents and t orques appe ar in Fig. 21-1. Th e angle cp = n /4 corresponds t o a time lead of t = CP/COi = T l /8. To sum up , it may be argued th at for unidirectional energy conversion, a sin gle-windi ng machine must carry a current at angular frequency COl equa l to h alf th e angular frequency of variations in the self-inductance: CO l
= co/2 = P zQ/2
The direction of energy conversion dep ends on the phase ang le between the cur ren t and the self-inductance. When cp = sil«, the machine will be operating as a motor. When cp = - n/4, it will be operating as a generator. Th e ang ula r veloci ty of the ma chi ne is proportional to the angular frequency of the cur rent in t he win ding connected t o the electrical sys tem Q = 2COl/P z (21-8) A machine whose angular velocity is proportion al to the angular frequency of the electrical system will be called a sunchronous machine . A ma chine whos e angular velocity does no t satisfy this rela t ion will be call ed an asynchronous one . From Eq . (21-8) it follo ws then that all single-winding a .c. machines are synchronous machines. 21-2
Two-Winding M achines
Th e electromag net ic t orque developed by a two-winding machine is giv en by
T = i 1i z dL i2/d 1,
+ ~. i~ dLii/dl' + ~ i ~ dLz)dl'
irrespective of the winding arrangement.
(21 -9)
Ch. 21 Unidire ctional En ergy Conversion
231
A major contribution to the electromagnetic torque comes from var iat ions in the mutual inductance between the windings and is represented by the first term in Eq . (21-9)* . Therefore, we may li mi t our ana lysis t o the first term. As in the pr evious case , we may li mi t ourselves to the peri odic component of the self-inductance with an angular peri od YI> (or Yz) and a time period T = 2rr,/PoQ = 2rr,/w where P o
P is the number of pole pairs on t he heteropolar windings of the stator and rotor [see Eqs . (20-2) t hrough (20-4)] p z = Z is t he number of teeth per pole on the rotor of a machine with two stator windings [see Eq . (20-8)] As in the previous case, the m utual in duct ance varies with time as L 12 = L 12 0 L 12 m cos tot (21-10) =
+
wher e ro = Qpo = 2rr,/T. The deriva ti ve of the mutual inductance with respect to the angula r position of the rotor is
dL1Z/ dl' = (dL 12 /dt) (dt/dy) = - PoL 12 m sin wt
(21-11)
In t he gen era l case, the windings carry alternating currenls** . Limiting ourselves to the fundamenta l components as contri buting most to the electr om agnet ic torque, we may write t, = J Im COS (WIt (PI) (21-12) 12 = J 2m COS (W2t cr2)
+ +
In Fig. 21-2, WI = 4w, W 2 = 3w, cri = 0, and (P 2 = - rr,/2 . Now le t us find the va lues of WI' w 2 , (PI' andtp, that will lead to a maximum mean electromagnetic' torque in Eq . (21-1), with all oth er conditions being equal.
* Th e other components of the mea n torque can be found as for a single-winding ma chine. As follows from Eq . (21-6), th e mean torque of t his kind m ay be non-z ero at WI = w/2 or W 2 = w/2. Th en i t will be a maximum a t (jlI = ±rtl4 or rp 2 = ±'It/4. * * A m achin e with, say , the second winding carrying d .c, is a specia l case for whi ch W 2 = 0 and i 2 = canst .
232
Part Two. En ergy Conversion by Electrical Machines
The mean torque is given by T
To=-(Po[lm[2mLI2mIT).\ cos ((Oit+CJli) cos ((02t+ (P2) sin wt dt o The products of the cosines in t he integrand may be rewritten 1
1
"2 cos [((01 + (02) t + (PI + CJl2] + 2" cos [(Wi - (02) t + CJlI - CJl2]
+
where the first term varies with a frequency W I 00 2 , and the second term with a frequency WI - (02' If one of these frequencies is the same as the frequency of variations in the mutual inductance, that is, WI
+ 002 =
00
or
WI -
002
= (0
(21-13)
then the mean torque will be nonzero. To demonstrate, on replacing the products
~ cos [(WI + (02) t +
(Pi + CJl2] sin wt
and 1
2" cos [(001- (02) t + CJli- CJl2] sin wt by a sum of four trigonometric functions, we get T
To = (Po[imI2mLi2 m/4T) .\ sin r«(Oi + (02- (0) t + (P I + CJl2] dt o Hence, on satisfying the condition defin ed by Eq. (21-13), we obt ain (21-14) where the " +" sign applies when WI + (02= 00, and the "-" sign applies when WI - (02 = (0. The integrals of the remaining three terms of the sum, varying at frequencies (01 + (02 + (0 =1= 0 WI -
(01
are equal to zero.
+
002
+ 00 =1= 0
002 -
00
=1= 0
233
Ch. 21 Unidirectiona l Energy Conversion
As is seen from Eq. (2'1-'14), in the motor mode of operation (To > 0), the mean to rque is a maximum when qJl + qJ2 = n/2 ; in t he generato r mo de of oper ati on (T o < 0), this happens when qJl + qJ2 = - n/2 . Unidirectional energy conversion by a t wo-winding machine is illustrated in Fig. 2'1-2. With the frequencies and ph ase adopted in the figure, and qJl -
CjJ 2
= n /2
so the resultant torque is nonzero . To sum up, for unidirectional energy conv ersion by a twowinding a.c. machine, it is essen tial that t he sum or the difference of the angular frequencies of the ' currents in the windings be equal to the angular fr equency of variations in t he mutual inductance between the windings. The direction of energy conversion is determined by the magnitude of the sum or difference of the phase angles of cur rent s with respect t o the mutual inductance. Whe n 0 < ((PI + qJ2) < n/2 , electric energy is convert ed to Fig. 21-2 Condi tio ns for unid imechanical; when - n/2 < rectional energy conversion in (rpt + qJ2) < 0, mechanical a two-w inding machi ne ene rgy is converted to electric. Accor din g as t he ro t at ion al frequency of t he rotor does or does not change with variations in the ext ernal torque, there may be asynchronous machines and synchronous machines. In a synchronous ma chine, both wind ings carry currents whose ang ul ar fr equ encies are fixed in ad vance. In the gener al case, t he m achine conv erts th e electri c energy fed into two windings. Therefore , such u n i Ls are also call ed doublejed machines. With COl and CO2 held constan t, the angular velocity of t he rotor in a synchronous machine rema ins con -
Part Two. En ergy Conversion by Ele ctrical Machin es
234
stant, irrespective of t he t orque on its shaft * Q
W
Po
W I ±W
2 ----'----=Po
= constan t
Most frequently , synchro nous machines are built with a three-phase heteropolar win ding on t he sta to r and a singleph ase heteropolar winding on the rotor. If we pu t W 2 = 0 and CjJ 2 = 0, all t he relati ons derived above will full y apply to such a ma chine. If not otherwise qualified, the te rm "syn chronous" refers exactly to the above type of machine. In an asynchronous machine (primarily, a motor) only one winding, sa y 1, is connected t o a lin e whose frequency , say WI is fixed in advance . The other winding is either shortcircuited or connected acro ss an impedance , and the cur rent i 2 in this winding is produced by electromagnetic induction. Accordingly, asynchronous ma chines are more frequently called induction machines . . The mutual emf is given by e12= -d'P'12/ dt = - { i 1m L 12m (W1-W) sin [(W1- W)
+~
i1mL12m (W1 + w) sin [(W1
where
'1'1 2 = i I L 12 = i l m cos
( W It
t+ {pd
+ o) t + {Ptl + (PI) L I 2m cos to t
is the mu tual flu x li nk age. The frequ ency W2 = Wi - W of i 2 is a fu nction of the angular velocity of the rot or (W2 = WI - Qp o) and satisfies the condition for unidirectio nal energy conversio n defined in Eq . (21-13). Most frequently , asynchronous (induction) m achines are built wi th a three-phase het eropolar a .c. winding on the stator, and a three-ph ase (or poly-phase) heteropolar shortcircuited winding on the. rotor . If not otherwise qualified , t he term "induction ma chine" refers to the above type of m achine.
* Vari ation s in th e load on th e shaft brin g about only cha nges in the amplitu de and phase of it and i 2 •
235
Ch. 22 Windings for A.C. Machines
22 .
W indings for A.C. M achines
22- 1
introductory Notes
Our discussion will be limited t o heteropolar cyli ndr ical windings since they ar e used most frequently in electrical machines. The coils of such windings are usually laid out in slots on the stator or rot or. The arrangement of single-layer (Fig. 22-1a) and tw o-layer (Fig . 22-1b) he teropolar windings along with that of a sim ple polyphase single-layer wi nding has been explained in Sec . 19-2. Therefore, our discussion here will only be concerned with pol yphase two-layer windings with m» 1 phases, since they are used most freq uently in a.c. m achines . 22-2
The Structure of a Polyphase Two-layer W inding
I"
A two-layer m-phase wind ing is designed for connection to an m-phase ba lanced a .c. li ne or system . In the case of a three-phase system, they can be connected in a star or . a delta (Fig . 22-2a and b). .. . For the phase currents l A' I B, and I c to form a ba lanced set, it is essentia l that the ph ase windings should have t he same inductive reactances . Thi s req uirement will be satisfied if the axes of the phase windings are displaced from one ano ther through an ang le equa l to 11m of the angular period of t he field (the pole pitch ang le)* ,,?plm
=
2nlpm
The core of a three-phase , two- la yer winding (m = 3) is shown in Fig. 22-3. Each ph ase winding consists of several coils (Fig . 22-4a), one coil side lying in the top half of a slot, and the other in the bottom half of another slot about one pole pitch away . Each coil may have one turn (we = 1) or several turns (we > 1) insulated from
* Thi s equa tion applies whon lire displaced through Yp/4 .
111
>
2. Wh en
In
=
2, th e ph ases
I~ '
236
Part Two. Energy Conversion by Electrical Machin es
Fig. 22-1 Windings: (a) single-layer and (b) doubl e-layer
(a)
~
:Fa
Fig. 22-2 Three-pha se winding: (al star-connected and (b) delt aconnecte d
.c-: B
--'
'f
238
Part Two. Energy Conversion by Elect rical Machin es
one another and from the slot sides (the coil in Fig. 22-4a has two turns). The coils may be lap-wound or wave-wound. In a lap winding, each coil is connected to the next adjacent coil in series. In a wave winding, each coil is conn ected to a coil two pol e pitches farther away than the nex t adjacent coil. Each coil has two leads. Let the lead on the upper coil sid e be t he st art (8) of the coil. Th en the lead on its lower side will be its fin ish (F). As a rule, the lead s of a coil are
~ "/'
.-...,, ~
•
•
F
S
-
W
-S 1 tL
I
...J .................
~)
~)
Fig. 22-4 Coils of a double-layer lap winding (solid lines) and a double-layer wave winding (dashed lines): (a) actual arrangement ; (b) sketch
made long enough for direct connection to another coil. According to the manner in which the coils are interconnected within a phase, the leads may be differently shaped and proportioned. In diagrams, coils are usually shown as single-turn loops. As is seen in Fig. 22-4b , a wave-wound coil differs from a lap-wound coil only in that the leads are sh aped differently (the leads of a wave winding ar e shown by the dashed lines). The coil pitch y equal to the distance between the coil sides, may be equal to a pole pitch (y = r), or it may be somewhat shorter than one pole pitch, or chorded (usually, y = 0.8'];). Accordingly, there may be a full-pitched or a short-pitched (or chorded) winding. The pole pitch and the coil pitch may be measured in terms of the distance along t he periphery of t he air gap or in
Ch. 22 Windings for
A.e.
239
Machines
tooth pitches
= Z/2p (tooth pitches) y = yc/tz (tooth pitches)
1:
where Yc tz
(22-1) (22-2)
= coil (or slot) span (see Fig. 22-4) = 2'JtR/Z = tooth pitch
Z = number of teeth (slots) on the core Taking as an example a three-phase, four-pole, shortpitched (chorded) winding with y = 7 and Z = 36, Fig . 22-3 shows how the coils should be distributed among the phases, the coil sides laid out in slots, and the positive currents directed in the coil conductors of polyphase, two-layer windings. The total number of coils in the winding is equal to the number of slots . So each phase contains
Z/m = 36/3 =
1~'
coils
To establish a four-pole field, the coils in each phase should be divided into 2p = 4 groups uniformly distributed all the way around the circumference (one group per pole pitch). Each group contains q = Z/2pm adjacent coils. The number q is equal to the number of slots per pole per phase, q = Z/2pm = 36 -;- (2 X 2 X 3) = 3 (22··3) Let us designate coils by the Nos. of the slots in which their top sides are laid. Then phase A will include the following coil groups: (1, 2, 3), (10, 11, 12), (19, 20, 21), and (28, 29, 30).
Adjacent groups in a phase are displaced from one another by one pole pitch 1:
= Z/2p = 36/4 = 9 slots
For the resultant field to be periodically varying, all the coils in each phase must carry identical currents reversing in direction as they pass from one pole pitch to the next. Assuming that the current in phase A (see Fig. 22-2) is in the positive direction, the currents in the top sides of coil group (1, 2, 3) will he flowing "inwards" (away from the reader), the currents in the top conductors of coil group (10, 11, 12) will be flowing "outwards" (towards the reader), etc. To facilitate design work, the coil groups in which the
240
Pa rt Two. Energy Co~version by Electrical Machines
top conductors carry curr en ts flowing away from the reader are assigned the index of the start of a given phase, A, and the coil groups in which t he top conductors carry curren ts flowing towards t he read er are ass igned t he index of the finish of the sam e phase, X. Given the same positive directions of cur rents, the patterns of coils and currents in the remaining ph ases will be t he same as in ph ase A. Th e only difference will be that phase B will be displaced from phase A counter-clockwise by an angle . Yplm = 2nlpm = nl3 (22-4) that is, through 2'r:!m = 18/3 = 6 slots. In turn , phase C will be displaced through the same angle from phase B. If the coils of a phase are divided into a iden tical parallel paths (circuits) within each of which they are connected in series, then each parallel pa th will carry a current equal to IIa. Referring to Fig. 22-3, it is seen that the currents in both the top and bottom con ductors of a phase set up patterns repeated every four poles, th at is with a period p = 2, so that in a short-pitched winding the currents in the bottom layer are replicas of the currents in the top layer, displaced by 't - Y = 9 - 7 = 2 slo ts clockwise. If the winding were full-pitched (y = 't), the layers would not be displaced from each other, the currents in the top and bottom conductors in all the 'slots would be in t he same direction and the conductors of a given phase would take up q slots per phase. In short-pitched (chorded) windings (see Fig. 22-3), the phase conductors are laid out in q (r - y) = 3 9 - 7 = 5 slots per pole. Chording results in an expanded belt occupied by the phase conductors within each pole pitch and, as will be explained in Sec. 24-5, makes the air gap field more sinusoidal.
+
22-3
+
Connection of Coils in a Lap Winding. The Number of Paths and Turns per Phase
As already noted, the coils of a winding may be lap-wound or wave-wound. In a lap winding, each of the q coils within a given pole pitch is connected to the next adjacent coil ill
Ch. 22 Windings for A.C. Machines
241
series ai ding to form a coil group. For example, connecting the finish of coil 1 to the start of coil 2, and the finish of coil 2 to the start of coil 3 (Fig. 22-5) produces a phase A B
Z
_ ~~ - .:::::::::: c ~
.
/',.
~ ....--.-
....:--,.
-C:>
-~
~ ~ Z B Fig. 22-5 Coil connection in a lap win di ng (Z q = 3 , 1" = 9, !J = 7, a = 1)
=
36, p
=
2, m
=
3,
coil group consis ting of coils 1, 2 and 3. The other coil groups in phase A, (10, 11, 12), (19,20,21), and (28, 29, 30), are formed in a similar manner. The start of a coil group is the start of the lowest- numbered coil, and the finish of this coil group is t he finish of the highest-numbered coil. For example, the starts of the coil groups listed just above are the starts of coils 1, 10, 19 , and 28, whereas the finishes of the coil groups are the finishes of coils 3, 12, 21, and 30. 16- 0169
242
Part Two. Energy Conversion by Electrical Machines
The term "lap" refers to the fact that, in going around from the start of a coil group towards its finish, the previous coil overlaps , as it were, the next adjacent one (see Fig. 22-5). The leads of the coils in a la p winding provi ded for connection to the next adjacent coils are; shown in Fig. 22-4. In the simplest case, when t here is only one path (or circuit) per phase (a = 1), the coil groups in a lap winding are connected in series. This arrangement is shown in Fig. 22-5. For this winding, the numbers of coils, phases and pole pairs have been chosen the same as for the winding in Fig. 22-3. For better presentation , the top coil sides are shown displaced counter-clockwise from t he bottom coil sides l aid' in the same slots . (The coil No . is the same as that of the slot where t he top sid e is lai d.) For proper periodic variations in the currents carried by the coil sides of the ph ase, coil gro up (1, 2, 3), bearing the index A, is conn ected in seri es opposition with coil group (10 ,11,12), bearing the index X . The finish of group A is connected to the finish of group X . The start of group (10, 11, 12), bearing the index X , is connected to the start of group (19, 20, 21) bearing t he index A, and so on . If the positive directions of currents in the coils are chosen in advance (see Fig. 22-3), it is an easy ma tter to establish the sequence of connection for the coils . In the other phases, the coils are interconnected in the same manner as in phase A . The coil groups in the lap winding of Fig. 22-5 can be interconnected in a simpler way . Referring t o Fig. 22-6, the coil groups are shown as sectors spanning an angle yp l2m subtended by the top coil sides. The numbering of the coil groups is given within the sectors. Each group has two leads. The start of a coil group is the lead of the lowestnumbered coil in t he group. When a . 1, the coil grou ps in Fig . 22-6a are connected in the same manner as in Fig . 22-5. The arrangement in Fig. 22-6b differs in t ha t the same coil groups within each phase form the largest possible numb er of paths, a = 2p = 4. Each path in a phase is formed onl y by one coil gr oup . The positive directions of cur rent in the coil groups are indicated by arr ows at the cur ren ts I Ala , I »!«. and I cia (pointi ng away from the finish towards the start of the forward groups A, Band C, and from the start towards the finish ill the backward groups X, Y,- and Z ).
Ch. 22 Windings for A.C. Machines
243
~ IT o
obtain the desired positive directions for the coil currents the starts of groups A, B, and C must be connected to the starts of the respective phases (A, B, and C), their finishes to the finishes of phases X, Y, and Z, respectively.
Fig. 22-6 Coil group connection in the lap winding of Fig. 22-3 for various numbers of paths: (a) for a = 1, (b) for a = 2p = 4
Conversely, the starts of groups X, Y and Z must be connected to the finishes of the respective phases, X, Y and Z, whereas the finishes of these groups to the starts of phases A, Band C. There is also a way of connecting the coils in a winding where a ranges between 2p and unity. Now, the coils are connected in series-parallel. For example, in the arrangement of Fig. 22-3, with a = 2, each path will contain two coil groups. Generally, the number of coil groups per path is 2p/a. This number must always be an integer. For the current to be equally shared among the paths, the latter must be completely identical (that is, present the same resistance and inductive reactance). This requirement is satisfied, if the paths are assembled from the same number of properly interconnected coil groups and have the same number of series-connected turns W = (2p/a) qWe (22-5) where We = number of turns per coil qWe = number of turns per coil group 2p/a = number of series-connected coil groups per path. 16 *
244
22-4
Part Two. En ergy Conversion by Electrical Machines
Coil Connection in the W ave W inding
The distribution of coils among the ph ases and t he choice of positive directions for the currents in the coils ar e independent of whether the coils are connected in a lap or in a wave winding. Therefore, with given Z , p, y , and In, the coil structure (say, the one shown in Fig. 22-3) is equally
~7-f--~ ~~A I
26
f /
/ 41 / 25 ;If / \ --...,..:::><~:-f /2 4 -1/ \ " jt "23 / \ " ,;: "if
(22/ .,;j' 2-(..' I Q
Fig. 22-7 Coil connection in a wave winding (Z' = 36, p 3, q = 3, 1: = 9, y = 7)
=
=
2,
111
=
applicable to both a lap and a wave winding. In a wave win ding, however, t he phase coils are in terconnected differently. Here, the winding progresses around the cor e by passing successively under each pole before again approaching the st ar t ing point as shown in Fig. 22-7. The coils in Fig. 22-7 differ from those in a lap winding only in t he sh ape of the leads. The diagram shows all t he connections between the coils of phase A. In forming this phase, let us start at coil 1. During the first tour, the finish of coil 1 is connected to the start of coil 19 whi ch is
Ch, 22 Windings for A.G. Machines
245
displaced from coil 1 by 21: = 2 X 9 = 18 slots or an angle ,,?p = 2n/p = 1800 • One complete passage round the core will encompass p coils. Thus, within one tour, a given phase contains p coils connected series-aiding. In our case, one tour encompases two coils. The first tour or wave must be followed by the second, third, etc., to give a total of q waves in the same direction. For the winding not to close upon its elf at the end of the passage, the spacing between the last coil in a previous wave (say , coil 19 in the first wave) and t he firs t coil of the next wave (say, coil 2 of the second wave) must be' 21: -+ 1, rather than 21:. For the arrangement in Fig. 22-7, this spacing is 21:
+1=
2 X 9
+1 =
19
On completing q waves, we shall have obtained the first part of the winding, Al- Xl, containing in our example q = 3 waves (the first wave consists of coils 1 and 19, the second of coils 2 and 20, and the thir d of coils 3 and 21). The second part of the winding is formed in a similar manner, starting at coil 10 which is displaced from coil 1 by 1: '= 9 slots or on angle ,,?p/2 = 900 • The start A2 of the second part of the winding will be the start of coil 10 , and its finish X2 will be that of coil 30. The two parts of the winding are perfectly identical, because they have the same number of coils connected in series ai ding (each part contains pq coils) . 'I'herefore, they ma y be conn ected not only in series, but in parallel as well. When connecte d in series, they form a winding with one path (or circuit) per phase (a = 1). When connected in parallel, they form a winding with two paths (a = 2). When a = 1 (see Fig. 22-7), for t he currents to flow in the chosen posi t ive directions, t he two par ts of the winding must be connected in oppo si tion , that is , the finish Xl of the firs t part must be connected t o t he finish X2 of t he second par t by a jumper. The star t A l of t he first part is the phase start A , and the start A 2 of t he second part is the phas e finish X . When a = 2, t he two parts are connect ed in parallel. The phase start A is connected to the start Al of the first part and the finish X2 of the second part; the phase finish X is connected to the finish X l of the first part and the start A2 of the second part. .
\
246
Part Two. En ergy Conversi on by Electrical Machin es
The other two phases (Fig. 22-7 shows only the first and last coil s) are formed in a similar wa y . The number of turns per path is foun d, as before, from Eq. (22-5). 22-5
The Selection of a W inding Type and W indin9 Characteristics
If a lap wind ing and a wave win din g have the same number
of coils (wound with wires of the same cross-sectional area, with the same number of turns per coil We and with the same coil span y), the same number of phases In and the same number of paths (or circuits) a, and intended t o generate a magnetic field with the same number of pole pairs p, they will be fully identical electromagnetically, because, given the same current , the phases set up identical magnetic fields. They onl y diff er in the total length of wire required t o make coils and coil conn ections. With a la rge numb er of tu rn s per coil and a lar ge number of slots per pol e per ph ase, the effect of coil ends is insignificant, and the total length of wir e is practically t he same in either case. With a small number of slots per pol e per phase, q ~ 2 or 3, a large number of pole pairs, an d a small number of turns per coil, especially when We = 1, a wave winding is mor e attractive. Then the saving in con duct or material may be as high as 5 % to 10 %. The larger figure applies to ma chi nes with a relative core length equa l t o ll : ~ 1. 5, and t he smaller figure to machines with a rela tive core length equal to ab out 3.0 . With a sm all numb er of pole pairs (say, p = 1 or 2), wher e t he len gth of jumpers between the two parts of a ph ase winding (A I -Xl and A2-X2) is large in com par ison wi th I the to t al length of coil ends , t he use of a wave wind in g \ offers no advantages. Practically, a single-turn coil is made by soldering, brazing or wel ding together two halves, called bars. Win dings in whi ch all t he coil sides carry t he same current Iw e/a = the sam e are equal as regards the pr odu ction of a magnetic fi eld. At the same t ime, t hey may have a differen t number of turns per coil, We, an d a different numb er of pa ths (or circuits) per phase, a . As an exa mple, Fi g. 22-8 shows the coil sides of three windings identical in terms of the magnetic field produced. (It is assumed that in all other respects these windings
247
Ch. 22 Windings for A.C. Machines
do not differ from one another. Notably, they have the same number of slots , the same number of pole pairs, the same number of phases, the same coil span, and the sam e current density in the conductors .) When the current per coil side is the same, the numher of cir cui ts and the number of turns are chosen t o su it the reliability requirements and to simplify t he manufacture. If
}~"
2
3
(!=:
if
(a)
l ..;
fWc ~
a=2
(b)
3
a«: (e)
Fig. 22 -8 Windings with the same current per coil si de, I wela: I - active conductor carryi ng f la; 2-turn insulation betwe en act ive conducto rs; 3-ground insulation
the number of circuits can be chosen such that the resultant coil will be a single-turn one, We = I , it is usual to pick the arrangement shown in Fig . 22-8c, because in a single-turn, two-layer coil (usually made of bars soldered, brazed or weld ed tog ether), t he ground insulation also doubles as t he turn insulation . As a result, its manufacture requires a smaller quantity of insulating materials, and the coil takes up a sm aller space in the slot , whereas the overall reliability is markedly improved . For a comparison of t he quantity of insulation require d, see Fi gs. 22-8a and b. ' . 22-6
A Two-Pole Model of a Win ding. !Electrical Angles between W inding Elements
If we go roun d t he peripher y of a multipole, polyphase winding, we shall see that its structure is a repetition of some basic patternvand this repetition occurs in .an angular dist ance equ al t o a pole pitch. For example, the pat-
Part Two. Ene rgy Conversion by Electrical Machi nes
248
tern tha t th e winding in Fig. 22-3 has betw een slots 1 and 18 is fully repeated between slots 19 an d 36 (in going coun t er-clockwise from pole to pole). In such a windin g, the cur rent s in the slots displaced from one another by 2'tk ar e a whole number k of pole pairs and numb ered N
+
.
X
X
~
~ A
~
(a)
Fi g . 22-9 Two-pole models of double-layer windings = 3, m
(a) for the winding in Fig. 22-3 (1' = 9, y = 7, q a win din g wit h r = Y = 9, q = 3, m = 3
=
3); (b) for
always the sam e. For exam ple, in slo ts N = 6 and N 2ruk = 6 2 X 9 X 1 = 24, the currents of ph ases C and B are of the same magnitude and flowing in the same directio n. To form a complete idea about the winding structure, it suffices to consider its pattern between an y pair of adjacent poles. Consi deration of its structure between other pair s of adj acent poles will add nothing new to our knowledge. A t wo-pole mod el of a multipole winding is the winding of a t wo-pole machine h aving the same number In of phases, t he same number q of poles per pole per phase, the same pole pi tch 't, t he same coil sp an y, and the sam e number of turns per coil, WlJ ' A two-pole model of the winding in Fig. 22-3 appears in Fi g. 22-9. It is an easy matter to see that it h as the same
+
+
249
Ch, 22 Windings for A.C. Machines
structure as the winding shown in Fig. 22-3 between an y pair of adjacent poles. (The currents in slot N of the model 2'tk of the prototype are the same as t he currents in slot N winding, where k = 0 or 1, and 't = 9.) The model equally applies to any pair of adjacent pol es (that is , between slots 1 and 18, or betwe en slots 19 and 36). From acorn parison of the prototype winding shown in Fig. 22-3, with its mo del shown in Fig. 22-9a, it can be seen that the angular period of the model, ex p = 2n, is p times as great as that of t he prototype winding given by
+
YP Hence,
=
2n/p
exp = PYp = 2n Th e angles between the winding elements in the model increase by the same factor as compared with the corresponding angles in the prototype (22-6)
ex = PY
where Y = angle between some elements of the prototype ex = angle between the same elements in the model. In the theory of electrical machines, the angle y between some elements in t he prototype is referred to as the mechanical angle*, whereas the angle ex = py between the corresponding elements in the two-pole model is called the electrical angle. The electrical angles bet ween the characteristic elements of a winding (the angles in the model) determine the fundamental properties of the winding (irrespective of the number of pole pairs on it). The angular period of a 2p-pole winding corresponds to an electrical angle
exp
= 2n
The too th (or slot ) pitch in a model spans an angle
ex z
= 2n /2mq = nlmq
(22-7)
In a multipole win ding, one t oot h pitch spans an angle
yz
=
2n /Z
=
2n/2pmq = nlpmq
'" In this te xt , it is referred to simply as the note,
angle. ~ Tra nsla to r'$
250
Part Two. Ene rg y Conve rsion by Electrical Machines
In the model, the phase belt occupied by t he phase conductors (in one layer) wi thin each pole pi tch spans an angle (22-8) that is, '1/m part of a pole pitch . (In a multipole winding, this belt spans an angle Yq = qyZ = nl pm.) The coil in a model spans an angle ay
=
(Yc!T:)n
(22-9)
In a multipole winding, the coil spans an ang le
Yy
=
(y!T:) (Yp/2)
=
(yh) (nip)
The procedure for deve lop ing a two-pole winding model does not differ from that set forth above for a multipole winding. As an example , Fig. 22-9b shows a mo del winding which differs from that in Fig. 22-3 onl y in having a full pitch, that is, y = 1:". As is seen, when y = 1:" , the top and bottom l ayers of the winding are not disp laced from each other, so that all the pha se cond uct ors within a given pole pitch are only laid in q slo ts (in our case, q = 3). Comp ar e this with Fig. 22-9a, where y = 7, 1:" = 9, and the phase (1:-Y) conductors within a pole pitch ar e la id in q = 3 9 - 7 = 5 slots.
+
+
22-7
Two-Layer, Fractional-Slot W indings
The two-layer . win din gs exa mi ned above are integra l-slot wind ings. This means t hat they h ave an int egr al number q of slots per pole per phase, and t he number q of coils in each phase remains the same from pole pitch to pole pitch (in Figs. 22-3, 22-5, and 22-7, q = 3 coil s in each ph ase). This is the most commo nl y used vari et y of tw o-layer win ding. Ano ther is wha t is calle d the fra ct ional-slot tw o- layer wi ndi ng . In such a winding, the poles of, say, the rotor are design ed to occupy only a part (fraction) of the st a tor sector t hat bounds three slots (in the case of three-ph ase machi nes) of the stator winding , or less t han one slot per ph ase per pole. Th a t is why t hey are call ed fract iona l-slot win dings. With a small number of slots per pol e per phase (q < 3) an d a large number of pole pairs , such windings offer a numb er of ad vantages over integral-slot windings.
Ch, 22 Windings for A.G. Machines
251
To obtain a fractional-slot winding, it is necessary that there be one coil more in some phase coil groups than in others. Those with one coil more (the l arger or major phase groups) will then have (b 1) coils each, and the others (the smaller or minor phase groups) will have b coils each. Becau se of this, the winding does not r epeat itself every pole pitch , but does so in a basic pole or coil pattern frequ ently called the r epeatable (pole or coil) group . Denoting the number of maj or ph ase groups in one basic pattern as c and the numb er of m ajor and minor phase groups (or poles) in one basic pattern as d, the number of slots per pole per phase for fractional-slot winding may be written
+
q= b
+ cld
As a rule, use is mad e of fractional-slot windings with b ~ 1. In them, each phase has 2p coil groups (one group ' per pole). Of this numb er , the m ajor groups will be under n = 2p/(c/d) poles, and the minor groups under (2p - n) poles. Then the number of slots per pole per phase for a fractional-slot winding ma y be written
q = n (b+1 )+(2p-n) b =b+ c/d 2p
As alread y noted, the denominator of t he fraction is the number of pole pitches in one basic pattern . In a com. pl ete winding, there may be 2 pld such basic patterns. For a winding to be feasible, 2 pld must be an integer. A further requirements is th at in a symmetrical pol yphase winding the denomina tor should not be a multiple of the number of ph ases. Since th e winding . must produce a periodic field , it mu st have an even numb er of poles. Ther efor e, the repea table group in a fr actional-slot winding t ak es up d pole pitches wh en d is even and 2d pol e pitches when d is odd . When d is even, t he winding h as 2 pld basic patterns; when d is odd , it h as 2p/2d basic pattern s. To sum up , a fractiona l-slot winding h as "m inor" phase groups of b coils each and "m ajor" ph ase gr oups of b "1 coils each . These groups alternate in a sequence whi ch depends on the m agnitude of the fra ctional part of the number q. Th e denominator is the number of all phase coil groups in which the. sequence of major and minor groups is repeated. One s equence made up of d coil groups contains d - c
+
252
Part Two. Energy Conversion by Electrical Machines
minor groups and e major groups. Each pole pitch corresponds to one phase group (with q > 1). In a three-phase winding the total number of phase groups is 6p , so th e sequence rep eats itself 6 pld. times. When b = 0, the winding will consist solely of major groups, with one coil each. The maximum number of circuits (paths) in a phase winding is a max = 2pld. The lowes t possib le number of circuits is such that 2p lad is an integer. .A simple procedure to construct a frac tional-slot, twolayer lap winding is as follows . 1. Det ermine the number of coils in a minor group, b, and 1. in a ma jor group , b 2. Write a series of e numbers: die , 2dle, 3dle, ... ,
+
ed/e
=
d.
