Induction Motor Analysis

Master Thesis

Time-stepping FE-analysis of a 22kW Variable Impedance Induction Motor (VZIM) and thermal analysis

Mehdi Hajinoroozi

Advisors: Prof. Dr.-Ing. habil. h.c. Andreas Binder Dipl.-Ing. Hooshang Gholizad

Novemebr 2011

Summary Variable Impedance Induction Motor consists of a sectionalized rotor, with 3 different sections and each section has its own bars and end-rings which have different geometry and materials, in order to achieve high starting torque and high efficiency at the rated operating point together with high breakdown torque. In order to investigate the electromagnetic and thermal characteristics of this motor (22 kW VZIM), three different motors VZIM1, VZIM2 and VZIM3 with the same stator but different rotors have been considered and analyzed. Electromagnetic and thermal analyses of a 22kW Variable Impedance (Z) Induction Motor (VZIM) are the aim of this master thesis. Finite element method is a precise and useful approach to analyze electrical machines, therefore the electromagnetic analysis of the VZIM motor is done using Flux2D software and the thermal analysis is done by ANSYS software. Furthermore all the calculation results obtained by finite element method and relevant tools are compared with analytical calculations, carried out by KLASYS program. In order to do steady state AC and time stepping electromagnetic analysis of the 22kW VZIM motor, first the geometries of three assumed independent motors VZIM1, VZIM2 and VZIM3 are generated in Flux2D and with assigning the materials and meshing the geometries, the models are solved and relevant torque-slip, fundamental stator phase current-slip, input power-slip, efficiency-slip and power factor-slip characteristics are calculated by Flux2D and compared with the results of the KLASYS software. In addition, for thermal calculations due to the existence of the measured values of the 5.5kW VZIM motor, first the 5.5kW motor is analyzed with the ANSYS tool and the results are compared with measured values, afterwards the 22kW VZIM motor is analyzed with ANSYS tool and the results are compared with a simplified thermal equivalent circuit model results. To carry out the simulation with ANSYS, first the geometries of three assumed independent motors VZIM1, VZIM2 and VZIM3 are generated and after assigning the materials and meshing the geometries, the models are solved with assigning the loss densities of different parts of the motors as heat sources. At the end the values of analytical calculations and finite element method are compared to make sure that the temperature rise in the stator winding does not exceed the thermal limit of the insulations.

ii

Contents Table of Contents List of Symbols List of Figures List of Tables

1

Introduction

1.1 1.2 1.3 1.4 1.5

Preface Electromagnetic and thermal analysis of the 22kW VZIM Introduction to finite element method Electrical and mechanical parameters and dimensions of the 22kW VZIM Procedure of the project

1 3 3 6 10

2 2.1 2.2 2.3 2.4 2.5 2.6 2.6.1 2.6.2 2.6.3 2.6.4 2.6.5 2.6.6

Geometry and mesh generation with Flux2D Introduction Geometry creation Meshing the geometry Material assignment Electrical circuit for the motor Calculation of values of circuit’s elements Stator resistance Inductance of stator winding overhang Rotor end-ring leakage inductance Resistance of the end-Ring Segment VZIM1 Resistance of the end-Ring Segment VZIM2 Resistance of the end-Ring Segment VZIM3

11 11 12 14 15 16 16 16 17 17 17 18

3 3.1 3.2 3.2.1 3.2.2 3.2.3

Steady state electromagnetic analysis Introduction Steady state AC analysis of the 22kW VZIM Steady state AC analysis of the VZIM1 Steady state AC analysis of the VZIM2 Steady state AC analysis of the VZIM3

20 21 21 26 29

4 4.1 4.2 4.3 4.4 4.5 4.6 4.6.1 4.6.2 4.6.3

Time stepping analysis of VZIM Introduction Stator and rotor field spatial harmonics Asynchronous harmonic torques Synchronous harmonic torques Derived slips used for time stepping analysis Time-stepping analysis of induction motors Time-stepping analysis of VZIM1 Time-stepping analysis of VZIM2 Time-stepping analysis of VZIM3

32 32 32 33 34 35 35 40 45

iii

4.7 4.8 4.9 4.9.1

Instantaneous Torque wave forms at different rotor speeds Synchronous harmonic torque assessment in VZIM Power losses in different parts of the motor Losses of VZIM at different speeds

50 54 61 63

5 5.1 5.2 5.2-1 5.2-2 5.3 5.3.1 5.3.2 5.3.3 5.3.4

Thermal analysis Preface Temperature rise calculation by thermal equivalent circuits Calculation of οߴ஼௨ and οߴி௘ for VZIM2 (22kW) Calculation of οߴ஼௨ and οߴி௘ for VZIM3 (22kW) Numerical calculation of temperature rise Thermal analysis of 5.5kW VZIM2 Thermal analysis of 5.5kW VZIM3 Thermal analysis of 22kW VZIM2 Thermal analysis of 22kW VZIM3

65 65 69 70 71 72 75 79 82

6

Conclusion

87

7

Bibliography

88

8 8.1 8.2 8.3 8.4

Appendix Appendix I Appendix II Appendix III Appendix IV

89 91 109 111

iv

List of Symbols ܽ ܽ௜ B H A ݀௦௜ f h ݇ௗ ݇௙ ݇௣ ݇௪ ߬௣ l L L ߣ௕ ݈௕ m ݀ாோ௠ ‫ܴܧ‬௟ ‫ܣ‬ாோ ‫ܴܧ‬௧௛ ‫ܴܧ‬௛ M n ܰ௦ ܰ௖ p P q Q R s ‫ݏ‬ொ t u,U ‫ݑ‬ሷூ ,‫ݑ‬ሷ௎ V ‫ݔ‬ ߜ ˍ ߩ ߩ ߤ ߤ ‫ݒ‬

T A/m Vs/m m Hz m m m m m m m m m m m N.m. 1/s W Ohm m s V A m m S/m Τଷ Ohm.m Vs/(Am ) -

number of parallel baranches of winding parallel wires per turn magnetic flux density magnetic field strength Magnetic vector potential inner stator diameter electric frequency height of stator slot opening distribution factor slot fill factor, frequency coefficient pitching factor winding factor Pole pitch axial length self-inductance overall length permeance /unit length length of the winding overhang number of phases mean diameter of the end-ring length of the end-ring segment cross section area of end-ring end-ring thickness end-ring height torque rotational speed number of turns per phase number of turns per coil number of pole pairs power number of slots per pole and phase number of slots electric resistance and thermal resistance slip slot opening time electric voltage current/voltage transformation ratio magnetic voltage (m.m.f.) circumference co-ordinate air gap width electric conductivity density electric resistivity ordinal number of rotor space harmonics magnetic permeability ordinal number of stator space harmonics v

߱ ܴ௧௛ ߴ οߴ ܲ஼௨ ܲி௘ ߙ j i ‫݌‬ሺ‫ݐ‬ሻ ݇௛ ݇௘ ‫ܤ‬௠ ‫ݒ‬

d ߣ௧௛

1/s K/W Ԩ K W W W/(ଶ ) A/ଶ A W Ws/( ଶ ଷ ) 1/(Ohm.m)) T m/s m W/(m.)

electric angular frequency Heat resistance temperature temperature rise Copper losses iron losses heat transfer coefficient current density current volume density of instantaneous power loss hysteresis coefficient coefficient of losses in excess peak value of the magnetic flux density wind speed over stator outer surface and winding overhang thickness of slot insulation and stator and rotor lamination Thermal conductivity

vi

List of Figures 1.1-1 1.1-2 1.4-1 1.4-2 1.4-3 2.2-1 2.2-2 2.2-3 2.3-1 2.3-2 2.4-1 2.5-1 3.1-1 3.1-2 3.2.1-1

3.2.1-2 3.2.1-3 3.2.1-4 3.2.1-5 3.2.2-1 3.2.2-2 3.2.2-3 3.2.2-4 3.2.3-1 3.2.3-2 3.2.3-3 3.2.3-4 4.3-1 4.4-1

Cross-section of variable impedance induction motor, rotor position at stand still [6] Cross-section of variable impedance induction motor rotor position at nominal speed [6] Cross-section of Variable Impedance Induction motor (VZIM1), Prepared by ANSYS Cross-section of Variable Impedance Induction motor (VZIM2), Prepared by ANSYS Cross-section of Variable Impedance Induction motor (VZIM3), Prepared by ANSYS Created geometry of VZIM1, by Flux2D Created geometry of VZIM2, by Flux2D Created geometry of VZIM3, by Flux2D Generated mesh of VZIM1 in the air-gap using Flux2D Generated mesh of VZIM1 using Flux2D B-H characteristic of M270-50A iron sheets, which are used in stator and rotor lamination [7] Star connected stator electrical circuit of the motor, by Flux2D Nonlinearity relation of B and H [5] Static and equivalent B-H curves for different cases [5] Torque-slip characteristics of VZIM1, comparison of the steady state calculation results of Flux2D and the analytical calculations by KLASYS Torque-slip characteristics of VZIM1, comparison of skewed and unskewed rotor, obtained by KLASYS Phase current-slip characteristic of VZIM1, comparison of the steady state calculation results of Flux2D and the analytical results of KLASYS

2

Normal component of air-gap flux density for one pole pair of VZIM1 at a slip of 0.0253, calculated by Flux2D Numerically calculated flux lines in VZIM1 at a slip of 0.0253, obtained by Flux2D

24

Torque-slip characteristics of VZIM2, comparison of the steady state calculation results of Flux2D and the analytical calculations by KLASYS Phase current-slip characteristic of VZIM2, comparison of the steady state calculation results of Flux2D and the analytical results of KLASYS

26


27

Torque-slip characteristics of VZIM3, comparison of the steady state calculation results of Flux2D and the analytical calculations by KLASYS Phase current-slip characteristic of VZIM3, comparison of the steady state calculation results of Flux2D and the analytical results of KLASYS

29


30

Asynchronous harmonic torques of the 5th and 7th stator field harmonics, which are superimposed on fundamental asynchronous torque [2] Typical effects of synchronous and asynchronous harmonic torques in induction machines [2]

33

vii

2 8 9 9 11 11 12 12 13 14 15 20 21 22 23 23

25

27

28

30

31

34

4.6.1-1

Torque-slip characteristic of unskewed VZIM1, calculated with Flux2D timestepping, the effects of the synchronous harmonic torques at slips‫ ݏ‬ൌ

36

ͲǤͺͷ͹, ‫ ݏ‬ൌ ͳ and ‫ ݏ‬ൌ ͳǤͲ͹ͳ of the the motor are observable 4.6.1-2

Torque-slip characteristic of unskewed VZIM1, calculated with KLASYS, the 36

effects of the synchronous harmonic torques at slips‫ ݏ‬ൌ ͲǤͺͷ͹, ‫ ݏ‬ൌ ͳ and ‫ ݏ‬ൌ ͳǤͲ͹ͳ of the the motor are observable 4.6.1-3

4.6.1-4 4.6.1-5 4.6.1-6 4.6.1-7 4.6.1-8 4.6.2-1

Torque-slip characteristic of unskewed VZIM1, time-stepping and AC analysis results obtained by Flux2D, the harmonic torque effects at lower speeds of the motor are obvious Stator phase current of unskewed VZIM1, time stepping analysis results calculated by Flux2D and KLASYS Input power-slip of VZIM1, comparison of the analysis results calculated by Flux2D and KLASYS Output power-slip of VZIM1, comparison of the analysis results calculated by Flux2D and KLASYS Efficiency-slip of VZIM1, comparison of the analysis results calculated by Flux2D and KLASYS Power factor-slip of VZIM1, comparison of the analysis results calculated by Flux2D and KLASYS Torque-slip characteristic of unskewed VZIM2, obtained by KLASYS and Flux2D time-stepping calculations, the harmonic torque effects at lower speeds of the