Re place each fractional number by the nearest integer number so as to obtain a ser ies of e numbers: N I , N 2' N 3' ... , d. These are the Nos. of ma jor coil groups arranged in the same sequence as the coil groups of all the phases are arranged around the core periphery for one repeat able group . 3. Assign numbers 1, 2, . . "., N I - 1 to the minor coil groups of b coils each. The next N 1 th coil group, a major 1 coils. In a similar way, form the other one, consists of b coil groups, assigning numbers N 2 ' N 3 ' . . • , d to major groups; the rem aining groups will be minor ones. Follow the same sequence in forming the rem aining 6pld - 1 rep eatabl e groups (a three-phase winding is meant) . 4. Distribute t he coil groups am ong the phases. E ach of the phases ta kes every third coil gr oup . Choose the posi tive direct ion s of currents so t hat in ad jacent ) coil groups (belonging t o different ph ases) t hey are in oppo site senses. Connect in series opposition t he adj acent coil groups in each phase. 5. One pa th (or circuit) can be formed from d seri esconnected phase coil groups. 6. Once the coils of phase A hav e been connected (the ph ase start can conveniently be combined with that of the first coil group in the firs t repeatable group), the leads and connections between the paths in phases Band C may be chosen in anyone of several ways, namely: (i) all connections between the coil groups in phase B may be made similar to those j n ph ase A , with a displac ement equal to two coil groups. The n the connections in
+
Ch, 22 Windings for A.C. Machines
253
phase C will repeat those in phase A, with a displacement equal to four coil groups; (ii) all connections between the coil groups in phase B (C) are displaced from the connections in phase A by one basic pattern (that is, by d coil groups) if d is even, or by two basic patterns (that is, by 2d coil groups) if d is odd. In phase C (B), the connections should be displaced from those in phase B (C) again by as many coil groups. In case (i), the phases are identical only as regards the production of the magnetic field; the sequence in which the minor and major coil groups are connected in the phases is different. In case (ii), the phases are nearly identical in regard to both the generation of the magnetic field and the sequence of coil groups. (Phase B can be formed by turning phase A through an appropriate angle.) Example 22-1. Given: Z = 42, 2p = 8, In = 3, q (b = 1, e = 3, d = 4), y = 4, a = 1. The number of coil groups in the basic pattern is
=
13 / 4
3d = 3 X 4 = 12
The number of coil groups in the basic pattern per phase is d = 4. Each minor group has b = 1 coil. Each major group has b 1 = 2 coils. The major coil groups in one basic pattern are numbered
+
die
=
4/3
=
F/ 3 , 2dle = 2 2/ 3" 3dle
=
4
Rounding them off to the nearest integers, we get: 2, 3, 4. The number of alternations in the entire winding is 6pld = 6 X 4 --;- 4
= 6
The manner in which the coil groups are distributed along the core periphery is as follows (the numeral indicates the number of coils in a coil group; the vertical spaces separate the basic patterns): 122212221222122212221222 AZBX
CYAZ
BXCY
AZBX
CYAZ
BXCY
The letters A, Band C mark the forward coil groups, and the letters X, Y, and Z, the backward coil groups in the respec-
254
Part Two. En ergy Conversion by Electrical Machin es
tive phases. The starts and finishes of the first basic phase patterns, labelled as advised in (ii) above, are underscored once ; the starts and finishes of the second basic phase patterns are underscored twice (the star t of a basic pattern
Fig. 22-10 Fractional-slot , double-layer lap winding = 42, p = 4, In = 3, y = 4, a = 1, q = 1 3 / 4 )
(Z
is the start of a forward coil group, and the finish of a basic pattern is the finish of a backward coil group) . Schematically, the resultant winding is shown in Fig. 22-10. As is seen, the pattern of the winding (and that of the fie ld established by t he currents) per iodically repe ats itself, its "period" being equal t o the length of the basic pattern which spans 2p/k pole pitches or an angle 2n/k, where k is the greatest common divisor for Z and p . In our example, Z = 42, p = 4, and k = 2. Therefore, the
255
Gh. 22 Windings for A.C. Machines
repeatable group repeats itself every 2plk = 2 X 412 = 4 pole pitches, 2pr;lk = Zpmqlk = 2 X 4 X 3 X 131 4 -;- 2 = 21 slot pitches, or ever y 2nlk = 180°. In Fig. 22-10, the pattern of currents and coils in the belt extending from slot 1 to slot 22 is fully repeated in the belt extending from slot 22 to slot 42. In a frac tional-slot winding, the pattern of currents (and of the resultant field) is especiall y clearly seen to be recurring about every two pole pitches, 2r;, or in an angle equal to "(p = 2nlp. For example, in phase A the groups of coil sides carrying currents flowing in alternate directions recur in about an angle "(p12 = (2nlp)/2 = (2n /4)/2 = 45°, as is shown in the figure . The group of coil sides carrying currents flowing in the same direction recur in the angle "(po As is seen, these groups do not con tain an exactly same number of coil sid es (three sides ar e laid in slots 41,42, and 1, three sides in slots 5,6, an d 7, four sides in slots 10,11,12, four sides in slots 15,16, and 17, etc. (In the general case, the number of sides is either 2 (b 1) or 2b 1.) Yet, the pattern of currents is periodic enough for the production of a magnetic field with the desired number of pole pairs p. No two-pole mod el can be built for a fractional-slot winding. A model representing a complete cycle of change in the pattern must contain 2p' = 2plk pole pitches, that is , as man y as is occupied by the basic pattern.
-+
22-8
-+
Field W indings
The function of a field winding is to set up a heteropolar exciting magnetic field. It is a single-phase, heteropolar winding energized by direct current. The two basic designs for t his winding have been examined in Sec. 19-2. Figure 19-2c shows the arrangement of a concentrated field winding used on salient-pole cores. It is fabricated in the same manner as a single-phase, two-layer winding, but has only one coil per group, so t hat the number of coils per pole per phase is q = 1. Also, this is a fullpitch winding (Yc = r), and its coil sides are laid in slots next to each other, taking up a half slot width each (see Fig. 19-2c). The arrangement of a concentrated, two-layer field winding laid in slo ts 1 through 8 between poles is shown
256
Part Two. Energy Conversion by Electric al Machines
in Fig. 22-11. The construction of a rotor carrying this type of winding is discussed in Sec. 51-3. In a concentrated single-circuit fi eld winding the number of series-connected turns is io = 2pll'c' where Wc is t he number of t ur ns per coil.
3
Fig. 22-11 Concentrated field winding (p = 4, .. = y , q = 2)
Fig. 22-12 Distributed single-laye r
field winding (p
=
1, q
=
6)
An alternate design of the fi eld win ding is shown in Fig. 19-2d. This is a distributed fi eld wind ing, and it is used on round (cylindrical ) cores. In con trast to a concentrated field winding which t ak es up only one slot within each pole pitch (q = 1), this winding is laid in s > 1 slo ts per
Ch. 23 Calculation of Magnetic Field
257
pole, and each slot receives only one coil side. Therefore, it may be treated as a single-layer winding. As a rule, the winding has q slots per pole per phase. Accordingly, within each pole pitch there are q/2 concentrically arranged coils. As is seen from Fig. 22-12, the coilsaof a distributed field winding differ in pitch. For design purposes the coil pitch in such a winding is equal to the pole pitch, Yc = ... To make the magnetic field set up by the winding as nearly sinusoidal as practicable, the slots carrying conductors occupy 2/3 of a pole pitch. For example, the winding in Fig. 22-12 is laid out in 12 slots. The construction of the rotor carrying a distributed field winding is examined in Sec. 51-4. In a single-circuit, single-layer distributed field winding, the number of series-connected turns is w = pqw c ' where We is the turns per coil.
23
Calculation of the Magnetic Field in an Electrical Machine
23-1
The Statement of the Problem
Energy conversion in an electrical machine operating by electromagnetic induction is based on its magnetic field. Therefore, the calculation of the magnetic field established by the currents flowing in the machine's windings is a major problem in the theory of electrical machines. In the general case, the problem reduces to finding the magnetic induction (magnetic flux density) B from the specified density current J in the windings of the machine (Fig. 23-1), and it can be solved by the theory of the electromagnetic field. The magnetic field strength (magnetic intensity) vector must satisfy Maxwell's first equation curl H = J
(23-1)
the equation connecting the magnetic induction and the magnetic field strength (23-2) 17-0169 ·
258
Part Two. Energy Conversion by Electrical Machin es
where ~L a is the absolute perme ability of the medium, and the continuity equation div B = 0 (23-3) implyin g that t he lines of magn etic flux ar e always closed. In most cases, the current density vector J is uniformly distribut ed over th e cross-sectional area Q of a conductor J = I/Q
and points along the axis of the conduct or in the dir ection where the current I is flowi ng (see Fi g. 23-1).
Fig. 23-1 Produc ti on of th e magneti c field
Ordinarily, the winding conductors are laid in slots on the stator and rotor cores, and the magnetic field exists in a space taken up by the two cores, in the nonmagnetic gap separat ing them , and around the coil ends (or overhangs) (Fig. 23-2). In many cases, this field even threads the magnetic and conducting structural parts of the machine (the shaft, frame, end-shields, and so on). To be able to calculate the magnetic field , the general field equations (23-1) through (23-3) should be extended to include the equations fik
(x, y, z) = 0
(23-4)
describing the surfaces separating the media i and If. differing in relative permeabilit y ,·· /-Ir. i =1= ~r. l, (above all , the
259
Ch. 23 Calculation of Magnetic Field
equations describing the surfaces bounding the cores), and also boundary conditions for the tangential and normal components of the magnetic Iield vectors at the surfaces
y
x
Fig. 23-2 Magnetic field set up by a basic coil set
separating the ith medium from the kth medium (23-5)
Ht;i = Ht,k En,i
where
=
/-Lr,i Hn, i =
/-Lr,h H
ll
,l<
=
En,h
tangential components of the magnetic field strength on the boundary En, i, En , l< = normal components of the magnetic flux density at the same points on the houndary. In cases where the permeability of the cores, /-Li',C' cannot be deemed infinitely large in comparison with that of the areas taken up by air, insulating materials , and winding conductors, it is essential to take into account the nonlinear magnetic properties of the ferromagnetic materials, that is, the dependence of the relative permeability on the magne17*
Ht,i' Ht,l< =
260
Part Two. Energy Conversion by Electrical Machines
tic field strength fLr,c =
f (II)
(23-6)
Equations (23-1) through (23-6) uniquely describe the magnetic field in a machine, but they cannot in most cases be solv ed analytically by electromagnetic field theory. This is, above all, because the surfaces bounding the cores and current-carrying conductors are intricate in shape and also because one would have to consider the nonlinear mag- . netic properties of the ferromagnetic materials. Further difficulties arise because the relative position of the cores and current-carrying conductors is changing all the time, and the solution would have to be sought for all the likely positions. 23-2
Assumptions Made in Calculating the M agnetic Field
rn the theory of electrical machines, several simplifying assum pt ions are made so that the magnetic field of a machine and the relevant winding characteristics could be determined analytically. 1. Because the current pattern in the windings repeats itself periodically (see Sec. 22-2), the field pattern is likewise repeated periodically every two poles. Therefore, in calculating the magnetic field of a machine, it will suffice to consider its variations over a pole-pitch angle 'Vp or even over a half of the pole -pitch angle, 'Vp/2 . On an enlarged scale, Fig. 23-3 shows the magnetic field over a half of a pole pitch in the machine shown in Fig. 23-2. 2. It is assumed that the ferromagnetic cores have an infinitely large permeability, fLr,c' in comparison with that of free space . Because at a magnetic induction of 1.5 to 2.0 T the relative permeability of the core is several tens or hundreds , this assumption does not introduce any appreciable error in the calculation of the field. Also, one may allow for the finite value of fLa,c and of the reluctance of the ferromagnetic parts of the magnetic circuit at a later stage, in practical calculations. 3. Once the assumption in (2) above is made, we may use the principle of superposition and treat the magnetic field of the machine as the sum of the fields set up by each of its windings. In turn, the field due to a winding may be treated
Ch, 23 Calculat ion of Magne tic Field
261
as the sum of the fields established by basic or repeatable coil sets* . The term "basic coil set" refers to a set of 2p coils un iformly distributed all t he way round the circle , disp laced from one another by a half pole pitch , and in terconn ect ed so as to form a per iodic current pattern with p pole-pairs.
Fig . 23-3 Magneti c field from Fig. 23-2 show n enlarged withi n a polo pitch
In Figs . 22-3, 22-5, and 22-7, such a ba sic coil set is shown by heavy lines. In these figures, the to p coil sides are laid in slots 1 , 10, 19, and 28. Referring to these figures, it is an easy matter to see t hat any winding may be decomposed into a multiplicity of basic coil sets. A winding phase is formed by q basic coil sets displaced from one another in space by one tooth pitch . The entire winding has mq such basi c coil sets .
* We
consider only integral-slot win dings (see Sec . 22-2).
262
Part Two. Energy Conversion by Electrical Machines
To determine the total field, it will suffice to calculate the magnetic field due to one basic coil set carrying a unit current, I = 1, to find the fields of all the basic coil sets by scaling up the field due to a unit current, and to combine these fields subject to their relative position in space. In this way, the problem of finding the magnetic field set up by the currents in all the windings of a machine reduces to calculating the magnetic field established by a basic coil set repeated periodically in the winding structure, assuming that the cores have a relative permeability of infinity. 23-3
The Spatial Pattern of the Magnetic Field Set Up by a Polyphase Winding
An idea about the spatial pattern of the magnetic field established by a polyphase winding can be gleaned from reference to the field established by the basic coil set (see Fig. 23-2). This is a four-pole field. Each two-pole interval contains one short-pitched coil. In moving through a half the polepitch angle, ,,?p/2, the field pattern repeats itself, but with the signs reversed. Assuming that the permeability of the cores is infinitely large, the magnetic field within the cores need not be considered because its energy is zero (in Fig. 23-3 it is shown by dashed lines). In the nonmagnetic areas, the magnetic field may be decomposed into three components, namely: (i) the field in the air gap (ii) the field in the wound slots (iii) the field around the coil ends (overhangs). In machine design, the most important factor is the air gap field , that is, the field in the clearance between the cores. In terms of energy, this field exceeds the other flux by a wide margin , which is why we shall give it most of the treatment in our further discussion. In Fig. 23-3, the flux lines in the air gap are shown by solid and heavier lines. The distinctions of this field may be summed up as follows. Firstly, within the core length l its lines lie in planes at right angles to the z-axis, and the flux pattern repeats itself in each of these planes, so we may call it a planar (or twodimensional) field. Secondly, all flux lines cross the air gap and determine the flux linkage and mutual inductance between the winding in question and the windings laid on the other core, for which reason we may call it a mutual field.
Ch. 23 Calculation of Magnetic Field
263
Thirdly, no distributed currents exist within the region taken up by this field, so in calculating it we may invoke the concept of a scalar magnetic potential (see Sec. 23-4). On an enlarged scale, the field in a wound slot, that is, one enclosing current-carrying conductors, is shown in Fig. 23-4. Its lines link only with the conductors of the winding in question. They never cross the air gap, nor do they link with the windings laid on the other core. Such field ( :are called leokagejields. The region taken up by the slot leakage field is separated from that occupied by the mutual air-gap field by characteristic field lines 01 and 04 which pass through point o on the surface of the other core. On a closer examination, the slot leakage field is seen to be the sum of a leakage field in the slot (that is, one existing inside the slot as far as line 23), and a leakage field in the tooth, whose lines extend into If the air gap and exist within the region 012340. Within the core length, the o slot leakage field is planar. Fig. 23-4 Enlarged element of Its pattern repeats itself at the magnetic field in Fig. 23-2, any section of the machine and around a current-carrying slots its lines lie in the section planes. The slot leakage field is more difficult to calculate than the air gap field, because of the distributed current existing in the slot region. (The current density J within the cross-section of a coil may be taken constant and directed along the z-axis of the machine.) In the general case, the slot leakage field can be found, using a general description of a magnetostatic field [see Eqs. (23-1) through (23-5)1. Still, although the field calculation is materially simplified because the field is two-dimensional, an analytical solution can only be obtained for some particular cases (say, for rectangular or circular slots). Even then the analytical solution is too unwieldy, and
264
Part Two. Energy Conversion by Electrical Machinos
practical calculations for slots of any shape are based on the approximate solutions deduced by idealizing the field pattern. Such an approximate solution will be considered in connection with the slot leakage inductance (see Sec. 28-7). The coil-end (overhang) field refers to that around the coilend connections and outside the cores. Its lines are closed around the coil ends and form a complex spatial pattern. For the coil-end field to be determined accurately, one would have to use a complete description of the magnetostatic field such as set forth at the beginning of this section. The solution of the problem is complicated by the fact that the field is three-dimensional and the coil overhangs are very complex in shape. Also, the field may to some extent be affected by the ferromagnetic parts of the machine, such as the end shields, frame, shaft, etc. On the other hand, we are free to neglect the angular position of the rotor relative to the stator. The coil-end field is very low in energy. Therefore, the accuracy in calculating the coil-end field and inductance (see Sec. 8-7) need not be very exacting. It is important to note that some lines of the coil-end field link with the coil ends of the windings on the other core and contribute to the mutual inductance between the Windings. Therefore, the coil-end field proper refers only to a fraction of the total coil-end field. The contribution of the coil-end field to the mutual inductance between the windings on different cores is very small (in comparison with the effect produced by the mutual air-gap field) and may be safely ignored. 23-4
Calculation of the Mutual Magnetic Field for a Polyphase Winding
On the assumption made in Sec . 23-2, the mutual field of a polyphase winding is planar (two-dimensional), and its energy is concentrated in the air gap where distributed currents are non-existent. Its strength 11 may be expressed as the gradient of a scalar magnetic potential, qJm = qJ 11 = - grad 'P
(23-7)
On substituting Eq. (23-7) into Eqs. (23-2) and (23-3), we can readily obtain for the scalar magnetic potential a secondorder p.artia~ differential e~uation/ known as the Laplace
Ch. 23 Calculation of Magnetic Field
265
equ ation: (23-8) To determine cp at any point (x, y) in the air gap, we must solve Eq. (23-8) subject to the boundary conditions corresponding to the instantaneous currents in the winding phases and existing on the ferromagnetic surfaces. The boundary conditions are specified by giving the dis tribution of the potential (p on the surfaces . The determination of this distribution is a problem in its own right, and it can be solved unambiguously, if we know the winding circuit and the instantaneous currents in the phases . Obviously, the solution becomes progressively more difficult to obtain as the winding grows more complicated in arrangement. Therefore, it" is advantageous to solve the problem first for the currents in one hasic coil set, and then to find the potential distribution for a polyphase winding by adding together the potentials of all the basic coil sets. With cp found by solving Eq. (23-8), the components of the air gap field are found hy Eq. (23-7): H x = - ocp/ox, 23-5
H
y
= - ocp/fJy
(23-9)
Effective Length of the Core
Figure 23-5 shows the machine of Fig. 23-2 cut lengthwise. One of the cores is divided into several pa ckets of length Z; each, separated by radial cooling ducts of width bd • As is seen from the gap field pattern, the air gap field is nearly uniform and constant in the region taken up by the core packets (in fact, within the cross-section passed through this region the fi eld ma y be regarded as pl anar); is somewhat weakened in the ducts, and gradually collapses on emerging from the core faces and on leaving the air gap. All this has a well-defined effect on the distribution of the radial (normal) components of the air gap flux density, B. To sim plify fur ther cal culations wi thout mistreating energy conversion hy the machine , we may replace the field varying along the length of the machine hy a uniform field with a maximum flux density B m in the packets. In doing so, we also assume that this uniform field exists over the design
266
Part Two. Energy Conversion by Electrical Machin es
or effective core length lo such that +00 cJ)
= .\' B dz = B ml o -00
Hence, +00
lo =
- '1 -
Em
J' B dz -00
It can be shown that (23-10)
where bd = Cd = Co = Co =
CoCd6 (b d/co6)/(5 b d/co6) 1, if ducts are made in the stator (or rotor) only 0.5, if ducts are made in both the stator and rot or .
+
Fig . 23-5 Det ermining the design length of the core
Also, if the ga p is very small (6 ~ b d ), the design or effect ive width of a duet is b;l ~ b d • If the gap is ver y l arge (8 ~ b d ), the design duct wi dth is ba ~ O. It may be added that, as oft en as not , lo stan ds for the axial gap length ,
267
Ch, 24 Mutual Magn etic Fi eld of Phase Winding
24
The Mutual Magnetic Field of a Phase Winding and Its Elem ents
24-1
The Magnetic Field and MMF Due to a Basic Set of Currents
A basic set of currents periodically alternating in dire ction every t wo pole pit ches, 21:, is shown in Fig. 24-1. The currents iw c and -iwc are carried in slots on core 'Cl . A displacement through a pole pitch, 1:, causes the direction of current flow in a slot to reverse. The excited core Cl is separat ed C2
iwc -2-
a
-
/'f2=O
;z:
iw c --2-
Fi g. 24-1 Repeat able. pattern of currents
from the unexcited core C2 by an air gap of wid th tJ . Because it is sma ll in comparison with R , the mean radius of t he air gap , we ma y neglect t he effect of t he curvature and replace the annular air gap by a "develope d" or fl attened- out gap (see Fig. 24-1). A reference point in the developed gap can conveniently be located by t he dis t ance x from the slot ax is, whi ch is connected to the angular coordinate of t he point in the annula r air ga p, y, by a simple relation x = yR To simplify the analysis , it is advan t ageous t o replace t he distribute d slot curren t , as shown in Fig. 24-1, say iwc ·, by an equa l linear current, i s = iw c , concentra t ed at t he axi s and nea l' the bottom of t he slot (Fig. 24-2) .
Part Two. Energy Conversion by Electrical Machines
268
The magnetic potential existing at the boundaries of the air gap, coinciding with the surfaces of the cores C1 and C2 can be found by applying Ampere's circuital law to the loop 1'-2'-2-1-1' which is symmst. II H rical about the slot axis: C2
2
»>
2'
~,
~ Hz dl
=
iio;
Assuming that the permeability of the core is infinite and the magnetic flux1f density B within the core is finite, we may write
1.'1
H = B/[La = 0 so that the magnetic potential experiences no drop within the cores
iwel2
:c
-twe /2 _ .
I
1
2,'
I Hz dl
Fig. 24-2 Mutual field and mmf in and around a wound slot (see Fig. 24-1)
I
=
2
\
tt.ei = 0
(24-1)
l'
Therefore, the circulation of the vector II may be written as the sum of variations in the air gap potential within portions 1'-2' and 2-1:
~
i
2'
HI dl =
1
Hz ell
+ .I HI dl = j
l'
2
Hz ell = iur;
(24-2)
1'2,'21
If we recall that the magnetic field is symmetrical about the slot axis, so that 2'
I
.\ Hz dl = \ Hz dl = iw c /2 l' 3
(24-3)
and set equal to zero the magnetic potential of the unexcited core, (pz = 0, then the magnetic potential of the excited core to the right of the slot axis (with x > 0) will be 2'
(PI ~
(P2
+ J Hz dl = iw,)2 I'
269
Ch , 24 Mutual Magnetic Field of phase Winding
and to the left of the slot axis (at x
<
0),
I
(Pt =
CP2 -
(24-4)
.\ HI dl = - iw c /2 2
It follow s from Eq. (24-4) that the potential of the excited core to the left of t he slot axis differs from t hat to the right by the slot current, iw c ' (The current flowing outwards , that is towards the reader, is t aken to be positive .) The air gap field dep ends on the difference in magnetic potential produced between the surfaces of the two cores by the currents in the respective windings. In the theory of electrical machines, this difference in magnetic potential; equal to t he linear integral of the air gap field intensity or the to tal air gap current, is usually called the magnetomotive [orce, or mm i for short. Choosing as positive for the air gap field and mmf the direction away from the inner core, C1, t owards the outer core, C2, we may define the mmf as
F = CPl - CP2 In our case, C1 is excite d, C2 is unexcited , and the mmf of C1 is F 1 = CPl - CP2 = CPl If C2 were ecxited and C1 unexcited , and CPl the excited core would be
(P 2
(24-5) = 0, so
= 0, the mmf of
F 2 = CPl - CP2 = - CP2 Wi th the boundary conditions given by Eqs. (24-3) and (24-4), the magnetic poten tial in t he air gap can be found by Eq . (23-8). If the potential distribution is known, t he field intensity can be found by Eq . (23-9) . At some distan ce from a slot, however, t he magnetic field strength can be found in a simpler way. As follows from Fig. 24-1 , the lines of the magneti c field are complex in sh ape only near a slo t, wher eas at some distance from the slot, I x I > 5, the field becom es practically uniform ; its lines run normal t o the core surfac e, and it s in tensity is t he sam e at all t he poi nts wi thin the air gap. Choosin g the path of integration (1'-2' or 1-2) t o run along a field line (where the field is uniform) and noting that
Hz = H y = H
=
constant
270
Part Two, Energy Conversion by Electrical Machines
we get 2
Fi=CPmi -CPm2=
~
0
HI dl =
1
JH yd y =H8 0
The magnetic field strength in the gap is
H
= F l /5 = Fllv
where Iv = 1/5 is the permeance of the air gap within a region containing a uniform magnetic fi eld. The magnetic flux density in the air gap is given by (24-6) In an electrical machine with a saturated core , the mmf is a sum of several components each of which balances the magnetic potential difference within a certain por tion of the magnetic circuit. These component mmfs are found by calculation. 24-2
The Effect of Core Saliency. The Carter Coefficient
Figure 24-3 shows t he magnetic field set up by a basic periodically repeatable set of currents, iw c and -iwc , carried in some of the slots of core Cl. The figure shows one slot carry ing iw c (a wound core) and several slots carrying no current (unwound cores). The slot width b; is assumed to be comparable with the gap width 5. Then the field in the region of the unwound slots is markedly reduced, and its strength is substantially smaller than it is in the teeth . The magnetic flux across the air gap can be expressed in terms of the normal component of the air gap field intensit y, fln = H y ' For example, the flux across the area bounded by tooth pitch 3-4 is 4
cD 34 =
~ A34
l-toHn dA = .
Jl-toHyl~ dz 3
where l o is the effective core (or axial gap) length . In many cases, however, one need not know t he exact dis tribution of the normal field component over each tooth pitch . Instead, one may limit oneself to the distribution of
Ch, 24 Mutual Magnetic Field of Phase Winding
271
the mean normal component, H o' which is taken to be such that the magnetic flux across the area bounded by a tooth pitch (unwound) remains unaffected: 4
H0 =
(J)3 4//100tZl {) =
J H y dx/t z
(24-7)
3
A detailed study into the air gap field will show that when the toothed core Cl is replaced by a smooth surface !J
C2
Fi g . 24-3 The effect of core salie ncy on the magneti c fiel d nea r an unwound slot
separated from C2 by a distance 6 0 >6 , the mean normal component of the air gap field, H o' will remain the same as it was with a toothed core, provided 60 = 6k{) (24-R) Here, k () is the Carter coefficient, named the airgap factor in t he USSR . When the actual air gap 0 is multiplied by k (), the product gives the effective gap width, 0 0 , . The air gap factor is given by · .' k"o = tz/(tz .- cso) (24-9)
272
Part Two. Energy Conversion by Electrical Machines
where Cs
= (bs/6)2/(5
+ bs/5)
In cases where both C1 and C2 are toothed, the effect of their teeth can be accounted for by applying the compound air gap factor k 6 = k 6 1k 0 2 (24-10)
where k 01 and k 62 are the air gap factors of C1 and C2, respectively. Each is found by Eq . (24-9), assuming that the other core has a smooth surface: k 01
t z 1/ (t Zl -
= (bS l/6)2/(5
C S1
!c 0 2 CS 2
24-3
·
=
t Z 2/ (t Z 2 -
= (bS 2 /5)2/(5
cs 15)
+b
S l/8)
c s 2 5)
+ bS2/8)
The MMF Due to a Basic Coil Set
A basic coil set is the simplest repeatable element of a phase winding (see Sec . 23-2) . Therefore, prior to determining the magnetic field of a phase, we should find the mmf due to a basic coil set carrying the phase path (circuit) current, ia. The instantaneous current in a phase path (circuit) is given by i a = i/a =
V2 I a cos (rot)
(24-11)
where i is the instantaneous phase current, I is the rms phase current, and I a = IIa is the rms path current. The pattern of currents carried by the coils in a basic set is repeated every two poles. Therefore, it will suffice to consider the mmf and field due to this set over two pol e pitches, as shown in Fig. 24-4. Each pole pitch is seen to contain one coil of the basic set . The instantaneous phase current is assumed to be flowing in the positive direction (that is, from its finish to its start). Its direction at the coil sections is shown in the figure. The mmf due to a basic coil set can be visualized as the sum of the mmfs due to two periodically recurring sets of currents, namely F' due to the currents in the odd-numbered slots (1, 3, 5, etc.), and F" due to the currents in the evennumbered slots (2, 4, 6, etc.). The mmfs due to periodically
273
ClI. 2/1 Mutual Magn et ic Field of Phase Winding
rocurrlng set s of cur rents ha ve been defi ned in Sec. 24-1. Graphi cally, th ey are combined in Fig. 24-4. In side a coil pit ch, Ye, · F = F' F " = + iawe
+
Between coils, F = O. On mo vin g in the positive direction, t he mm f at the slot axis is in crement ed by the slot cur ren t i alV e if th e cur ren t is
f ig . 24-4 MMF du e to a basic coil set
fl owin g towards t he reader , or decrem en t ed by the same amount if the current is flowing away from the reader. The mm f is thus seen to vary peri odically with a period equa l t o two pole pit ches , 2• . Th erefore, a displacement of 2. leaves the mmf with it s origina l sign F (x
+ 2. ) = F
(x)
whereas a disp lacement of • causes it to change sign F (x
+ .) = - F
(x)
(2f [-13)
Assuming that the positive directi on for the fi eld an d the mmf is fr om the exc ited to the u nexcited core and taking 18-01 0 0
274
Part Two. Energy Conversion by Electrical Machin es
as th o origin the axis of t he coil setting up a positive phase mmf when th e cur rent is flowing in t he positive direction,
we ma y write the following equation for the mmf over one pole pitch: F e = ialVc = F em cos wt for -Ye/2 < x < Ye/2
F
=
{
0
for - . /2 for . /2
<
x
>x>
<
-Ye/2 and
(24-14)
Ye /2
where Fern = V2Ia lV c is the peak mmf due:to the basic coil set. Thus, when the basic coil set is carrying a sinusoidally varying current i a , the resultant mmf is a wave stationary 1
t
2.8
I I
-oh -;
x
I
3
7
If.6
L - rL..L:~
-
- --( J
Fig. 24-5 Br ea thi ng mrnf waveform
in space and pulsating at an angular frequency co = 2nf. The position of the wave in space depends on the arrangement of the coils, and the magnitude of the mmf is determined by the value of i a • Figure 24-5 shows the "breathing" mmf waveform during one cycle of change in the current. Equations (24-12) through (24-1 4) completely describe the mmf all the way round the periphery of the air gap having p pol e pairs. For the ma chi ne of Fig. 22-3 in which p = 2 , the dist ri bution of the mmf along t he periphery of the air gap is shown in Fig. 24-6a . The position of an ar bitra ry point in the air gap can be specifi ed by giving eit her the distance x from the origin along the periphery of the air gap, as indicated on the developed ("unfolded") view of the annul ar gap (see Fig . 24-4 and elsewhere), or the angle I' from the origin to the point
..