38 38 38 39 39 40 41

motor are observable 4.6.2-2



4.6.2-4 4.6.2-5 4.6.2-6 4.6.2-7 4.6.2-8 4.6.3-1

Torque-slip characteristic of unskewed VZIM2, time-stepping and AC analysis results obtained by Flux2D, the harmonic torque effects at lower speeds of the motor are obvious Stator phase current of unskewed VZIM2, comparison of the time stepping analysis results calculated by Flux2D and KLASYS Input power-slip of VZIM2, comparison of the analysis results calculated by Flux2D and KLASYS Output power-slip of VZIM2, comparison of the analysis results calculated by Flux2D and KLASYS Efficiency-slip of VZIM2, comparison of the analysis results calculated by Flux2D and KLASYS Power factor-slip of VZIM2, comparison of the analysis results calculated by Flux2D and KLASYS Torque-slip characteristic of unskewed VZIM3, calculated with Flux2D timestepping, the effects of the synchronous harmonic torques at slips‫ ݏ‬ൌ

42 43 43 44 44 45 46

ͲǤͺͷ͹, ‫ ݏ‬ൌ ͳ and ‫ ݏ‬ൌ ͳǤͲ͹ͳ of the the motor are observable 4.6.3-2



4.6.3-4 4.6.3-5 4.6.3-6

Torque-slip characteristic of unskewed VZIM3, time-stepping and AC analysis results obtained by Flux2D, the harmonic torque effects at lower speeds of the motor are obvious Stator phase current of unskewed VZIM2, comparison of the analysis results calculated by Flux2D and KLASYS Input power-slip of VZIM3, comparison of the analysis results calculated by Flux2D and KLASYS Output power-slip of VZIM3, comparison of the analysis results calculated by Flux2D and KLASYS viii

47 47 48 48

4.6.3-7 4.6.3-8 4.7-1 4.7-2 4.7-3 4.7-4 4.7-5 4.7-6 4.7-7 4.8-1 4.8-2 4.8-3 4.8-4 4.8-5 4.8-6 4.8-7

4.8-8 4.8-9 4.8-10 4.9.1-1 4.9.1-2 4.9.1-3 5.2-1 5.2-2

5.2-3 5.3.1-1 5.3.1-2 5.3.1-3 5.3.1-4 5.3.1-5 5.3.2-1 5.3.2-2 5.3.2-3 5.3.2-4 5.3.2-5

Efficiency-slip of VZIM3, comparison of the analysis results calculated by Flux2D and KLASYS Power factor of VZIM3, comparison of the analysis results calculated by Flux2D and KLASYS Torque-time characteristic of VZIM2 at 1462rpm, calculated by Flux2D, torque oscillates around a constant mean value after reaching the steady state Torque-time characteristic of VZIM2 at 1462rpm, calculated by Flux2D, zoomed view after steady state is reached Torque-time characteristic of VZIM2 at 1200rpm, calculated by Flux2D, torque oscillates around a constant mean value after reaching the steady state Torque-time characteristic of VZIM2 at 1200rpm, calculated by Flux2D, zoomed view after a steady state is reached Torque-time characteristic of VZIM2 at 525rpm, calculated by Flux2D, torque oscillates around a constant mean value Torque-time characteristic of VZIM2 at 525rpm, calculated by Flux2D, zoomed view after a steady state is reached Torque-time of VZIM2 at 205.5rpm, calculated by Flux2D, where break down slip of the 7th asynchronous torque harmonic happens at slip 0.863 or 205.5rpm

49

First and second positions of the rotor bars with fixed stator position to calculate the synchronous harmonic torques, prepared by Flux2D Torque-time of unskewed VZIM2 at‫ ݏ‬ൌ ͲǤͺͷ͹, calculated by Flux2D, rotor

54

position step 1 Torque-time of unskewed VZIM2 at‫ ݏ‬ൌ ͲǤͺͷ͹, calculated by Flux2D, rotor position step 5 Torque-time of unskewed VZIM2 at‫ ݏ‬ൌ ͲǤͺͷ͹, calculated by Flux2D, rotor position step 7 Variation of synchronous torque at‫ ݏ‬ൌ ͲǤͺͷ͹ as a function of rotor position Variation of synchronous torque at ‫ ݏ‬ൌ ͳ a function of rotor position Variation of synchronous torque at ‫ ݏ‬ൌ ͳǤͲ͹ͳ as a function of rotor position Torque-time of unskewed VZIM2 at speed 224.28 rpm, calculated by Flux2D Torque-time of unskewed VZIM2 at speed 10 rpm, calculated by Flux2D Torque-time of unskewed VZIM2 at speed -97.14 rpm, calculated by Flux2D

49 50 51 51 52 52 53 53

54 55 55

56 56 56 58 59 59 Comparison between losses of VZIM1 at speed equal to 1390.9 calculated 63 with Flux2D and KLASYS Comparison between losses of VZIM2 at speed equal to 1455.8 calculated 64 with Flux2D and KLASYS Comparison between losses of VZIM3 at speed equal to 1474.65 calculated 64 with Flux2D and KLASYS The simplified thermal equivalent network for an induction motor 66 Output power-slip characteristics of 22kW VZIM2 and VZIM3 and the average of 67 the powers Calculated output power of the VZIM motor with the supper-position and equivalent circuit methods 2D model of the 5.5kW VZIM2 and meshing, by ANSYS 3D model of the 5.5kW VZIM2, by ANSYS 3D model meshing of the 5.5kW VZIM2, by ANSYS The thermal solution of the 5.5kW VZIM2 under nominal operation, by ANSYS The calculated temperature in the winding overhang of VZIM2, by ANSYS 2D model of the 5.5kW VZIM3 and meshing, by ANSYS 3D model of the 5.5kW VZIM3, by ANSYS 3D model meshing of the 5.5kW VZIM3, by ANSYS The thermal solution of the 5.5kW VZIM3 under nominal operation, by ANSYS The calculated temperature in winding overhang of 5.5kW VZIM3, by ANSYS ix

68 72 72 73 74 75 75 76 76 78 78

5.3.3-1 5.3.3-2 5.3.3-3 5.3.3-4 5.3.3-5

5.3.4-1 5.3.4-2 5.3.4-3 5.3.4-4 5.3.4-5

2D model of the 22kW VZIM2 and meshing, by ANSYS 3D model of the 22kW VZIM2, by ANSYS 3D model meshing of the 22kW VZIM2, by ANSYS The thermal solution of the 22kW VZIM2 under nominal operation, by ANSYS The calculated temperature in the winding overhang of 22kW VZIM2, by ANSYS 2D model of the 22kW VZIM3 and meshing, by ANSYS 3D model of the 22kW VZIM3, by ANSYS 3D model meshing of the 22kW VZIM3, by ANSYS The thermal solution of the 22kW VZIM3 under nominal operation, by ANSYS The calculated temperature in the winding overhang of 22kW VZIM3, by ANSYS

x

79 79 80 81 82 82 83 83 85 85

List of Tables 1.4-1 1.4-2 1.4-3 1.4-4 1.4-5 1.4-6 2.4-1 2.6-1 4.8-1

4.8-2 4.8-3 4.8-4 4.8-5 4.8-6 4.8-7 4.8-8 4.8-9 4.8-10

4.8-11

4.8-12

4.8-13

4.8-14

4.9-1

Stator dimensions Stator winding details The rotor dimensions and parameters of VZIM1 The rotor dimensions and parameters of VZIM2 The rotor dimensions and parameters of VZIM3 VZIM material details B-H characteristic data of the iron sheet type M270-50A [7] Electrical values of VZIM stator circuit Maximum, minimum and peak to peak values of synchronous torque for VZIM2 in two different methods of calculation (rotation and oscillation methods) for the slip ‫ ݏ‬ൌ ͲǤͺͷ͹ Maximum, minimum and peak to peak values of synchronous torque for VZIM2 in two different methods of calculation (rotation and oscillation methods) for the slip ‫ ݏ‬ൌ ͳ Maximum, minimum and peak to peak values of synchronous torque for VZIM2 in two different methods of calculation (rotation and oscillation methods) for the slip ‫ ݏ‬ൌ ͳǤͲ͹ͳ Maximum, minimum and peak to peak values of synchronous torque for VZIM1 (oscillation method) Maximum, minimum and peak to peak values of synchronous torque for VZIM3 (oscillation method) Maximum, minimum and peak to peak values of synchronous torque for VZIM1, at the slip ‫ ݏ‬ൌ ͲǤͺͷ͹, the comparison of the calculated values by KLASYS and Flux2D Maximum, minimum and peak to peak values of synchronous torque for VZIM1, at the slip ‫ ݏ‬ൌ ͳ, the comparison of the calculated values by KLASYS and Flux2D Maximum, minimum and peak to peak values of synchronous torque for VZIM1, at the slip ‫ ݏ‬ൌ ͳǤͲ͹ͳ, the comparison of the calculated values by KLASYS and Flux2D Maximum, minimum and peak to peak values of synchronous torque for VZIM2, at the slip ‫ ݏ‬ൌ ͲǤͺͷ͹, the comparison of the calculated values by KLASYS and Flux2D Maximum, minimum and peak to peak values of synchronous torque for VZIM2, at the slip ‫ ݏ‬ൌ ͳ, the comparison of the calculated values by KLASYS and Flux2D Maximum, minimum and peak to peak values of synchronous torque for VZIM2, at the slip ‫ ݏ‬ൌ ͳǤͲ͹ͳ, the comparison of the calculated values by KLASYS and Flux2D

6 6 7 7 7 8 14 19 57 57 58 58 58 60 60 60 60 60 60

Maximum, minimum and peak to peak values of synchronous torque for 61 VZIM3, at the slip ‫ ݏ‬ൌ ͲǤͺͷ͹, the comparison of the calculated values by KLASYS and Flux2D Maximum, minimum and peak to peak values of synchronous torque for 61 VZIM3, at the slip ‫ ݏ‬ൌ ͳ, the comparison of the calculated values by KLASYS and Flux2D Maximum, minimum and peak to peak values of synchronous torque for 61 VZIM3, at the slip ‫ ݏ‬ൌ ͳǤͲ͹ͳ, the comparison of the calculated values by KLASYS and Flux2D

Values of specific total losses according to the data sheet of the M270- 62 50A [7], and calculated power losses, in frequencies 50Hz, 100Hz and 200Hz xi

5.2-1 5.2-2

5.2.1-1 5.2.2-1 5.2.2-2

5.3.1-1 5.3.1-2

5.3.2-1 5.3.2-2

5.3.2-3 5.3.3-1 5.3.3-2 5.3.4-1 5.3.4-2 5.3.4-3 5.3.4-4 Appendix II Appendix II Appendix II Appendix II Appendix II Appendix II Appendix II Appendix III Appendix III Appendix III

Heat transfer coefficient for one side of the winding overhang close to the centrifugal mechanism Losses in different parts of the VZIM2 and VZIM3 in the speed equal to 1467.36 rpm Thermal conductivities of the materials used in VZIM2 thermal model Thermal conductivities of the materials used in VZIM3 thermal model Temperature rise in the stator winding of VZIM2 and VZIM3 calculated based on simplified thermal equivalent circuit in the nominal operating speed equal to 1467.36 rpm Loss densities in different parts of the 5.5kW VZIM2 Heat transfer coefficient at different boundary conditions of the 5.5kW VZIM2 Loss densities in different parts of the 5.5kW VZIM3 Heat transfer coefficient at different boundary conditions of the 5.5kW VZIM3 Temperature rise in stator slot of 5.5kW VZIM2 and VZIM3 calculated by ANSYS Loss densities in different parts of the 22kW VZIM2 Heat transfer coefficient at different boundary conditions of the 22kW VZIM2 Loss densities in different parts of the 22kW VZIM3 Heat transfer coefficient at different boundary conditions of the 22kW VZIM3 Temperature rise in stator slot of 22kW VZIM2 and VZIM3 calculated by ANSYS Temperature rise in stator slot of 22kW VZIM2 and VZIM3 calculated by ANSYS Synchronous harmonic torque slip

Stator ordinal numbers and related asynchronous harmonic torque slips Stator ordinal numbers and related asynchronous harmonic torque slips Stator ordinal numbers and related asynchronous harmonic torque slips Slips which are used to perform time-stepping analysis of VZIM1 Slips which are used to perform time-stepping analysis of VZIM2 Slips which are used to perform time-stepping analysis of VZIM3 Losses in VZIM1 calculated by Flux2D and KLASYS Losses in VZIM2 calculated by Flux2D and KLASYS Losses in VZIM3 calculated by Flux2D and KLASYS

xii

67 68 69 70 71 73 74 76 77 78 80 81 83 84 85 86 91 92 96 101 106 107 108 109 109 110

Chapter 1: Introduction

1.1 Preface Due to the robustness and reliability with low production and maintenance costs of the line operated induction motors, this type of machines are used in more than 80% of motor applications worldwide. Making use of current displacement effect by designing induction motors with deep bar or double cage rotors results in desirable starting torque and efficiency at nominal speed operation, but reduction of breakdown torque is not favorable which happens in this case because of increased rotor leakage reactance. One possible solution is Variable Impedance Induction Motor (VZIM). The Variable Impedance Induction Motor has high starting torque, in addition to high efficiency at the nominal operating point. This type of induction motor includes a sectionalized cage rotor with different bar shapes for each sub-cage section is used. At standstill, sub-cages with high resistances are placed inside the stator bore to increase the starting torque and reduce the starting current. By increasing the rotational speed, a centrifugal mechanism moves the rotor in the axial direction and brings the other sections of the rotor cage with low resistances inside the stator bore to reduce the rotor losses at rated speed. This leads to increased efficiency at nominal operation [6]. Figures 1.1-1 and 1.1-2 show the cross section view of a 3-section VZIM motor at standstill and nominal speed. The rotor length is 1.5 times the stator length, and it is composed of three different sub-cages (“A”: starting sub-cage, “B”: middle sub-cage and “C”: final sub-cage). At standstill, the subcages “A” and “B” are inside the rotor bore; means they are active at start-up of the motor). By increasing the speed, a centrifugal axial rotor shifter mechanism pulls the rotor in the axial direction and brings the third section of the rotor cage inside the stator bore. At the nominal speed the sub-cage “A” is put completely out of the stator bore and only the sub-cages “B” and “C” are active.

-1-

Fig 1.1-1: Cross-section of variable impedance induction motor, rotor position at stand still [6]

Fig 1.1-2: Cross-section of variable impedance induction motor rotor position at nominal speed [6]

-2-

1.2 Electromagnetic and thermal analysis of the 22kW VZIM A 4 pole, 22kW VZIM operating with a nominal voltage of 400V and a nominal frequency of 50Hz has been designed. The rotor has 3 sections with different rotor bar shapes. The first rotor section has round bars, the middle cage has deep bars and the final cage has wedge bars. The electromagnetic characteristics and thermal effects of the electrical losses in this motor has been analyzed and investigated. With the help of Flux2D and ANSYS which are suitable tools for electromagnetic and thermal analysis and are based on finite element method, the motor’s characteristics and features are calculated and the results of the electromagnetic analysis prepared by Flux2D are compared with analytical calculations which mostly are obtained by KLASYS tool. In this analysis it has been assumed that there are three motors with the same stator but different rotor bars, therefore three motors VZIM1, VZIM2 and VZIM3 are analyzed. 1.3 Introduction to finite element method [4] In engineering and science there are many physical phenomena which can be described with Partial Differential Equations (PDE), solving these equations with analytical methods for arbitrary shapes is almost impossible. The Finite Element Method (FEM) is a numerical approach by which these PDE can be approximately solved. Solving a PDE by FEM method is done by dividing the calculation domain in finite elements in which a known variation of the physical values is assumed. This variation is usually a polynomial variation with an arbitrary power but in practical applications the maximal degree Three is used. In conclusion the field distribution is assumed to be a polynomial variation of first order in most of the cases, of second order in more rare situation and extremely rare of the third order. To have a good FEM calculation some general rules must be followed. After dividing the calculation domain in finite elements with meshing the area of calculation, first of all the finite elements have to cover the entire calculation domain, the elements must not overlap and the nodes of one element cannot be found on the lines of the adjacent elements but must have the same position with the adjacent elements nodes. In case of magneto static problems, the field is described by the following Maxwell equations, ݀݅‫ ࡮ݒ‬ൌ ͲሺͳǤ͵ െ ͳሻ ‫ ࡴݐ݋ݎ‬ൌ ࡶሺͳǤ͵ െ ʹሻ and the constitutive law for material, ࡮ ൌ ߤ ή ࡴሺͳǤ͵ െ ͵ሻ From ሺͳǤ͵ െ ͳሻ we can write B as, ࡮ ൌ ‫࡭ݐ݋ݎ‬ሺͳǤ͵ െ Ͷሻ because it is always true݀݅‫ݒ‬ሺ‫࡭ݐ݋ݎ‬ሻ ൌ Ͳ, where ࡭ is called the magnetic vector potential.

FromሺͳǤ͵ െ ʹሻ, ሺͳǤ͵ െ ͵ሻ and ሺͳǤ͵ െ Ͷሻ the differential equation that describes a magneto static problem can be obtained, -3-

‫ݐ݋ݎ‬ሺఓ ‫࡭ݐ݋ݎ‬ሻ ൌ ࡶሺͳǤ͵ െ ͷሻ ଵ

Each field is defined by its sources݀݅‫ ࡲݒ‬ൌ ܽ, and by related eddies ‫ ࡲݐ݋ݎ‬ൌ ܾ, so ݀݅‫ܣݒ‬ may be chosen free. The coulomb gauge is often used as ݀݅‫ ࡭ݒ‬ൌ ͲǤሺͳǤ͵ െ Ͷܽሻ

Further, if a constant vector ࡭௖ is added to࡭ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݖ‬ሻ, still ሺͳǤ͵ െ Ͷሻ and ሺͳǤ͵ െ Ͷܽሻ are valid, so the magnetic vector potential ࡭ is defined by choosing the value for one node from domain. This is necessary in order to obtain a correct solution. The unknown of the problem is the value of ࡭ in the grid nodes and the field source is given by the current density in the elements volumes (areas for 2D). After solving the linear system of equations the values of the flux density B are obtained. Afterwards the field strength H due to ሺͳǤ͵ െ ͵ሻ is calculated and by integration over the domain the energy and generated force can also be calculated. In FEM the stored magnetic energy is calculated by integrating the energy density as a volume integral for 3D-FEM or as an area integral for 2D-FEM. The volume energy density is defined as:‫ ݓ‬ൌ ሺ࡮ ή ࡴሻΤʹ . ௡೐

The energy density can be integrated on the calculation domain for the 3D-FEM as it follows

࡮Ǥ ࡴ ࡮Ǥ ࡴ ܹ ൌ න ‫ ߗ݀ ݓ‬ൌ න ܸ݀ ൌ ෍ න ܸ݀ ǤሺͳǤ͵ െ ͷሻ ʹ ʹ ௘ୀଵ

As ࡮ ൌ ‫ ࡭ݐ݋ݎ‬is calculated as derivative of࡭, ࡮ is constant within each element. Via ࡮ ൌ ߤ ή ࡴ also H is constant within each element. So considering that inside one element the values for B and H are constant, ሺͳǤ͵ െ ͷሻ can be transformed into: ௡೐

௡೐

௡೐

‫ܤ‬௘ଶ ࡮Ǥ ࡴ ‫ܤ‬௘ ή ‫ܪ‬௘ ή ܸ௘ ൌ෍ ή ܸ݀ ǤሺͳǤ͵ െ ͸ሻ ܹ ൌ ෍න ܸ݀ ൌ ෍ ʹ ʹ ʹ ή ߤ௘ ௘ୀଵ

௘ୀଵ

௡೐

௡೐

௘ୀଵ

For 2D-FEM the equation ሺͳǤ͵ െ ͸ሻ is valid, if we consider that we have a length of the model in the z direction equal with݈ ൌ ͳ:

ܹ ࡮Ǥ ࡴ ܸ݀ ‫ܤ‬௘ଶ ൌ ෍න ൌ ෍ ή ‫ ܣ‬ǤሺͳǤ͵ െ ͹ሻ ݈ ʹ ݈ ʹ ή ߤ௘ ௘ ௘ୀଵ

௘ୀଵ

The energy in ሺͳǤ͵ െ ͹ሻ is measured in Ȁ, whereas the energy in ሺͳǤ͵ െ ͸ሻ is measured in

. In order to get the value of the stored magnetic energy in a volume from a 2D-FEM calculation, one has to multiply the calculated value with the length of the model in the zdirection. For the magneto static calculation the magnetic field is considered to be constant and no eddy currents are induced in the conductive materials. This stationary situation is sufficient to describe only the problems with slow varying fields and problems where the effects of the variation of the fields are considered negligible. For the rest of the problems consideration of -4-

the variation of the fields is necessary and calculation in time domain or in frequency domain is applied. The time domain calculation is actually a succession of static calculations for different time moments. The time variable cannot be considered continuously and has to be considered using a certain time step. Each static calculation that is performed during a transient calculation is called a non-linear iteration and for each iteration the results from the previous iteration are considered as starting point. The variation of magnetic field between two iteration steps determines the variation speed of the magnetic field. In order to calculate this influence the Maxwell equations must be written in a form that considers the magnetic field variation in time ݀݅‫ ࡮ݒ‬ൌ ͲሺͳǤ͵ െ ͺሻ ‫ ࡴݐ݋ݎ‬ൌ ࡶሺͳǤ͵ െ ͻሻ ߲࡮ ‫ ࡱݐ݋ݎ‬ൌ െ ሺͳǤ͵ െ ͳͲሻ ߲‫ݐ‬ and the constitutive law for material ࡮ ൌ ߤ ή ࡴሺͳǤ͵ െ ͳͳሻ ࡶ ൌ ˍ ή ࡱǤሺͳǤ͵ െ ͳʹሻ

Like for magneto static problems we can write the flux density as a curl of vector potential࡭ ࡮ ൌ ‫࡭ݐ݋ݎ‬ሺͳǤ͵ െ ͳ͵ሻ Using ሺͳǤ͵ െ ͳ͵ሻ in ሺͳǤ͵ െ ͳͳሻ we obtain ߲ ߲࡭ ‫ ࡱݐ݋ݎ‬ൌ െ ‫ ࡱ ֜ ࡭ݐ݋ݎ‬ൌ െ ሺͳǤ͵ െ ͳͶሻ ߲‫ݐ‬ ߲‫ݐ‬ which is the electric field strength determined by the time-varying field. It represents the electromotive force induced in the conductor due to the magnetic field variation. From ሺͳǤ͵ െ ͳͶሻ we can calculate the current density induced by the magnetic field variation usingሺͳǤ͵ െ ͳʹሻ ߲࡭ ࡶࢋ ൌ െˍ ή ሺͳǤ͵ െ ͳͷሻ ߲‫ݐ‬ The current density ࡶ ൌ ࡶௌ ൅ ࡶ௘ has 2 components: -The source component ࡶௌ that is given as entry data for the problem at the beginning of the calculation. It is represented by the current density in the conductors that are exciting the primary magnetic field model. -The component that is induced due to the magnetic field variationࡶ௘ . Considering these two components results in ͳ ߲࡭ ‫ ݐ݋ݎ‬൬ ‫࡭ݐ݋ݎ‬൰ ൌ ࡶௌ െ ˍ ή ǤሺͳǤ͵ െ ͳ͸ሻ ߤ ߲‫ݐ‬

This is the differential equation to be solved by the FE program, in a time-step solution. In practice with usage of a finite element tool there are some steps to solve the problem. The geometry is prepared, then geometry must be meshed and afterwards material are assigned to the regions, finally by linking an external electrical circuit the model is ready to be solved. To create the geometry, symmetry and periodicity must be taken into account. If the geometry -5-

has periodicity, type of the periodicity (symmetric or anti-symmetric conditions) must be defined. 1.4 Electrical and mechanical parameters and dimensions of the 22kW VZIM Due to three rotor sections the names VZIM1 for the first cage, VZIM2 for the middle cage and VZIM3 for the last cage, are assigned to each part of the rotor. Stator dimensions are given in Table 1.4-1. The stator winding details, which has a star connection are listed in Table 1.4-2. Tables 1.4-3, 1.4-4 and 1.4-5 list the rotor dimensions and parameters of the VZIM1, VZIM2 and VZIM3. The rotor sub-cages are not skewed. Table 1.4-6 consists of the material details used in VZIM. Table 1.4-1: Stator dimensions Description Stack length Inner stator diameter Outer stator diameter Air gap width Number of stator slots Width of stator slot opening Height of stator slot opening Stator slot heit Stator slot width Radius of stator slot opening