Gh.
24 rvilltuai Magneti c Ficld of Phase Winding
in question:
'\, = xlR = »sd-cp
275
(24.-15)
where R = 1:pln is t he me an radius of the air gap circumference . Because t he mmf pa ttern is rep eated every two poles, all that is necessary to know about the mutual field in a machine can be gleaned from its two-pole model. This model should retain the winding arrangement, as does the F
Fig. 24-6 Distribution of the mmf along the periph ery of the air gap in (a) a 2p-pole machine and (b) its two-pole model
two-pole model of the winding (see Sec. 22-6), the slot and t ooth dimensions along the periphery of the air gap (bs' t z , 1:, and Yo), and also the radial gap length B and the effective axial gap length lo. The length of the gap circumference in a two-pole model of the machine is 21:. As compared with t he length of the air gap circumference in the prototype machine , it is reduced by a factor of p. Therefore, the radius of t he air gap in the model is likewise 1lp of that of the actual gap , th at is, Rip. In the model, as in the prototype machine, a given point within a pole pitch takes up a position defin ed by t he same distance x from the origin O. The angle ex. specifying the position of t he similar point in the model, called the electrical angle, is p times the mechanical angle in the prototype i 8*
Part Two. Energy Conversion by Eiectricai Machines
machine. Since x = 'VB = o.Rlp
and subject to Eq. (24-15), it follows that a = P'V = (aIr;) rr
(24-16)
In going from a proto type machine to its mod el, the angles between any characteristic machine elements within a pole pitch are multiplied by the same factor: a1' = P'V]J = 2n a y = P'Vy = ynIT:
To sum up, the electrical angle a between any two machine elements within a pole pitch is thought of as the angle be tween the same elements in the two-pole model of the machine in which a cycle of change in the field is completed within an angle of 2n . 24-4
Expansion of the Periodic MMF due to a Basic Coil Set into a Fourier Series. The Pitch Factor
Let ui?: expand t he r" mmf due: to a repeatable coil set into a Fourier series for t = 0, when the current in a parallel path is a positive maximum ia =
Then the mmf . in will be
V 2I a
the coil region F crn =
(-Y c/2
<
x
<
Yc/2)
V 2 I aw c
The mmf waveform for t = 0 is shown in Fig. 24-7. As is seen, the mmf is an even function about the axis passing through the middle of the coil; therefore, the Fourier series will only consist of cosine terms. Also, during the next halfcycle of change t he waveform repeats itself, but with its sign reversed. For this reason , the series can only contain odd harmonics [14]. Figure 24-7 shows the fundamental mmf of peak value F C1 which completes a half-cycle of change in a time equal to 't1 , and the vth harmonic mmf of peak value F c v which completes a half-cycle of change in a time equal to 'tv = xl» ,
Gil. 24 Mu tu al Magn etic Fi eld of Ph aso Winding
277
Th e mmf can be represented as a sum of h armonic te rms 00
Ft=o = ~ F c v m cos (v xlo:) :rt
(24-17)
v= 1
where v = 1 + 2c = 1, 3, 5, 7, etc ., an d c = 0, 1, 2, 3, et c. By compar ing t he arguments of the cosines with Eq. (24-16), it is seen t hat t hey are t he electrica l angles locating
t (Jt)
7:(Ji)
F ig . 24-7 Exp an sion of the 111111f du o to tho basi c coil sot in t o a Fourie r ser ies (t = 0, i a = V 2I a )
the position of the point x multiplied by v (vx h ) :rt = va = a v In other word s, the arguments of the cosines are equal t o the electrical angles a v for the vth h ar monic with a period taken equal t o 2:rt: (vxh) :rt = (xh v ) :rt = a v Subject to t he qu alifica tion s made as rega rds t he mmf, t he coefficients, or the amplitudes, of t he var ious harmonic s are given by
+, / 2 F c vm
= 2T
Jr
Ft=o cos (vxh) :rt dx = _ 4_ F e m k l'v ~
- , /2
where
F t=o = F e m F t=o = 0
for for
- Ye/2 < x < Yc/2 0:12 > I x I > Yc/2
(24-18)
Part Two. En ergy Conversion by Electrical Machines
278
Equation (24-18) contains what is known as the pitch factor, k p , defined by k p 'V = sin (vY c'It/2T:) = sin (va y/2) It characterizes the effect ' that the coil pitch Yc and the chording angle a y have on the peak value of a harmonic mmf . For the fund ament al, t hat is , for v = 1,
(24-20) With a full pitch, that is, when Yc = T:, the pitch factor for the fundamental is equal t o unity kp1 = kp = 1
For the higher harmonics (v kp 'V =
>
1), it may take values
sin (v'It/2) = + 1
(24-21)
The sign of the pitch factor gives the sign of the harmonic mmf at the coil axis where x = 0 (in Fig. 24-7, k p 1 and F c 1m ar e positive , whereas k p 3 and F c 3m are negative) .
Fe; I
/
Ye = ~ t ---,........
-,
Fe l m
\
~~
I'
t
Fig . 24-8 The effect of pitch-shortening (chording) on the mml harmonics
In the li ght of t he for egoing, the pitch factor may be construed as the ratio of the peak value of a harmonic mmf in a given coil to the peak value of t he same harmonic in the case of a full coil pitch , that is, when Yc = T:. It. foll ows from E C[ . (24-19) that the effect of the pi tch factor on the mm f vari es with th e elect rical angle spanned by the coil and the ord er (or number) of the harmonic. This effect is a maximum for the fundam ental whose peak
Gh. 24 Mutual Magnetic Fi eld of Phase Winding
279
value is F c 1m
= 4Fc mk p 1/n
(24-22)
The higher harmonic mmfs have substantially lower peak values (their absolute values are meant)
I r-; III r-: I = I k p v Ilv I k p 1 I If we choose the coil pitch such that
Yc = (v - 1) xl» the vth harmonic mmf will be nonexistent. This can be proved from Eq. (24-19) on recalling that v is an odd number and, as a consequence, v - 1 is always an even number. With the coil pitch thus chosen, the pitch factor for the vth harmonic will be zero k . ' (v-1):rt nv = SIn 'V'L v2,; : =sinkn=O 0.2 where k is an integer. As an example, when v-1
-O.~
Yc=--'L =2'L/3 v
the third harmonic mmf will, as is seen in Fig. 24-8, be nonexistent (F c 3 m = If,P3 = 0). -1.0 . For a better performance of Fig. 24-9 Plots of kpv as a the machine, it is desired that function of Yc the mmf should be sinusoidally or cosinusoidally distributed in space. Therefore, the pitch factor should preferably be chosen such that the higher harmonic mmfs are minimized. This cannot, however, be done for all the higher harmonic mmfs at the same time, because in order t o eliminate any particular harmonic, the pitch factor must have a particular value. The best that can be done is to strike a balance by choosing Yc ranging between 0.82'L and 0.85'L. Then, as is seen from Fig. 24-9, the fundamental mmf will remain about the same as with a full pitch (k p 1 = 0.96 to 0.98), whereas the fifth and seventh harmonics will be substanti-0.6
-0.8
280
Par t Two. En ergy Conversion by Ele ctri cal Machin es
ally att enua t ed (k p 5 = 0.16 to 0.35, and k p 7 = 0.35 t o 0.08). Unfortunately, t he third harmonic still retains a mark ed value, but it can be eliminated from the resultant mmf by other means (see Sec. 25-4). As reg ards the st ill hi gher harmonics (the 11th, 13th, 15th, etc .), they ar e substa nt ially lower in peak value than the fu nd am ental. E ven wit h high values of the pitch factor, c1ll 7• F e l m -- I.F 7/ F CVm _4F - - - /i·pv ~ 1 e m ,e P I :rt :n:V
24- 5
(24.-23)
The Phase MMF. The Distribution Factor
With an arbitrary number q of coils per group , a phase of a winding ma y be im agined consisting of q basic coil sets (elementary phases) in which there is only one coil per pole pitch (see Sec. 24-3). Such a basic coil set (an elementary pha se) has q = 1. For example , phase A in the winding of Fig. 22-3 consists of q = 3 basic coil sets (elem entary phases), namely: t he basic set of coils 1, 10 , 19, and 28 (shown by heavy lines), the basic set of coils 2, 11 , 20, and 29, and th e basi c set of coils 3, 12, 21, an d 30. Therefore , with an arbitra ry q, the ph ase mmf can be found as the sum of the mmfs due t o t he var ious ba sic coil sets. . For this sum to be taken analytically, it is convenient first t o find the various harmonic compo nents of the ph ase mmf as the sum s of t he respect iv e harmonic compo nents of mmfs due to the basic coil sets . Let us take this sum, beginning with the fun damental com ponent, t hen for the hi gher order harmonic components of the phase mmf on the assumpt ion , as before, that the phase curr ent is a maximum ia =
l!2Ja
Th e fundam ental mmfs for q basic coil sets with peak values F cI m are sh own in Fi g. 24.-10 . Th e b asic coil set labelled "1" is made up of the firs t coil s within each pole pitch. The basic coil set numbered "2" consists of the second coil s, and so on. Neighbouring basi c coil sets are displaced from one an other by a tooth . (or slot) pitch t z along the periphery of the air gap , and t he fu ndam ental mm fs in the elementary phases are displaced from one another by the electrical slot (or t oot h) angle a z = t z:rth: = (2:rttZ) /p .
Gh. 24 Mutual Magn etic Fi eld of Phase Winding
281
To sim pli fy the matters, the figure shows full-pitch (unchorded) coils. In t aking the sum of the fundam ental components of mmf set up by the basic coil sets, it should be rem emb ered that they are displaced from each other by the tooth angle a z - and their axes 1 , 2, 3, and 4 passed through Phase a xis
Phase-axis 1 2 s Ii(h) (g)
(+)
'1:(:n:)
Fig . 24-10 The funda mental mm fs due to t he basic coil sels making up a ph ase with q = 4
the peaks of the cosinusoidally distributed mmf are displa ced by the angles a 0 1 , a 0 2 ' 0.:0 3 = aO It and ao~ = a oq from the phase axis (the latter being the axis of symmetry of the coil group within a given pole pitch) . The electrical angle between the axis of the nth basic coil set and the phase axis is that is,
a OIt
= az (n - 1) - a z (q - 1)/2 a Ol
a 02
= - 3a z /2 = -o.: z/2
0.:0 3
a o~
= =
(24-24)
a z /2 3a z/2
Th e pha se mmf equal to t he sum of the mmfs due to the basic coil sets is cosinusoidally distributed over a pole pitch F = F ph 1m cos a The phase mmf has a peak value, F phlm , at the phase axis
282
Part Two. Energy Conversion by Electrical Machin es
and can he found as the sum of the mmfs produced hy the various basic coil sets at the phase axis: q
F phlm = ~ F elm cos CX On
(24-25)
n=1
In developing an analytical expression for the peak value of the phase mmf, it is convenient to write the mmfs due to the basic coil sets as complex amplitudes
Fnm =
F e 1 m exp (jcxon) (24-26) shown for n = 1, 2, 3, 4, ... , q in Fig. 24-10. If we align the phase axis with the real axis of the complex plane, the mmf due to the nth basic coil set at the phase axis will he equal to the real part of the complex amplitude, F nm : F e 1 m cos
CX on
because
= Re {F nm}
(24-27)
+
exp (jcx on) = cos CX on j sin CX On Hence the peak value of the phase mmf will he q.
F p h 1m= ~ Re{Fnm}=Re n=l
q
r 2J
n=l
•
•
Fnml =Re{F p h 1m }
As is seen, the complex amplitude of the phase mmf •
q.
Fphlm=liFnm n=l
is the phasor sum of the component mmfs due to the basic coil sets. In Fig. 24-11, their sum is taken on a reduced scale. Noting that the polygon formed hy the complex amplitudes of mmfs being summed can he inscribed in a circle of radius OA = OB F e 1m/2 sin (cxz/2) we can find the peak value of the phase mmf, F p h i m- from the right-angled triangle ODA
+
F p h 1m = 2 (OA) sin (qcxz/2) = qF elmkdl = 2
yex Iwk o1/np
(24-28) where I = ala = rms phase ;~current w = 2pweq/a = turns per phase path kW 1 = kp1k d 1 = winding factor for -the fundamental component of mmf.
283
Ch. 24 Mutual Magn etic Field of Phase Winding
E quation (24-28) contains wh at is known as the di stribution factor for the fundamental component of phase mmf: k d1
sin (qaz/2)/q sin (az/2)
=
(24-29)
It is t he ratio of the peak value of the fundamental mmf of a phase to the arithmet ic sum of the peak fundamental mmfs Phase axi s A
Phase asis (+)
o -)
0 000 f Z J If
r(Ji)
Fig . 24-11 Combi nin g th e mmfs due to the basic coil sets in Fig. 24-10, ma king up a phase
clue t o the basic coil sets in that phase k d1
=
F P lil m/ q F c 1m
In finding the peak value of the vth harmonic component of the phase mmf, it sh oul d be rememb ered t ha t the respecti ve angles, a z v , are v t imes as great as for t he funclamen tal: (24-30) Th erefore, the peak value of the v th ha rmonic of the phase mmf will be (24-3'1)
where si II (i'qa z /2)
k cl\, = qsin (v IXz/2) (24-32) is known as t he di stri bution factor for the vt h harm oni c.
Part Two. En ergy Conversion by Electrical Machines
284
In an m-phase symmetrical winding, a coil group spans 1/mth fraction of a pole pitch or the electrica l angle sclm. = qaz determined for the fund amental. Therefore , k = s in (v n /2m) (24-33) dv
q si n (v n /2mq)
Thus, for a three-phase winding, where m. = 3, k d v = sin (vJt/6)/q sin (vJt/6q) Using Eq s. (24-31) and (24-33) , it is an easy ma tter t o trace how an inc rease in q, the number of coils per group,
t.o 0.0
J>=f
0.6
J S 7 9
IJ =1
ff
.3
ts
S
7 8 ff
ts
Fig . 24-12 Diagrams of in g)
k dv
as a fun cti on of q (for a t hree-phase wind-
can affect the phase mmf waveform . Fi gur e 24-12 gives the distribution factors for the fund amental and higher-order harmonic components of phase mmf for several values of q. As is seen, when q = 1, all distribution factors are unity. As q is increased , k d 1 decreases insignificantly (when q = 2, it is 0.969, and when q = 00 , k d 1 = 0.9 55). In contrast , the distribution factors for t he higher harmonics go down abruptly as q is increased, so that wh en q = 00 , the distribution factors for the triplen harmonics becom e I k d v I = 2k d / v
6h. 24 ~iutual Magnetic Field of Phase Winding
285
and for all the other harmonics,
I kd'V I =
kdl/V
As is seen, even with moderate values of q (say, 3 or 4), the distribution factors are about the same as they are for q = 00. The only exception is the so-called slot harmonics (slot ripple) whose order (number) is given by v = kZ/p + 1 = 2mqk + 1 (24-34) where k is any integer. For them, the distribution factor is equal to that for the fundamental component, kd'V = k d l *. For example, when q = 2, this property is manifested by the harmonics of order v = 2mqk + 1 = 2 X 3 X 2k + 1 = 11, 13, 23, and 25. This can readily be verified by reference to Fig. 24-12 where the distribution factors for slot harmonics are shown shaded. As q goes up, the higher harmonic components contribute progressively less to the phase mmf (the only exception being the slot harmonics). Importantly, in the phase mmf their effect is less noticeable than in the mmf due to a basic coil set [see Eqs. (24-29) and (24-31)1,
In the limit, for a uniformly distributed winding (with v other than a multiple of three, and also for other than slot harmonics) Because in the phase mmf the slut harmonics are present to the same extent as in the mmf due to the basic coil set [see Eqs. (24-26) and (24-31)],
To minimize their effect, it will be a good plan to avoid t he values of q that are less than three. However, already at q = 3 the order of slot harmonics, v
=
2mq
+ 1= 2
X 3 X 3
+ 1 = 17 or 19
* For slot harmonics (slot ripple), the pitch factor, too, is the same as for the fundamental, that is 1cp 'V = k p i-
286
Part Two. Energy Conversion by Electrical Machin es
is so high that even with k d v = k d 1 and k p v = k p 1 thes e harmonics are only slightly present in the phase mmf F ph17m
=
F phlm/17
F ph19m = F Phlm/19 With an appropriately chosen va lue of Ye and a sufficiently large number of coils per pole pel' phase , the phase mmf can be made sinusoidal very nearly. When the degree of chording (short-pitching) is taken equal to its recommended value, yeh: ~ 0.8, the phase mmf may contain a fairly noticeable third harmonic . This is, however, of minor importance Phase axis
Fig. 24-13 The pha se mmf of a three-ph ase winding (m = 3, q = 4, !felT:
=
0.83 5)
.
because the resultant mmf of a three-phase winding contains no third harmonic. Figure 24-13 shows the phase mmf and its harmonics for a three-phase winding with q = 4 and yeh: = 0.835. Using Eq. (211-31), the peak value of the phase mmf and of its harmonics may be expressed in terms of the peak value of the coil-side current , 11 "21aWe, that is , as
F p h 11l = q 11 "2 I aWe (24-35)
281
CII . 24 Mutual Magnetic Field of Phase Winding
The peak values of the mmfs found for the condition s specified in Fig. 24-13 are as follows: F ph m
=
4"V2" I aWe
F phlm = 4.741/ 2" Iaw e
F ph 3m = - 0.80S 1/2 t,»; F ph 5m = +0.OS1/2 I a we F ph 7m = -0 .0281/2 I aw e 24·6
Pulsating Harmonics of the Phase MMF
In the previous section, we have seen how the phase mmf is distributed in space at t ime t = 0, when the ph ase current is a maximum, i = V"2 I . Because the phase current varies cosinusoidally Phase axis
i
"1/ 2" I
cos rot
it is clear that at any other point x (a) in the air gap the
phase mmf will be proportional to the inst ant aneous phase current . Obviously the spa t ia l distribution pat t ern of the phase mmf will be the same as at t = a (see ab ove) . The solid line in Fig . 24-14 shows FIg . 24-14 Ripp le in [the funthe fundamental component damental component of the of the phase mmf at time t = phas e mmf = O. For t ime t l , it is shown by a dashed line. Spatial variations in the fundamental component of the phase mmf can be described by the following equation
F (a,t) = F (0, t) cos a = F phlm cos rot cos a
(24-36)
Here, a = xsd-: [see Eq . (24-16)] is the electrical angle defining the position of a given point relative to the phase axis, F (0, t) is the mmf on the phase axis at a = a and at time t : F (0, t) = F phlm cos rot
288
Part Two. Energy Conversion by Electrical Machines
Equation (2/1-36) is the equation of a pulsating wave; it enables us to de termine the fundamental cornponen t of the mmf at any point along the ail' gap and at any time . For the vth harmonic of the mmf, this equation is written similarly F (a, t) = F phvm cos wl cos a v (24-37) where a v = xnIr: v , The axis of the pulsating mmf remains stationary in space and coincides with the phase axis (see Fig. 25-3).
25
The Mutual Magnetic Field of a Polyphase Winding
25-1
Presentation of the Pulsating Harmonics of the Phase MMF as the Sum of Rotating MMFs
The mutual magnetic field of an In-phase winding is produced by the sum of the phase mmfs . The pulsating harmonics of the phase mmfs can be presented as the sum of revolving mmf waves . If we write the product of cosines in Eq . (24-36) as the sum of cosines, we get F (a, t) = 1/ 2F ph1m cos (wt - a) 1/ 2 F p h 1m COS (wt a) = F~I11m cos (wt - a) + F~h 1m COS (wt a) = F' (a, t) Fit (a, t) (25-'1)
+
+
+
+
The first term in Eq. (25-1) is a forward revolving mmf wave , and the second term is a backward revolving mml wave. The revolving mmf waves are written with reference to the phase axis which is assumed to be stationary in space. To get insight into the basic properties of these waves, let lis re-write Eq . (25-1) in a rotating system of coordinates. The state of the forward rotating mmf wave relative to i ts axis, which also rotates at angular velocity wand coincides at t ime t = 0 with the phase axis (Fig. 25-'1a), is defined by the angle au = a - wt, and its equation m ay be written as (25-2) F' (a, t) = F~ hl m cos (-au) = F~ 111m cos au
Ch. 25 Mutual Magn etic Field of Polyphase Winding
28[1
At time t, the forward rotating mmf Wave is shown in Fig . 25-1a. From Eq. (25-2) it follows that the forward rotating mmf is a maximum at a o = 0, th at is
F' (a , t)
=
F~h lm
It will remain unchanged at any point displaced by an angle a o from the mmfaxis. In other words, the forward rotating mmf wave remains sta tionary relative t o the mmf Phase axis
Fig. 25-1 Forw ard rotating componen t of the phas e mmf in (a) th e model of a machine and (b) in th e machine its elf for p = 2
axis and rotates together with this axis at an angular velocity CD in the positive direction (which is counterclockwise). At t = 0, the positive m aximum of the mmf wave occurs at the ph ase axis, CDt = 0. Figure 25-1b shows a rotating mmf wave in a four-pole machine. Therefore, all the angles are halved, that is, reduced, by a factor of p, and the angular velocity of the mmf is . Q~
=
Q
= CDlp
(25-3)
Or, in words, the angular velocity of the mmf is 1lp of its electrical angular velocity which is equal to the angular frequency of the phase cur rent . 19-0169
Par t Two. Ene rgy Conversion
Phase axil: (w t - a) __--1--_
MMF(iXLS
Fig. 25-2 Backward rotating mmf in the model of a machine
Phase axis
Ji/2
Fig: 25-3 Pulsat in g harmonic of the . phase mmf as the sum ' of th e forward and backward ro tating uuuf wa ves
by
Eiectrica i MachInes
The rotating mmf completes one cycle of change in the time span equal to the circumferential period of the winding) 21: = 2nR/p The circumferential linear velocit y of the forward rotating mmf wave is u = u~ = QR = 211: (25-4) where 1 is the frequency of the cur rent . The electrical angle is given by ex.
=
yp
= (xh)
rt
where x (y) is the distance from the phase -axis. Reasoning as" abo ve and writing an equation for the backward mmf, F" (cc, t) rotating about its axis which in turn rot at es in direction t he negative (which is clockwise) at a velocity (0 and is displaced from the phase axis by an angle ust, it can be shown that F" (cc, t) is a backward mmf wave which has properties similar to those of the forward mmf wave (Fig. 25-2). To be more specific, both waves have the same peak value F~ h 1m = F~h
1m
.
F ph 1m/ 2
and rotate at angular velocities Q~ = (0/p and Q~ = - (0/p,
Ch, 25 Mutual Magnetic Field of Polyphase Winding
291
Their electrical angular velocities are likewise the same in magnitude and are equal to the angular frequency of the phase current coi = Qip = co co; = Qip = -co At t ime t = 0, both waves are coincident in space with the phase axis (Fig. 25-3a). From that instant on, the forward wave travels in the positive direction, and the backward wave in the negative direction. Figure 25-3b and c shows the positions of the two waves for cot = n/6 and cot = n12, respectively. 25-2
Presentation of Phase MMF Harmonics as Complex Time- Space Functions
The mmf at point a and time t, that is, 1'. (a , t), may be treated as the real part of the sum of some complex timespace functions .
F' (a, t) + F" (a, t) = Re [F~hlm exp (jcot) exp (-ja)] + Re [F~hlm exp (-jcot) exp (-ja)] The complex time-space function
F (a, t)
=
F~hl
=
(25-5)
F~hlm exp (jcot)
describes the forward wave of the phase mmf. The complex time-space function
Fp h1 =
F p h1m
exp (-jcot)
describes the backward wave of the phase mmf. Therefore, Eq. (25-5) may alternatively be written as F (ct, t) = F' (a, t) F" (e, t)
+
=
Re [F~hlm exp (-ja)]
+
Re
LFp hl m exp
(~ja)]
If we plot the complex functions F~hlm and Fp h1m on the space-time plane of a ' two-pole mod el , ·in which the real axis runs along the phase axis in Fig. 25-4, and the imaginary axis is turned through n/2 counterclockwise, we shall see that the angle between the point at the angle ct and F~hlm is (cot - a). Likewise, the angle between the point at the 19*
Part Two. Energy Conversion by Electrical Machin es
angle a and F~Jh1m is (-rot - a). Therefore, as stems from Eq. (25-1), a projection of F~h 1m or 1m on a direction at the ,angle a will, respectively, give the forward mmf, F' (a, t), or the backward mmf, F" (a, t), at the point in question. The complex function F~h 1m rotates in the forward direction (in the direction of positive angles) at angular velocity
n,
Phase axis
~
t
CtJt
_II CtJt _ - - Fphlm
-
CtJ
+j
I --- {2[ COSCtJt
Fig. 25-4 Representation of the phas e mmf on the complex pl ane of a two-pole model oi, whereas the complex function F~h 1m does so in the backward direction at the same angular velocity co. This form of presentation applies when the angle a is reckoned from the phase axis aligned with the real axis of the complex plane, and time is counted from t = 0 when the phase current is a maximum, i =V2I. In the final analysis, however, we are interested in the mmf of a polyphase winding, and it can be found by adding together the mmfs of the individual phases. To tackle this problem, we should learn to write the equation of the mmf for an arbitrary phase whose axis makes an angle 'aph with the real axis of the complex plane (Fig. 25-5), and whose current is given by
i=
V2 I
cos (rot- (PPh)
so that at t = 0, the current is i=
V2" cos ( -
CPPh)
293
Ch. 25 Mutual Magnetic Field of Polyphase Winding
The equation for an arbitrary phase mmf can be written in trigonometric or complex form by analogy with Eq. (25-1) or Eq. (25-5), noting that the angle cat is now replaced by Phase axis
t
Fig. 25-5 Representation of an arbitrary phase mmf with an arbitrary ph ase current on the complex plane of a two-pole model
(wt - (Pph), and the angle a by (a - aph) reckoned from the phase axis,
F (a, t)
=
F~h 1m COS
+
or
[(wt - (Pph) - (a - aph)] F~h 1m COS [-(wt - (Pph) - (a - aph)]
Re {F~h1m exp [j (wt - (PPh)] exp (japh) X exp (-jan Re {F~ll1m exp [-j (wt - CPPh)] exp (japh) X exp (-jan F (a, t) = Re LF~ 111m exp (-ja)]
F (a, t)
=
+
= Re [F~hlm exp (-ja)1
+ Re
LF~ll1m exp (-ja)] (25-6) A plot of an arbitrary phase mmi on the complex plane of the model is shown in Fig. 25-5. The complex function
Fph1m
=
F~ll1m
+ F~ll1m
describes the phase mmf, the complex function
17~h1m I
, .
=
F~ll1m exp [j (wt - (P Ph) exp (japh)] I
, :
I
I
'
I
'
Part Two. Energy Conversion by Electrical Machines
294
describes the forward rotating wave of the phase mmf, and the complex function
describes the backward rotating wave of the phase mmf. 25-3
Time and Space-Time Complex Quantities * and Functions of the Quantities Involved in Operation of a Polyphase Machine
Scalars sinusoidally varying in time (currents, voltages, emfs, and flux linkages) are customarily represented as complex functions whose projections on the time axis give (t)
wt- 25r 3
Axis A (t)
wi
Fig . 25-6 Representation of currents in a three-phase mac hine on the time complex plane (on the left) and on the spac e complex plane (on the right) of a two-pole model
instantaneous va lues of those quantities. For example, on the left of Fig . 25-6, the inst ant aneous value of the phase A current, reduced by a factor of y2, is equal to the project ion of the complex harmonic function of the phase A current, fA = I A exp (jwt), on the time (t) ax is aligned with the rea l ax is of the time comp lex plane or, to state this differently, to the rea l part of the complex current
* Time varying quantities are usually called pha sors . Spatially distributed quantities are true vectors. Frequently, they are plotted to~eLhel' on combined phasor-voctor dia gram .i-- Translator's Ilo ! ~.
Ch. 25 Mutual Magnetic Field of Polyphase Winding
295
function i A /V "2 = Re (TAl = I A cos rot
where I A
rms current in phase A in coniplex notation I A = rms current in phase A Under balanced conditions, the quantities in the other phases of a three-phase machine can be described by complex functions displaced by -2n/3 (for phase B) and -4n/3 (for phase C) from those associated with phase A. For example, the currents in phases Band C are written in complex-function notation as =
I A exp (jO)
=
In
=
j
n exp
t;
=
ie
(jrot)
exp (jrot)
where j n = In exp (-j2n/3) and j e = Ie exp (-j4n/3) are the complex rms currents in phase Band C, respectively. Since the rms phase currents are the same, the magnitudes of the complex currents are likewise the same
IA=In=Ie=I The instantaneous phase currents can be found from the following equations: i A /l/ "2 = Re
rf,d =
Re [j exp (jrot)J
i n/y
[Tnl
Re {j exp [j (rot- 2n/3)]}
2= i e /V2 =
Re
=
(25-7)
Re [Iel = Re {j exp [j (rot- 4n/3)]}
where j = j A' and are each a projection of the respective complex current function on the real axis of the time complex plane (Fig. 25-6). The theory of electrical machines uses another form of representation for the quantities existing in polyphase systems under balanced conditions. More specifically, scalars (currents, voltages, and so on) associated with the various phases are depicted on the space complex plane of a twopole model as a complex function common to all the phases. For the phase currents defined by Eq. (25-7) and shown on the left of Fi g. 25-G , such a complex function has the
296
Part Two. En ergy Conversion by Electrical Machines
form
T= j
exp (jwt)
wh ere J = J A' an d is pl ot t ed on the space com plex plane of a two -pole model as shown on the right of Fig. 25-6. In a two-pole model , t he phase windings are shown each as a coil traversed by a positive current. The axes of ph ases A, B, and C are drawn through the centres of the coil groups represented by a single coil. Because the event s in phase B lag behind those in phase A, the axis of ph ase B is di splaced from that of phase A by an electrical angle equal to 2n /3 in the positive direction (counterclockwise), and that of phase C by an angle equa l to 4n/3 in t he same direction . The instantaneous phase curre nt (reduced by a fac tor of }/ 2) is given by a pr ojection of the complex function I on t he respective phase axis. Because the complex current fu nction I takes up t he same position relative to the axis of a given phase as the complex current function of the same phase re lative to the real ax is of the time com plex plane (on the left of Fig. 25-6), ei ther form of representation gives t he sam e inst an t an eous phase current . To demonstrate, proj ections of t he com plex current function on the respective phase axes on the sp ace complex plane
i A /V'2 = Re [/]
iB/V
'2 = Re
=
Re [j exp (jwt)]
a
exp ( - j2n/3]
= Re {I exp [j (wt - 2n /3)J)
(25-8)
ic/V 2" = Re [/ exp ( - j4 n /3)] = Re {j exp [j (wt - 4n/3)J)
r..
are the same as proj ections of th e complex functi ons and I e on the time axis (see Eq . (25-7) and the plot on the left of Fig . 25-6). Similarly , we can depict on t he sp ace complex pl an e of t he model t he emfs , voltages and flux linkages asso ciated with t he va rious phases . These quantities, too, will be represented by t he re spective compl ex functions common to all th e ph ases . E arlier, we d isc ussed the represen ta tion of spatially di stributed, time-varyin g sca l ars in the form of space-time complex functions on the space complex pl ane of a model. .. . , . .