Table 1.4-2: Stator winding details Description Number of phases Number of pole pairs Number of slots per pole and phase Number of turns per coil Number of parallel branches Type of winding Winding pitch Number of turns per phase Length of overhang Number of parallel wires per turn Diameter of conductor Winding connection

parameter ݈ி௘ ݀௦௜ ݀௦௢ ߜ ܳ௦ ‫ݏ‬ொ ݄௦ସ ݄ொ௦ ܾொ௦ ‫ݎ‬௦ொ

parameter m p q ܰ௖ a ܰ௦ ݈௕ ܽ௜ ݀஼௨ -

-6-

Dimension mm mm mm mm mm mm mm mm mm

Value 240 170 270 0.45 36 3.1 0.737 22.6 10.277 3.606

Dimension mm mm -

Value 3 2 3 12 1 Single layer Full pitch 72 226.6 9 1 Star

Table 1.4-3: The rotor dimensions and parameters of VZIM1 Description parameter Outer rotor diameter ݀௥௢ Inner rotor diameter ݀௥௜ Number of rotor bars ܳ௥ Bar shape Bar material Width of rotor slot opening ‫ݎ‬ொ Height of rotor slot opening ݄௥ସ Rotor bar diameter ݀ொ௥ End-ring thickness ܾ௘௥ End-ring height ݄௘௥

Dimension mm mm mm mm mm mm mm

Value 169.1 80 28 Round Bronze 4.5 1 10.2 4.3 44

Table 1.4-4: The rotor dimensions and parameters of VZIM2 Description parameter Outer rotor diameter ݀௥௢ Inner rotor diameter ݀௥௜ Number of rotor bars ܳ௥ Bar shape Bar material Width of rotor slot opening ‫ݎ‬ொ Height of rotor slot opening ݄௥ସ Height of rotor bar ݀ொ௥ Width of rotor bar ܾொ௥ End-ring thickness ܾ௘௥ End-ring height ݄௘௥

Dimension mm mm mm mm mm mm mm mm

Value 169.1 80 28 Deep Copper 4 1 8.2 9.2 4.6 44

Table 1.4-5: The rotor dimensions and parameters of VZIM3 Description parameter Outer rotor diameter ݀௥௢ Inner rotor diameter ݀௥௜ Number of rotor bars ܳ௥ Bar shape Bar material Width of rotor slot opening ‫ݎ‬ொ Height of rotor slot opening ݄௥ସ Height of rotor bar ݀ொ௥ Width of rotor bar ܾொ௥ End-ring thickness ܾ௘௥ End-ring height ݄௘௥

Dimension mm mm mm mm mm mm mm mm

Value 169.1 80 28 Inverse wedge Copper 2 1 17.7 10.2 6 44

-7-

Table 1.4-6: VZIM material details Description Conductivity of bronze at 110Ԩ Conductivity of copper at 110Ԩ Lamination type Conductivity of lamination Loss at 50Hz, 1T Loss at 50Hz, 1.5T

parameter ˍ஻௥௢௡௭௘ ˍ௖௢௣௣௘௥ ˍ௟௔௠௜௡ ߥଵ଴ ߥଵହ

Dimension Sିଵ Sିଵ Sିଵ W/kg W/kg

Value 1.69E+7 4.22E+7 M270-50A 1.818E+6 1.07 2.52

Cross-sections of the motors have been depicted in figures 1.4-1, 1.4-2 and 1.4-3. The first motor VZIM1 has round rotor bars made of bronze and second and third motor VZIM2 and VZIM3 have rotor bars made of copper.

Fig 1.4-1: Cross-section of Variable Impedance Induction motor (VZIM1), prepared by ANSYS

-8-

Fig 1.4-2: Cross-section of Variable Impedance Induction motor (VZIM2), prepared by ANSYS

Fig 1.4-3: Cross-section of Variable Impedance Induction motor (VZIM3), prepared by ANSYS -9-

1.5 Procedure of the project In order to simplify the simulation, three separate induction motors have been considered, in this simulation VZIM1, VZIM2 and VZIM3 each one has a full length rotor and stator. With the help of Flux2D, electromagnetic analysis and with ANSYS, thermal analysis has been done. Two types of electromagnetic analysis have been carried out: -Steady state magnetic AC analysis -Time stepping analysis. All results from Flux2D have been compared with the analytical results of KLASYS tool.

- 10 -

Chapter 2: Geometry and mesh generation with Flux2D

2.1 Introduction Electromagnetic analysis based on finite element method is done with Flux2D. In this chapter the creation of geometry, mesh generation, electrical circuit determination and material assignment is explained. 2.2 Geometry creation The variable impedance induction motor which is analyzed has 4-poles, 36-stator slots and 28 rotor bars, three different motors with the same stator but different rotors are being considered. Based on this consideration 3 models has been created. Because of simplicity and according to Anti-cyclic boundary condition, ¼ of the motor is modeled. Three models which have been prepared as VZIM1, VZIM2, and VZIM3 are depicted in Fig 2.2-1, Fig 2.2-2 and Fig 2.2-3.

Fig 2.2-1: Created geometry of VZIM1, by Flux2D

- 11 -



2.3 Meshing the geometry An automatic mesh generator is in charge of meshing the faces. With the help of the mesh points, it is possible to have fine and dense meshing in the areas which the higher accuracy is needed. Fine mesh in the air-gap, at the top of the rotor bars, and the stator teeth, is necessary because of accuracy of the results, air-gap must be fine meshed, to have accurate torque calculation. The top of the rotor bars near the air gap are densely meshed, to taking into the account the current displacement effect. In addition the stator teeth must be meshed fine enough to consider the saturation effect because of high flux density in this area. In Fig.2.3-1 and Fig.2.3-2 generated mesh for VZIM1 is shown. The fine mesh in the air-gap is observable in Fig.2.3-1.

Fig 2.3-1: Generated mesh of VZIM1 in the air-gap using Flux2D

- 12 -

Fig 2.3-2: Generated mesh of VZIM1 using Flux2D

- 13 -

2.4 Material assignment Stator and rotor lamination in 22kW VZIM are made of M270-50A [7]. B (H) curve and values are shown in Fig.2.4-1 and table 2.4-1. Table 2.4-1: B-H characteristic data of the iron sheet type M270-50A [7] B/Tesla H/(A/‫)ܕ‬ H/(A/‫)ܕ‬ 0 2119.027 0 21.73374 0.454861 2668.102 43.46748 0.7016061 3354.447 73.65323 0.902243 4212.377 111.3854 1.048599 5284.791 158.5506 1.157766 6625.307 217.5072 1.240593 8300.953 291.2029 1.304297 10395.51 383.3224 1.353865 13013.71 498.472 1.392849 16286.45 642.4088 1.42385 20377.38 822.3299 1.448816 25491.05 1047.231 1.469241 31883.13 1328.358 1.486302 43860.55

B/Tesla 1.516379 1.534915 1.556988 1.582977 1.613156 1.647613 1.686154 1.728191 1.772635 1.817807 1.861388 1.900418 1.931347 1.955117

B-H Characteristic of M270-50A iron sheets 2.5

2

B/Tesla

1.5

1

0.5

0 0

5000

10000

15000

20000

25000

30000

35000

40000

45000

50000

H/A‫(^ܕ‬−૚) Fig.2.4-1: B-H characteristic of M270-50A iron sheets, which are used in stator and rotor lamination [7] - 14 -

VZIM1 has rotor bars which are made of bronze, VZIM2 and VZIM3 have rotor bars made of copper. This special alloy of bronze has the electrical resistivity equal to 0.588E-07 Ohm.m. Copper has the electrical resistivity equal to 0.237E-07 Ohm.m. 2.5 Electrical circuit of the motor Three phase winding in 22kW VZIM is star connected. The electrical circuit which is created in Flux2D, is depicted in Fig.2.5-1.

Fig.2.5-1: Star connected stator electrical circuit of the motor, by Flux2D

The resistances R1, R2 and R3 represent the winding overhang resistances; also the inductances L1, L2 and L3 represent the overhang inductances. In addition stator phase windings under stator slots are shown as BPA, BPB and BMC. VA, VB and VC are three phase voltage sources with 230 V and 120 degree phase shift with each other; moreover Q1 represents the squirrel cage rotor which is connected to common point of voltage sources.

- 15 -

2.6 Calculation of values of circuit’s elements The values of the circuit components are calculated and assigned to the circuit. In this section the values of the stator resistance, inductance of stator winding overhang, rotor end-ring leakage inductance and resistance of the end-ring segments are calculated. 2.6.1 Stator resistance Stator winding resistance per phase ܴ௦ is sum of resistance of the coil under stator slots ܴ஻௉஺ (alsoܴ஻௉஻ ܴ஻ெ஼ ) and winding overhang ܴଵ (alsoܴଶ andܴଷ ). Resistance per phase is calculated according to [1]: ͳ ܰ௦ ൉ ʹ ൉ ሺ݈ி௘ ൅ ݈௕ ሻ ܴ௦ ൌ ሺʹǤ͸Ǥͳ െ ͳሻ ˍ௖௨ ‫ܣ‬௖௨ ൉ ܽ ߩሺߴሻ ൌ ߩሺʹͲԨሻ ή ሺͳ ൅ ߙణ ή οߴሻǡ ߙణ ൌ ͳΤʹͷͷ ିଵ ǡ οߴ ൌ ߴ െ ʹͲԨǡ ߩ ൌ ͳΤˍ ǤሺʹǤ͸Ǥͳ െ ʹሻ Resistance of one phase under stator slot at 20Ԩ equals to: ͳ ܰ௦ ൉ ʹ ൉ ݈ி௘ ͳ ͹ʹ ൉ ʹ ൉ ʹͶͲ ή ͳͲିଷ ͳ ܰ௦ ൉ ʹ ൉ ݈ி௘ ൌ ൌ ܴ஻௉஺ǡଶ଴Ԩ ൌ ଶ ήܽ ˍ௖௨ ߨ ή ݀௖௨ ͷ͹ ή ͳͲି଺ ߨ ή ͳଶ ή ͳͲି଺ ή ͻ ή ͳ ˍ௖௨ ‫ܣ‬௖௨ ൉ ܽ Ͷ Ͷ ൌ ͲǤͲͺͷ͹͹ߗ and the resistance of one phase under slot of stator in 110Ԩ is given by: ʹ͵ͷ ൅ ͳͳͲ ܴ஻௉஺ǡଵଵ଴Ԩ ൌ ܴ ൌ ͲǤͳͳ͸ȳǤ ʹ͵ͷ ൅ ʹͲ ஻௉஺ǡଶ଴Ԩ

Resistance of the winding overhang at 20Ԩ is calculated as below: ͳ ܰ௦ ൉ ʹ ൉ ݈௕ ͳ ͹ʹ ൉ ʹ ൉ ʹʹ͸Ǥ͸ ή ͳͲିଷ ͳ ܰ௦ ൉ ʹ ൉ ݈௕ ൌ ൌ ܴଵǡଶ଴Ԩ ൌ ଶ ήܽ ˍ௖௨ ߨ ή ݀௖௨ ͷ͹ ή ͳͲି଺ ߨ ή ͳଶ ή ͳͲି଺ ή ͻ ή ͳ ˍ௖௨ ‫ܣ‬௖௨ ൉ ܽ Ͷ Ͷ ൌ ͲǤͲͺͲͻͺߗ and the resistance of winding overhang at 110Ԩ is equal to: ʹ͵ͷ ൅ ͳͳͲ ܴଵǡଵଵ଴Ԩ ൌ ܴ ൌ ͲǤͳͲͻͷȳǤ ʹ͵ͷ ൅ ʹͲ ଵǡଶ଴Ԩ 2.6.2 Inductance of stator winding overhang

Inductance of the winding overhang is calculated according to [1], using following equation: ʹ ‫ܮ‬ఙ௕ ൌ ߤ଴ ܰ௦ଶ ߣ௕ ݈௕ ሺʹǤ͸Ǥʹ െ ͳሻ ‫݌‬ where ݈௕ is the length of the winding overhang and ߣ௕ is the permeance /unit length which is calculated as below: ݈௕ െ ͳͲ ή ݀௪ ൨ሺʹǤ͸Ǥʹ െ ʹሻ ߣ௕ ൌ ͲǤ͵͹ ൉ ܿ ൉ ൤ͳ ൅ ͲǤͻ ൉ ݈௕ ൅ ͳͲ ή ݀௪ where ݀௪ is the diagonal of the cross-section of the coils in the winding overhang and is calculated as following: - 16 -

݀௪ ൌ ඨ

ଶ ܰ ܽ௜ ݀௖௨ ͻ ή ͳଶ ή ͳʹ ௖ ൌඨ ൌ ͳͷǤͻሺʹǤ͸Ǥʹ െ ͵ሻ ݇௙ ͲǤͶʹ͸

According to [1] for a single phase winding with q=3, c is equal to 0.67 and ߣ௕ is calculated as below: ʹʹ͸Ǥ͸ െ ͳͲ ή ͳͷǤͻ ൨ ൌ ͲǤʹͺ͹Ǥ ߣ௕ ൌ ͲǤ͵͹ ൉ ͲǤ͸͹ ൉ ൤ͳ ൅ ͲǤͻ ൉ ʹʹ͸Ǥ͸ ൅ ͳͲ ή ͳͷǤͻ Therefore the inductance of the winding overhang equals to: ʹ ‫ܮ‬ఙ௕ ൌ Ͷ ή ߨ ή ͳͲି଻ ή ͹ʹଶ ʹʹ͸Ǥ͸ ή ͳͲିଷ ή ͲǤʹͺ͹ ൌ ͲǤͶʹͶǤ ʹ

2.6.3 Rotor end-ring leakage inductance

Rotor end-ring leakage inductance is calculated according to [1] as following: ʹ ‫ܮ‬ఙ௕௥ ൌ ߤ଴ ܰ௥ଶ ߣ௕ ݈௕ ǤሺʹǤ͸Ǥ͵ െ ͳሻ ‫݌‬ where ߣ௕ ൌ ͲǤͳʹ and ݈௕ ൌ ߬௣ ൌ ‫ܮ‬ఙ௕௥ ൌ Ͷ ή ߨ ή ͳͲ

ି଻

ௗೞ೔ గ ଶ௣

ൌ

ଵ଻଴గ ସ

ൌ ͳ͵͵Ǥͷ, hence:

ͳ ଶ ʹ ή ൬ ൰ ή ή ͲǤͳʹ ή ͳ͵͵Ǥͷ ή ͳͲିଷ ൌ ͷǤͲ͵ ή ͳͲିଽ Ǥ ʹ ʹ

2.6.4 Resistance of the end-ring segment of VZIM1

First of all the mean diameter of the end-ring ݀ாோ௠ is calculated which is the mean value of the inner diameter݀௜௥ and the outer diameter ݀௢௥ of the end-ring: ݀௢௥ ൅ ݀௜௥ ͳ͸ͻǤͳ ൅ ͺͳ ݀ாோ௠ ൌ ൌ ൌ ͳʹͷǤ ʹ ʹ With ܳ௥ the number of the rotor bars and ݀ாோ௠ the mean diameter of end-ring, the length of the end-ring segment is calculated as below: ߨ݀ாோ௠ ߨ ή ͳʹͷ ൌ ൌ ͳͶǤͲʹǡ ‫ܴܧ‬௟ ൌ ʹͺ ܳ௥ the end-ring thickens is ‫ܴܧ‬௧௛ ൌ ͶǤ͵ and the end-ring height is ‫ܴܧ‬௛ ൌ ͶͶǡusing endring thickness and height, the cross section area of end-ring is calculated as: ‫ܣ‬ாோ ൌ ‫ܴܧ‬௧௛ ή ‫ܴܧ‬௛ ൌ ͳͺͻǤʹଶ . Then resistance of the end-ring is calculated as: ͳͶǤͲʹ ή ͳͲିଷ ‫ܴܧ‬௟ ൌ ൌ ͶǤͳ͸ ή ͳͲି଺ ȳǤ οܴாோ ൌ ͳ͹Ǥͺ ή ͳͲ଺ ή ͳͺͻǤʹ ή ͳͲି଺ ፺஻௥ǡଵଵ଴Ԩ ‫ܣ‬ாோ 2.6.5 Resistance of the end-Ring Segment VZIM2

The mean diameter of the end-ring ݀ாோ௠ is calculated using the mean value of the inner diameter݀௜௥ and the outer diameter ݀௢௥ of the end-ring: - 17 -

݀௢௥ ൅ ݀௜௥ ͳ͸ͻǤͳ ൅ ͺͳ ൌ ൌ ͳʹͷ ʹ ʹ With ܳ௥ the number of the rotor slots and ݀ாோ௠ the mean diameter of end-ring, the length of the end-ring segment is calculated as: ߨ ή ͳʹͷ ߨ݀ாோ௠ ൌ ൌ ͳͶǤͲʹ ‫ܴܧ‬௟ ൌ ʹͺ ܳ௥ end-ring thickens is, ‫ܴܧ‬௧௛ ൌ ͶǤ͸ and the end-ring height is ‫ܴܧ‬௛ ൌ ͶͶ .using endring thickness and height, the cross section area of end-ring is calculated as: ‫ܣ‬ாோ ൌ ‫ܴܧ‬௧௛ ή ‫ܴܧ‬௛ ൌ ʹͲʹǤͶଶ conductivity of copper in 110Ԩ is calculated as: ʹ͵ͷ ൅ ʹͲ ʹͷͷ ˍ௖௨ǡଵଵ଴Ԩ ൌ ˍ௖௨ǡଶ଴Ԩ ൌ ͷ͹ ή ͳͲ଺ ൌ ͶʹǤͳ͵ ή ͳͲ଺ ൗ ʹ͵ͷ ൅ ͳͳͲ ͵Ͷͷ Then resistance of the end-ring is calculated as: ‫ܴܧ‬௟ ͳͶǤͲʹ ή ͳͲିଷ οܴாோ ൌ ൌ ൌ ͳǤ͸ͶͶ ή ͳͲି଺ ȳ ፺௖௨ǡଵଵ଴Ԩ ‫ܣ‬ாோ ͶʹǤͳ͵ ή ͳͲ଺ ή ʹͲʹǤͶ ή ͳͲି଺ ݀ாோ௠ ൌ

2.6.6 Resistance of the end-ring segment VZIM3

The mean diameter of the end-ring ݀ாோ௠ is calculated using the mean value of the inner diameter݀௜௥ and the outer diameter ݀௢௥ of the end-ring, ݀௢௥ ൅ ݀௜௥ ͳ͸ͻǤͳ ൅ ͺͳ ൌ ൌ ͳʹͷ ݀ாோ௠ ൌ ʹ ʹ with ܳ௥ the number of the rotor slots and ݀ாோ௠ the mean diameter of end-ring, the length of the end-ring segment is calculated as: ߨ݀ாோ௠ ߨ ή ͳʹͷ ‫ܴܧ‬௟ ൌ ൌ ൌ ͳͶǤͲʹ ʹͺ ܳ௥ end-ring thickens is ‫ܴܧ‬௧௛ ൌ ͸ and the end-ring height is ‫ܴܧ‬௛ ൌ ͶͶ .using end-ring thickness and height, the cross section area of end-ring is calculated as: ‫ܣ‬ாோ ൌ ‫ܴܧ‬௧௛ ή ‫ܴܧ‬௛ ൌ ʹ͸Ͷଶ conductivity of copper in 110Ԩ is calculated as: ʹ͵ͷ ൅ ʹͲ ʹͷͷ ˍ௖௨ǡଶ଴Ԩ ൌ ͷ͹ ή ͳͲ଺ ൌ ͶʹǤͳ͵ ή ͳͲ଺ ൗ ˍ௖௨ǡଵଵ଴Ԩ ൌ ʹ͵ͷ ൅ ͳͳͲ ͵Ͷͷ Then resistance of the End-Ring is calculated as: ‫ܴܧ‬௟ ͳͶǤͲʹ ή ͳͲିଷ οܴாோ ൌ ൌ ൌ ͳǤʹ͸ͳ ή ͳͲି଺ ȳ ፺௖௨ǡଵଵ଴Ԩ ‫ܣ‬ாோ ͶʹǤͳ͵ ή ͳͲ଺ ή ʹ͸Ͷ ή ͳͲି଺ All the values derived from formulas in sections 2.6.4 to 2.6.6 are shown in Table 2.6-1.

- 18 -

Table 2.6-1: Electrical values of VZIM stator circuit Parameter value description 0.116 Ω Resistance per phase of the coil under stator slot ܴ஻௉஺ , ܴ஻௉஻ ǡ ܴ஻ெ஼ 0.1095 Ω Resistance per phase of winding overhang ܴଵ , ܴଶ ,ܴଷ 0.424 mH Inductance pre phase of winding overhang ‫ܮ‬ଵ , ‫ܮ‬ଶ ,‫ܮ‬ଷ End-ring leakage inductance ‫ܮ‬ఙ௕௥ ͷǤͲ͵ ή ͳͲିଽ ି଺ Resistance of the end-ring segment, VZIM1 οܴாோଵ ͶǤͳ͸ ή ͳͲ ȳ ି଺ Resistance of the end-ring segment, VZIM2 οܴாோଶ ͳǤ͸ͶͶ ή ͳͲ ȳ Resistance of the end-ring segment, VZIM3 οܴாோଷ ͳǤʹ͸ͳ ή ͳͲି଺ ȳ

- 19 -

Chapter 3: Steady state electromagnetic analysis

3.1 Introduction In steady state electromagnetic analysis, it is assumed that variables are changing purely sinusoidal with time and there are no harmonics. Due to the saturation effect of magnetic material, electromagnetic flux density B and electromagnetic field strength H, could not be sinusoidal at the same time. If the electric circuit had voltage source the B would be considered as sinusoidal variable and therefore because of B (H) curve, H would not be a sinusoidal variable with time. Fig.3.1-1 shows that how B and H curves change if one of them is sinusoidal.

Fig. 3.1-1: Nonlinearity relation of B and H [5]

The above contradictions are dealt by using an equivalent B(H) characteristic, which is different for the original B(H) characteristic and based on the energy equivalence method, as shown in Fig.3.1-2. For the two extreme cases of B sinusoidal and H sinusoidal the B(H) curve is modified and shown in Fig.3.1-2 based on the above method. The results obtained with the equivalent curves calculated in the two extreme cases most often include the exact result. The equivalent curve can equally be calculated by means of a linear combination between these two extreme cases as shown in Fig.3.1-2. Only the numerical values of the post-processed quantities, that depend on energy, are correct. The instantaneous values computed by this analysis are approximations as they are sinusoidal. - 20 -

Fig. 3.1-2: Static and equivalent B-H curves for different cases [5]

3.2 Steady state AC analysis of the 22kW VZIM As previously was mentioned the VZIM motor is considered as three different motors with the same stator and different rotor bars. Therefore three different motors VZIM1, VZIM2 and VZIM3 are analyzed and the results are compared with analytical calculations obtained from KLASYS tool. 3.2.1 Steady state AC analysis of the VZIM1 Fig.3.2.1-1 shows the torque-slip characteristic of VZIM1, which compares the steady state AC analysis results calculated by Flux2D and KLASYS. It is apparent that the values calculated by KLASYS are higher than values obtained using Flux2D. In lower slips the difference between the values of KLASYS and Flux2D are less than 10% and in slips higher than 0.3 the difference reaches to around 12%. In addition the breakdown torque in both calculation methods happens around slip equal to 0.65, for the unskewed rotor cage. The calculation results for the skewed rotor with one stator slot pitch obtained by KLASYS shows that the breakdown slip is close to one as depicted in Fig.3.2.1.2. As the rotor is equivalently skewed by shifting the sub-cages in the circumferential direction with respect to each other by half a stator slot pitch, Ƚ௦௞௘௪ ൌ ͵͸ͲȀ͵͸Ȁʹ ൌ ͷι, which is equivalent to one sator slot pitch skewing, therefore in the design phase, the optimization has been done for the skewed rotor case to achieve the maximum starting torque at stand still.