1 B,
Ch. 25 Mutua l Magnetic Field of Polyphase Winding
297
We did this for the rotating mmf wave which is a scalar quantity sinusoidally varying with time and space. The va lue of the mmf at a given point in the air gap, say, F' (a , t), displaced by an angle a from the origin , was found for each instant of time as a projection of the rotating complex function F~ll1m on the direction at the ang le a (see Figs . 25-4 and 25-5). . Now we have depicted phase sca lar quantities as complex functions on the same space complex plane . In contrast to the comp lex functions depicting spatially distributed sca lars (mmfs, and , as we shall see later , the normal component of the air-gap magnetic flux dens ity), h owever , t he complex functions representing phase quantities can only be projected on the phase axes . Their projections on an arbitrary reference direction have no physical meaning. To stress this difference, the complex functions of spatially distribu ted, time-varying quantities (mmfs and the normal component of the airgap magnetic flux density) will be called time-space complex functions . The comp lex func tions of the phase quantities which only vary with time (currents, voltages, emfs, and flux li nk ages) will be referred to as t ime complex functions. 25-4
The MMF of a Polyphase Winding. Its Rotating Harmonics
Consider a symmetrical m-phase winding . To simplify the matter, let m. be equal to 3. We set out to find the mmf of this three-phase winding as the sum of the mmfs in the individual phases. In doing so, we shall remember that ' the phase axes are disp laced from one another in space by an electrical angle 2n/m = 2n /3, and that the phase currents are disp laced from one another by the same ang le in time. Suppose that the phases carry a balanced set of PPS currents. Such currents are shown in Fig. 25-6 and can be found by Eq. (25-7) or (25-8) . As will be reca lled, in a balanced set of PPS currents, the phase B current lags behind the phase A current by 2n /3, and the index "B" is assigned to the phase whose axis is displaced from that of ax is A by an electrica l ang le a A B = 2n/3 in the positive direction, (see Fig. 25-7). To combine the ph ase mmfs in complex form, the ax is of phase A must be aligned with the real axis of the complex
298
Part Two. Energy Conversion by Electrical Machines
pl ane, and the positive angles cx. must be counted counterclockwise . Resolving the phase mmfs into the forward and backward components and noting the phase shift f[Jph between the currents and the spatial shift cx.ph between the
Axis A
F,m=~~
(+)
Fig . 25-7 Produc tion of a rot at ing mrn f by a three-phase winding carrying PPS currents
phase axes from Eq . (25-6), we obtain the total forward mmf for a t hree-phase winding as
r.; =
Fim =
F~Im
+ FBIm+ Feim
= F~ll1m
+ +
exp [j (rot - 0)] exp (jO) F~ll1m exp [j (rot - 2n/3) ] exp (j2n/3) F~ll1m exp [j (rot - 4n/3)I exp (j4n/3)
= 3F~hIm = 3F~hIm exp (jrot) =
Fi m
exp (jrot)
As is seen, all phase mmfs are identical and are depicted graphically by the same complex function . The peak mmf of an m-phase winding is
Fim = Fim = mF~hlm = mF p h i n/ 2 where F p In m is the peak va lu e of the pulsating phase mmf wave . . Upon suitable substitution s [see Eq. (24-28)1, we get
Flm= Fl m =
(rn. 11
2/n) (Iwk p1k ,lj/p)
(25-9)
Ch. 25 Mutual Magnetic Field of Polyphase Winding
299
The backward phase mmf waves sum to zero [see Fig . 25-7 and Eq . (25-6)1
---
~
F;m = FAl m
=
+ F1JI m+ FeInt
F~ll1m
....."
exp [- j (rot - 0)1 exp (j0) [-j (rot - 2:rt/3)1 exp (j2:rt/3) [- j (rot - 4:rt/3)l exp (j4:rt/3)
+ F~ll1m exp + F~hlnt exp
= 0 To sum up , the fundamental mmf of a three-phase (or, generally, a polyphase) winding carrying a set of PPS currents is the forward rotating mmf with the peak value given by Eq . (25-9). It rotates at an electrical angular velocity ro and a mechanical angular velocity Q l = ro/p in the positive direction (counter-clockwise). On the space complex plane, this mmf runs in the same direction as the complex functi on 1 depicting the PPS phase currents (see Fig. 25-7). Recalling that t his mmf is proportional to current , F l m = kF I, we may re-write Eq, (25-9) in complex form as
r.;
where k»
= kF1
= m V2 wkt/:rtp
The distribution of the fundamental mmf set up by the PPS currents along the periphery of the air gap [see Eqs. (25-'1) and (25-5)] can be descr ibed by an equation of the form F (ex, t) = F l m cos (rot - ex) = Re [J,\m exp (-jex)] (25-'10) where ex = yp = xsd -t = the electrical angle defining the position of a given point along the periphery of the air gap y = the mechanical angle from the origin (from the axis of the main phase A carrying a current i A = V 2/ X cos rot) to the point in question x = the distance along the periphery of the air gap from the axis of the main phase to the point of int erest L = the pole pitch for the fun damental component p = the number of pole pairs along the peri phery of air gap for the fundamenta l component
300
Part Two. Energy Conversion by Electrical Machines
If a polyphase winding carries NPS currents, the backward phase mmfs will be represented by the same complex function, and the forward mmfs will cancel out . This results in a backward rotating mmf which can be described by an equation of the form F (a, t) = F 1 m cos (-wt - a) = Re [F 1 m exp (-jwt) exp (-ja)] (25-1'1)
Acting in a similar way, we can combine th e higher harmonic components of the forward and backward phase mmfs . In combining, either the forward or the backward vth harmonic waves, or both, may cancel out . Because of this, the resultant mmf may only contain some of the harmonics whose order is given by v
= 2mc + 1
where c = 0, 1, 2, 3, .. . For a three-phase winding, the order of the resultant rotating mmf waves will he v = 1, 5, 7, 11, 13, etc. The vth harmonic component of the resultant rotating field will rotate in the forward direction (clockwise), if the sign is adopted in finding the order of the harmonic by the above equation, and in the backward direction, if the "-" sign is adopted . The peak value of the rotating vth harmonic mmf can be found by an equation similar to Eq. (25-11)
"+"
F
_
m l(Z Iwkdvkpv :nvp
vm -
(25-12)
where vp = Pv is the number of pole pairs for the vth harmonic component. The mechanical angular velocity of the vth harmonic mmf is given by (25-'13) Q" = «lp ; = « lvp The angular velocity Wv
=
Q"pv
= w
(25-14)
defined as the product of the mechanical speed by the number of pole pairs for the vth harmonic field component (which is also true of the electrical angle for the vth harmonic, a ; = ,\,p,,) may be looked upon as the electrical angular velocity of the vth harmonic mmf . It is the velocity at which
Ch. 25 Mutual Magnetic Field of Polyphase Wind ing
30'1
the complex function of the vth harmonic mmf rotates [see Eq. (25-10)] Fv m = F v m exp (+j(j)t) The mechanical angular velocity of the vth harmonic is tlv of that of the fundamental component of the mmf, Qv
=
Qlh
The distribution of the vth harmonic mmf along the periphery of the airgap is descr ibed by an equation set up by analogy with Eq. (25-10) (if the harmonic is rotating in the forward direction) or Eq . (25-11) (if the harmon ic is rotating in the backward direction): F v (cc , t) = F vrn cos (+(j)t - Clv) = Re [F vm exp (+j(j)t) exp ( - j ClvH (25-15)
+"
sign applies when the harmonics are rotating where the" in the positive (forward) direction, and the "-" sign, when the harmonics are rotating in the negative (backward) direction. Clv
= Pv"? = vp,,? = (xh v ) n = vastl :
is the electrical angle defining the position of a given point in the fie ld set up by the vth harmonic component Pv = pv
is the number of pole pairs along the periphery of the airgap for the vth harmonic component, and
"v =
xl»
is the pole pitch for the vth harmonic component. As a ru le, the vth harmonic is small in peak value, because the winding factor kdvk p v is only a few hundredths of unity, whereas for the fundamental component it is close to unity. Also, many harmonic components cancel out (for example, this is true of the triplen harmonics in the case of a threephase winding, that is, those whose order is 3, 9, 15, etc .). Therefore, with a judicious choice of the coil pitch (Yc = = 0.83,,) and of the coils per group (q;;;;;;: 2), the mmf of a polyphase (three-phase) winding will differ but little from the fundamental mmf, because the higher harmonics it contains are insignificant in their effect. In fact, it may be treated as the rotating fundamental wave with a peak value
Pa rt Two. Energy Conversion by Electrical Machines
given by Eq. (25-9) and with a mechanical angular velocity Q = Q 1 = « [p, If the winding carries PPS currents and the current in phase A is iA
= V'Z I
cos rot
the peak value of the fundamental mmf at time t will be displaced by a mechanical angle '\' = iatlp from the axis of phase A (or by an electrical angle ex. = rot). The above dis tinctions of the mmf induced in a threephase winding are depicted in Fig. 25-8. The phases of the
2VZlaW c
tz(!XIz)
~I Fig. 25-8 MMF of a three-pha se winding (q = 4, yclr: = 0.835)
winding shown in the figure do not differ from those in Fig. 24-13. From a comparison of the mmf in a three-phase winding with that of anyone phase in the same winding (see Fig. 24-13), it is readily seen that the waveform of the mmf is improved appreciably, and it appears sinusoidal very nearly. The third harmonics, rather pronounced in the phase mmfs (see Fig. 24-13), cancel one another upon com-
eh. 25
Mutu al Magnetic
Fieid
of Polyphase Winding
30fl
bining. It is to be noted that in Fig. 25-8 the mmf F is found at t = 0, when the current in phase A is a maximum iA
= V2 I a cos rot = V2'I a
and the currents in phases Band C are the same iB =
11 2 I a cos (rot -
ic =
V 2 I a cos (rot -
V 2" I a/2 4n13) = - V 2 I a/2 2n13) = -
As will be recalled, I a = IIa is the current in a parallel path (circuit) of the winding. The waveform of the mmf can be plotted as for the mmf of one phase in Fig. 24-13, if we note that the currents in phases Band C are flowing in the reverse direction relative to the positive direction of the phase currents. The peak value of a harmonic mmf can be expressed.in terms of the peak values of the coil-side currents, using the equation derived from Eq. (24-35): m 2qmk pv k dv FVm=TFphvm=
:n:v
~
-
(l/2I a w e )
(25-16)
By the above equation, it is an easy matter to get
r.; = 7.12 11 2' t,»; r.; = 0.075 V2" Iaw e F =-0.042 V2i,»; 7m
25-5
The Fundamental Component of the Magnetic Flux Density in a Polyphase Winding (The Rotating Field)
As a rule, the pole pitch for the fundamental mmf of a polyphase winding, 't1 = 't, is many times the tooth (or slot) pitch of the cores, t ZI and t Z2: 'tIlt ZI
= qlml
't 1It z 2
»
»
'1
1
Therefore, in calculating the field set up by the fundamental mmf, F (CG, t) = F 1m cos (rot - CG)
304
Part Two. Energy Conv ersion by Electrical Machines
we are in a position to allow for the effect of saliency on the average by using the air gap factor It6=k61k62
from Eq. (24-10) . Then the mean airgap permeance is the same as in the homopolar case and is given by
Ao
11M 6
=
(25-'17)
By definition (see Sec. 24-3), the fundamental radial com. ponent of the gap magnetic flux density can be written as B (cc, t) = floAoF (Ct, t) = B l m cos (wt - «)
(25-18)
where B 1 m = ~loAoF1m is the peak value of the fundamental component of the m agnetic flux density . In a polyphase machine, the fundamental component of the airgap magnetic flux densi ty is a rotating wave travelling at the same mechanical an gular velocity and having the same pole pitch and t he same pole-pitch angle as the fundamental mmf (Fig . 25-9): Q
=
Q1
L
=
L1
I'p
= wlp
= 2nlp
Like the mmf, the magneti c flux density B (Ct, t) can be depicted on a two-pole model (Fig . 25-10) either as a cosinusoidal rotating wave (F ig. 25-10a) or as a space-t ime complex function (25-19) which is in phase with F1 and I. The projection of iJ1 on an arbi tra ry direction at an electrical angle Ct = PI' from the origin in the model (Fig. 25-1 Ob) is equal to the radial component of t he fundamental magnetic flux density at a mechanical angl e I' from the origin in the prototype machine (see Fig. 25-9). The origin is usually taken to be the axis of the main phase (phase A) in which the current is a maximum iA = I a cos wt at t = O.
V2
Ch. 25 Mutual Magnetic Field of Polyp hase Winding
ilOfi
Fig . 25· 9 Rotating magnetic field in t he ai rgap of a polyp hase 2p-po le m achi ne
(a )
Fig. 25-10 Itoprosont ation of th e rotating magnetic fie ld in th o model 01 Fig. 25-9
20- 0169
306
25-6
Part Two. Energy Conversion by -Electrical Machines
Magnetic Flux Density Harmonics in the Rotating Magnetic Field of a Polyphase Wi nding
The magnetic flux density harmonics present in the field set up by a polyphase winding are functions of both the spatial distribution of the mmf in . the winding and the saliency of the cores. Unfortunately, the saliency affects different harmonics differently, and the air gap fac tor used in calculating the fundamental component (see Sec. 25-5) does not permit the higher harmonics to be found wi th sufficient accuracy. This is the reason why the first step in determining the magnetic field set up by a polyphase winding is to construct a stepped waveform for F, the phase mmf. Prior to that, we must calculate the instantaneous-phase currents i A , i B , and i c, the coil-side currents +t~wc/a, +iBwC/a, and +icwc/a, and the slot currents is(k-l), is(k)' and ' is(k+l)' This can readily be done, once the winding circuit is known. The stepped F waveform shown in Fig. 25-8 has been constructed within a pole pitch, 'to In Fig. 25-11, a similar mmf waveform is constructed for a half pole-pit ch , 't/2. In this case, we assign an arbitrary magnetic potential, say, qJi, = 0, to one of the tooth (or slot) pitches, say, k , Then the potential at the next adjacent tooth pitch, k 1, 1)th slot with a cur rent is(It+l) will be following the (k
+
+
qJit+l
=
qJit
+ is(k+l)
(As will be recalled, the current in a slot is assumed to be positive when it is flowing outwards, that is, toward the reader.) Once the potentials in the tooth pitches lying between two adjacent poles, or within two pole pitches, have been found , t he next step is t o determine the ave rage potential n qJav =
1 ""\:1
II: Li
I
qJ"
h= 1
and the mmf, F , of the pol yphase winding for each of t he t ooth pitches. It is measured over and above the average potential, qJav, and is equal to (PIt-l =
qJk-l -
qJav
!lO'l
Ch. 25 Mutua l Magnetic Field of Polyphase Windin g
in the (k - 1)th tooth pitch, t
'Ph = 'Ph
~
(Pav
in the kth tooth pitch, and (PhH :- (PhH -
!Pav
in the (k + 1)th tooth pitch, and so on. The mmf thus found is shown in Fig. 25-1'1. It ma y be taken equal to t he sum of tooth-pitch mmfs, (Ph (x) . Accord ,"/2
0
[!]
[±]
C C C1 {}"
f)-
ez X
x/<
x /
--I
Fig. 25-11 The magnetic field of a polyphase winding and the specified distribution of instanta neous currents among th e phases
ingly, the magnetic field set up by a polyphase winding may be defined as the sum of the elementary fields estahlished by the tooth-pitch mmfs (Ph (x). The distribution of the mmf for th e kth tooth pitch is shown separately in 20*
/
308
Part: Two, Enr.rgy Conversion bv Elec tr ical Machin es
Fig . 25-12. As is seen, it is rect angular in shape, being (Ph (x) = lPh inside the tooth pitch, and (Ph (x) = 0 outside t.he tooth pitch. This implies that in calculating the field t.he following boundary conditions must. be assum-
Fig . 2fi-t2 The field set up by the mmf of t he kth to oth (slot) pitch
ed : !P = 0 for core 2, !P = !Ph for the kth too th pitch on core 1 1)th and (k - 1)th and all the and !P = 0 for the (k other too th pitches on core 1. The fiel d thus obtained is marked ly affected by the shape of the slots and air gap. To simplify calculations, onl y t.he shape of the excited core, 1 , is accurately reproduced,
+
Ch. 25 Mutual Magnetic Field of Polyphase Win ding
309
because its slots carry the winding in question . The unexcited core , 2, is replaced by a smooth one, and the effect of its saliency on the air gap permeance is accounted for approximately b y introducing an equ ivalent ail' gap
5" = tJk 6 2 [see Eq. (24-10)1. The scalar magnetic potential
The plot of BI< (x) is shown in Fi g. 25-12. Its shape depends on two ratios, namely bs/ 5" and t z /5". Because of this, the waveform of magnetic flux density for any other tooth pitch , say, Bh+ 1 (x) or B k _ 1 (z), is sim ilar in shape to t hat of B k (x). Thei r ordinates, however, are multip lied by
B k (x ) = B k • a v
+ v=1 2.j Bl
where v = 1, 2, 3, .... The peak va lue of t he vth harmon ic in the Fourier series is given by 't'
rt V ,'fh- (I Xk B fJ vm = -1 ~ B 11. (X ) cos 't' 't' -'t"
31~
Part Two. Energy Conversion by Electrical Machines
Unfortunately, the analytical expression for Bh. (x) is so elaborate that the integral can only be evaluated numerically on a digital computer. To avoid cumbersome computat ions in engineering applications, Soroker'" has proposed to express the peak value of the vth harmonic, Bh.vm, in the Fourier expansion of Bh. (x) in terms of the vth harmonic, B h ov m, taken from the Fourier expansion of an idealized rectangular magnetic flux density Bh.o( x) with a peak value Bh.o
=
f!oCJJh. I6"
Owing to the symmetry of the B h O (x) waveform about the centre of the tooth pi tch (Fig. 25-12), its Fourier expansion likewise contains only a constant and cosine terms 00
B
/{o
( X0) = B
u». av
+
"'" LJ B /{Ovm
cos - vx/{lt 1:-
,
v=1
,\,=1, 2, 3, .. . The peak values of the expansion terms can readily be found analytically
, JI'
1 B h Ov m = -:r xh
=-,
B hO () d (I>:.") x cos -VX/llt 1:- X h = f!o u
{Phvm
where
is the peak value of the vth harmonic mmf in the kth tooth pitch, CJJh (x) . The vth harmonics of Ell. (x) and B h O (x) are shown in Fig. 25-12. As is seen, they have the same pole pitches, = .. Iv, but different peak values, E h v m and B h.ovm' The ratio of the values of the magnetic flux density harmonics found with and without allowance for the effect of saliency is termed the slot factor for the vth harmonic
"v
(25-20)
*
Soroker T.G., Electrotechnlcky Obzor, 1972, 10.
311
Ch, 25 .Mutual Magnetic Field of Polyphase Winding
It is the same for all tooth pitches of a given core and solely depends on its relative dimensions and the harmonic number
c;
=
t (bsU),',
bsltz, Z/vp)
where Z = number of teeth on the core p = number of pole pairs for the fundamental component bs = slo t width at the air gap From statistical analysis of the numerica l values of C v found for various relative dimensions, t he following approximate procedure has been proposed for its calculation . The slot fa ctor for the vth harmonic is npv
C; = Dv-Av/tan -
z-
(25-21)
where Il ; and A v are found, subject t o th e r atio bs/6" an d the value of Cv= -
(i) For e,
~
txpv bs - - Z tz
2, Ev A v= 1+51l"/ bs ('1 -
2
S:: "lb s )
' (PmACv U
where
(Pm A
=
0.4.845 -
= 1-
Dv an d
_ 05
crmB -
(ii) For
Cv
.
>
I
T
0.0255 bsW'
+ 0.014.2 (bs/6")2
cr mBe~ (1 -
(PmBc U6)
2 (6"lb)2 3" s -
1
3 (1 +0.08bs /o")
2,
A v = exp (-1.46 c v6"l bs ) sin (0.95 e v D v = exp (-1.4.6 c,,6"l bs ) cos (0.95 e v
-
(PmC)
-
cr mc)
where
(Pm C
=
0 .7484. - 0.05037 bs/6"
+ 0 .001195 (bs/6")2
As an exam pl e, we shall trace the ca lc u la tion of C v for [he cores in Fig. 25-11 or 25-12 :
8 = '1 mm, bs = bS l = 5 mm, bS 2 = 3 .75 mm tz
=
t Z1
=
10 mm , t Z 2
=
7.5, mm, Z
=
24, p
=
J
312
Part Two. Energy Conversion by Electrical Machines
We shall carry out the calculations for the first tooth (slot) harmonic of order v = Zip - '1 = 24/'1 - '1 = 23 The airgap factor for the second core [see Eq . (24-'10)) is k 0 2 = 7.5/(7 .5-'1.607 X '1) = '1.272 The term "'1.607" is given by '\' = (3.75/'1)2/(5 3.75/'1)
=
The equivalent air gap is 6" = 6k 02 = 1 X '1.272
'1.272 mm
+
=
'1.607
The values of the other quantities, as Iouud by l.lie equations given above, are as follows: Cv =
tan (npvIZ) CJJmA
=
'1 .5053 < 2 -0 .13'1G5
= 0.60367
0.28955 A v = 0.43'198
4JmB =
D v = 0.4'1564 C" ~ 3.7'13
Once C; is found, it is an easy matter to determine j he peak value of the vth magnetic flux density harmonic due to the mmf of the kth tooth pitch, (Ph (x), with allowance fOI' the saliency of the core (25-22)
where (Phvm is the peak value of the vth harmonic mmf over the kth tooth pitch. Knowing the spatial distribution of the vth harmonic muif over the kth tooth pitch (Phv
(x) =
(Phvm
cos (vxllnh:)
and using Eq . (25-22), we can readily write an equation for the distribution of the vth harmonic of the magnetic flux density with allowance for saliency B hv (x)
= Bhvmcos
(vxhnh:)
Ch, 25 Mutual Magnetic Fi eld of Pol yphase Winding
313
So that we coul d take the sum of the i ndivid ual magnetic flux densities, we must write the above equation in a coordinate system common to all the loops and having its origin on the axis of phase A (see Figs . 25-1'1 and 25-12):
B k v (x)
=
B h v m cos (x -
Xh O)
(vrr/»)
Here , x - X h O = X k is the distance from t he axis of the kth t oot h to a given point in t he air gap, X h ll is t he dis tan ce from the axis of phase A to the ax is of the kth tooth, and x is the distance from the axis of phase A to the point in question. Noting that xsd-t = a is the electrical angle from the axis of phase A to the point in qu estion for the fundamental com ponent, and X h On h = a hO is t he elect rical angle from the ax is of phase A to t he axis of t he kth tooth , we ma y rewri te t he equat ions for t he spatial disribution of t he v th harmonics of the mmf and magneti c flu x density as (P" v (a) = (Phvm cos v (a - a"o) B h v (a) = B h v m cos v (a - a"o)
(25-23)
Because the currents traversing t he ph ases of the winding have an angula r fr equen cy (1), we may argue that the mmf over t he kth too th pitch vari es intime wi th t he same frequency (P" = (P"m cos (wt - Bk) where (P'lm is the time peak value of the mmf over the Hh tooth pitch , an d ~" is t he time phase of the mmf over th e kth tooth pitch . The peak va lues of t he space harmon ics of the mmf an d magn eti c flux density in the kth tooth pi tch will vary in t ime in t he sa me manner:
B II vm = where CPhvm
=
qJ/ivmm
=
/loC v
~ (Ph vm
(2(p,,/vn) sin (vt zn /2l:) is the peak value of the vth h armonic mmf at time t, and (2(Pkm/v n) sin (vt zn /2l:) is th e peak value of t he vth ha rm onic mm f at th e time when
(P" (t)
=
(Pkm
Noting that in Eq. (25-23) both CP"vm a nd B "vm are fun ctions of time, we may write the following equations for the vth harmonics of the mm f and magnetic flux density at any
314
Part Two . Ene rgy Conversion by Electrical Machines
point in the air gap at an angle a to t he phase axis at any instant of time t: (wt :-- ~ 'l ) cos v (a - a/1o) (2.5-24 ) = (/loCvI8")(Pllvmm cos (cut - ~h) cos v (a - a/1o) On comparing the above equations with Eq .(24-37) , it ca n be seen that they describe pulsat ing waves. The resu ltan t vth harmonic of the magnetic flux dens ity in a polyphase winding is obtained by combining the magnet ic flux densities due to the mmfs over the tooth pitches : z z B ,. (a, t ) = :lj B "v (a , t) = !!6~v 2j CP /1" (a , t) (a , t) B/1 v (a, t) CP/1v
=
CP1lvmm cos
/1 = 1
/1 = 1
The sum of the vth harmonic mmfs of all t he tooth pitches z
2j
CP1lv
(a, t) , is equa l to the vth harmonic mmf of th e
~=I
polyphase winding , F v (a, t ). Earlier (see Sec . 25-4), it has been sh own that if v = 2 me + + 1 (where e = O. 1, 2, 3, ... ), an m-phase winding will generate the vth harmonic mmf as a rotating wave which can be described by Eq . (25-1!1). Therefore , z ~
,,=1
CP';v
(a , t) = F v (a , t)
= F vm cos (+cut - va)
As a conse quence', the vth harm oni c of t he magnetic flux density in a polyphase winding is given by B,. (a , t)
=
B vm cos (+ wt - vee)
(25-25)
where B v m = ~loC vFvn,l8" is the peak value of th e r ot at ing wave of the vth harmonic magne t ic fl ux density in a polyphase wind ing . To sum up, the magnet ic flux dens ity in a polyphase winding , as found with allowance for the effect of slots, contains the harmonies of the sa me order as that of the mmf harmonics. Th e effect of slot s on the peak value of the magnetic flux density harmonics is accoun ted for by th e factor C v calculat ed by Eq . (25-21). The slot factor may be positive or negat iv e . Accordingly, the magnetic flu x dens ity wa ve may be in phase or in antiphase with the mmf wa ve. For harmonics with l arge pol e
Ch. 25 Mutual Magne tic Field of Polyp ha se Winding
315
pitches and sa Lisfy ing the condition Tv = T/V » t Z 1 ' C; ~ tlk o! ' Since for the fundamen tal component this condition is usually satisfied T1
»
tZ
=
T/ m 1q1
th e peak va l ue of the associate d mag netic Ilu x densi t y as given by Eq . (25-25) is
B1
=
F17Il ~tO Cl/ 6" = Flm ~to/ 8kolk 62
which chec ks with Eq. (25-18) . The vth harmoni cs of th e magnetic flux density in the case of a rotating fie ld have the same number of pole /lxi~' A·
t
AXi s A
S?I t
.. I
Ji/Sp
ff/p
Jtlp
I; ig . 25-13 The effect of higher harmonics on th e waveform of th e rot at ing field set up by a polyphase windi ng: (a) field at t = 0, (b) field at t = n /2w
pairs , the same pole pitch, the same sense of ro tation , and the same electrical angular frequenc y as the vth harmonic mmfs [see Eq . (25-13)J. From the fundamenta l fl ux dens i ty the y only differ in the much smaller peak values, the num ber of pole pairs , and the me chanical frequen cy of rotation . Because each harmonic component of the fie ld travels at its own m ech an ical angular veloc ity Q v , their relative position is changing all t he Lime, and th e res ultant fie ld patt ern goes t hro ugh a cycle of change per io dica lly. This proper ty of the fie ld is ill ustra te d in Fig . 25-13 where the mag netic flux dens ity is shown as the sum of t he fun damen tal
316
Part Two. Energy Conversion by Electrical Machin es
and the fifth harmonic . The magnetic flux dens ity is given for two in st ants , na mely t = 0, whe n the current in phase A of the t hree-phase winning is a m aximum (see Fig. 25-13a), and t = n/2w , when the phase A current is zero (see Fig . 25-13b). At t = 0, t he peak va lu es of the harmon ics occur on the ph ase ax is . During the t ime t = n /2w the Iunrlament al wave travels in the positive direction th rough a mechanical ang le Q i t = (wlp ) (n /2w) = n /2p or an electrical angle n /2, whe reas the 5th harmonic wave t ravels in the opposite direction through a mec han ica l angle Q 5 t = (wI5p) (n I2w) = n/2 (5p) or an electrical angle (5p) (Q5 t ) = n/2 Referring to the figure, the magnetic flux density waveform at t = differs fr om t hat at t = n /2w because the relativ e positi on of the harmonics is cha ng ing all t he t im e. (For convenience, the fifth harm oni c is shown enlarged fiv efold.) When the contribution fro m t he h ighe r ha rmonics is insignificant , th is change in sha pe is negligible . The pr operties of the h igher harmonics listed above are typical of the rotating fields produced by polyphase windings car ry ing ba lanced sets of PPS or NPS curre n ts with a circ ular freq uency ui. To sum up, t he harmon ic components of the field set up by a pol yphase winding rotate all at the same electrical velocity W v = ro which is the same as the ci rcu la r frequen cy of the currents , but with different mechan ical angular velocities, Q" = talvp .
°
26
The Magnetic Field of a Rotating Field Winding
26-1
The Magnetic Field of a Concentrated Field Winding
Another way of pr oducing a rotating field is to place the field win ding on the r ot or of a mach in e. When this win ding is energized with d. c. , it establishes a magne tic fie ld s ta tio-
317
Ch. ' 2fl Magn eti c Fielrl of Rotating Fip.ld Winding
I,
i
nary relative to the ro tor, with a radial component of magnetic Ilux den sity B (Fig . 26-'1). If, now, t he ro tor is made to rotate at me chanical angular velo city Q , the magnetic field set up by the rotor winding will likewise rotate with the same angular velocity. The mmf F produced by a concentrated field winding can be depicted by a rectangular waveform (see Sec . 24-1) . It remains constant and equal to Pm = iw c over a pole pitch. At the slot axis, it changes r--r-----,r--~~F--.., abruptly by an amount equal t o the slot current , 2iwc , and , reverses in polarit y, t urn ing t o - F m - The peak value of t he mmf can be found as for a single-phase, double-layer, full-pitched winding for which q = 1 and Yc = T and which carries a direct current,
= V 2"
fa:
F m = q (V
2. fa)
Wc
=
ito;
The air gap field set up by F is cal culated over a half-
pole pitch by the Laplace equa t ion , (23-8), for a scalar magnetic potential under t he Iollowing boundary con ditions: the potential at the surface of the pole-shoe is rp; the Fig. 26 -1 The magn etic fi eld potential at the surface of the of a concen tra ted field winding smooth core and at the slot axis is zero . , The shape of t he waveform depicting t he radial component of magnetic flux density at the surface of a smooth core , B, depends on th e pole en clo sure ex = b"IT, the relative air gap at the pol e tip y = 8m /8, and the re la ti ve air gap at the pole axis, e = BIT. The magne tic flux density waveform shown in Fig. 26-1 has been plo tted for ex = 0.55, Y = 2, and e = 0.01 . It is usual to generate magnetic flux density wav eforms on a computer for various valu es of ex, 1', and f , and to sub ject them to Fourier analysis . The peak values of the various
Part Two. Energy Conversion by Elect rica l Machines
318
k'f 1.1
1.0 0.9
/,k'f 0.8
0.3
0.7 0.25 0.6
0·2
0.5 0.15
G.'I
0·1
0.3 0.05 0.2
+
a
0.1 /).05 0.3
F ig . 26-2 The coefficients of the excitat ion field determinin g the mean mag netic flux density: af= at at = Bmean/Bm' and its funda mental: kf = kt kt
IX
os
0.5 0.6
0.7 0.8
0.9 1.0
+
F ig . 26·3 Coefficients of the excitation field determining the hig her harmonics of magnetic flux densi ty : k fV = k fv + k/v
eh. 26· Magnetic Field of Rotating Field Winding
3-H1
harmonics are then expressed as fractions of the maxim um flux density, B m , called the harmonic coefficients of the excitation field: k f = Blm /E m for the fundamental (26-1)
kf v
= B vm/ B m for the vth harmonic
(2G .2)
Here, B m = ~toF m/8 is assumed to be the magnetic flux density set up ill a uniform air gap 6by a constant mmf, Fm*The most accurate values for k f and k j v for v = 1, 3, 5, 7, 9, tl , 13, 15, and 17 can be found in [38]. We will only give those required to calculate the harmonic coefficient for the fundamental , k f (Fig . 26-2) and for the 3rd and 5th harmonics, k f s and k f 5 (Fig. 26-3) . Referring to the figures, we can find the components of the respective harmonic coefficients, namely kj and kj, kjs and ki ;j , and kj5 and ki5(As is seen, the figures give yki, yki 3 , yk j5') The harmonic coefficients are found by combining their components for the specified va lues of a, y, and B: kf = kj
+ ki,
kf v = kjv
+ kiv
The B waveform differs in shape from the mmf waveform and , with a judicious-choice of the relative air-gap (limensions, it can be made sinusoidal very nearly . The magnetic flux density waveform can be expanded into a Fourier series where the equation for the vth harmonic about the winding axis is
B (a) = B v m cos a ov
(26-3)
where a ov = va o = VPYo a o = electrical ang le defining the position of a given point relative to the winding axis, and Yo = mechanical angle defining the position of the same point relative to the winding axis 26-2
The Magnetic Field of a Distributed Field Winding
The mmf produced by a distributed winding can be depicted by a stepped wa veform (Fig. 26-4) sim il ar t o that for the phase mmf of a double-layer winding . For a single-layer distributed winding with a slot current hUe and with q * In thi s cas e, the scalar ma gneti c potential is 'P = Fill'
320
Pnrt Two. Energy Conversion hy Electricnl Mnchi nes
woun d slots per pole, the pea k va lues of the harmon ic mmfs can he found by Eq . (24-35) der ived for a phase of a doub lelayer wind ing, assuming that the winding is fu ll-pi tched (Ye = L)* and that the maximum coil current in a doub leWinding axis
F ig . 26- 4 Th e ma gneti c field set u p by a distributed field winding (q = fl, bIT = 2/3)
layer winding, V 2f aWe ' is equal to half the sl ot current in the fie ld win ding, tw e . Not ing that for Ye = r, the harmonic pi t ch fac tor is u ni ty, k p ,. = 1, Eq . (24-35) can be re-written 1.0 give the following expressicn for the peak value of the vth harmoni c mm f: Fv
=
2qlcd v · ----m;--110 c =
4kd v . ---nv uo
(26-4)
where
10 = we q/2 = turns per pole of the fie l d win ding k clv = Si ~(q(v'\'z//22» = distrib ut ion factor for the vl.h har-
qSITI V'\'z
monic' l' z =tz;rr,/-r = p;rr, /q = electrical ang le bet ween adja cent wound slots
- - --
* As regard s the gener a t ion or a ma gn et ic field , th e field winding may be treated as a full-pitched winding, becau se the distance between adjacent groups of wound slots is equal to the po le pitch.