- 21 -

The stator phase current-slip characteristics of VZIM1 which are prepared by Flux2D and KLASYS are depicted in Fig.3.2.1-3. Maximum deviation between results of the KLASYS and Flux2D is less than 4%, which shows a good precision. In Fig.3.2.1-4 the normal component of the air-gap flux density of VZIM1 at a slip of 0.0253 for one pole pair or 180ι mechanical degrees has been depicted. Fig.3.2.1-5 shows the numerically calculated flux lines in VZIM1 by 2 pole pairs at a slip of 0.0253 or 1462.05ିଵ, obtained by Flux2D.

Fig.3.2.1-1: Torque-slip characteristics of unskewed VZIM1, comparison of the steady state calculation results of Flux2D and the analytical calculations by KLASYS

- 22 -

Fig.3.2.1-2: Torque-slip characteristics of VZIM1, comparison of skewed and unskewed rotor, obtained by KLASYS

Fig.3.2.1-3: Phase current-slip characteristic of unskewed VZIM1, comparison of the steady state calculation results of Flux2D and the analytical results of KLASYS

- 23 -

Fig.3.2.1-4: Normal component of air gap flux density in the center of the air gap, for one pole pair of VZIM1 at a slip of 0.0253, calculated by Flux2D

- 24 -

Fig.3.2.1-5: Numerically calculated flux lines in VZIM1 at a slip of 0.0253, obtained by Flux2D

- 25 -

3.2.2 Steady state AC analysis of the VZIM2 Fig.3.2.2-1 shows the torque-slip characteristic of VZIM2, which compares the steady state AC analysis results calculated by Flux2D and KLASYS. It is apparent that the values calculated by KLASYS are higher than values obtained using Flux2D. The difference between values of KLASYS and Flux2D are around 15%. In addition the breakdown torque in both calculation methods happens at around a slip of 0.3. In Fig.3.2.2-2 stator phase current-slip characteristics of VZIM2 which are calculated by Flux2D and KLASYS are shown. In lower slips the difference of the values in KLASYS and Flux2D is around 12%, but in higher values of slips the difference is lower than 4%. In Fig.3.2.2-3 the normal component of air-gap flux density of VZIM2 at a slip of 0.0294 for one pole pair or 180ι mechanical degrees has been depicted. Fig.3.2.2-4 shows the numerically calculated flux lines in VZIM2 at a slip of 0.0294 or 1455.8ିଵ, obtained by Flux2D.


- 26 -


Fig.3.2.2-3: Normal component of air gap flux density in the center of the air gap, for one pole

pair of VZIM2 at a slip of 0.0253, calculated by Flux2D

- 27 -

Fig.3.2.2-4: Numerically calculated flux lines in VZIM2 at a slip of 0.0253, obtained by

Flux2D

- 28 -

3.2.3 Steady state AC analysis of the VZIM3 Fig.3.2.3-1 shows the torque-slip characteristic of VZIM3, which compares the steady state AC analysis results calculated by Flux2D and KLASYS. It is apparent that the values calculated by KLASYS are higher than values obtained using Flux2D. The difference between values of KLASYS and Flux2D are around 12%. In addition the breakdown torque in both calculation methods happens at around a slip of 0.2. In Fig.3.2.3-2 stator phase current-slip characteristics of VZIM3 which are prepared by Flux2D and KLASYS are shown. In lower slips the difference of the values in KLASYS and Flux2D is around 11%, but in higher values of slips the difference is lower than 3%. In Fig.3.2.3-3 the normal component of the air gap flux density of VZIM3 at a slip of 0.0169 for one pole pair or 180ι mechanical degrees has been depicted. Fig.3.2.3-4 shows the numerically calculated flux lines in VZIM3 at a slip of 0.0169 or 1474.65ିଵ, obtained by Flux2D.


- 29 -


Fig.3.2.3-3: Normal component of air gap flux density in the center of the air gap, for one pole

pair of VZIM3 at a slip of 0.0253, calculated by Flux2D

- 30 -

Fig.3.2.3-4: Numerically calculated flux lines in VZIM3 at a slip of 0.0253, obtained by

Flux2D

- 31 -

Chapter 4: Time stepping analysis

4.1 Introduction Despite the fact that stator voltage and current are sinusoidal, spatial distribution of flux density in the air gap is not sinusoidal but step-like, as winding is located in slots. In steady state analysis only the fundamental sine wave of air gap flux density distribution was considered which gives a rough estimation of motor characteristics, but more precise and accurate results are obtained by consideration of higher space harmonics of the stator and rotor field distribution. 4.2 Stator and rotor field spatial harmonics The step-like air gap flux density distribution generated by stator, ‫ܤ‬ఋǡ௦ ሺ‫ݔ‬௦ ǡ ‫ݐ‬ሻ along stator circumference co-ordinate ‫ݔ‬௦ is represented by Fourier series [2] ‫ܤ‬ఋǡ௦ ሺ‫ݔ‬௦ ǡ ‫ݐ‬ሻ ൌ

ஶ

෍

௩ୀଵǡିହǡ଻ǡǥ

‫ݔߨݒ‬௦ ‫ܤ‬ఋǡ௩ ή ቆ െ ߱௦ ‫ݐ‬ቇሺͶǤʹ െ ͳሻ ߬௣

‫ ݒ‬ൌ ͳ ൅ ʹ݉௦ ݃ ൌ ͳ ൅ ͸݃ ൌ ͳǡ െͷǡ͹ǡ െͳͳǡͳ͵ǡ െͳ͹ǡͳͻǡ െʹ͵ǡʹͷǡ െʹͻǡ͵ͳǡሺͶǤʹ െ ʹሻ ݃ǣ ݅݊‫ݎܾ݁݉ݑ݊ݎ݁݃݁ݐ‬ǣ ݃ ൌ Ͳǡ േͳǡ േʹǡ േ͵ǡ ǥ

Furthermore the step-like air gap flux density distribution generated by rotor ‫ܤ‬ఋǡ௥ ሺ‫ݔ‬௥ ǡ ‫ݐ‬ሻ along rotor circumference co-ordinate ‫ݔ‬௥ is represented by Fourier series [2] ஶ

ߤߨ‫ݔ‬௥ െ ߱௥ ‫ݐ‬ቇሺͶǤʹ െ ͵ሻ ‫ܤ‬ఋǡ௦ ሺ‫ݔ‬௦ ǡ ‫ݐ‬ሻ ൌ ෍ ‫ܤ‬ఋǡఓ ή ቆ ߬௣ ఓୀଵ

ߤ ൌ ͳ ൅ ሺܳ௥ Τ‫݌‬ሻ ή ݃ሺͶǤʹ െ Ͷሻ ݃ǣ ݅݊‫ݎܾ݁݉ݑ݊ݎ݁݃݁ݐ‬ǣ ݃ ൌ Ͳǡ േͳǡ േʹǡ േ͵ǡ ǥ . 4.3 Asynchronous harmonic torques Rotor harmonic currents ‫ܫ‬௥௩ produce not only additional cage losses, but also due to Lorentz forces with stator field harmonic ‫ܤ‬ఋ௦௩ additional torque, which is called asynchronous harmonic torque. For the special case ‫ ݒ‬ൌ ͳ this is the asynchronous torque of KLOSS - 32 -

function. The stator harmonic field induces the rotor harmonic current and the rotor harmonic current produces torque with the stator harmonic field. Asynchronous harmonic torque follows a KLOSS function, depending on harmonic slip ‫ݏ‬௩ , at ‫ݏ‬௩ ൌ Ͳ this torque is zero. At ‫ݏ‬௩ ൐ Ͳ torque is positive and adds to fundamental torque. At ‫ݏ‬௩ ൏ Ͳ the torque is negative and is breaking the machine. At harmonic break down slip: ܴ௥ ሺͶǤ͵ െ ͳሻ ‫ݏ‬௩௕ ൌ േ ሺ‫ܮ‬ ߱௦ ௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ torque reaches positive and negative maximum values. As ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ‫ܴ ب‬௥ harmonic break down slip is small.

Fig.4.3-1: Asynchronous harmonic torques of the 5th and 7th stator field harmonics, which are superimposed on fundamental asynchronous torque [2]

4.4 Synchronous harmonic torques Rotor field harmonic of step like air gap flux density distribution, excited by rotor fundamental current‫ܫ‬௥ , Will also produce parasitic torque with the stator field harmonics. Like asynchronous harmonic torque, the condition for constant torque generation is: -the same wave length -the same velocity (means also same direction of movement) of the stator and rotor field wave. Therefore conditions for slip where synchronous harmonic torque occurs are as following: ‫ ݒ‬ൌ ߤ ֜ ‫ כ ݏ‬ൌ ͳ - 33 -

‫ ݒ‬ൌ െߤ ֜ ‫ כ ݏ‬ൌ

‫ݒ‬െͳ Ǥ ‫ݒ‬൅ͳ

Fig.4.4-1: Typical effects of synchronous and asynchronous harmonic torques in induction machines [2]

Time stepping simulation of the motors is done in the slip range from 0 to 2. In between of these 2 operating points some important slips corresponding to synchronous and asynchronous torques are taken into account. Asynchronous and synchronous harmonic torques happen in slips that have been calculated in Appendix II and they are considered in the simulation of the motors (VZIM1, VZIM2, and VZIM3). Fig.4.4-1 shows typical effects of synchronous and asynchronous harmonic torques in induction machines. 4.5 Derived slips used for time stepping analysis The sets of slips to perform the time stepping analysis of VZIM1, VZIM2 and VZIM3 motors have been derived and are given in Table V, Table VI and Table VII in Appendix II. In these sets of slips the asynchronous and synchronous harmonic torque slips and breakdown slips of asynchronous harmonic torques are included.

- 34 -

4.6 Time-stepping analysis of induction motors Time-stepping analysis of the VZIM motor has been done to observe the transients and harmonic effects on the characteristics of the motor. When the time step is in the order of 1/32th of the electrical cycle, fairly accurate and good results are provided. In our analysis the motor is line fed and the frequency is 50Hz, and the corresponding electrical cycle is 1/50=0.02s; therefore with consideration of 40 time steps per one cycle, ο‫ ݐ‬ൌ ͲǤͲʹΤͶͲ ൌ ͷ݁ ିସ would be an acceptable time step. Time stepping analysis has been done for some speeds of the motor, in the range of -1500rpm to 1500rpm corresponding to the slips given in Table II, Table III and Table IV in Appendix II. In this analysis the friction and windage losses and load inertia are neglected. 4.6.1 Time-stepping analysis of VZIM1 Time stepping analysis of the VZIM1 motor has been done in different speeds which are given in Table V in Appendix II. The results consist of the output torque, input phase current, input power, efficiency, and power factor which have been compared with the results of the KLASYS tool. The synchronous harmonic torques which happen at slips, ‫ ݏ‬ൌ ͲǤͺͷ͹, ‫ ݏ‬ൌ ͳ and ‫ ݏ‬ൌ ͳǤͲ͹ͳ are calculated with Flux2D as explained in the section 4.8 of this chapter.

In Fig.4.6.1-1 the torque slip characteristic of VZIM1, derived from time stepping analysis with Flux2D, is shown. The effects of the synchronous harmonic torques at slips‫ ݏ‬ൌ ͲǤͺͷ͹, ‫ ݏ‬ൌ ͳ and ‫ ݏ‬ൌ ͳǤͲ͹ͳ are clearly observed. In Fig.4.6.1-2 the torque slip characteristic of VZIM1, derived from KLASYS, is shown. The effects of the synchronous harmonic torques at slips‫ ݏ‬ൌ ͲǤͺͷ͹, ‫ ݏ‬ൌ ͳ and ‫ ݏ‬ൌ ͳǤͲ͹ͳ are clearly observed.