Ch. 25 Magn etic Field of Rotating Field Winding
321
t z = tooth (slot) pitch
P
= bl» = enclosure of the wound part of a pole
b = length of the wou nd par t of a pole pitch
In this case, the air gap permeanc e may be deemed constant and equal to over the entire length of the pole pitch. Therefore, t he magnetic flux density waveform , B = f.toFAo' is the same in shape as the mmf waveform, and the peak values of the har , monic flux densities are proportional to those of the harmonic mm fs B; = f.toFvAo The equation for the vth magnetic flux densit y harmonic, referred to the winding axis, does not differ from that for a concentrated winding, Eq. (26-3). 26-3
The Rotating Harmonics of the Excitation Field
As the rotor rotates at mechanical angular velocity Q , the excitation fi eld an d its harmonics (Fig . 26-5 shows onl y the fundamental and the 5th harmonic) ro tate all at the same mechanical angular velocity Q . This is the reason why, in contrast t o the ro ta ting fi eld set up by a polyphase winding, the field established by the field winding rem ains unchanged in shape as it rotates. In contrast, the electrical angular velocities of the various harmonics are all different
As is seen, it increases with the harmonic. order. (Compare it with the field set up by a polyphase winding, where t he electrical angular velocities are the same, but the mechanical angul ar velocities ar e different.) An equation for th e v th ha rmoni c of the rotating magneti c flux density wav e pro duced by t he field winding , referred t o a station ar y reference axis, may be derived from Eq. (26-3) for the same ha rmo nic ..\ ~~llIne that at t = 0 the axis of th e winding rotating at mechan ica l angular velocity Q run s along the refere nce axis (see Fig . 2G-5a). On this 21 - 0169
a22
Part Two. Energy Conversion by Electrical Machine,
assumption, the angular coor dina tes of an arbitrary poin: relative to the winding axis, Yo , and relative to t he re ferene, axis, 1', at an arbitrary instant of time will be connsctso by an equation of the form
'\' = ~o
+ Qt
Consid ering the above equation together with Eq. (26-3), Winding axis
W inding axis
Stationary
St ati qnary
52
ax es
axes
1i/5p
tri p
(6)
(a )
Fig. 26-5 Hi gher harmonics of th e excitation field (a) at t an arbi trary ti me t
=
0, (b) at
we obtain the equation for the vth harmoni c of t he ro t ating flux density wave
B; (a, t) = B v m cos (vpQt - vp,\,) = B v m cos
(CDvt - va)
(26-5)
Outwardly , Eq. (26-5) is the same as Eq , (25-10) or (25-20) for t he vth h arm oni c of the flux density wave produced by a polyphase winding . The coeffi cien t of I' in this equation is the number of pole pairs for the harmonic in que stion, vp= Pv' The coeffi cien t of t is t he el ect r ical angular velocity of the harmonic, vpQ = Pv Q = CD v ' The ratio of the two coefficients is th e me chanical angul ar velocity vp Q /vp = Q
tho 27 Flux Linkages and EMF's
27
Flux Linkages of and EMFs Induced by Rotating Fields
27-1
Introductory Motes
When energized, the windings of an electrical machine set up magnetic fields varying in time and space . As has been shown in Chapters 25 and 26, the air gap magnetic fl ux density, no matter how it is produced, can be expanded into a Fourier series and presented as the sum of rotating fields differing in the peak va lue of the r adial component, E vm, the number of pole pairs, Pv' and the mechanical angular velocity, Q v ' An import ant problem in the theory of electrical machines is to determine the flux linkages with, and the emfs induced in, the phase winding by the rotating fie lds . Because polyphase wind ings and rotating fi el d windings are always designed so that the h igher harmonics rapid ly dim in ish in amplitude with increasing order, the winding fie ld can, to a good approximation, be represented by the first term (v = 1) of the Fourier series. The flux density wave of such a rotating field , with a peak va lue E 1 m , is shown, for example, in Figs. 25-9, 25-13, an d 26-1 . Re lative to a stationary reference ax is, the flux density of the forward rotating field is given by Eq . (25-18) as
E (ex , t ) = Elm cos (wt - ex) = Elm cos (pQt - py) (27-1) Th e emf in duced in a phase winding by a rotating field can be found as the sum of the emfs in its coils. Therefore, we shall begin by finding the flux linkage and emf for one coil. 27-2
The Flux linkage and ~ MF of a Coil
Consider a coil displaced from the origin 0 by a distance Xc along the per iphery of the cor e. The axis of this coil is turned by a me chanical ang le Yc = xci R from the stationary reference axi s . H ere, R = xpls: is the mean radius of the air ga p periph ery (Fig. 27-'1). In the general case, the coil pitch Yc is taken to be shorter than the pole pitch 't o The me2 1*
I
324
Part Two. Energy Conversion by Elec trical Machines
chanical angle spanned by the coil or the coil pitch angle is "(y = yc/ R The rotating wave of flux density described by Eq . (27-1) travels relative to the coil at mechanical angular velocity Q. At time t, the axis of the rotating field is displaced from the reference axis by an angle Qt and takes up the position shown in the figure. The radial component of B at any point Fi eld axe s
Coil axi s
Statiqn ary ans
o
Fig. 27-1 To determining the flu x linkage of a coil t urn
on the circle, with all angular coordinate "( relative to the reference axis and at t ime t , can be found by Eq (27-1). The magnetic flu x links wi th the coil t urn s through an area A y of a cylindrical surface of radius R; it sp ans an arc Yc and extends along the generator of the cylinder for a distance equal to the axial gap length, lo (see Sec. 23-5), or mathematicall y, cD =
~ Ay
B n dA = .\ dcD Ay
Rec alling tha t over t he axial gap length t he flux density at the ax is of the machine rem ains constan t and t ha t in a cylindrical syst em of coordin ates t he normal component at a cylindrical sur face is equal to t he radial component,
Bn = B R = B
325
Ch. 27 Flux Linkages and EMFs
we may replace integration over a surface by integration over a circle on which the position of a point is defined by the angular coordinate y. An elementary area dA may be expressed in terms of an elementary length along the circle, dx = Rdy, as follows: dA
= lodx=loRdy
Then, an elementary flux will be given by
= BloRdy
del>
and the integral will have to be taken over the coil pitch, that is, from y~ = Yc - yyl2 to y~ = Yc "Y,,12:
+
'l'c"
el>(t=const) =
J de!) = JBhR dy 'l'~
Ay
'l'~
=BimloR
Jcos(rot-py)dy I
'l'c
=
B 1m i.u . U SIll (py -
rot) I'l'~
p
I
'l'c
Upon substituting the limits of integration and expanding the sines of the sum and difference of angles, namely -
rot) +a y/21
sin [(ac -
rot) - a y/21
sin [(a c
and we get
el> = el>ym cos (rot - a c ) = lcpel>m cos (rot - a c ) (27-2) where el>ynt el>m
=
lcpcP m = maximum flux that can link with a given coil of coil pitch Yc
= ~ -rl"B 1 m = maximum flux linking with a fullpitched coil, Yc = -r
kp
=
sin (a!l/2) = sin (ycn/2T:) = pitch factor for the fundamental component .. . , . . of the. field ." .
326
Part Two. Energy Conversion by Electrical Machines
a!J = PY!J = ycnh; = electrical angle spanned by the coil a c = PYc = ,Tcnh; = electrical angle defining the position of the coil axis relative to the reference axis (origin) It is seen from Eq . (27-2) that the flux linking the coil turns varies with an angular frequ ency w = pQ, equal to
Fig. 27-2 The effect of pitch-shortening (chording) on the maximum flux linking the coil
the electrical angular velocity of the wave . The frequency of the flux is given by I = w/2n = QpI2n Accordingly, the time period of the flux is T = ill = 2nlQp = yp/Q It is also seen from Eq . (27-2) (see Fig . 27-2 as well) that the flux linking a turn passes through a positive maximum cD =
when the axis of the field aligns itself wi th the coil axis Q
t = Q (Yc/Q) = Yc
The amount by which the flux lags behind depends OIl th e electrical angle a c = PYc defining the position of the coil relal.ive to the reference axis. The maximum coil flux is equal to the shaded area in C) . " . . · F. l~. ' -~a : 2~
GIL 27 Flux Linkages and EMFs
327
The flux linkage of the rotating field with the coil is found by multiplying the flux defined in Eq. (27-2) by the number of coil turns We 1p'=
10e
cD = Wem cos (wt - (X) e
(27-3)
where is the peak or maximum flux linkage with tho coil. The instantaneous emf induced in the coil is given by
e = - dWIdt = w1p'em sin (wt - (Xc)
=
1/2 E e sin (wt- (Xc)
. (27-4.)
The rms value of the coil emf is Ee =
WWemlV
:2 =
10
ekpcDmw/V 2
(27-5)
Both the flux linkage and tho emf can be portrayed on a time vector (phasor) diagram (Fig. 27-3) as complex functions We m and Ee whose pro(+) jections on the real axis of the complex plane aligned with the time axis give the respective instantaneous values:
('¥ c) Re {Wem exp [j (wt - (Xc)])
'If = Re =
e=Re(V2Ee ) e
V2
=Re{V:2Ee
n/2)]} (27-6) The positions that the above phasors take up in Fig. 27-3 correspond to the magnetic field shown in Fig. 27-1. Here, W > 0, because the flux is directed with the coil axis, whereas e < which implies that it is directed against the positive direction in the coil, in accord with the right-hand screw rule. . The coil emf is X
exp (j (wt -
(Xc -
Fig. 27-3 Phasor diagram 0 the coil flux linkage and emf
°
~27-7~
328
Pa rt Two. Energy Conversion by Electrical Machines
27-3
The Flux Linkage and EMF of a Coil Group
Each pole pi tch of a double-layer winding has q coils of a given ph ase (in Fig . 27-4, q = 3). Th e wav eform of the flu x linkages and emfs for the coil group shown in Fig. 27-4, plotted by Eq. (27-3), (27-4) or (27-6), appears in Fig. 27-5. Because the coils in the group ar e displaced from each ot her by an electri cal angle CG z
=
Pl' z
=
(tzh:) rt = CG C 2
-
=
CG CI
CG C 3
-
= . ..
CG C 2
the flux linkage and emf ph asors are likewise displaced from each other by t he same angle . 1)th (say, second) coil lag behind The events in the (k those in the kth (say, first) coil by the t ime required for the flux density wave to mo ve through a mechanical angl e I'z, t ha t is , .
+
t
=
yz/Q
=
=
pyz/pQ
CGz/ro
This lag must be all owed for in combining the flux linkages (and emfs) within a gi ven coil grou p. The coil -group flu x linkage and emf phasors, '¥g and Eg , are each the ph asor sum of the coil linkages and emfs, \fc l ' lfC2 ' 1fc 3 and ECl> Ec2' EC3. Going back to Eq. (24-29) and Figs. 24-10 and 24-11 in Sec. 24-5, it will be recall ed that the problem of combining several phasors equal in magnitude, '¥c lm
or E CI
= 1¥c2m =
1¥ c3m
=
'f c m
= E C2 = E C3 = E c
and displaced from each other by the same angle CG z has already been sol ved in determining the mmf of t he win ding. Therefore, the coil-group flu x linkage and emf may be wri tten 'I' g m = q1¥cmk ct = qWck pkctfP m (27-8) E g = qEck ct The coil-group flu x linkage phasor is directed alon g t he ax is of symmetry of the coil ph asors and, as is seen from a comparison of Figs. 27-5 and 27-4, is tur ned through an angle (rot - CG p h) from the real axis of the complex plane. H ere, CGph = Pyph a is the e~ectri9Rl ~ ,
1
I
1
•
320
Ch. 27 Flux Linkages and EMFs
Fig . 27·4 EMF induced in a coil group
(+)
Fig . 27-5 Phas or dia gram of flux linkage and emf for a coil grou{J
330
Part Two. Ene rg y Conversi on by El ectri cal Machines
angle of a coil group or the ph ase axis. It is to be noted that t he ax is of a coil group (the ph ase ax is) is the axis of symmetry for the coil group . The an gle defining the position of th is axis is foun d as the ar ithmetic mean of t he angles defini ng th e positions of the coils in the group "'Ph =
YCl +'\'C2+ · · · + YCq
o.C1 +o.C2+ · · · +'Xcq
q
pq
aph/ P
(27-9) The coil-gro up emf lags behi nd th e coil -gr oup flux linkage by n /2, and ma y be written as (27-10) 27-4
The Flux Linkage and EMF of a Phase
A phase of a winding is ma de up of coil gr oups connect ed in series-parallel (see Sec . 22-3) A ph ase of a double-la yer winding ha s 2p id en t ical coil groups (one group per pole Group II
1ph
if ('f)
Group X
Group II
¥('f)
F ig . 27-6 EMF induced in th e coil field
Group X
¥(f)
groups of a ph ase by a rot ating
pitch). As an exa mple, Fig. 27-6 shows t he coil gr oups of pha se A in a four-pole, three-phase win ding (2p = 4). Its comp le te circu it diagra m is sh own in Fi g. 22-5 . A d J a ceI~~
331
Gil. 27 Flux Linkages and EMFs
coil groups in t he ph ase are displaced from one another by one pole pi tch 1: or by a h alf of the pole-pitch angle
,,? p/2 = 2rrJ2p = nip The respective el ec tri cal angle is r:t. p /2
= p,,?p/2 = n
Therefor e, the flux linkages and emfs of the backward coil groups in the sam e phase, 'ijf gXm and if g X, are in antiphase with t hose of the forward (+) coil groups, 'ijf g Am and Jj; g Ao If a phase h as a parallel pa ths (circuits) , t hen each pa th cont ains 2pla coil groups. T11e forw ard coil gr oups are connected aiding (with their finishes t o t he phase finish), wh ereas the backward coil groups are connec ted in opposition (with t heir starts to the ph ase fi nish). Exactly this form of connection of coil groups in parallel paths is sh own in F igs . 22-5 an d 22-6 . Now the positive direction around a parallel path Fig. 27-7 Ph asor di agram of (from its fi ni sh X to wards its flux li nk age and emf for a phase start A ) is t h e same as the in a doub le-l ayer winding positive direction around a coil gr ou p (fr om its fin ish F t owards its star t S) in all t he forward gr ou ps connected a iding (A) and is opposite to the positive direction around all the backward coil groups (X) . With this arrangeme n t, t he flux linkages and emfs of the coil gr oups are combined ar ithme t ica lly wi thin a parti cular path , and the flux linkages an d emfs in all t he paths are the same (Fig. 27-7) . The ph ase fl ux li nk age an d ph ase emf are resp ectively equ al to th e flu x linkage and emf of a path ~
{if. ph m -_
~
p' l' g -,1 III -p 'l' g .K m _ ? a
-
~P
1~¥
gA m
I
a
(27-'11) ~2 7- t 2~
332
Part Two. Energy Conversion by Electrical Machines
. The phase emf can be expressed in terms of the phase flux li nkage directly
Ep h = 2pEgAIa =
- j2PU/ ¥gAmla V 2"
=-
jro'f p h m lV 2
(27-13)
The phase flux linkage (see Figs. 27-6 and 27-7) is in the same direction as the flux linkage of the main coil group whose axis is taken as t he phase axis and makes an angle
(+)
~ Eg z CU
(V
£s
r; iEc=Egc
Es=?g,.
(6) Fig . 27-8 Phasor diagrams of flux linkages and emfs for the phases and coil groups of a three-phase winding: (a) for two paths (circuits) in a phase; (b) for four paths (circuits) in a phase
= PYph with the ori gin. The phase emf lags behind the phase flux linkage by rrJ2 (see Figs. 27-7 and 27-8). The magnitude of the phase flux linkage or ph ase emf is 2pla times the magnitude of the coil-group flux linkage or emf. The ph asor diagrams in Figs. 27-7 and 27-8 are plotted for phase A consisting of the coil groups shown in Fig. 27-6. In Fig. 27-6, the number of paths is a = 1, so 2pla = 4 (see the dashed connections in Figs . 27-6 and 22-5) . In Fig. 27-8a, the number of paths is a = 2, so 2pla = 2. In Fig. 27-8b1 G = 4, so 2pla = 1 (see Fig. 22-6b) , ' . . . ' Chph
333
Ch. 27 Flux Linkages and EMF's
In accord with Eqs. (27-8) and (27-11), the peak value of the phase flux linkage is where
1¥phm = 2p1¥gm/a = wkwCP m
w = 2pwcq/a k w = kp/k d
(27-14)
= number of series turns per phase = phase winding factor (for the funda-
mental component of the field) cD m = peak value of the magnetic flux over a pole pitch The rms value of phase emf given by Eq. (27-13) is E ph = ro1¥PhmlV2= 2njwkw Cl)m/V '2 27-5
(27-15)
The Flux Linkages and EMFs of a Polyphase Winding. A Space-Time Diagram of Flux Linkages and EMFs
All the phases in a symmetrical polyphase winding are identical in arrangement. Adjacent phases, say, phases A and B, whose axes make mechanical angles I' A and I' B wi th the stationary reference axis, are displaced from each other by a mechanical angle (see Fig. 22-6) I' BA = I' B - 1 ' A = Zsilmp = I'p/ m or by an electrical angle CGBA
=
CGB -
CG A
= PI' BA = 2n/m
Therefore, the phase flux linkages and phase emfs are the same in magnitude (Fig. 27-8):
1¥ Am = 1¥ Bm = 1¥ Cm = 1¥m EA=EB=Ec=E
In the case of a forward rotating field, that is, one moving from phase A to phase B to phase C, the flux linkages and emfs of a polyphase winding form on the complex plane an m-ray star in which the adjacent arms are displaced from each other by an angle 2n 1m (for a three-phase winding, this angle is 2n/3, see Fig. 27-8). Let the axis of phase A run along the stationary reference axis. Mathematically, this will be written as CGA
=PI'A = 0
3M HUrl
Part Two. Energy Conversion by Electrical Machines
the instantaneous phase flux linkages will be 1p" A = 1p" Am COS (wt - C(.A) = 'f A m COS tat 1]1 B= 1p" Bm. COS (wt - CXBA) = 1p" Bm COS (wt - 2:n:/3) lJF C = 1p" c m COS (wt - CXCA) = 1p" Cm COS (wt - 4:n:/3)
Or, in complex notation,
= Re
1p" A 1p" B = 1p" C
Re
=Re
[W Am] =
["If Bm] = Re
[lY em]
= Re
Re
["0/A m
exp (jwt)l
[1¥.4.m exp (-j2:n: /3)]
[1f Am exp
(-j4:n:/3)]
The instantaneous phase emfs can be written in a similar way:
v:2 E cos (wt - :n:/2) = V:2 En cos (wt- :n:/2-cx nA ) = V:2 E cos (wt - :n:/2 - 2:n:/3)
eA = eB
A
B
Or, in complex notation, eA
=
Re
en = Re
lV2 EA ] =
rv :2E
B]
Re {V2 EA exp [j (wt - n/2)]}
= Re
[V 2" En exp (- j2n/3)]
By analogy with the phase currents (see Sec . 25-3 and Fig . 25-6), the phase flux linkages (phase emfs) can he depi cted on the complex plane of a two-pole model as complex functions common to all the ph ases . For the three-phase winding whose flux linkages and emfs are shown on the time complex plane (Fig . 27-8a), the flux linkage phasors
-qr m =
1p" m
exp (jwt)
and the emf phasors
E=
E p h exp [j(wt - :n:/2)]
corresponding to the respective phase quantities are shown on the space complex plane of the model in Fig. 27-9. In the two-pole model, the phase windings are each shown for clarity as a single coil; the positive direction is shown in the sectional view drawn in the same figure. The phase axes are drawn through the centres of the coil groups represented by
335
Ch. 27 Flux Linkag es and EMF's
one coil. T he instantaneous ph ase flux linkages (or the ins tan taneous ph ase emfs r educed by a factor of 11 2) are given by pro jections of the respective phasors on the axi s of t he respective phase. Because the position of the flux linkage 01' of the emf relative t o the ax is of a given phase in Fig. 27-9 is the same as tha t of th e flux linkage (01' emf) of that ph ase relative to the real axis of t he time complex plane, their instantaneous flux linkage (01' the ins tantaneous emf) is the same in either case. The sp ace complex plane in Fig. 27-9 also shows the complex funct ion
13 1m
= E lm
exp (jwt)
depicting the magnetic flux den sity of t he rotat ing fi eld we are considering [see Eq . Fi g. 27-9 Rot ating-field flux density, ph ase flux linkages and (27-1)]. It has been plotted in phase emfs shown on th e space exactly t he same way as in complex pl ane of a two-pole Fig . 25-10. (It will be recalled model tha t the axis of ph ase A has been assumed to run alon g the stationa r y reference axis .) As follows from Fig. 27-9 and the applicable equations, the complex functions depi cting the m agn etic flu x densi ty of a rotating field and the flu x linkage pro duced by th at field are both in t he same direct ion . This is because the phase flux linkage is a maximum at t he instant when t he magnetic flux density at t he phase axis is a maximum (see abo ve). Axis c
27-6
the Flux Linkages and EMFs due to . the Harmonics of a Nonsinusoidal Rotating Magnetic Field
As h as been explained in Chapters 25 and 26, a rotating fie ld ma y, in addition to t he fun dam ental com pone nt, con tain an amount of harmonics . A rotating field cont aining harmonics is nonsinusoidal. The flu x linkages and emfs produced by the
336
Pa rt 'Two. Energy Conversion by Electrical Machines
harmonics can be fou nd by the equations deri ved for the fundamental component, if they are re-arranged to include the respec tive harmonic qu antities, such as B vm, 'Lv, and Q v ' From Eq. (27-14.) it follows that the vth harmonic component of a rotating field gives rise t o a flux linkage wi th each phase winding, defined (the peak value is meant) by "Ifphvm =
where k..vv =
(27-16)
wkwvc))vm
:;= phase winding fac tor for the vth harmonic k p v = pitch factor for the vth harmonic, Eq. (24.-27) k d v = distribution factor for the vth harmonic, Eqs. (24.-32) and (24-33) cD vm = (2/1£)'L v l oBvm= magnetic flux due t o the vth harmonic over a pole pitch, Eq. (27-2) It follows from Eq. (27-15) that the rms value of the ph ase emf induced by the vth harmonic of the magnetic field is given by
kpvk d v
E ph v = (wv/ V 2) cD ph vm = (21£lV
= 2 VZ!vwkwv('L/'V) where
Wv
loB vm
2) !vwkwv(Pvm (27-17)
= QvPv = Qvpv = electrical angular velocity of
the harmonic, equal t o the circular frequency of the induced emf ! v = w)21£ = fr equency of t he induced emf The emfs induced by the harmonics are superimposed on the emf induced by the fu ndamental component and . affect the resultant phase emf and, in the final analysis, the performance of the m achine. This effect va ries accor ding as the nonsinusoidal rotating field is produced. Consider two cases which are most typical of all , namely: t he magnetic field produced by a polyphase winding (see Chap. 25), and the magnetic field produced by a rotating fie ld wind ing (see Chap. 26). 1. Typically, t he waveform of the nonsinusoidal ma gnetic field set up by a polyphase win ding is continually varying in sh ap e, because its rotating h armonics tr avel at different mechanical angular velo cities (see Fig. 25-13 an d Sec. 25-6): Q.. .
=
cslp ;
=
»lp v
where ro = 21£1 is the circular fr equency of t he currents in the polyphase win di ng.
ci, 27 Flux Linkages and EMFs
337
It is readily seen [see Eq. (27-17)] that the rotating harmonics of this field induce emfs of the same frequency equal to the frequency of the current in the winding
=
CO v
2nfv
=
QvPv
=
co
=
2nf
A more detailed analysis would show th at . the emfs due to the harmonics are in phase with the fundamental emf and are added to it arithmetically. The harmonics do not affect the waveform of the fundamental emf, and the resultant emf is sinusoidal. The effect of the vth harmonic on the rms emf depends on the ratio EphvlEph
=
kwvBvm/vkwBlm
=
kwvcDvmlkw(J)m
Therefore, even for a concentrated (q = 1), full-pitched (Yc = 't) Winding, when kwv = k w = 1, this effect is 1/v of that produced by the vth harmonic of magnetic flux density (with a peak value B vm) on the fundamental flux density (with a peak value B l m). Still, the total emf induced by all the harmonics Ea,rms=
LJ
v*1
E phv=2 V2f w'tl.-!
LJ
v*l
kwoBvm/v
may be fairly large in magnitude, especially in a fullpitched winding and with small values of q. In practical machines, the ratio Ea,rmslEph =
I
LJ
v*1
EphvlEph =
LJ
v'!=l
kwvBvm/vkwB1m
may range anywhere between 0.005 and 0.05. The smaller values apply to short-pitched windings for which Yc ~ ~0.83't, and q~1, so that k wv/vk w
2 2-0169
~art
33B
'I'wo, Energy Conversion by Elect rical Machines
is accordingly called the differential (or difference) leakage emf and treated separately from the emf induced by the fundamental field (see Sec. 28-7). 2. For the nonsinusoidal magnetic fi eld esta blished by a rotating fi eld winding, it is characteristic that the magn etic flux density waveform remains unch ange d as t he fiel d rotates (see Fig. 26-5 in Sec. 26-3). Because of this, all of its harm onic s rotate at the sam e mechanical angular velocity equal to the mechanical angular velocity of the field winding Q"
=
Q
The frequency of the emfs in duced by the fi eld harmonics is propor tional to the order (number) of the harmonics [see Eq. (27-17)] co"
=
2rr,f"
=
Q"pv
=
Qpv
=
COy
=
2rr,fv
where co = 2rr,f is the circular frequency of t he emf induced by the fundamen tal fi eld. Thus, the emf induced by t he vth sp ace harmonic of the field is the vth time harmonic of the emf. The contributio n by the var ious harmonics depends on t he ra tio Eph ,,/E ph
= kw"B vm/kwBlm
(27-18) -
where, as will be recalled, t; = v] , The higher values of t he above ratio correspond t o a lar ger departure of the result an t emf from the sinusoid al waveform . On the other hand, for energy conversion b y elec trical machines an d transform ers t o be most economical (to suffer a minimum of loss) , it is essential that the voltages , emfs and currents involved be as close to a sinusoidal waveform as practi cable. One of the causes of the increased losses associated with a nonsinusoidal voltage waveform is the circulating currents pro duced by harmonic emfs and flowing between t he machines when several of them are connected for parallel operation. In designing an electrical machine, ever y effort is ma de to make the winding volt ages as close t o sinusoidal as practicable . In Sovie t pr actice, this is assessed in terms of the devi ation factor of a voltage (current) wave defined as
l/ >; k (per cent ) =
v=l= l
Elm
E~m X
100
Cit. 27 Flux Linkages and EMF's
where Elm is the peak value of voltage at the fundamental frequency, and E vm is the peak value of the vth harmonic voltage. One way to achieve this goal is to make as sinusoidal as possible the waveform of the magnetic flux density due to the excitation field (we have already shown how this can be don e with a concentrated and a distributed fiel d winding in Chap. 26). Still, for all the measures taken, the distortion factor of the exci ta tion field may exceed the limit. In fact, if the tim e wavefo rm of the emf were allowed to follow that of magnetic flux density in space, the machine would not be able to perform its designated function. Fortunately, this only happens (compare Figs. 27-10a and b) in a concentrated phase winding with one coil pel' group (q = 1) and wound wi th a full pitch (Yc = 1:). In the circumstances, kw = kwv = 1 and, as follows from Eq. (27-18), the ratio of the harmonic emfs to t he fundamental is the same as the ratio of t he harmonic flux density to t he fundamental component, or mathematically Eph v/Eph = Bvm/B l m
I \
When t he phases of a three-phase winding are st ar-connected, the line volta ge is free from triplen harmonics of emf, t hat is , t hose for which v = 3k = 3, 9, 15, etc. , where k = 1, 3, 5, et c. This is so becaus e (see Part One of this te xt) t he ha rm onic emfs of such an order are the same in all the ph ases (eA V = env = ec v), and cancel one another in the line emfs found as the difference of the phase emfs:
eAR, ~ eM - en> ~ 0 When a t hree-phase winding is delta-con nect ed, the line voltage is again free from the triplen harmonics but for a differen t reason (see Par t I of the t ext). The poin t is that around a delta circuit t he t riplen harmonics are added together arithmetically, giving rise to a circula ting current
.
.
I v = 3E Avl3Z p h
so that the respecti ve harmonics of line voltage add to zero:
VA V = EA V
-
ZplJv
= 0
Thus, as we have seen, the waveform of liue voltage in a three-phase winding is impro ved as (:0 111 pared with the 22*
340
Pari Two. i!;iiergy Conversion by Electrical Machin es
waveform of magnetic flux density (see Fig. 27-10c) even in the case of the least perfect winding configura t ion (Yc = 't , q= 1). A further improvement in the wave form of bo th phase and line voltage in t hree -phase windings can be obtained
(e) 1---
-
---'----->-1
Fig. 27-10 Effect of three-ph ase winding arra ngement on the waveform of phase and line emfs: (a) waveform of the excitat ion-field flux densi ty; (b) phase emf for Yc = .., q = 1; (c) line emf for Yc = .. and q = 1; (d) phase emf for Yc = 0.83.., q = 2; (e) line emf for Yc = 0.83r , q = 2
by using dis tribut ed windings ts > 1) woun d with a shor t pi t ch (Yc ~ 0.83"1"). In such windings, for all harm onics, except the too th harmonics (slot ripple), as has been shown in Figs. 24-9 and 24-12, we get
kw vlkw = (k pvkdvlkpkd) ~ 1
Ch. 28 Ind uct anc es of Polyphase Windings
341
Hence, Ephv/Eph =
kwvBvm/kwB lm ~ Bvm /B 1m
This implies that t he waveform of emf is more sinusoid al than t hat of magnetic flu x den sity (see Fig. 27-'1Od an d e). As is seen , when Yc ~ 831: and q?;3:- 2 , the emf is practically sinusoidal , even though th e waveform of magnetic flux density due to t he exci t ation field is substantia ll y nonsinusoidal. It should be add ed t hat in such windings the rms value of phase or line emf does not practically diff er from the rms value of t he fun da mental emf E E ph Z= -.V,rE 2 + E 2ph 3 + E 2p h 5 + E 2ph7 + • . . ,...., ,...., ph ph
28
T he Inductances of Polyphase W indings
28-1
The Useful Field and the Leakage Field
Let us consider the magnetic field in an elect ric al machine with two pol yphase windings one of which is wound on the stator, and the other on the roto r . Assuming that the relative permeability of the stator and ro tor cores is infinit ely large (u, = 00) , the st ead y-sta te magnetic field of such a ma chin e can be visuali zed as consisting of two components , namely t he us eful field an d the leakage field. As will be recalled the useful m agnetic f ield is t hat which is associated wi th t he fu ndamental component of t he ra dial magnetic flux density in the air gap . This field plays the decisive part in energy conversion . When fLr = 00, the useful field may be im agined as form ed by tw o fi elds which are stationary rel ative t o each other, namely the useful stator field set up by the currents in the stator winding, and the useful rotor field set up by the currents in the rotor winding. Of course , the ai r gap flux density due to each of t hese field s contains onl y t he fund ament al componen t . In turn, the useful stator (rotor) field may be visualized as the sum of the usefuJ fields established by the various phases of the stator (rotor) winding.
342
Pa rt Two . Energy Conversion by Electrical Machines
The leakage f ield is that which is es tabli shed by the sets of currents in t he s tator and rot or windings tha t do not contribute to th e useful field . In other word s, when the fu nd amen tal fluxes of the st ator and ro tor fiel ds cancel out , the leakage field only exists in the machine . The total flux linkage of a polyph ase winding may he visualized as the sum of the useful flu x linkage and th e leakage flux linkage. Th e form er is associated with the useful fie ld whose lines close via the air gap and link both windings of the machine . Forithis reason, it is called the mutual field . The leakage flu x linkage is associa te d with that par t of the leakage field which links only one (stator or rotor) winding. 28-2
The Main Self-Inductance of a Phase
The main phase self-inductance is associat ed with the mutual flux linkage produced by the respective phase current. Let us find the main self-inductance of phase A in t 11f' sta tor winding. Suppose that the phase winding carries by ~ nositive current whose peak value is iA
=V 2/ A
In Fig. 28-1, phase A is shown for clarity as a single coil. The fundamental component of t he phase mmf with the peak value given by Eq. (24-28) is
F phlm = 2 V 2 / AWlk Wl!np In the air gap, it gives rise t o a cosi nusoidally distributed mu tual magnetic field whos e flu x densi ty at the phase ax is , according to Eq. (25-18), is B
-
1m -
~to
F
ph l m
'A 0-
2
V2 rrp6k I Aw1kw1llo o
The fund amental component of the phase mmf and t he norm al component of the air gap flux density are sho wn in Fig. 28-1a, and the ph ase magnetic field pa ttern in Fig. 28-1b. Rec alling that t he axis of the magnetic fi eld runs along the phase axis , its flu x linkage with the phase winding [see Eqs. (27-2), (27-13), and (27-14)] can be written as 'I'AAm =
.
2
wl k wl<1>m = - T:l!'J WlkwIB lm 11:
343
Gh. 28 Inductances of Polyphase Windings
(.4"R This flux linkage is proportional to the number of mutual field lines that cut the surface which spans the contour of coil AX representing phase A actually consisting of many coils. Axis If
Axis A
(a )
(b)
Fig. 28-1 The mutual magnetic field of phase A in a two-pole model: (a) distribution of the normal component of the phase flux density; (b) magnetic field pattern for phase A
By definition , the main self-inductance of a phase A is )2 -r l 6 L AA = ur r AAm [i~A = -4~lO - 2 ( Wi k Wi"""--k pIT.
u 0
28 ( - '1)
It is seen from Eq. (28-1) that the main self-inductance of a phase depends on the air gap dimensions (lo, 't, 6, k 0), the magnetic properties of the air gap (/lo), and the characteristics of the stator winding (p, w, kW 1 ) ' In our example, the air gap is uniform, so the main self-inductance is independent of the relative position of the rotor and stator. 28...s
The Main Mutual Inductance between the Phases
The main mutual inductance between the phases varies with the electrical angle between the phase axes. To find the main mutual inductance between phases A and B of a poly- . phase winding, with their axes displaced from each other by
3<1 4
Part Two. En e rgy Conversion by Electrica l Machines
an electri cal ang le a BA. = PYB A, we should first find the flux linkage between t he useful field of phase A (shown in Fig. 28-1a and b) and th at of phase B. It has been shown in Sec. 27-2 that the flu x linkage of a rotating field with a phase is proportional t o the cosine of the electrical angle between the field axis and the pha se axis. (In Eq. (27-13) and in Fig. 27-7, this angle is ro t - ap h.) Therefore , the flux linkage of the phase A fi eld (for i A . = V 2 I A) with that of phase B is
'P'BA m = 'P'A A m
COS aBA
By definition, the main mutual inductance between phases A and B is L E A = lJf B Amli A = ('P'A A.m1i A) cosaBA = L A. A. cos a BA (28-2) For the three-phase winding in Fig. 28-1, a B.'!.
Therefore ,
=
2rrJ3,
a CA
=
4:n:/3
cos aB A = cos a C A = -1/2 and the main mu tual inductances between the phases are negative L B A = L C A = - L A A I2 It is seen from Fig. 28-1 b that the plane of the phase B coil is cut by half as many field lines as t he plane of the phase A coil. Also, whereas the plane of the phase A coil is cut by fi eld lin es in t he positive direction (with the axis of phase A ), the plane of the phase B coil is cut by field li nes in the nega tive direction (against t he axis of ph ase B). This difference in flu x linkages controls the magnitude and sign of t he mu tual inductance. 28-4
The M ain Mutu al Inductance Between a stator Phase and a Rotor Phase
As in the previous section , this mu tual inductance is a fu nct ion of the cosine of the elec trical angle between the axes of the stator and rot or ph ases considered. Also, in finding t he flux linkage of primary (say, stator) phase A (the primary phases have upper-case letters in their indexes) with, say , secondary (say, rotor) phase b (the secondary phases have lower-case let t ers in their ind exes) , it . is important to re-
Ch. 28 Inductances of Polyphase Windings
345
member that a secondary phase h as a different numb er of turns, w 2 , and a different winding factor , 1~w2 ' If the electrica l angle between the axes of phases A and b (see Fig. 28-1) at a given instant is equal to CX b A' then the flux li nkage with phase b is given by
Accordingly, the ma in mutual inductance between ph ases A and b is (28-3) wher e 4110 7 k ) 'tl(\ L m
= -P1C".-
( W 1/~W 1 w'" "-k U Ii
is the peak value of mutual inductance between a pr imary phase and a secondary phase (say, phases A and a, when their axes coincide and t he electrica l angle between them is zero). As is seen from Eq. (28-3) , when the rotor rotates at a 'mechanical angular veloc ity Q , the angle CXbA = Qpt inc reases in a linear fashion, and L b A is varying harmonically . In Fig. 28-1b, the mutua l inductance is positive, because the field lines cut th e coil pl an e by in the positive direction (along its axis). 28-5
The Main Self-Inductance of the Complete Winding
In add ition to the self and mutual inductances examined in the previous sections, which are found by definition, it is convenient in the theory of electrical ma chines to introduce the concept of the self -inductance of t he complete winding. It can be defined as the self-inductance of a ph ase (say , phase A ) which is in tur n defined as the ratio between the ma ximum flux linkage due to all t he primary phases with phase A , and the peak value of the phase A cur rent * . For a three-phase winding, L
11
=WAmIV21A =
'Y A Am+¥ABm+WAcm , I_ ~
V 2IA
* This self-inductance is th e same for balanced sets of PPS and NPS currents, but is different for ZPS currents.
34.6
Part Two. Energy Conversion by Electrical Machines
On expressing the flux linkages in terms of currents, main self- and mutual inductances,
1V AA m = V'2"JA L A A , 1jf ABm = V 2I B L A B , 1VA C m = V 2Jc L A C and recalling that, in accord with Eq . (28-2) , LAB
=
LAC
= -
L A A
/2
and also no ting t hat for balanced sets of PPS and NPS currents IB+l c = - I A we can see that the m ai n self-induct ance of t he complete winding can be written in terms of t he main self-inductance of a phase as 3
L I1 = ZL A A
= (6Ilo/p:n;2) (w 1kw 1)2 (Tlo/ako)
In the .general case, for an Tnt-phase winding, the main self-inductance is
L I1 -:- (Tnt/2) L A A
=
(2TnlIl0/p:n;2) (WtkWl)2 (Tl r,/8k{j) (28-4)
As an alternative , the main self-induct an ce of t he com plete winding can be fou nd from t he peak flux linkage of the fundamental ro ta ting field set up by all the ph ases , wi th one of the phases. In accord with Eq. (25-9), the peak value of the fundament al mmf of the Tnt-phase prima ry win ding that sets up the field is
F
_ mt If'Z IAwt kwt rtp
1(O m -
and, in accord with E q. (24-17), the peak value of magn etic flux density is B t (l ) m
=
~to Flm/6k{j
As follows from the above equations, t he peak flux linkage of a ro ta tin g field wi th a phase is
'Y..l m = (2/:n;) Tl {j lV 1kw 1B
1(l )m
and the main self-inductance of the primary winding is
L I1 = '¥ Am/V 2 t , Naturally, the result -is t he same as that given b y E q. (28-4):
Ch.:28 In ductanc es of Polyphase Windings
28-6
347
The M ain Mutual Inductance between
a Primary Phase and the Secondary Win ding
The main mutual induct ance b etween phases of different windings is likewise found from the peak flux linkage of all the second ary phases (or , in other words, due t o t h e ro tating seconda r y field) with a primary phase. It is equal t o the ratio of t h is flux linkage to the peak value of secondary current. The flux linkage with phase A is a m aximum when the axis of this ph as e is aligned with that of phase a, t h e axis of the ro t ating field is aligned with the axis of phase A , and t he current in ph ase a is at its peak value. The peak fund ame n tal m mf of the m 2-phase ' secondary win ding which set s up t he fie l d is giv en by Eq. (25-9)
F l( 2 ) m
=
(m zl f2/:n: ) (I a W2 kw2/ p)
The peak value of t he associated fundamental flux density is
Bl(2)m
=
~LoFI (2)m/5k ~
T he pe ak flux linkage of t he ro t atin g field with the primary phase A is '1'Am ~ (2/:n:)T:l oWikwIBI( 2)m and the main mutual inductance between a primary phase and a secondary phase is
L i2m = \f' Am/ )/ 2" I a = (2m2~O/p:n:2) (wikwiW 2kw2) (T:l~ /Oko) (28-5) I t is an easy matter t o see t h at t h is parame ter is connected t o the peak mutual inductance between a primary phase and a secondary phase by a simple relation of the form
L I2m = m 2Lm/2 which . is similar t o Eq. (28-4). The ma in mutual induct an ce between a se condary phase and the primary winding is gi ve n by L 21m
= m]L m/2
and, if mI :;6 m 2 , it differs from the mutual inductance between a primary phase and the secondary winding, L I2 m .
Part Two. Energy Conversion by Electrical Machines
348
28·7
The Leakage Inductance of the Complete Winding
By definition (see Sec. 28-2), a leakage field exists when the fundamental components of the air gap magnetic fields due to the currents in the stator and rotor windings cancel out: Bj(l)m
=
Bj (z)m
If, to simplify the argument, we assume that the rotor winding is stationary relative to the stator winding, that the axis of phase A is aligned with that of phase a (see Fig. 28-2), Axis A
IAxis a I
1
•
BI (I ) m
~1------- ---........
'PUn, , 2m ~82
Fig. 28-2 Leakage magnetic field due to the polyphase windings on the stator and rotor ('If g stands for 'If t. and 'If line for 'If e)
and that the air gap is uniform , then the fundamental component of the air gap magnetic field will vanish when the fundamental mmfs of the two windings are equal and opposite in peak value. Mathematicall y, this condition may he written as
This condition will he satisfied if the secondary phase currents are appropriately related to the primary phase currents
349
eh. 28 inductances of Polyph ase Windings
The oth er primar y (or seconda ry ) pha se cur rents and th e ph ase A (or a) current form between t hem a bal an ced set of PPS or NPS currents. The flux linkage of t he le ak age fiel d wi th prim ary phase A s a m aximum when the current in that pha se is a ma ximum, i A = V2 lA, whereas the current in secondary phase a must be such that ia =
-
i A (mtwtkwtlm2 w 2kw2) =
- V 2" I a
Precisely such currents in phases A and a, an d appropri ate currents in the other primary and secondary ph ases set up the magnetic field shown in Fi g. 28-2. Given a set of curren ts , the leakage field can be fou nd by electrical-field equa tions (see Chap. 23). Then one finds the leakage flux linkage with phase A in the various parts (ll'sl m, 'l'tlm, 'l' el m, and 'l'd 1m), the tot al leakage flu x linkage with a phase
'Y al m = 'Y sl m+ 1Yum + 'Yel m + 'Y dt m and the leak age self-induc t an ce of a ph ase
= 1Ya1m ll/ 2" I A
(28-6) In this way, the leakage flux linkage of a phase is found with allowance for t he effect of t he other ph ases on t he stator and rotor. The leakage flux linkage of the secondary winding is calculated in a similar way La!
L a2 =
11' a2m l V 2" I a =
\¥S2m
+ ll't2m + ll'ezm +'If d zm V 2I a
The field lines contributing to the slot leakage flux linkage ('l'slm and 'l's2m) , the too th leakage flux linkage ('Yu m and 'l't2m) , and the coil-en d leak age flux linkage ('l' el m and 'l' e2m) are sho wn in Fig. 28-2 (see also Chap. 23). The differential leakage flux linkage ('l'd l m and 'Y d 2 m ) is . also taken in to consideration. Th e stator and rotor (primary and secondary ) windings always differ in t he number of phases, the number of slots per pole per ph ase , and so on. As a result , the stator and ro tor windin g factors are differen t even for t he harmonic mmfs of the same order, and th e harmo ni cs themselv es ro t ate at diffe rent angular velocities. This is t he reason why th e harmon ic mmfs do not canc el one another , although t he fundam ental mmfs do.
Part Two. Energy Conversion hy Electrical Machines
The differentia l leak age emf, E od ' induced by t he higher harmonics has been discussed in Sec . 27-6. In it s terms, the differential leakage flux linkage for the primary winding may be written lJfod m =
V 2 E od/2nj = 1/2
~ E phvl2nj
,,*1
-\1 = (2/n ) w(t l{j .:....: kW1vBvWmiv
where B vm = EV(1)mflo /6k {j. Omitting the details, we sh all only give an equation for the leakage self-inductance of a primary phase, stemming from Eq . (28-6) (28-7)
+
+
+
where A{j1 = AS1 Au Ae1 Ad1 is the permeance for the leakage flux linkage (a dimensionless quantity). The terms of the sum above are the permeances of the various leak age fie ld s, defined per unit of coil-side design length . The higher a given permeance, the larger the associated leakage inductance . The magnitude of a permeance depends on the dimensions governing the respective leakage field. To facilitate computation, equations giving the various permeances have been developed on making certain assumptions as regards the leakage field pattern and taking the permeability of the ferromagnetic parts of a machine to be infinitely large. For a three-phase, double-layer winding, the various permeances can, in a first approximation, be fonnd by the following equations. (i) The slot leakage permeance (for the rectangular slot of Fig. 23-4) AS1
= (helb o
+ hr/3b s) (3B + 1)/4
where li, he bs bo
=
radial depth taken up by conductors in a slot
= clearance between . conductors and airgap =
slot. wid th
= width of opening towards the airgap
B = Yeh = chording (pitch-shortening) factor (ii) The tooth leakage permeance (see Fig . 23-4) At1 =[1.1 (6'/b o)-O.35(6'/b eF'-O.26] (3~t1)
Ch. 29 Electromagnetic Torque
351
where B = k oB = effective radial length air gap (iii) The coil-end leakage perm eance (see Fig. t
A.e i
28-2)
= 0.3 4 11 ~ro.i + 0.1 (~lTql/l0) ~ 0.3~lLql/l0
where ~ro'l ~l
= lro.l/Ycl = =
.!Jc1h
lro .l/~lL
= rela tive coil-end overhang
= ch ord ing (pitch-s hor tening) factor
(iv) The differential leak age perm eanc e (see Fig. 24-3) A.d l = (0.7 to 1.0) (tz/12 B k o) where t z = to oth (slot) pitch B = radial air gap length The factor 0.7 to 1.0 in t he equation for A.dl depends on the degree of pitch shortening (chording), relative slot opening (bo /t z and bo/B), the damping effect of currents induced in the secondary winding, etc. For the secondary three-phase winding, the above permeances can be found by the same equations on replacing the in dex "1" with "2".
29
The Electromagnetic Torque
29-1
The Torque Expressed in Terms of Variat ions in the Energy of the Magnetic Field
Let us consi der an in duction or a synchro nous a .c. machine wi th a un iform air gap. ' Ve shall repl ace the t oot hed cores by smo oth ones an d in tro duc e a n equivalent air gap leng th Bo = ko B where k 0 is the slot fa ct or acco unting for t he effect of cor e saliency on the permeabili ty of the air ga p [see Eq. (24-10)J, Suppose that the stator is wound with a symmetrical polyphase winding with In l 2, an d that t he ro tor carr ies 2, or ei ther a symmetrical pol yphase winding with 1n 2 a single-phase field wind ing (this applies to a synchronous machine). Let the stator winding carry a set of PPS currents , II, varying with an angular fr equency WI' and the rotor carry either a set of PPS currents 12 (in the cas e of a polyphase winding) varying with an angular fr equency W 2 , or a direct
>
>
352
Part 'Two. Energy Conversion by Electrical Machines
current 1 2m (in the case of a single-phase field winding), for which WQ = O. As has been shown in Sec. 21-2, such a machine will be capable of unidirectional energy conversion only if the frequency of stator currents, WI' of rotor currents, W 2, and of mutual inductance, W = pQ, satisfy a certain condition. More specifically, it is required that WI + w2 = W = pQ N
If this condition is met (to make the matter more specific, let W < WI and W2 = WI - «i), then, as can readily be shown, the fundamental components of mmfs (or of the rotating fields) due to the balanced sets of currents in the primary and secondary windings will be rotating relative to the stator at the same mechanical angular frequency 0. 1 = Wl/P
Referring to the two-pole model of a machine (Fig. 29-1) whose polyphase (three-phase) windings carry PPS currents, it can be seen that the fundamental mmf of the primary winding
or the fundamental component of the air gap magnetic flux density
ii.; = f!oFim/8o = V 2" lnif!oIikwiwi/8onp rotates at an electrical angular velocity WI = pQ 1 [see Eq . (25-3)1 in the positive direction (this is, from phase A towards phase B). The fundamental mmf of the secondary winding
Fzm = V2 InzI2kwzw2/np or the fundamental component of the air gap magnetic flux density
rotates relative to the rotor at an electrical angular velocity W2 equal to the angular frequency of the current in the rotor winding, and does so likewise in the positive direc-
353
ell. 29 Electromagnetic Torque
tion, that is, from phase a to phase b. (In Fig. 29-'1, the velocity of the mmf relative to the rotor, w2 , ·is shown relative to the rotor). To find the angular velocity of the secondary mmf F _ relative t o the stator, w'2 , it should be recalled that 9
F ig . 29-1 Relative position of the fund amental mrnls and flux lin . ' kages in the primary and secondary wi~dings
in the model the rotorrotatesatan electrical angular velocity W = Qp in the positive' direction, so this velocity must be added t o that of F z relative to the rot or w~ =
W
z
+ oi
(In Fig . 29-'1, W z is shown likewise relative to the rotor). Since the condition for t he velocity of rotor cur rent is satisfied, we may wri te -' w ~ =
WI -
W
+ W = ' WI
= QI l'
To sum up, unidirectional energy conversion can be performed only if the rotor mmf rotates with the sameelectrical angular velocity WI = QIP in t he model and with the same mechanical angula r velo city Q I in the prototype machi23 -0169
3!i4
Part Two. Energy Conversion by Electrical Machin es
ne as the stator mmf does. (In Fig. 29-1, cu ~ = CUI is shown relat ive to t he sta tor .) • Th e conv erted energy and t he period-aver aged elec tromagnetic torque depend , as will be shown la te r, on the ele ctrical angle a12 betw een t he ax es of t he stator and ro t or fields stationary relative to each other. This angle is connecte d to the mechanical angle 1'12 betw een t he sam e axes in the prototype machine by a known relation, a 12 = PY1 2' The positive direction for a 1 2 (or 1'12) and for the torque act in g on the roto r is counterclockw ise from the ax is of the rotor field . Because under steady-state conditions a 1 2 is constant, the t orque will likewise remain cons t ant over a rev olu tion, and the period-averaged electromagnetic torque can be fonn d by Eq. (18-6) T em
= aw/ay
lin = constant
for the arbitrary relative position of the sta tor and rotor shown in Fig. 29-1. (The angle a between the axes of the main rotor and stator ph ases is arbitrary .) In order to find the electromagnetic torque, we should first determine the energy of the air gap magnetic fiel d as a function of B l m, B 2m, and a 12. Th e energy of the magnetic field in an elementary volume dV of the air gap is
dW = (Bg/2flo) dV where B o = B om cos (pcp) = magnetic flux density in t he elementary volume dV = l o<'3 oRd cp B om = V Brm + B~m + 2B lmB2m cos a 1 2 = peak flu x densit y of the resultant air gap fi eld cp = angle defining t he position of the element ary volume relat ive t o t he resultan t fi eld R = mean air gap radius The energy of the air gap magnetic fi eld is found by takin g t he integral over the volume, V = 2nRlo8o. It is
W
=
= where
1:
r (B~/2~to)
2n
dV =
) (B5m/2flo) loooR cos- (pcp) drp
v 0 2 2 pO:Oolo (B B 2B 1m B 2m COS a 12) 2 1m + 2m + flo
= nR/p is the pole pitch.
355
eh. 29 Electromagneti c Torque
. . Now we turn the rotor through a small angle aI', deeming the current constant, and find dW/dy. As will be recalled, the mechanical angle I' (or the corresponding electrical angle a = yp) is the angle, say, between the axis of the stator phase A and the rotor phase a (as reckoned from the stator phase A in the positive direction , that is, counterclockwise) . As the rotor turns through a small angle dy = da/p in the positive direction (with t he phase currents held constant) the rotor mmf and field move along with the rotor, whereas the stator mmf and field remain stationary (see Fig. 29-1). In the process, the angle 1'12 = a 12 /p between the rotor and stator mmfs decreases in the same proportion as the angle I' increases. (We have assumed that t he angle 1'12 is reckoned from a rotor phase to a stator phase, that is, in the revers e direction from that for the angle I' or cc .) As a consequence, the changes in I' and 1'12 only differ in sign dy = da/p = - dl'12 = - da12/P dee = - da l2 H ence , it is legitimate t o write the derivative as follows: dW dW T em = -d- = - P -d- = y
p2-r:ool r,B1mB 2m · ~to
0: 1 2
B lm
= const,
B 2m
SIn a 12
(29-1)
= const
Expressing the magnetic flux densities in terms of currents and recalling Eq . (28-3), we get
T em =
mlm2P 2
I 12m I L SIn . a12
where L m is the maximum mutual inductance between the stator and rotor phases as defined by Eq . (28-3) . The elec tromagnetic torque acting on the rotor is positive (that is, is directed counterclockwise) when 0 < a12 < rr, and negative when n < a l2 < 2n (or 0 > a1 2 > - rr). On expressing B 2 in terms of 1 2 and noting that 'P'2lm
=
2Blm'J:lr,W2kw2/n
is the peak flux linkage of the stator field with a rotor phase, we may write the electromagnetic torque in t erms of current and flux linkage as 2P I nr . (29-2) T em = mV2 2 T 21m SIn a12 23*
356
Part Two. Energy Conversion by Electrical Machines
or, in complex notation , I [nr 1~*] T em= m2P l fz m T 21 2 where ~' is the complex conjugate of the secondary (rotor) current. Using the above equations, it is an easy matter t o show tha t the interaction of 1 2 with its own tield or flux linkage qr 22 produces no electromagne tic torque . To demonstrate, by Eq. (29-2) , this to rque is zero: ' ¥ z 12HI 22m SIn CG 22 =
m2P
i
0
where qr 22m = 2 B 2 m L loW 2kw 2/ :rr, is the peak flux linkage of the useful self-field with the rotor turns, and CG 2 2 = 0 is the angle between qr 22m and 1 2 , Now we are in a position t o express the electromagnetic torque in terms of the total flux linkage with a given winding, If 20m' that is, in terms of the flu x linkage produced by both the external field , If 21m' and the self-fi el d, qr 22m' For t his purpose, we add to the right-hand side of Eq. (29-2) the zero torque associated with the self-flux linkage T e m --
m 2P I
¥z
= V~
m
[in
TU m
1~']
Im [(W21m
I
2 - -
m 2P I
¥Z
m
[lIf 22m ~ 1* ] 2
+ 1Jf22m ) 11]
1~'] = m¥2 Pz I m [ill T20m 2
-_ m2P ¥z
HI i
1~' 20m 2 sin CG20
+
(29-3)
ijr 22m -qr 21m = peak t ota l flux linkage 'of the main field with the ro tor winding CX 20 = angle between 1; (or B2 ) and "If 20 m (see . Fig. 29-1) A torque, equal in magnitude but opposite in direction, is also acting on the stator. It can be found by Eq . (29-1) or Eq. (29-2), recalling that the t or que at the stator is deemed positive when it is acting clockwise (that is, against the sense of rotation). Alternatively, the electromagnetic torque at the stator may be expressed in terms of sta tor
where
"if 20 m =
357
Ch. 29 Electroma gne tic Torque
qu antit ies. To t hi s end we write B I in E q. (29-1) in terms of II an d recall t hat Th en ,
t ern = ».» ' a1" , r - 1If12m I 1 SIn
"
v 2
=
»,» I m ['If*10 1-] ------r=1 1/ 2 -
(29-4)
where 1Jf 12m is the peak flu x li nkage of the rotor field with a stator phase. Since the in teraction of II with the self-field or self-flux linkage pro duces no electromag net ic torque, i. e.
it is an easy matter t o express t he electromagnet ic t orque at t he stator in terms of t he to t al flux linkage of the useful fi eld with a stator phase
T em
=
»-»
ur
- r = - r10m
1/ 2
I 1 S I. n
a10 =
mlP ,rV 2
Im ['If*10 m I~1]
(29-5)
+
1jJ12m if11m = peak t otal flux li nk age of t he useful field with a st ator phase a I O = angle between -Plom (which is in phase with -p 20m) and L (as reckoned counterclo ckwise from flu x linkage to wards cur rent). Equations (29-4) an d (29-5) are equally applicable t o a polyphase ro tor winding and a d.c.-en ergi zed single-phase rot or winding suc h as used in synchronous machi nes. ' Equations (29-3) and (29-5) where the elect roma gnet ic torque is expre ssed in terms of 1ff 20 m or 0/ 10m also hold for sa turable ma chines with nonlinear cores. Deri ving t hem subject t o t he remarks ma de in Sec. 18-2, it will be seen that to fin d t he elect romagne t ic t orque in such a case it will suffice t o sub st itute into Eq. (29-3) or (29-5) the peak values of the fun dam ent al flux li nk ages found with allowa nce for the nonlinearity of the ma gneti c circuit,
wher e
W10m
=
::1:
Part Two. Energy Conversion by Electrical Machines
358
29-2
The Electromagnetic Torque Expressed in Terms of Electromagnetic Forces
In the previous section, we have found the electromagnetic torque from the law of conservation of energy. It can be determined in other ways as well. For example, we could combine the torques due to the electromagnetic forces which arise when a rotating magnetic field interacts with the elementary currents and elementary surfaces of the magnetized cores. We could then have obtained a more detailed picture about the distribution of electromagnetic forces throughout the active parts, the flows of energy converted by a given machine, and their directions. Unfortunately, the mathematics involved would be prohibitive out of an y proportion. Therefore, if we are only interested in the main electromagnetic torque associated with the fundamental mmf and airgap magnetic flux density, it is convenient to use the concept of the surface current which replaces the currents in the core slots. (i) Surface Current and Its Fourier Expansion We obtain the surface current on replacing the toothed core by a smooth one and spreading each slot currents i s /n over the core surface as a thin sheet with a linear density given by Ash = ish/b s The replacement of slot currents by a surface current is illustrated in Fig. 29-2 which repeats the winding and current patterns shown in Fig. 25-8 for i'.4. = V:2 I a and in = i c = - 112 I a/2. Shown below the sectional view of a slot layer in a toothed core is an equivalent smooth core. The air gap is enlarged k 6 times, and the currents are shown spread outside the slots but within the slot boundaries as thin sheets of density ASh. The slot current ish is the sum of alternating currents in the conductors laid out in the kth slot. For example, the current in slot 2 of Fig. 29-2 enclosing the forward conductors of phase A and the reverse conductors of phase C is i S2
=
iAlOC -
iclO c
At the instant of time shown in Fig. 29-2, when iA = i A m a x =
-V 21a
i p=i 9 = -1I2I~/2
359
Gh. 29 Electromagn etic Torque
the current in the second slot will be i S2 =
V2" Iawc + V2" I awc/2 =
(3/2)
V2" I awe
The current ish and the corresponding surface current density for the kth slot, A sh = ish/b s ' are t aken t o be nega tive if the cur rent is shown flowing "inwards" (away from -the read er), Ax i s A
Y
z . . . . . . .. A
;
PI. or 2
rt:'
2
Fig . 29-2 Surface current densi ty and mmfs of a pol yph ase winding = 3, q = 4, i A = V2I a , in = i c = - lffl a/2)
(m
that is , with the Z-axis. Th er efore in , sa y, slot 9 the current at the instant of t im e in question is negative \ iso= i nwc+i nwc= -l/ZIa we
Betwe en slots, the surface current density is zero . Th e cycle of change in the surfa ce curren t density is the same as for slot currents. R ound the peripher y of the air gap , it changes p t imes (wher e p is the numb er of pole pairs) . A~
360
Part Two. Ene rgy Conversion by Electrical Machines
slot cur rents va ry , the wa veform of surface current densi t y also varies in .a continuous fashion. On expressing slot currents as functions of time and expanding the spatial distribution of surface current density in to a Fourier series, we could find th e fundamental surface current density with 2p poles round the periphery an d with peak va lue A Im, as shown in Fig. 29-2. Unfortunately, the above procedure is too time- and effor t consuming. ' A far simpler appro ach is to expr ess thes urface
Phase axls A
Z Fig. 29-3 Relationship between the tangential field intensity and surface curr ent density (axial component) in polar coordinates
current density in terms of the mmf or of its harmonic COIlJponents. sPrior t o that , it is neces sary t o find t he tangential component,H'\', of the airgap magnetic in tensity on the smoo th cor e surface where t he surface current is distribut ed as an infinite thi n shee t of thickness 11 = 0 an d of a linear density A (Fig. 29-3) . In polar coordinates , the surface curr ent density A , and al so the slo t currents, are directed along th e Z-axis, and A = A z - Let us find the current l1i for a surface element of length R 111' . , l1i = A x . R 111' Enclose t h is curren t by a rec tangular loop labelled 1-2-3-4 an d having a r adi al dimension h and a tangential dimension R (111') . Applying Ampere's circuital law t o the circulation Of t he ma gnetic in tensity round the loop 1-2-[/-4 where h -+ 9
36'1
Ch. 29 El ectromagnetic Torque
and noting that the magn etic intensity on sid e 1-4 lying within the core of an infinite permeability is zero, we get
~
n, dl = H1'R!1y = !1 i
On passing to the limit with !1y --+ 0, we ohtain
H l' = H t = lim (!1i/R !11') = A This is the expression for H 1" t he magnetic intensity on the surface of the inner core . Applying the same procedure , we can obtain an expression for the magnetic intensity on t he surface of the outer core, H'I' = - A. To sum up , the tangential component of the magn etic intensity, H 'I' = H t, on the surface of the core with ~ta = 00 is equal in absolute value to the current density on that surface , I A !. Proceeding from Eq . (23-9) and setting dx = Rd'\' in polar coordinates, we can write the tangential magnetic intensity as a derivative of the scalar potential on the core surface H 'I' = -
drp/R d'\'
Finally, using Eq . (24-5) . and the accompanying relations between the mmf and the distrihution of rp on the excited core surface (F = qJ when the inner core is excited , and F = - qJ when the outer core is excited), we can express the surface current densit y on the inner or outer core as a derivative of the mmf A
=
-
dF/R dy
(29-6)
This relation can be applied not only t o the surface current density as a whole , bu t t o its harmonics as well, so that for each harmonic mmf there will be a surface current density harmonic of its own. Now we set out to find the fundamental surface current density when a pol yphase winding car ries PPS curren ts of frequency CD . The fundamental mmf is a rotating wave described by Eq. (25-10). Proceeding from Eqs. (29-6) an d (25-10) , we have A = -
dF/R d'\'
=
A I m cos (CDt -I- n/2 -
a)
(29-7)
In terms of the peak value of t he fundamental mmf, the peak value of the fundamental surface current density; A ln~ !
362
Part Two. Energy Conversion by Electrical Machines
can be written (29-8) If we use Eq . (25-16) for F i m, then Aim can be expressed in t erms of the sum of rms slot currents , 2wc I a = I s as
AIm =
11 2" Aokpkd
(29-9)
where A o = 2wcIait z = Is lt z is the line load current found from the sum of slot curren ts . Axi~
A
Fig . 29-4 Fundamental compone nt of surf ace current density (in a two-pole mod el)
At a point at an angle a to the origin , t he surface curren t density can be presented as the projection of the compl ex surfa ce current densi t y Aim
=
v.; (nIT) =
Ai m exp [j (wt
+ n /2) ]
all the axis at an angle a, t hat is, A = Re [A i m exp (-ja) ] Using Eqs . (29-7) and (29-9), it is an easy matter to plo t the fundamental component of surface current density on the same scal e as th e slot current densities. This plo t is shown ill Figs. 29-2 and 29-4. Figure 29-.(' al so shows the surface cur ren t den si t y phasor Aim.