The values of synchronous torque calculated by Flux2D and KLASYS at slips‫ ݏ‬ൌ ͲǤͺͷ͹, ‫ ݏ‬ൌ ͳ and‫ ݏ‬ൌ ͳǤͲ͹ͳ, are compared in Table 4.8-6, Table 4.8-7 and Table 4.8-8. It is noticeable that the synchronous torque values calculated by KLASYS are in most of the cases much higher than the values calculated by Flux2D. The values of Flux2D are more reasonable and acceptable.

- 35 -

effects of the synchronous harmonic torques at slips‫ ݏ‬ൌ ͲǤͺͷ͹, ‫ ݏ‬ൌ ͳ and ‫ ݏ‬ൌ ͳǤͲ͹ͳ of the the motor are observable Fig.4.6.1-1: Torque-slip characteristic of unskewed VZIM1, calculated with Flux2D time-stepping, the

Fig.4.6.1-2: Torque-slip characteristic of unskewed VZIM1, calculated with KLASYS, the effects of the synchronous harmonic torques at slips‫ ݏ‬ൌ ͲǤͺͷ͹, ‫ ݏ‬ൌ ͳ and ‫ ݏ‬ൌ ͳǤͲ͹ͳ of the the motor are observable

In Fig.4.6.1-3 the torque slip characteristics calculated by time stepping and steady state AC analysis are compared. The effects of the field harmonics lead to considerable deviations in the calculated values. In Fig.4.6.1-4 the fundamental phase current prepared by Flux2D and KLASYS are compared. The results match very well and maximum difference is less than 5.5%. - 36 -

Fig.4.6.1-5 shows the input power-slip characteristic of VZIM1 obtained from Flux2D and KLASYS. At lower slips the calculated results of KLASYS are lower than Flux2D and at higher slips results of KLASYS are higher. The values match well and the maximum difference is less than 5%. In addition Fig.4.6.1-6 shows the output power-slip characteristic of VZIM1 obtained from Flux2D and KLASYS. Fig.4.6.1-7 shows the efficiency calculated by Flux2D and KLASYS. Maximum efficiency achieved from Flux2D results is 91.41% at a slip of 0.3 and maximum efficiency calculated by KLASYS is 90.5% at the slip equal to 0.3. The results match very well. Fig.4.6.1-8 compares the power factor values obtained from Flux2D and KLASYS. The maximum power factor calculated by KLASYS equals 0.924 which at slip equal to 0.12 happens, and in Flux2D maximum power factor is equal to 0.962 which at slip equal to 0.1 happens, which is higher than KLASYS maximum power factor.

Fig.4.6.1-3: Torque-slip characteristic of unskewed VZIM1, time-stepping and AC analysis results obtained by Flux2D, the harmonic torque effects at lower speeds of the motor are obvious

- 37 -

Fig.4.6.1-4: Stator phase current of unskewed VZIM1, time stepping analysis results calculated by Flux2D and KLASYS

Fig.4.6.1-5: Input power-slip of unskewed VZIM1, comparison of the analysis results calculated by Flux2D and KLASYS

- 38 -

Fig.4.6.1-6: Output power-slip of unskewed VZIM1, comparison of the analysis results calculated by Flux2D and KLASYS

Fig.4.6.1-7: Efficiency-slip of unskewed VZIM1, comparison of the analysis results calculated by Flux2D and KLASYS

- 39 -

Fig.4.6.1-8: Power factor-slip of unskewed VZIM1, comparison of the analysis results calculated by Flux2D and KLASYS

4.6.2 Time-stepping analysis of VZIM2 Similar to VZIM1, time-stepping analysis of VZIM2 has been done with Flux2D and results have been compared with results of the KLASYS tool. In this analysis the considered slips similarly vary from 0 to 2 and in between additional 50 slips have been considered for calculations, which have been given in table VI in Appendix II. The synchronous harmonic torques which happen at slips, ‫ ݏ‬ൌ ͲǤͺͷ͹, ‫ ݏ‬ൌ ͳ and ‫ ݏ‬ൌ ͳǤͲ͹ͳ are calculated with Flux2D as explained in the section 4.8 of this chapter. In Fig.4.6.2-1 the torque slip characteristic of VZIM2, derived from time stepping analysis with Flux2D, is shown. The effects of the synchronous harmonic torques at slips‫ ݏ‬ൌ ͲǤͺͷ͹, ‫ ݏ‬ൌ ͳ and ‫ ݏ‬ൌ ͳǤͲ͹ͳ are clearly observed. In Fig.4.6.2-2 the torque slip characteristic of VZIM2, derived from KLASYS, has been shown. The effects of the synchronous harmonic torques at slips‫ ݏ‬ൌ ͲǤͺͷ͹, ‫ ݏ‬ൌ ͳ and ‫ ݏ‬ൌ ͳǤͲ͹ͳ are clearly observed.

The values of synchronous torque calculated by Flux2D and KLASYS at slips‫ ݏ‬ൌ ͲǤͺͷ͹, ‫ ݏ‬ൌ ͳ and‫ ݏ‬ൌ ͳǤͲ͹ͳ, are compared in Table 4.8-9, Table 4.8-10 and Table 4.8-11. It is noticeable that the synchronous torque values calculated by KLASYS are in most of the cases much higher than the values calculated by Flux2D. The values of Flux2D are more reasonable and acceptable.

- 40 -

Fig.4.6.2-1: Torque-slip characteristic of unskewed VZIM2, obtained by KLASYS and Flux2D timestepping calculations, the harmonic torque effects at lower speeds of the motor are observable

the synchronous harmonic torques at slips‫ ݏ‬ൌ ͲǤͺͷ͹, ‫ ݏ‬ൌ ͳ and ‫ ݏ‬ൌ ͳǤͲ͹ͳ of the the motor are observable Fig.4.6.2-2: Torque-slip characteristic of unskewed VZIM2, calculated with KLASYS, the effects of

To observe the effect of the harmonics, the torque-slip characteristic of VZIM2 prepared from time-stepping and steady state AC analysis have been depicted in Fig.4.6.2-3. A considerable

- 41 -

difference happens at the slip 0.857 and 1.071 due to the synchronous harmonics at these points. Fig.4.6.2-4 depicts the comparison between fundamental stator phase current calculated by KLASY and Flux2D. The difference between results is below 10% which mostly happens at high values of slip (‫ ݏ‬൐ ͳ). Fig.4.6.2-5 shows the input power-slip characteristic of VZIM2, calculated by KLASYS and Flux2D. At higher slips the values of Flux2D are around 11% lower than values of KLASYS. In addition Fig.4.6.2-6 shows the output power-slip characteristic of VZIM2 obtained from Flux2D and KLASYS Fig.4.6.2-7 depicts the efficiency-slip characteristic of VZIM2, prepared by KLASYS and Flux2D. The efficiency calculated by Flux2D at rated speed of VZIM2 (1455.8 rpm) is equal to 0.9104 is higher than that of KLASYS which is equal to 0.905. Fig.4.6.2-8 depicts the power factor-slip characteristic of VZIM2, prepared by KLASYS and Flux2D.


- 42 -

Fig.4.6.2-4: Stator phase current of unskewed VZIM2, comparison of the time stepping analysis results calculated by Flux2D and KLASYS


- 43 -



- 44 -

Fig.4.6.2-8: Power factor-slip of unskewed VZIM2, comparison of the analysis results calculated by Flux2D and KLASYS

4.6.3 Time-stepping analysis of VZIM3 Similar to VZIM1 and VZIM2, time-stepping analysis of VZIM3 has been done with Flux2D and results have been compared with the results of the KLASYS tool. In this analysis the considered slips similarly vary from 0 to 2 and in between additional 50 slips have been considered for calculations, which are given in table VII in Appendix II. The synchronous harmonic torques which happen at slips, ‫ ݏ‬ൌ ͲǤͺͷ͹, ‫ ݏ‬ൌ ͳ and ‫ ݏ‬ൌ ͳǤͲ͹ͳ are calculated with Flux2D as explained in the section 4.8 of this chapter. In Fig.4.6.3-1 the torque slip characteristic of VZIM3, derived from time stepping analysis with Flux2D, is shown. The effects of the synchronous harmonic torques at slips‫ ݏ‬ൌ ͲǤͺͷ͹, ‫ ݏ‬ൌ ͳ and ‫ ݏ‬ൌ ͳǤͲ͹ͳ are clearly observed. In Fig.4.6.3-2 the torque slip characteristic of VZIM3, derived from KLASYS, is shown. The effects of the synchronous harmonic torques at slips‫ ݏ‬ൌ ͲǤͺͷ͹, ‫ ݏ‬ൌ ͳ and ‫ ݏ‬ൌ ͳǤͲ͹ͳ are clearly observed.

The values of synchronous torque calculated by Flux2D and KLASYS at slips‫ ݏ‬ൌ ͲǤͺͷ͹, ‫ ݏ‬ൌ ͳ and‫ ݏ‬ൌ ͳǤͲ͹ͳ, are compared in Table 4.8-12, Table 4.8-13 and Table 4.8-14. It is noticeable that the synchronous torque values calculated by KLASYS are in most of the cases much higher than the values calculated by Flux2D. The values of Flux2D are more reasonable and acceptable. To observe the effect of the harmonics, the torque-slip characteristic of VZIM3 prepared from time-stepping and steady state AC analysis have been depicted in Fig.4.6.2-2. A considerable difference happens at the slip equal to 0.857 and 1.071 due to the synchronous harmonics at these points. Fig.4.6.3.3 shows the comparison between fundamental phase current prepared by KLASYS and Flux2D. The difference between the results is below 5% which mostly happens at high slip values (‫ ݏ‬൐ ͳ). - 45 -

effects of the synchronous harmonic torques at slips‫ ݏ‬ൌ ͲǤͺͷ͹, ‫ ݏ‬ൌ ͳ and ‫ ݏ‬ൌ ͳǤͲ͹ͳ of the the motor are observable Fig.4.6.3-1: Torque-slip characteristic of unskewed VZIM3, calculated with Flux2D time-stepping, the

Fig.4.6.3-2: Torque-slip characteristic of unskewed VZIM3, calculated with KLASYS, the effects of the synchronous harmonic torques at slips‫ ݏ‬ൌ ͲǤͺͷ͹, ‫ ݏ‬ൌ ͳ and ‫ ݏ‬ൌ ͳǤͲ͹ͳ of the the motor are observable

- 46 -


Fig.4.6.3-4: Stator phase current of unskewed VZIM2, comparison of the analysis results calculated by Flux2D and KLASYS

Fig.4.6.3-4 depicts the input power-slip characteristic of VZIM3, prepared by KLASYS and Flux2D. At higher slips the values of Flux2D are around 4% lower than values of KLASYS. In addition Fig.4.6.3-5 shows the output power-slip characteristic of VZIM3 obtained from Flux2D and KLASYS.

- 47 -



Fig.4.10.3-6 depicts the efficiency-slip characteristic of VZIM3, prepared by KLASYS and Flux2D. The efficiency calculated by Flux2D at rated speed of VZIM3 (1474.65 rpm) is equal to 0.912 is lower than that of KLASYS which is equal to 0.925.

- 48 -


Fig.4.6.3-7 depicts the input power factor-slip characteristic of VZIM3, prepared by KLASYS and Flux2D.The maximum power factor is equal to 0.956 which happens at a slip equal to 0.025 in Flux2D calculated results and in case of KLASYS the maximum power factor is equal to 0.937 which happens at a slip equal to 0.03.