363
Ch. 29 Electromagnetic Torque
From the foregoing, we may conclude that the fund am ental component of linear surface current den si ty is a rotating wave with period 2'1: and amplitude A ]m, t ra vell ing at the same mechanical angular velo city Q (or an elect rical angular velocity CD in the model) as the fundamental mmf. Irrespective of the direction in which the mmf is ro tating, the
Fig . 29-5 P attern of th e ma gnetic field set up by currents in the inner core of a machine
surface current densi t y wave always makes an angle of n/2 with the mmf wave or is displaced through r /2 counterclockwise. The above rel ation between t he tangential magn etic in t ensity and t he surface current density, Il , = H 1, = + A , also holds for their fundamental component s. Therefore , t he air gap magnetic field set up by t he slo t cur ren ts (or by the fundamental component of the equivalent surface current density) always has the tangential as well as the radial com ponent . Th e radial component of t he magnetic fl ux densi t y can be written in terms of t he mmf using Eq. (25-18) as
n, =
lkoF/k 1i6
(29-10)
The tangentia l magnetic flux densit y on the surface of t he excited core may be written directly in t erm s of the surface current as B t = B i' = ~t oH l' = + ~toA (29-11) Sin ce no tangenti al magneLic flux dens ity exists on the surf ace of the unexcited core , t he air gap fi eld set up by the funda ment al component has t he pattern shown in Fig. 29-5,
364
Part Two. Energy Conversion by Electri cal Machines
It should be noted, t hough, that the figure shows the field for a very large relativo gap , 6IL , when the radia l component is comp arable with the tangential component; compare Eqs . (29-10) an d (29'-11). For the small values of oIL usu all y encountered in practice , the radial component is sub stantially larger t han the tangential com ponent , so the field is essentially a radial one. If, however, we neglected the tangential components, H I" due to t he surfa ce current of
Fig. 29-6 Relative position of rotating waves of radial magnetic flux dens ity B ; and surface current density A
density A, we would be unabl e t o get proper insigh t into the generation of electromagnetic forces and transfer of energy across the air gap when an electrica l machine is running.
(ii ) Elec tromagnetic Torque Let us find the electromagnetic torque acting on the rotor rot at ing at a mechanica l angular velocity Q (or at an electrical angula r velocity ro in the model of Fig. 29-6) . Sup pose that the poly phase rotor wind ing carries a current /2 with angular frequency w2 , wh ich gives rise to t he fundamental surface curren t densi t y wave of peak va lue A 2 m • The mech anical angular velocity of the surface current density wave relative to t he rotor is proportional to the frequency of the
Gh. 29 Electromagnetic Torque
current
365
Q 2 = w 21p The electrical angular velocit y of th is wave relative t o the rotor in the model is t he same as that of t he cur rent and is equal to W 2 • Relative to the st at ionary frame of ref eren ce, the surface current wave ro tates at a mechanical angular velocity QI = Q + Q2 ' The electrical ang ular velocity of the wave relative to t he stator in the model, WI = W + W 2, in the case of uni directional energy conv ersion (see Sec . 29-1) is always the same as the angular frequency WI of currents in the pol yphase stator winding. Therefore, the fundamen t al stator mmf rotates at a mechanical angular velocity QI = wIlp and produces, toge ther with the fundamental component of the rotor surface current rotating at t he same mecha nical angular velocity, t he useful rotating magnetic field in whi ch the radial magnetic flux density has a peak value given by B I (O)m = B om . Let the angle between B o and A 2 z/ (or between Bom and A2m ) be Fig . 29-7 Electromagnetic fordenoted by P02 and counted ces acting on an element of from A~2m towards jj om' surface current The electromagnetic t or que can be defined as t he sum of the torques developed by the electromagnetic forces dN acting on elementary surface currents di = A 2 (R dy). Assume t hat each elementary current extends along the machine axis for a distance equal to a unit of length and is lying in a magnetic field in which the radial magnetic flux density is B r • Then t he elec tromagnetic force that is acting on that elementary current in a tangential direction ma y be written as dN = B ; di = B r A 2R dy The direction of dN can be found by t he left-hand rule. An elementary surface current at an angle l' to the origin and the force acting on it ar e sho wn in Fig. 29-7.
Part Two. Energy Conversion
by Electrical
Machines
The electromagnetic torque acting on the surface current and transmitted to the shaft by the mechanically stressed rotor core can be found by adding together the torques produced by the elementary forces dT em = R dN over the entire rotor surface facing the air gap (from '\, 2:n: along the effective core length lo)
=
0 to
'\, =
2n
2n
T em = lo 1 RdN = loRz1 BrAzd,\, o 0 Using Eq. (25-18) for a rotating magnetic flux densit.y wave B; = B om cos (WIt - a)
and Eq. (29-7) for a rotating surface current wave
A z = A zm cos (WIt - a -
Poz)
where a = py, and on setting for simplicity Wit= 0, because the integration yields the same result for any instant of time, we obtain p2,;2
Tern = :n:RzloBomA2m cos P02 = -n- loBomAzm cos Poz
(29-12)
Noting that the peak value of the linear surface current density can be expressed in terms of I z [see Eqs. (29-8) and (25-9)],
A zm = :n:Fzm/-t =
ev 2 m2/-t) (Izwzkwz/
p)
and the peak value of the magnetic flux density defined by Eq. (27-2) can be expressed in terms of magnetic flux B om = :n:cI>m/2'tlo the electromagnetic torque can be written as a function of the magnetic flux and winding current
Tern = (pmz/V 2) (wzkwzcI>m) I z cos Poz
(29-13)
If we recall that the peak value of the total flux linkage with a phase given by Eq. (27-14) is
"IfZOm = w2k wzcI>m
and replace Poz by sin a oz = sin (n/2 + Poz) = cos P02 [see Fig. 29-6 and Eq. (29-7)] 1 the expression for the torque
367
Ch. 20 Ele ctromagne tic Torque
will be analogous with Eq. (29-3): Tern = (pmz/V
2) '¥zom1z sin a oz
where a 0 2 is the electrical angle (see Fig. 29-6) between the flux linkage phasor"lf 20m and the current phasor 1 2 (or the mmf phasor F 2)' As already noted in Sec. 29-1, the torque equation, Eq. (29-3), also holds for saturated machines with a nonlinear magnetic circuit. A further proof of that statement is the fact that it is analogous to the equation derived here for a saturated machine.
.
29-3
.
Electromagnetic Force Distribution in a Wound Slot
In finding the electromagnetic torque acting on the rotor, we replaced the toothed rotor whose slots carried certain currents is by a smooth core, mo ved the currents is to the surface, and distributed them in the slot regions as an infini te thin sheet of linear density A s = i s/ b s (see Fig. 24-2). The replacement of a toothed by a smooth core will leave unchanged the tangential electromagnet ic force N acting on a slot, if the mean air gap magnetic flux density due to the external magnetic field is as found with all owance for the saliency of the stator by the equation Bo,mean = fJ-o F/{) k 01k oz
where F
=
k 01 = k oz =
() =
external mmf produced by the stator currents at t he axis of the slot in question stator air gap factor rotor air gap factor radial gap length.
The total tangential electromagnetic force per unit length of air gap can be found as the sum of the for ces dN
=
B o,mean Asdx
applied t o the elementary surface currents +bs/ Z
N
=
J
-b s/2
s; rneanAs dz = s; meanAsbs = B o• meani s
3G8
Part Two. Energy Conversion by Electrical Machines
Let us now see how this force is distributed in the slot region. The resultant magnetic field in the slot region may be visualized as the sum of the external field (Fig. 29-8a) and the self-field set up by the slot current (Fig. 29-8b). To determine the tangential forces acting on the slot sides and the slot current, it will suffice to find the respective flux densities in the slot sides and at the centro of a currentcarrying conductor. As has been found for real slots and an infinite core permeability (~~I'C = (0) , nearly all field lines entering a slot terminate in the slot sides and only a small fraction of the lines reaches the current-carrying conductor. If the flux per unit slo t length (see Fig. 29-8a) is + bs/ Z
cD =
J' B I1(y=o) d l : = B o, meanbs -l>s/2
then the flux passing through the slot section at the level of the current-carrying conductor (y = hi) will he +l>s/2
ccIJ=
~ BY(Y=hj)dx=Bsb s - "s/~
where c ranges from 0.002 to 0.001 . . Because the magnetic field is symmetrical and continuous, the flux entering the slot side
cD o = (cIJ
ccD)/2
-
differs but little from cD/2. On moving away from the air gap, the external magnetic flux density on the slot sides, B 01 and B 02' rapidly diminishes so that level with the conductor top (y= h) it is zero very nearly. Therefore , the external flux enters the slot sides within the depth 12 h
h
cPo = cI) (1-c)/2 = ~ B 0 1 dy = ~ Bozdy o
0
From a comparison of the expressions for c~ and ccD the external flux density in the conduct or region can be written
B
=
cB 0 .
mean
369
Ch. 29 Electromagnetic Torque
7l.. ..".~~----<~~.."..
Bit
- BiZ
(h) ////////////////// / /!/I/ /4
Bz
(c) Fig. 29-8 Distri bution of elect romag netic forces in the reg ion of a wound slot: (a) external magne ti c field due to currents on the ot her core, I s = 0, B /i,mean=(= O; (b) magn etic field due to slot cur ren t , B/i,m ean=O , 1 s= = 0; (c) electromagnetic forces in and aro und a slot, I s =(= O, B/i,mean=O
24 -016 9
370
Part Two. Energy Conversion by Electrical Machines
The pattern of the field set up ;by the slot current and the distribution of the magnetic flux density in the slot sides are shown in Fig. 29-8b. Within the larger part of the space above th e conductor (0 < y < h), the magnetic flux density in the slot sides is nearly constant. Applying Ampere's circuital law to the loop enclosing the current and coincident with a field line over ' the slot width , the respective magnetic flux ' "densit y can aproximately be written as
B il
=
B i2 = ftoic/bs
Now we are in .a position to find the tangential electromagnetic forces asso ciated with the resultant magnetic field in the slot region (see Fig. 29-8e). The curren t-carr ying conductor is acted upon by a tangential electromagnetic force given by
Nc
=
Bci c
=
eB a,mean i c
=
eN
which is a small fraction of the total force N = B a,meanic
acting in the wound-slot region. The greater part of the total electromagnetic force is applied to the slot sides as a magnetic pull. Let us first find the specific magnetic pull, T (in pascals), th at is the force per unit area of a ferromagne tic surface (ftr ,Fe = 00) in a magnetic fi eld . It can be expressed in terms of the normal component of magnetic flux density, B n , whi ch acts at the surf ace of a ferromagnetic bod y in the same direction as the total magnetic flux density, and in terms of the permeability fto of the nonmagnetic medium surrounding that body: . T = B212fto The specific magnetic pull vector is always aligned with the normal ;; to the surface of the ferromagnetic body, directed towards the medium having the lower permeability T = ;; (B2/2fto) The specific magnetic pulls T 1 an d T 2 acting on the slot sides are normal to the sides. The dis tribution of
37'1
Cit. 29 Electromagnetic Torque
T I an d T z
T z = B;/2~Lo
T I = Bi / 2 f-lo,
over the slot depth is a funct.ion of
B1
B Ol -t- Bi,
=
on th e left-hand sid e, and of
B ; = B 02
Bi]
-
on the right-hand side . Because they are unbalanced only over the int erval 0 < y < h, the resultant tangential force
N ss
= JVS S I
N ssz
-
applied to the slot si des can be found b y combining the ele m en tar y forces on each side within the limits specified: h
N ss=NsSI-Nss2=
JT
h lcly
-
~
h
T 2dy =
00
J(Tt-T z)dy 0
Noting that we get
TI
-
T 2 = (2/ ~Lo) BolB il
In taking the integral , i t should be re called that B i l ~ f-loic /bs
=
cons tan t
for 0 < y < h (see above). So, on expressing the flux through a slo t side h
JB
Ol
dy
a in terms of the external flux, cD
= bsB 0 , mean'
h
N ss=
we get
It
J(Tt -T z) cly= (2/f-lo) Bi J B l
oldy
a a = (B ii/f-lO) (1- c)
372
Part Two. En ergy Conversion by Electrical Machines
N e , the force applied to the current-carrying conductor,
N = N ss
+Ne =
(1 -
e)B Il,mean ie
+ eB Il,mea nie
= B Il,meanie This for ce is equal t o the for ce ac ting on ie ' the slo t curran t shifted to the core surface. As we have learned , howe ver, t he greater proportion of this force acts on the slo t sides r aLh er th an on t he conductor in a slo t (e = 0 .00 1 to 0. 002). Our reasoning has been based on cer tain simplifying assumptions as regards the distribu tion of the ex ternal flu x density in t he air gap an d of t he flux densi t y du e to t he slot current in the slot sides. However, th e rigorous appr oach would lead t o t he sa m e solution - a fac t of impor tant practical significance. Because electr omagnet ic forces l argel y act on the slot sides (or the core t eeth), the conductor insula tion m ay be designed as me ch anically s tr ong as m ay be necessary t o t ransfer N'; = eN which is a very small quan tity. To sum up, owing to the shielding action of teeth on a toothed core, the exter n al fi eld in the region taken up by current-carryin g con ductors is substanti ally reduced , and the requirements for the mechanical st re ng th of insulation m ay be less str ingen t .
30
Energy Conversion by a Rotating Magnetic Field
30- 1
Electromagnetic, Electric and Magnetic Power
The elect r om agneti c power en taring a 1'0 tor surface element dS = '1 X Rely from the ail' ga p is a fu n cti on of the powe r dev elop ed by an eleme n t of tor que as an ele me nt of current rotates at a m echanical angular vel ocity Q 1 (see Sec. 29-2) (30-1) Le t us writ e t he me ch anical angul ar ve locity of a surface current el em en t as t he sum of Q , the rotational angular veloci t y due to the rotat ion of t he 1'0101', an d Q 2' the ang ular velo ci t y of t he curre nt sheet rel a ti ve to the rotor body,
373
Ch. 30 Energy Conversion by Rotating Field
associa ted with period Ic variations in the curren ts of the rotor winding: Q] = Q
+Q
2
Then the electromagnetic power entering a rotor surface element can he written as the sum of the mechanical power
dP m = Q dT transferred via this surface element to the shaft, and the electrical power entering the sur fac e current elemen t or the winding to which it is equivalent : dP e m = Q 1 dT
= = ttl
QdT + Q 2 dT dP m
+ dP e (30-2)
The electromagnetic power flow per unit area (power flux density) and the direction of this flow may he defined in terms of the --+
radial component, 'fl' e m , of the Poynting vector [24] --+
:f?em
Fig . 30-t Flux density: elect ro magnetic power (P em), elect r ic power (P e)' and mechanical power ( Um ) (on th e su rfa ce of th e rotor core)
= E em X H I' = q zq i,E emH i ' = - q,.E emH y
which is seen (Fig. 30-1) to he the product of the axial com ponent of the electric in l ensi ty vector, E e rn = q zE em, and the tangential com ponen t of the magnetic in tensity vector, H I' = Cl I'H I" H er e , Cl i " (I " an d Cl ,. are the unit vectors along [he respective axes of a cylindrical coordina te system . If we recall that th e tangential magn etic intensity on the rotor surface is equal to the surface current density (see Sec. 29-2) HI' = A 2 ancl al so tlia L in defining the to tal energ y en tering a surface element we should take into account the electric intensity
374
Part Two. Energy Conver sion by Electri cal Machines
found from the total lineal' velo city of a surface current wave "1 = fJ VUI = q vRQlj
that is ,
E em
=
B, X " I
=
fJr X q vBrUI = q zBrRQ I
th en the radi al componen t of the Poynting vector may be written
-
~ em =
where
= -
,fT' em
+ fJ r ;P em
E emH v
(30-3)
= - B r R QIA 2
Naturall y, the electromagnetic power ent ering a r otor surface element , '1 X R dy, and expressed in terms of th e radial component of t he Po ynting vector -+
elPem - =. ~ emR dl' = - BrRQ IA 2 (R ell') = - Q I d T is the sam e as given by Eq . (30-1). The " _ " sign implies that the power flow is directed inside t he rotor, that is, oppos ite to the radial uni t vect or qr ' In a similar man ner, we can represent the electric power entering a surface curre n t eleme n t. In this case, however, the elect r ic intensi ty must be dedu ced from the' velocity of the surf ace current wav e, solel y related to vari ations in the winding current s: "2
=
dP e = ff'e R dl' = --
fJ vU2 = q VR Q 2 (B rRQ 2 ) A 2 R dl'
= -
Q 2d T
(30-4)
-+
where ff' e = E e X H v = qr ff' e, an d f? e = - E cH v = - B r R Q 2 • Here, the " _ " sign likewi se indicates tho directi on of the power fl ow (see Fig. 30-1). By th e same token , the mechanical power f1P m = Q dT ma y be expressed in te rms of th e radial componen t of th e Umo v vector , D , defined as the mechanical power flux en tering a surface element of a mechanically straine d body . Th e radial component of th e Umo v vec t or fo r a rot ating body is determined asthe product of the taug en tial mecha nical stress T v by the ' ,t angent ial linear di splacemen t velocity of a surface element , /I.., = u. = RQ Dr U';
=
fJrUr
=
-
TvUv
-
A-aBr R Q
(30-5)
375
Ch, 30 Energy Conversion by Rotating Field
The tangenti al mechanical stress 't v arises on the outer surface of the ro tor because it carr ies surface curren t elements and the tangential forces d N act ing on t hem 't v =
dN/R d'V =
(qzA2R dl') x qrBr R dl'
=
CI v't v
(30-6)
wh ere 't v = AzB r [N m "]. The mechanical power entering a rotor surface element (30-7)
is the same as given by Eq. (30-2) wh ere it is expressed in terms of the electromagnetic torque. By taking an integral over the rotor surface , we can find the electr oma gnet ic and electri c powers t ha t en tel' the rotor in th e process of energ y conversion: 2n
P ern = l o
Io
2
dP ern =
u.i, j
2n
Pe =lo
»; =
i,
dT
= Q 1T
0
-"
2
j dP e =Q j dT =Q T " 2l 0
2
o
0
2n
2n
j dP m = Ql o j"
o
(30-8)
dT = QT
0
Obviousl y, as follows from Eq. (30-2),
r.;
=
» ; + Pm
The electroma gne t ic power , P em ' is the total power tra nsferred by the ro tating fi eld t o the rotor (Fig. 30-2). Some of this power , P e- is dissipated as heat in t he ro tor windi ng or in t he line connec ted to that winding. This can be proved by re-arranging Eq. (30-8) wi th t he aid of E qs. (29-3) and (27-'1 5) and also rec alling tha t ~02 = ao z - n /2 is, at the same time, the angle hetween I z and E z in the rotor winding. Therefore , "
Part Two . Energy Conversion by Electrical Machines
where pQ 2 = (02 is the electrical angular frequency of E 2 induced in the rotor winding by the rotating fie ld .
Fig. 30-2 El ectromechani cal energ y con versi on by a machine (on t he left) and in its model (on the ri ght)
Finally, the remainder of input power is converted to mechanical power Pm = QT transmit ted by a mechanically loaded shaft (as the power flow defined by the Umov vector) .
* 30-2
Energy Conversion in an Electrical Machine and Its Model
As we have seen in the previous chapters , t he two-pole model of an electrical machine is a convenient and instructive tool with which to study what goes on in the machine itself. This is also true of th e even ts in vol ved in energy conversion . In the model shown in Fig. 30-2, we retain the prototype's pole pitch T, fundamental amplitudes of magnetic flux
Ch, 30 En ergy Conversion by Rotating Field
377
densi ty and surface current waves , B 0 111 and A z, and also phase displacement between these waves in fractions of a pol e pitch. Therefore , in the mod el the wa ves tr av el thr ough an angle ~oz which is p times as great as the angle between these waves, ~0 2Ip, in the machine itself . (On th e left. of Fig. 30-2 , t he m achine's field has four pol es.) Also, we retain in the mod el th e proto type's linear peri pheral velocities of the waves and of the cor e element , U 1 , U z and U, on the outer surfa ce of th e rotor. This is don e because in the model the r adius of this surface is 1lp of its value in the prototype, whereas the angu lar velocities are p times as high: U 1 = (R ip) QIP = Q1R liz = (R ip) Qzp = QzR U = (R ip) Qp = QR i
The t angential electromagne tic force acting on the surface current in the model's rotor is the same force as operates on the surface curren t over a pol e pitch in t he pro to t ype , that is N ip, which is 'lip as large as the tot al tangential force (see above) 2n
N = i,
J dN o
The electromagnetic t or que in the model is 'lIpz of its magnitude in the prototype, because it is given by th e product of th e tangen t ial for ce in the model , N ip , by t he radius of its ro to r , R ip : (N ip) (R ip) = N R lp? = I'lp ? Ho wever , the powers entering t he r ot or of t he model do not differ from t he respective powers exi sting: over a pol e pitch in the prototype machine. This can be demonstrated by applying Eq. (30-8) 't o the model: (Q1P) (T lpZ) = Q1T lp = P emlp (Qzp) (T lp Z) = QzT lp = P elp (Qp) (T lpZ) = QTlp = Pmlp The same relations can be obtained , if we re call that at the similar poi nts over a pole pitch in the pro to type and in its model (that is, the points wh ich are a t angles '\, and
378
Part Two. Energy Conversion by Electrical Machines
a= P"Y, respectively) , the radial magnetic flux densi ty B n
the linear surface current density A , and all the velocities are respectively the same. Accordingly, P em, P e and Pm' definable in terms of the radial components of the Po yn ting ane! Umov vectors, Eqs. (30-3) through (30-5), are .likewisa the same at th e similar poin ts in t he ma chine and its model. As is seen , the model is convenient not only in calculating th e air gap fiel d, mmf, emf, and flux linkage, and in plotting
Fig. 30-3 Electrical ma chine an d an equiva lent system of p twopole model s
th eir respective ph asor an d vector diagrams wh ere these qu antities app ear as phasors and/or vectors, but also in ana ly zing power fl ows . In doi ng so, it is import an t to hear in mind tha t the powers in the model are 1/p of their magnitu des in t he pro tot yp e ma chi ne. Therefore , as regard s .energy conv ersion, a 2p-pole machine maybe replaced by a system of p models ha ving a comm on shaft an d connecte d to the same lines and ro t ating at a speed multiplied p t imes*. Obviously, such a model system will handle the same powers. A four-pol e ma chine and an equi vale nt syst em consisting of two models are shown in Fi g. 30-3. The pro totype is a syn-
* In stead of combining p models wi th th e same core length l6 as in th e prototype ma chine, we ma y use one two-pole machine in whi ch . th e core length is p times as large'. .
Ch. 31 Energy Conversion Losses
379
chronous motor (see Part 5) in which the rotor winding is energized with direct current (W2 = 0) and the me ch an ical angular velocity of the field, Q l ' is the same as the mechani cal angular veloc ity of the rotor , Q . Therefore, all of t he electromagnetic power, P em = Q1T, applied to the rotor is converte d to mec han ica l power, P m = QT, and the elec tric power, P e , entering the rotor winding is zer o F ; = Q2T = 0 The directions uf the power fluxes , electromagnetic forces and torques hold for the motor mod e of operation . The external to rque acting on the motor shaft is denote d by Text .
31
Energy Conversion Losses and Eff iciency
31-1
Introductory Notes
As has been shown in Cha p . 2'1, for electric energy , P " to be conver ted into mechan ica l energy , P m' or back by a rotating electr ical machine , the following conditions must be satisfied. (i) The rotor whose shaft transmits mechanical energy must be rotating continuo usly . (ii ) The windings must carry currents whose frequencies are related to one another and to the mechanical angular velocity of the r otor in a certa in defin ite manner. (ii i) The m agnetic fl uxes li nki ng the wind ings that are responsib le for energy conversion must vary per iodi cally . As a consequence, some of t he energy han dl ed by an electrical machine is inevit abl y dissipat ed owing to fr iction between t he rotating parts; t his is what is known as mechanical losses. Another fracti on of the t otal energy is lost as currents t raverse the wind ing conductors; this is electrical losses. Still another fraction of the total energy is lost as the cores underg o cyclic m agn eti zati on ; t his is magnetic losses . All kinds of losses are customarily expressed in terms of the equ ivalent t hermal energy dissipated pel' unit time or the time rate of energy loss, "x'P . In our subsequent dis cussion, it will be collectively called the power losses.
:380
Part Two. En ergy Conversion by Ele ctrical Machines
From th e law of conserv ation of energ y , it follows that the useful output power from a machine is alwa ys smaller than its input power by an amount equal to the power losses . The ratio of output power to input power gives what is known as the effi ciency of an electrical machine, defined as 11
=
P e/PIll
=
'1 - 2:.P / (P e
+ 2:.P)
in the generator mode of operation, and as 11 =
r;»,
=
'1 - 2:.P/(P m
(31-1)
+ 2:.P)
for the motor mode of operation . It will have been noticed that the effi ciency is expressed as a fraction , th at is , on a per-unit basi s. It may as well be expressed 011 a percentage basis. The efficiency of an electrical ma chine is le ss than uni ty on a per-unit ba si s, or less t han '100 % on a percentage basis. Obviously, as the losses decrease th e effi ciency approach es unity (or '100%) . To prevent overheating, the heat dissipat ed in a machin e mu st be withdrawn a nd disch arged to th e surroundings b y a cooling system using a gas (mo st Irequent.lv, air) 01' a liquid as the cooli ng agent. 31-2
Electrical Losses
The electr ica l losses in a m achine can eJfect iv el y be redu ced by making its conductors of a material having a low resistivity , Pt . The best choice is sof t coppe r wire of circula r or rectangular cro ss-section with a low impurity content. Accordingly , this kind of l oss is trad i Lionally called th e copper loss. The second best choice is al um in ium which is cur ren tly used on a limited , bill, an ever increasing sca le . Its resistivi ty is, how ever, mu ch higher than tha t of copper. Bec au se the wind ings carry al terna ting current, we h ave to reckon with the skin effect. It gives rise to variations in the inductive impedance and , as a consequen ce, in th e distribution of current density over the cress-sec tion of conductors. Th ese variations ar e more noticeable in the conductors laid ou t in slots than when they are surroun ded by a nonmagnetic medium , say, ail'. This loss is well known FR loss , but R must be the effec tive resistance. The mea sured d .c. resi stance is only th e effecti ve resistance at low frequencies, when the current
381
CII . 31 En ergy Conversion Losses
dis tribution m ay be assumed to be uniform Ro =
where 2wl m ean
2wlme an
P t-~=
Sa
(31-2)
= leng th of series-connecte d wind ing (or
pha se) con duc tors lm ean = m ean length of a half-turn S (as b s) CaCb = cr oss-sect ional ar ea of th e effecti ve conductor a s, b s = dimensions of a rectangular strand in the slot height and width , r especti vely (see Fig . 31-1) Ca , Cb = number of strands in the slo t height and wid th , respecti vely a = number of circuit s in the winding Pt= P20 [(1 ex (t - 20°)] = r esistivity of the conduc tor at the design operating tem pera ture t ex = 0.00 4°C- 1 = temperature coefficient of resistance for copp er (or aluminium) For a. c. the copper loss has to be compu ted in te rms of R = k RR o, the resistance of th e winding with allowance for a nonuniform curre n t distribution over the cross-sec lion of the con ductors . The extent of variations in the a .c. di stribution over the con duct or cr oss-section depends on the magnitude of the slot leakage field. Becau se the lines of that fi eld ar e at right angles to the slot axis and are nearl y s tra ight lines in a rec tangular slo t (see Fig. 31-1), the slot le akage flux has just about the same linkage with any strands l ying at the same level in t he slo t height (say, str and s 1 and 2). Accor dingly, th e inducti ve impedanc es of su ch stra n ds are t he sa me, t oo. In contrast, the strands tak ing up different positions in the slot height differ in inductive impedan ce as well. As is seen from Fig. 31-'l a wh ere the slot is shown to con tain onl y one conductor, the inductive imped anc e (or flux linkage) of s tran d 1 which is nearer to the air ga p is sm aller than that of strand 3 lying closer to the slot bottom . This also explains \\ hv the curren t is dis tri Luted almost uniforml y across thE.'
+
382
Pa rt Two. Energy Conve rs ion by Electrical Machines
conductor width and less uniformly along the conduc tor height. 'When a slot contains only one conductor , th e cur rent. densi ty is higher in the s tr ands th a t are nearer t o the air gap (see the current disLribution curve in Fig. 31-1a). In that part of the cross-section, the cur rent den sity ma y substantially exceed the a ver age current density in the conduc tor J o = IIS = IIa~bs Th e part of the conductor section lying deeper in a slot carries onl y an in significant fraction of the total con rl uctor
(b) Fig. 31-1 Alternating cur re n t densi ty (J ) distribution over the cros ssecti on of an effect ive con duct or (a) one-piece effect ive con duc tor, Cn = Cb = 1, lt c = 1; (b) effective conductor subdivided into st ra n ds transposed in the slot depth , c" = 10, cb = 2, lt c = '1
current. As a result, t he useful cross-section of the conduc tor decreases, and its resist.ance goes up. This property is accounted for by wha t is known as the Fi eld coeffic ient defin ed as k
R
= ius,
As is seen from the foregoing, it is a function of th e heigh t and number of u nstranded effec tive conductors per slot, and is independen t of their width. Consider the most common forms of slot conductors. 1. Unstranded effective conductors. A slot with a single effective conductor which consists of 'only one s trand occupying the entire slot dep th (ea - 1) is shown in Fig. 31-1a. The height of the effective conductor is the same as that
383
Ch. 31 En ergy Conversion Losses
of the strand as. The number of effective conductors in the slot depth is the same as that of strands U c = m s = 1. The number of strands in t he slot wi dth, Cb, may be taken such that the total width of strands in the slot is b, = cbbs. (What is important is that t he effective conduct or is not stranded in the slot depth.) With this arrangement the magnitude of the skin effect is a fun ction of the rel ative height of a strand £ = a s / fJ. (31-3) which is defined as the ratio of t he height of a strand as, t o the skin depth * defined as fJ. = V2ptbs/CUflob1 (31-4) where bs = slot width bl = cbbs = conductor width in the slot co = 2rr,j = angular f zequency of the current flo = permeability of the conductor material (copper or aluminium). . Assuming that the skin effect is onl y observable within the active length of a half-tun: (the part enclosed in a slot) , that is , over the length lfll and is non- existent in the overhang, that is, over the length lmean - I f' ' we may wr it e for k R (3'1-5) k R = '1 (l o/Zmeari)( k R a - '1) wh ere
+
kR a
= cP
m+ ~ '1)(5) (k~m~ -
1)
(31-6)
+
7)/16 is the chording (pitch-shortening) where kiJ = (g~ factor (~ = ych:) for double-layer windings (for singlelayer windings, k~ = '1) , and cp (£) and '1 ) (£) ar e th e Emcle functions (see Fig. 3'1-2). For £~ '1, cp (£) = 1 For
5> 2,
+~
5~
'1) (£) = £4/3 cp (£) == £
'p (£) = 2£ * For a conductor carrying currents at a giv en frequency as a result of the electromagnetic waves incident on its surface this is th e depth below the surface at whi ch th e current density ha s decreased one neper below that at th e surface.-Transl at or's not e.
384
Part Two. En ergy Conversion by Electrical Machines
As is seen , the va lue of k R (and the winding loss) in creases with an increase in the relative conductor height and the numb er of effective conductors in a slot. As a consequence, when the effective con duct or in a slot is unstranded (ca = 1), the loss may be pro hibitively heavy. A way out is to strand 2.0 If
1. 6 1.2
6'
0.8
If
0.4
2
~
0
1
2
s
4-
0
Fig . 31-2 Emd e fun cti ons
it (in the slot depth) and to transpose the strands within . t he slot (case 2 below) or within the overhang (as in case 3 below)* . 2. Stranded effective conductors transposed within the slot. Ref erring to Fig . 3'1-'1b , each turn of th e effective conductor is made up of two bars joined (by sol deri ng, brazing or welding) at the ends and completely transposed within the slot. The construction of the bar is clear from Fig. 3'1 -3. Owing t o the tr ansposit ion , each stra n d suc cessi vel y takes up all th e likely positions (or levels) in a bar. As a result, the strands in the effective bar have the same inductive impedance , and the current in the effect ive bar is equally share d among all the strand s
Is
=
IIcac lJ
The current distribution may be other than uniform only within a given strand (see the J curve for strands). The nonuniformity is notice able in the str and sections lying nearer to the air gap where the leak age field is stronger. Even then,
*
f
St randing without transp osition would not redu ce th e losses. ,
385
Ch, 31 Energy Conversion Losses
the distribution in such sect ions is more uniform than it is in an unstranded effect ive conductor (compare Figs. 31-1a and b). The current densi ty at the peripher y of a strand diff ers bu t little from the av erage current density
r, =
Is /asb s
Therefore, the losses in a t ra nsposed str anded effective conductor are subs ta n tia lly smaller than they are in an f
2 34
5
8
I
III
I
I
I
I II
I
I
E=??Jd?eJ=fI!Id???z;:===J
d
0~-----,
Fig. 31-3 Bar conductor in which the strands are transposed within the slot
unstranded cond uctor of the same cross-sectional area. In such a case, k R can be found from Eqs. (31-3) through (31-6). The number of strands in the slot depth is where ue is the number of transposed effective conductors in the slot depth (as a rule, U e = 2). 3. Stranded effective conductors with strands transposed within the overhangs. The strands are transposed by twisting some of the effective conductors. Figure 31-4 shows a doubleturn coil with two strands per effective conductor (ca = 2, U c = 2). The strands are electrically joined (by soldering, brazing or welding) at the coil ends. Within the coil, they are insulated from each other. The inductive impedance of t he strands (say, 1 or 2) depends on the position they t ake up I n the depth of the slot where the coil is laid. In an "untwist:<: 5 - 0 16 9
386
Part Two. Energy Conversion by Electrical Machines
ed" coil (shown at the upper left of Fig . 31-4) an d in a "twisted" coil (shown at the bot t om ri ght 'of Fig. 31-4) where the effective conductor is turned 180 0 every turn, the strands t ake up different combinations of posi tions.
Fig. 31·4 Arrangement of strands in t he effective conductors of a coil. The sketch at the bottom right shows a part of a coil overhang with some of the st rand twist ed (tr ansp osed). The numerals in parentheses refer to the corresponding strands in the twi sted coil after each turn
In a "twisted" coil, t he difference in inductive impedance between the strands is less noticeable than in an "untwist ed" coil, and the current distribution among the strands is more uniform. For such a coil, k R is given by
,k R = cP (~c)
+ N (~c) /2 + L s (~4/6c~)
(lmean/lrJ (as/a;)2 (31-7}
357
Ch. 31 En erg y Conver sion Losses
= rela t ive he ight of the effec tive where Se .= ca S ~V/" as ~il6 mea n con ductor rp (Se), cP (Se) = the Emde func t ions for £e in Fig . 31-3 as = height of a bar e strand a, = height of an insulated strand L, L s = coeffic ients to be taken from the table according to type of transposition. Coefficien ts Land Ls L
Coil t y pe
Untwisted Twist ed eve ry turn
31-3
u~- 1 ) -1 ( 2
4
o
6u~
(
'
3)+ 61
4kfl-T
32
( ue2kfl' 1)
Magnetic Losses
Magn etic losses in the cores, or the core loss of ele ctrical m achines occur owing to periodic variations in t he magneti c field with t ime. Here, too , the core loss can be minimized by subdividing the core into electrically insulated elementary magneti c circuits. Th e required effect ive cross-sectional area of th ecore is obt aine d as the sum of the cross-sect ional areas of the elementar y magn etic circuits whi ch t ake the form of ferromagn etic lami na tion s in sulated from one ano ther and m ade in certain thicknesses . Th e material and thickness of t he lamin ations are ch osen accordin g to the frequency of cycl ic magn et ization. As has heen shown is Sec. 21-2, the frequency of cyclic magnetization for t he st ator is diff eren t from t hat for the rotor in the general case (WI =1= w z* ), eac h fre quency bein g
* Th e referen ce is to the most typica l a.c . mac hi ne in whic h oneof the win dings is laid in slots on th e sta t or core and t he other in slots on the rotor core. If the windings are carried by th e same core , two magnetic fields will exist within it , each var yin g at a frequency of its 0'""11, WI and 002' 25*
388
Part Two. Energy Conversion by Electrical Machines
the same as that of the current in the respective winding. In induction machines, the relation between the two frequencies depends on the rotational speed, and the lamination thickness must be chosen to suit the nominal speed of rotation. To obtain a uniform distribution of the magnetic flux over the cross-section of each lamination and to keep the core loss to an acceptable level as the frequency is increased, the laminations are made progressively thinner and from suitably alloyed electrical-sheet steels. The cores to be cyclically magnetized at about 50 Hz (which is true of, say, the _s tat ors of synchronous and induction machines) are assembled with laminations punched from hot-rolled electrical-sheet steel, usually 0.5 mm thick. The more recent trend has been towards cold-rolled, nonoriented-grain electri-cal-sheet steels which show a reduced specific iron loss. In large machines, the poles are fabricated from cold-rolled, grain-oriented steels having still better properties (specific loss and permeability) when magnetized in the direction of rolling; such punchings are made also 0.5 rum th ick. For higher frequencies (400 to '1 000 Hz), use is made of highalloy electrical-sheet steels in thicknesses of 0.35 mm, ,0 .2 mm , O. '1 mm , and 0.05 mm. For machines where the frequency of cyclic magnetization is several hertz or a fraction of a hertz (such as in the rotors of induction machines), or is zero (such as in the rotors of synchronous machines where the magnetic field remains constant in magnitude and direction) the cores may be assembled from structuralsteel laminations. The thickness for such laminations is usually chosen from manufacturing considerations (ease of stamping) and may be '1.0 mm , '1.5 mm , 2.0 mm , 4.0 mm, ,6 .0 mm , and more. As often as not, especially in mechanically strained rotors, the cores are made in one piece from steel forgings or steel (sometimes cast-iron) castings. In more detail, the properties 'Of electrical-sheet and structural steels are discussed in ['13].
(i) Cyclic Magnetization of the Core of an Electrical Machine Figure 3'1-5 shows the rotating magnetic field set up by the currents in the three-phase winding of an electrical machine. If we compare the field patterns spaced a quarter-cycle apart, we shall see that the field changes differently in the differ-
389
Ch. 31 Energy Conversion Losses
[a )
0(0) 7
I
8 1m
2
6
(6)
82m
0(8) ( e)
0(8}
1 7
8 1m
2
s
J 5 If
r
2
I '£
f
3
ofo) a li-
"·~' m, 0(')
82m
If.
.1 7
2 Ii J
If
(d )
0(8)
s li-
7 ~
)I
5
s
0.8 I 7 2 Ii
If I
31:
0(6) 7
2----.0---- Ii 3 If !i
Fig. 31-6 Magnetic flux density for various forms of cyclic ma gnetization: (a) pul sational magnetization; (b) rotational ma gnetization; (c) mixed or elliptical magnetiz ation (with the magnetic flux density vect or represented by the sum of two pulsational vectors) ; (d) mixed or ellipti cal magnetization (wit h the m agnetic flux density vector repr esented by the sum of a rotational and a pulsational vector)
390
Part Two. En ergy Conversion by Electrical Machines
ent parts of the magnetic circui t . In t he te eth (say, at point I) , the field lines are always directed radiall y, and the f ield only changes periodically in m agnitude as in a transformer. Accordingly , we may call it pulsational or transform er magnetization . In Fig. 3f-Ga , the magnetic flux density vector with a peak value B l m for pulsational m agnetiz ation is shown at instants 0, '1 , 2, . . . , 8 spaced TI8 ap ar t. In the yoke of the inner core, the m agneti c fiel d changes in a different manner. At point, say, II in Fig. 3'1-5, it r em ains un chang ed in m agnitude and onl y cha nges in direction. For this field, the magne tic flux den sity vector rotates at an angular velocity co = 2nf r elative to the yoke, while retaining its value . Accordingly , we ma y call it rotation al magnetization*. In t he case of rotational magnetiz ation , cha nges in t he magnetic flux density vector are shown in Fig. 3t-6b . Now the vector B can be depicted as the sum of two pulsa t ionaI magnetic flux density vectors equal in peak value to the magnitude of the ro tating B vector, that is , 1B I. To t his end, it will suffice to displace the ve ct or axes by n /2 and t o cause one of them to lag behind the other by a qu arter of a cycle or an angle n/2 (so t hat one of the pulsational vect ors is a maximum and the other vanishes). In the outer yoke , the magnetic flux density h as both a radial and a tangential component , su ch that at points where at a given instant the tangential component is a m aximum (at say, point IV in Fig . 3t-5a, wh ere B ", = B l m ) , the radial component is nonexistent iB, = 0). Conversely, a t points where at a given instant t he ra di al com ponent is a ma ximum (at, say, point III in Fig. 3t-5a, where B r = B z m ) , the tangential com ponent is nonexiste nt (B ", = 0). A quarter of a cycle later (see Fi g. 3t-5b) , t he r adial component becom es a maximum at point I V (B; = = B 2 m ) , and the tangenti al component vanishes (B ", = 0). The reverse cha nge occurs at poin ~ I I I (B l' = tIm , and B; = 0). The relative magnitudes of the radial and t an gentia l components, B l m an d B 2 m , va ry from point t o point roun d t he outer periphery of the yoke . The peak val ues of both are maximal (B l m = B 1m ax an d B z m = B zn,ax ) on the inner
* Rotat ion al mag neti za tion occurs in th e yoke of t he inner core only in the case of a t wo-p ole field (p = 1). Wh en p > 1, ellipti cal magnetization t akes pl ace in th e inner core (see below).
391
Ch. 31 Energy Conversion Losses
periphery of the yoke. As the radius of the yoke is increased, both components decrease in peak value , but B l m is reduced insignificantly, whereas B 2 m on the outer periphery of the yoke vanishes (if we assume that the field is confined within the limits of the yoke). Thus, on the outer periphery of the yoke, where only the t an gen t ial time-varying component exists , with a peak value B l m , the magnetization is puIs ational. On the inner periphery of the yoke, where two puIs ational components unequal in peak value exist , displaced from each other in time and space by an angle n /2, the magnetization is elliptical. Now the magnetic flux density vector not only rotates at a frequency whose mean value is
+ q yB 2m sincoz
(31-8)
where B l m is the peak value of the vector pulsating along the x-axis and B 2m is the peak value of the vector pulsating along the y-axis. As already noted, the locus of the B vector in the general case (for B l m:::/= B 2m) is an ellipse. In vector form, the ellipse is described by Eq. (31-8) where the parameter is time, t. For rotational magnetization, which is a special case, with B l m = B 2m, the locus of the B vector is a circle. This form of magnetization is quite aptly called circular. For pulsational magnetization, which is another special case where B 2m = or B l m = 0, the locus of the B vector degenerates t o a straight line which is aligned with the x - (or y-) axis. Elliptical magnetization may be treated as a hybrid form because it can be visualized as the superposition of rotational and pulsational magnetization. On re-writing Eq. (31-8) as
°
B
=
qx (B l m -
B 2m) cosroz
+ [qxB2m cos
+ q yB 2m sin rot] we can see that the magnetic flux density vector is obtained as the sum of a rotating vector whose magnitude is B 2m and a pulsational vector whose peak value is B l m - B 2m. Exactly this form is given to the field in Fig. 31-6d:
392
Part Two. Energy Conversion by Electrical Machines
(ii) Core Loss with PulsationaI and Rotational Magnetization The core loss associated with pulsational magnetization was considered in connection with transformers (see Sec . 2-7). The core loss associated wi th rotational magnetization differs from that of pulsational magnetization . A comparison of hysteresis loss for rotational and pu lsational magnetization is given in Fig. 31-7. For rotational magnetization, the loss is plotted as a function of the magnitude of the B vector, W/k!J 7 and for pulsational magnetizaPM rl6 tion, as a function of the peak 1/ \ va lue of the B vector. 5 I/ \ At B < 0.7 T, the core is If 1/ 2 slightly saturated (the perme3 V / ability is nearly constant), and 2 v / . / may treat rotational mag we 1 V netization as the superposition ~V 1.5 of two independent pulsationT o 0.5 al magnetizations along two F ig. 31-7 Hysteresis loss in (1) rotational magnetization and (2) mutually per pen dicul ar axes . For the va lue of B given above , pulsational magnetization (for electrical-sheet steel with the hysteresis l oss assoc iated 1.91 % Si) with rotational magnetization is ab out twice as gre at as that for pulsational magnetization. As the value of B is increased, the core is saturated pr ogressi vely mo re, and the pr inciple of superposition is no longer ap pli cable owing to a substantial nonlinearity of cyclic m agnet izat ion . As is seen from Fig . 31-7, the nonlinearity of the function B = f (H) at B > 0. 7 T ma nifests itself in that the ratio of hysteresis loss in rotationa l magnetizat ion to t hat in pulsational loss gradually decreases so that at B = 1.0 or 1.5 T it is from 1.45-to-1 to 1.65-to -1. When B is about 1. 7 T, hyst eresis loss is the same in either case, whereas a further increase in B leads to a sudden decrease in hysteresis loss associated with rotational magnetization, so that it is a fraction of that in pulsational magnetization . Eddy-current loss is solely a function of the magnetic flux density in the laminat ions and is independent of the field intensity. If we write the magnetic flux density in the case of rotational magnetization as the sum of two pulsation-
Ch. 31 Energy Conversion Losses
al components, we shall see t hat the eddy-current loss in this: case is twice as great as the eddy-current loss in pulsational magnetization for the same peak value of magnetic Ilux. density (that is, irrespective of the magnetic flux density) ; (iii) Magnetic Loss in the Magnetic Circuit Elements
In calculating the core loss in the cyclically magnetizedelectrical-machine components assembled with elecrical-sheet steel laminations insulated from one another, it is important. to consider the form of magnetization (pulsational or rotational) , the increase in the loss due to manufacturing factors; and also various additional losses. The point of departure in determining the magnetic loss. in magnetic-circuit elements is the total loss in 1 kg of laminations, assuming pulsational magnetization at 50 Hz and a magnetic flux density of 1 T. The total loss is measured bywhat is known as the Epstein apparatus and referred to as the specific loss designated by PLO/50' The values of PLO/5 0' for various steels are given in [13] . For other values of frequency and magnetic flux density (B~ '1.6 T), the specific loss can be found by the equation, Pm =
PI.O/50
(f150)1.3B2
(31-9)
if f ranges anywhere between 40 and 60 Hz. If, however, f va-ries over a broader range, the specific loss is found by aIL equation in which the hysteresis loss and the eddy-current loss are separated . This is usually accomplished by taking' advantage of the fact that for a given value of B m , thehysteresis loss varies directly with the frequency and the eddy-current loss with the square of the frequency. This. equation is Pm =
e. (lISa) B2
+
(J
(l150)2B2
(31-10) -
where e = specific hysteresis loss, W /kg, at B = 1 T and f = 50 Hz (j = specific eddy-current loss, W /kg, at B = 1 T ' and f = 50 Hz . The specific core loss is measured under carefully controlled conditions. Among other things, it is reqnired that theindividual core laminations should be ideally insulated from. one another and annealled after cutting or punching, and.
=394
Part Two. Energy Conversion by Electrical Machines
.suhject ed to a sinusoidally varying magnetic flux density. In commercial ma chines , however, the workmanship is not 'so perfect. In most cases, the laminations are not annealled aft er cu t ting and punching and this lead s t o an increased hysteresis loss. The insulation between the laminations is .oft en damaged by t he heavy pr essure us ed in the assem bly of ·cores. Nor is it alw ays possible to avoid electrical contact be tween the laminations and the frame (or shaft) and also through burrs on t he core teeth. Because of t his , addit ional .shor t-circuit ed pa ths are formed for eddy currents. This increase in core loss owing to manufacturing fac tors is accou nt-ed for by applying sui t able correction factors. In calculating t he loss in the v ar ious elemen ts of the magnetic circuit , one has also to r eckon wi th the fact t ha t the magnetic flux density in the case of pulsational magnet ization varies non sinusoidally (whereas in core-loss .measurements by t he Epstein apparatus , it is made to var y .sinusoida lly). If we expan d th e magnetic flux density into a Fouri er -ser ies, we shall see that, in addition t o the fundament al -com ponent, the seri es also contains higher harmonics . The .losses due to the higher harmonics are added to those asso ci a.t s d with the fundamental component , an d may quite aptly .be called additional core losses. If they are not calc ulat ed . .separ at ely , the additional core losses are accounted for by .applying addi tional-loss coefficients. I t should be no ted that the additional magnetic losses -can be caused even in the cores whose windings carry cur.rents at zero frequency (that is , direct current). The funda:m en t al magnetic flux density in such cores (for example, in ·t he poles of synchronous machines) varies at zero frequency .a ud, as a conse quence, t he basic core loss is non-existent. In .contrast, the addi tional losses associated with the higher .harm onics of ma gn etic flux density, varying at a high fre .qu ency, may be considerable. This is especially so when ·one core is assembled of heavy-gauge laminations or even nna de one-piece (because there is no cyclic magnetization at the fundamen t al frequency), and the other core (on the other .side of the air gap) ha s an appreciable saliency which gi ves ri se to no tice able pulsations in t he magnetic flux density -on its surface. For such cores, t he addit iona l magnetic .Iosses should be calc ula t ed separately and added to the to tal .loss.
C h, 31 Energy Conversion Losses
~ iv )
395
Magnet ic Loss in the Core Yoke
In calcula t ing th is loss, remember that the m anner of magnetization is ell ip tical at t he boundary with the toothed I a yer , and pulsational on the peripher y. For t he yoke, the -over all coefficient allowing for t he increase in loss due t o in accuracy in manufacture is taken as kad • a = 1.3 to 1.6, .an d the yoke loss is given by (31 -11) where ln a = mass of the yoke iron; Pma = spe cific yoke l oss by Eq . (31-9) or Eq. (3'1-'10) as m easured at B a and rthe fundamental frequency i: B a = peak value of the 'tangent ial magnetic flux density in the yoke as found for 'the fun damental magnetic flux densi ty b y ca lcula t ion of t he magnetic circuit for a given type of electrical machine (see P ar ts 4 and 5) . ( v ) Magnetic Loss in Core Teeth 'The cyclic magnetization of core teeth is pulsational. Therefore, account needs to be taken only of the in crease in losses owing to manufacturing fa ctors and of the higher time h arm onics . As com pared with the yoke, the teeth are sma lle r i n siz e, and the effect of cutting or stamping is noticeable o ver a larger too th area . So, the coeffi cient t aking care of t he increase in loss due to manufacturing factors is larger t han it is for the yoke . In addition t o the fundament al c om ponent of peak value B z- the magnetic flux densi t y in a tooth has substantial harmonics . Therefore, the additional 'loss coeffi cie nt for t eeth is k a d • z = 1.7 to 1.8 . The t oot h l oss is given by (31 -12) where In z = mass of the too th iron; Pm. Z = specific too th l oss as given by Eq . (31-9) or (3'1-'10) at B z and i, B z = pe ak value of ma gnetic flux densi ty in the mean tooth sect ion for th e fund am ental com ponent , as found by calculat ion of the magnetic circuit for a given machine (see Parts 4 an d 5); f = frequency of cyc lic m agne tization for t he fund am ental m agn et ic fl ux density. As in transformers, t he core loss is m ainly a fun ction of t he mutual flux. As is seen from E qs. (31 -9), (:1'1 -it) and
396
Part Two. Energy Conversion by Electrical Machines;
(31-12), the core loss is proportional to the voltage squared' and remains nearly constant as the winding currents vary. The losses associated with the cyclic magnetization by' winding leakage fields are poportional to the square of current . These losses are lumped with the additional load loss which is a function of the load current. 31-4
Mechanical Losses
These consist of hearing-friction loss, brush-gear loss, win- · dage loss , and cooling loss. Bearing-friction loss. Its magnitude varies with bearing design and the lubricant used. In small machines, the bearing-friction loss is kept to a tolerable level by using grease-packed ball and roller bearings. In larger machines, preference is given to sleeve bearings and thin lubricating oils . With all other conditions being equal, the hearingfriction loss builds up with increasing rpm, rotor mass, and shaft diameter at the bearings . Windage loss . Its amount increases with increasingdensity and viscosity of the medium inside the machine. The loss is a maximum when the rotor rotates in a liquid medium, and falls to a fraction when the rotor rotates in air. A further decrease by a factor of about ten is . obtained when hydrogen is used to fill the insides of a machine. To minimize the windage loss, the outer surface of the rotor must be made as smooth as practicable . For a given power rating and a given rpm, the 'win dage loss is larger in machines with a larger rotor diameter (with a larger ratioof rotor diameter to design length). Cooling loss. This loss is the power spent to drive fans or pumps supplying a circulating coolant in the cooling system of a machine. It is proportional to Q, the flow rate of coolant, and H, the head developed by the fan or pump, and increases with decreasing efficiency of the fan or pump. I t can be reduced by making the hydraulic system as perfect as possible and by choosing an appropriate coolant. For' liquid coolants it is lower than for gases because a liquid coolant has a higher specific heat capacity, and its flow rate may be kept at a lower level. The cooling loss is found as explained in Part. 3.
Bibliography
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1leCnll
To
Part One
15. Ilerpon r . H . Tpallcrjiop .ltamOpbl. OHTI1, Mocxna, 1934. 16 . TIIXOMUpOB n. M. Pncueni m pancqicpsui m o poe, 8HeprrrJI, Mocxaa, 1976. 17 . CarrOJImrrROB A. B. K oncmptj upoeauue mpallcrjiop .ltamOpoo. roci:lHeprOII3AaT, Mocxsa-Jlerraarpan, 1959 . 18. AJIeKCeeHKo r. B., AIIIpJITOB A. H ., Bepexreii E . B.,
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20. cDap6Man C. A . , Byu A. 10 ., Pa iixnn n H . ~ l. P esionni II sioilep-: n u ea uusi m pancifiop w am op oe , Mocxaa, 8 uep rllH, 1974 21. BaCIOTJInClHIii C. E. B onp ocu m eopuu u. p acuem a m paucrPop.llam opOB. JIeHIIHrpaA, 8 ueprIlH, 1970. To Part Two
22 . II OJIIIBan OB K. M. T'eop em unecxu e OC/WBbl ssesnipomexnu.n u, q. 1Jlnaenssre 8JIeliTpJIqeCKJIe lJ,eIIII C cocpenor ose nnsnnr IIOCTOHnnbIMII . 8BeprIIH, Mocxaa , 1972. 23. JRYXOBJIIJ;KIlii E . H ., Hernea nmmii H. E. Te op emunecsu e OC/IOB bl.·. es enmpomeznunu, q . 2. Jlanefimae 8JIeKTpIIQeCKIIe lJ,eIIII (IIPOAOJImenIIe). Hennaeiimae 8JIeKTpnQeCKJIe IJ;eIIJI. 8 neprIlH, Mocxaa, 1972. 24 . IIoJIlIBanoB K . M. T eopem uuecnue OCItOBbl enenmpomexnunu q . 3. Teopua 8JIeKTpOMarnJITnOrO IIOJIH . 8HeprJIH, Mocxsa , 1975. 25. Adkins , B. The Genera l Th eor y of Electrical Machines, Chapman and Ha ll, Lond on , 1959. 26. White, C., Woodson, H . El ectromechan ical Energy Conversion _ J. Wiley and Son ., In c., New. York, 1969. 27 . KOllbIJIO B H. II . 8 Aenmp o.H eXaltU eCnoe np eotipasoeanu e suepeuu : 8nepr IIH, M6cKBa , 1973. 28. HBanOB-CMOJIenCKllii A . B . 8AeKmp0l>ta3ItUmltble nOAR. u np ou eccu: B sne nm.puu ecnu x siauiu n ax u ux rPuaullecnoe .1100eAUp OBaItUe. 8 neprJIH, wlocKBa, 1969. 29 . Schuisky, W. Bere chnurig electrischer M as chinen, Springer , Wien .. 1960. 30. Ceprees II . C., BJInOrpaAoB H. B., I'opmnros <1). H . JIp oenmup oeau u e enenm.p uuecn ua: sui uuuc , 8 neprIlH, Mocxaa, 1969. 31. II ocTnIIKoB H. M . tt poenm u p oeau u e 3AenmpU 'leCI>UX .1LaUl!t It. re cTeXJIaAaT YCCP, KlIeB, 1960 . 32. Liw sch it z-Garik , M. W indin g Alterna tin g-C urrent Ma ch in es. 33. KYQep a H., f aIIJI H . Otiscomxu. sne nm puwecn ua: uaiuun , Il epen •. C nemcx . HaA-Bo AKaAeMJIII HaYK, Ilpara , 1963 . 34. 3 mIlIH B. H ., KaIIJIaH M. H ., Ilaaeii A . M. II np , Otiu omm; 3M nm.puuecnu a: suuu u n, 8 nepn JH, Jleanarpan, 1970. 35 . ,I!,aHIweBIIQ H . E ., KarnapcKIlll 8 . f . otiaeo-ucue n ome p u B 3/!8Kmpu uecnu x suuu unax, I'ocaaepronanar, Mocxaa-Jl ea na r pan, 1963. 36 . ,I!,aBIweBIIq H . E ., Kymm 10. A. T eopusi U p acueni oe.unrf!ep/t blx .o6.1Lomon cun xpo n tcux sia uuuc , HaA-Bo AH CCCP, Mocxsa, 1962. 37 . ,I!,anJIJIeBIIq H. E. , )J;oM6pOBCKllll B . B ., Hasoscxnii E . H . Il aposiem.p n. 9Aenmp U'leCIWX suuuun n epesi euu oeo m ona. Hayxa, Mocxsa .. 1965. 3 8. Ta JIaJIOB H . H . Il apnsi em p a U x ap anmepucmuxii R.BItOnOA IOClt N:11: cutcx p ou n u« suuuun . 8HeprIIH, Mocxaa , 1978. ll
.a
Index
Many subjects are not included in the Index because they arelisted separately in the Contents. Therefore, the reader will be welladvis ed first to consult the various sections under the indi vidua l; chapter headings th ere. In the Index, each topic has its own section, but subjects common to several topic s may be found separately listed. A ir gap fac tor, 270 Asyn chronous machine, 23 3 Au to t ransfo rm er , 125, 13 3 Ax ial gap l en gth , 266 Breathing mmf wave , 274
Field windings, 255 Flux, l eakage, 45 mutual, 45 Flux linkage, 32 3 coil, 323 coil group, 328 h a rmonic, 33 5 phase , 3 30 winding , 333
Carter coeffic ien t, 270 Coil pitch , 238 Coolin g systems for t ransfo rme r s, 186 Condu ctors , Generator, def., 13 stran ded , 381, t r an spo sed , 384 unstran de d, 382 Imp edance, Co re mutual in transformer, 117 cycli c m agn eti zation of, 38 8 short-circuit, 119 sali ent-pol e, 202 Inductance , Core l eng t h, 265 l eakage for complete winding , 348; Cyc li c magnetizat ion , 388 mutual between phases, 343 mutual between stator and phase., Distribution factor, 280 344 mutual in transformer, 51 Effec t iv e core l ength , 265 polyphase winding, 341 El ectri cal machines, Induction machine, 234 asynchron ous , 233 bas ic arran gement, 18 Load unbalance , 145 ba si c definitions, 13 Losses, basic designs, 207 bea ring-friction, 396 cl assific a ti on , 15, 192 cooling, 39 6 disc-typ e, 19 3 copper in transformer, 53 ge a re d , 195 co re, 392 genera l theory, 192 in t ransformer, 53 he teropo la r , 207 ele ct ri ca l, 37 9-80 h omopolar, 207 magnetic, 379, 387 inducti on, 234 m echanical, 379 , 396 inducto r , 218 n o-load in transformer, 53 lin ear fl at, 19 3 power, 379 linear t ubu la r , 194 win dage, 39 6 r eluctance, 2 14 synchronous , 2 33 Electromagneti c forces, 367 Ma gn etic fi eld, El ectromagnetic torque, 351, 364 calculation of, 257 EMF , from concentrated field winding, 31(;; rotational , 197 from di stributed fi eld winding, 319' transformer, 197 leakage, 34 1 Energy conversion, mutual, 342 by electrical machines , 18 of phase, 267 by rotating m agne ti c field, 372 of pol y ph ase winding, 288 by transformer, 15, 75 periodic, 201 conditions for, 201, 227 from rotating field winding, 316 efficiency, 379 . spa t ia l pattern of, 262 losses, 37 9 u seful, 341
400
Index
.Magn et izat ion , cycl ic, 388
Ml\IF , 269 phase , 280 Ml\IF eq uation for transrormer . 58 Mot or , def. , 13 P er-u ni t nota ti on, 69
Ph ase belt , 205 .P h ase sequence negative, 91 positi ve, 91 zero, 99
.P hasor diagra m . 65 Pi t ch fac tor , 276 P ole pairs, 202 ,P ol e pi tc h, 23 8 P ower, I el ectri c, 372 electr omagne ti c, 37, 2 magn eti c, 372 'R ot ary convert er, d,ef . , 14 Ro tor, der. , 18 :Self-in du ctance, comp lete win ding, 3/,5 phase , 31,2 :Slot s per pole, 205 St a t or, def. , 18 .Sur face cur rent , 358 .Synchronous m ac hi ne, 233 ' T emperat ur e l imits, t ransform er , 186 'Tempera t ur e rise, t ran sforme r , 18 5 ' Test , op en-circ uit , 99 shor t-circuit , 102 Tr ansform ati on, three-phase, 79 T ransfor ma t ion ratio , 46 ' Tr ansformer, der., 14 arc -wel ding, 180 at no load , 43 .au t o-, 125 , 133 ba sic a rra ngemen t , 15 , 27 bu t t- join t , :l4
no-l oa d cur rent, 49, 117 no-l oad losses, 52 n o-loa d' tes t , 99 on load , 56, 107 open-circuit curre nt , 117 open-circ uit test, 99 ov e rvol tages in, 171 .< .. pal:al!~le d, 138 pa rameter cal culation , 117 peak ing, 182 ph ase displ acement reference nura. bel's, . 82 phasor di agram, 65 shell -t ype, 33 sh ort -circui t impe dan ce, 119 < short-ctrcur t t est , 102 size re lati ons, 121 spe cial -p urpose, 177 strip-wound, 38 str uctur a l part s, 38 t ap -c h a ng in g , 112 off l oa d, 113 on load, ;I,J.4. _ th ree-ph ase, 8 3, 89 , ]1,5 t ra nsferring t he secondary quantit ies , 62 transients in , 164- · ·· · turn s ratio, 46 va ri a ble-volt age, 179 voltage, 182 Transform er core l oss, 53 Tr ansformer emf equa tion , 46 Tra nsformer eq uiva lent circui t , 68 Tra nsformer fittin gs, 4 1 Transformer fra me, 37 Tr ansformer I eads, 1, 1 Tra nsform er load unbalance , 145 Transformer mmf equation , 58 Tran sformer m utu al impedance, 117 Tr an sform er m utual in duct ance, 51 Transformer tan k accessori es, 4 1 Transf ormer t ermin al bus hin gs, 4 1 Transform er v olt age equa t ions, 1, 5, 60 Transformer voltage reg ulat ion , 107 Transform er Wind ing connections, 79 Transformer Win di ng in sulation , 39 Transfo rme r Win dings , 32 Transformer yo ke clam pin g, 36 Triplen harm onics, 91 Win d in gs , a .c . ma cbines, 236 ch or ded , 238 conc entrated, 205 cyli nd rical , 202 distri bu t ed , 205 dou ble-l a yer , 20 3 d rum . . 202
fi el d , 25 5 fr acti onal- slot, 250 full-pi t ched, 2 38 l ap , 238, 240 polyphase, 205 select ion of , 246 sbor t-pitched , 238 single-layer , 203 t wo-layer , 236 wave, 238 , 244
Printed in the Union of Soviet Socialist Republics
l ~
'
I
ABOUT THE AUTHOR I Professor Alexei V. IVANOV-SMO· ~' ', .1." ~~. :'~'.,... , ~ LE NS KY, D. Sc. (Tech.), is a leadin~ "'ilo- ~ . Soviet authority in his field. Currently, he is with the Moscow Power I nstl.' tute. He has written (individually anc -a s a coauthor) six books on electrici· ty, including the present one. '
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