Fig.4.6.3-8: Power factor of unskewed VZIM3, comparison of the analysis results calculated by Flux2D and KLASYS

- 49 -

4.7 Instantaneous torque wave forms at different rotor speeds In order to draw the torque-slip characteristic, the average value of the torque over one period of the stator electrical cycle has been considered. In order to observe the oscillations and ripple of the torque over time in some sample operating speeds of the motors, the torque-time characteristics are shown in Fig.4.7-1 to Fig.4.7-7. From these figures the influence of the harmonics on the torque characteristics is clearly observable, the lower the speed, the higher the influence of the harmonics and therefore the ripple on the torque-time characteristics increases. At the speed equal to 1462 in Fig.4.7-1 the average torque is lower than the average torque at the speed equal to 1200 in Fig.4.7-4 and the ripple has also lower amplitude. At the speed equal to 205.5 rpm in Fig.4.7-7 the influence of the synchronous harmonic torques is quite clear. This operating point where the peak of the 7th asynchronous harmonic torque occurs, is quite near the slip ‫ ݏ‬ൌ ͲǤͺͷ͹ where the synchronous harmonic torque happens. Therefore the result is a quite oscillating torque-time characteristic.

Fig.4.7-1: Torque-time characteristic of unskewed VZIM2 at 1462rpm, calculated by Flux2D, torque oscillates around a constant mean value after reaching the steady state

- 50 -

Fig.4.7-2: Torque-time characteristic of unskewed VZIM2 at 1462rpm, calculated by Flux2D, zoomed view after steady state is reached

Fig.4.7-3: Torque-time characteristic of unskewed VZIM2 at 1200rpm, calculated by Flux2D, torque oscillates around a constant mean value after reaching the steady state

- 51 -

Fig.4.7-4: Torque-time characteristic of unskewed VZIM2 at 1200rpm, calculated by Flux2D, zoomed view after a steady state is reached

Fig.4.7-5: Torque-time characteristic of unskewed VZIM2 at 525rpm, calculated by Flux2D, torque oscillates around a constant mean value

- 52 -

Fig.4.7-6: Torque-time characteristic of unskewed VZIM2 at 525rpm, calculated by Flux2D, zoomed view after a steady state is reached

Fig.4.7-7: Torque-time of unskewed VZIM2 at 205.5rpm, calculated by Flux2D, where break down slip of the 7th asynchronous torque harmonic happens at slip 0.863 or 205.5rpm

- 53 -

4.8 Synchronous harmonic torque assessment in VZIM

Synchronous harmonic torques of VZIM, happen at the slips, ‫ ݏ‬ൌ ͲǤͺͷ͹,‫ ݏ‬ൌ ͳ and ‫ ݏ‬ൌ ͳǤͲ͹ͳ. In this section the torque time characteristics of VZIM1, VZIM2 and VZIM3 at these slips are studied. For VZIM2 the synchronous harmonic torques are studied and derived by varying the rotor position with respect to a fixed stator position, for a rotor slot pitch (we call this method the rotation method). The step for each position is taken as one tenth of a rotor slot pitch i.e. = 360/28/10 = 1.2857 mechanical degrees. The average torque for a stator period is calculated for each of the eleven steps. Fig.4.8.1 shows first and second positions of the rotor bars with fixed stator position.

Fig.4.8-1: First and second positions of the rotor bars with fixed stator position to calculate the

synchronous harmonic torques, prepared by Flux2D Fig.4.8.2 to Fig.4.8.4 show the variation of the torque for the rotor position steps 1, 5 and 7 of VZIM2 at the slip ‫ ݏ‬ൌ ͲǤͺͷ͹ respectively. It is clear that the average value of the synchronous torque at each step is a constant value. Similar procedure has been done for VZIM2 at slips ‫ ݏ‬ൌ ͳ and ‫ ݏ‬ൌ ͳǤͲ͹ͳ, and the torque time values have been calculated for eleven positions. The calculated synchronous torques, for three different slips of VIM2 are shown in Fig.4.8-5 to Fig.4.8-7.

Fig.4.8-2: Torque-time of unskewed VZIM2 at‫ ݏ‬ൌ ͲǤͺͷ͹, calculated by Flux2D, rotor position step 1 - 54 -

Fig.4.8-3: Torque-time of unskewed VZIM2 at‫ ݏ‬ൌ ͲǤͺͷ͹, calculated by Flux2D, rotor position step 5

Fig.4.8-4: Torque-time of unskewed VZIM2 at‫ ݏ‬ൌ ͲǤͺͷ͹, calculated by Flux2D, rotor position step 7

- 55 -

500 400

Torque/N.m

300 200 100 0 0

20

40

60

80

100

-100

Fig.4.8-5: Variation of synchronous torque at‫ ݏ‬ൌ ͲǤͺͷ͹ as a function of rotor position Rotor position at percentage of rotor slot pitch

580

Torque/N.m

560 540 520 500 480 460 440 0

20

40

60

80

100

Rotor position at percentage of rotor slot pitch

Fig.4.8-6: Variation of synchronous torque at ‫ ݏ‬ൌ ͳ a function of rotor position 350

Torque/N.m

330 310 290 270 250 0

20

40

60

80

Rotor position at percentage of rotor slot pitch

100

Fig.4.8-7: Variation of synchronous torque at ‫ ݏ‬ൌ ͳǤͲ͹ͳ as a function of rotor position

- 56 -

Furthermore the synchronous torque of the VZIM2 has been calculated with another method (we call this method the oscillation method). The torque time characteristics of VZIM2 have been calculated in three different speeds and each speed is 10 rpm higher than the speed where synchronous harmonic torque happens. For VZIM2 the synchronous harmonic torques happen at 214.28 rpm (‫ ݏ‬ൌ ͲǤͺͷ͹), 0 rpm (‫ ݏ‬ൌ ͳ) and -107.1428 (‫ ݏ‬ൌ ͳǤͲ͹ͳ). Therefore the torque time characteristic of the VZIM2 at speeds 224.28 rpm, 10 rpm and -97.1428 are calculated. As the oscillation of the torque time characteristic occurs because of the synchronous harmonic torque; therefore the maximum and minimum of the variation of the average torque in the torque time characteristic, equals to the maximum and minimum values of the synchronous harmonic torques. Fig.4.8.8 to Fig.4.8.10 show the torque time characteristic of VZIM2 at three different speeds 224.28 rpm, 10 rpm and -97.1428. At speed 224.28 rpm the maximum value of the average torque is 438.4 Nm and the minimum value is -20.88 Nm and the peak to peak value is 459.28Nm. On the other hand the maximum value of synchronous harmonic torque calculated by changing the position of the rotor related to fixed stator method is 448.24 Nm and the minimum value is -20.8 Nm with the peak to peak value equal to 469.04 Nm. The difference between two methods of synchronous torque calculation of peak to peak value is around 2.1% at speed 224.28 rpm. The values are summarized in Table 4.8-1. The torque values for speeds 10 rpm and -97.1428 rpm are calculated in similar way and are shown in Table 4.8-2 and Table 4.8-3. Clearly the calculated values with two different methods (rotation and oscillation methods) have a maximum difference of 7.66%. Due to the reasonable differences of the results of two calculation methods for VZIM2, the calculation of the maximum and minimum and peak to peak values of synchronous torques at the slips where synchronous torques of VZIM1 and VZIM2 happen are done by the oscillation method due to the low calculation time of this method. Table 4.8-4 and Table 4.8-5 show the values of synchronous torques for VZIM1 and VZIM3, calculated by oscillation method in three different slips, where synchronous harmonic torques happen (‫ ݏ‬ൌ ͲǤͺͷ͹,‫ ݏ‬ൌ ͳ and ‫ ݏ‬ൌ ͳǤͲ͹ͳ).

Table 4.8-1: Maximum, minimum and peak to peak values of synchronous torque for VZIM2 in two different methods of calculation (rotation and oscillation methods) for the slip ‫ ݏ‬ൌ ͲǤͺͷ͹ Oscillation method Rotation method Difference % Maximum torque (N.m) 438.4 448.24 2.19% Minimum torque(N.m) -20.88 -20.8 0.3% Peak to peak value (N.m) 459.28 469.04 2.08% Table 4.8-2: Maximum, minimum and peak to peak values of synchronous torque for VZIM2 in two different methods of calculation (rotation and oscillation methods) for the slip ‫ ݏ‬ൌ ͳ Oscillation method Rotation method Difference % Maximum torque (N.m) 313.32 326.84 4.1% Minimum torque(N.m) 244.56 262.16 6.7% Peak to peak value (N.m) 68.76 64.68 5.9%

- 57 -

Table 4.8-3: Maximum, minimum and peak to peak values of synchronous torque for VZIM2 in two different methods of calculation (rotation and oscillation methods) for the slip ‫ ݏ‬ൌ ͳǤͲ͹ͳ Oscillation method Rotation method Difference % Maximum torque (N.m) 574.2 556.8 3.03% Minimum torque(N.m) 470.88 461.4 2.01% Peak to peak value (N.m) 103.32 95.4 7.66% Table 4.8-4: Maximum, minimum and peak to peak values of synchronous torque for VZIM1 (oscillation method) ࢙ ൌ ૙Ǥ ૡ૞ૠ ࢙ൌ૚ ࢙ ൌ ૚Ǥ ૙ૠ૚ Maximum torque (N.m) 605.56 456.24 629.72 Minimum torque(N.m) 130.8 406.28 499.8 Peak to peak value (N.m) 474.76 49.96 129.92 Table 4.8-5: Maximum, minimum and peak to peak values of synchronous torque for VZIM3 (oscillation method) ࢙ ൌ ૙Ǥ ૡ૞ૠ ࢙ൌ૚ ࢙ ൌ ૚Ǥ ૙ૠ૚ Maximum torque (N.m) 370 300.88 458.32 Minimum torque(N.m) -42.44 185.68 351.72 Peak to peak value (N.m) 412.44 114.32 107.6

Fig.4.8-8: Torque-time of unskewed VZIM2 at speed 224.28 rpm, calculated by Flux2D

- 58 -

Fig.4.8-9: Torque-time of unskewed VZIM2 at speed 10 rpm, calculated by Flux2D

Fig.4.8-10: Torque-time of unskewed VZIM2 at speed -97.14 rpm, calculated by Flux2D

In addition, the values of synchronous harmonic torques for VZIM1, VZIM2 and VZIM3 in three different slips, where synchronous harmonic torques happen are calculated by KLASYS. In Table 4.8-6 to Table 4.8-14 the calculated values with KLASYS and Flux2D are compared. It is noticeable that the synchronous torque values calculated by KLASYS are in the most of the cases much higher than the values calculated by Flux2D.The values of Flux2D are more reasonable and acceptable.

- 59 -

Table 4.8-6: Maximum, minimum and peak to peak values of synchronous torque for VZIM1, at the slip ‫ ݏ‬ൌ ͲǤͺͷ͹, the comparison of the calculated values by KLASYS and Flux2D Maximum torque Minimum torque Peak to peak value (N.m) (N.m) (N.m) Flux2D 606.56 130.8 474.76 KLASYS 1165.215 -208.695 1373.91 Difference 558.655 338.659 899.15 Table 4.8-7: Maximum, minimum and peak to peak values of synchronous torque for VZIM1, at the slip ‫ ݏ‬ൌ ͳ, the comparison of the calculated values by KLASYS and Flux2D Maximum torque Minimum torque Peak to peak value (N.m) (N.m) (N.m) Flux2D 456.24 406.28 49.96 KLASYS 526.085 473.915 52.17 Difference 69.845 67.635 2.21 Table 4.8-8: Maximum, minimum and peak to peak values of synchronous torque for VZIM1, at the slip ‫ ݏ‬ൌ ͳǤͲ͹ͳ, the comparison of the calculated values by KLASYS and Flux2D Maximum torque Minimum torque Peak to peak value (N.m) (N.m) (N.m) Flux2D 629.72 499.8 129.92 KLASYS 899.99 230.427 669.564 Difference 270.27 269.373 539.644 Table 4.8-9: Maximum, minimum and peak to peak values of synchronous torque for VZIM2, at the slip ‫ ݏ‬ൌ ͲǤͺͷ͹, the comparison of the calculated values by KLASYS and Flux2D Maximum torque Minimum torque Peak to peak value (N.m) (N.m) (N.m) Flux2D 438.4 -20.88 459.28 KLASYS 1083.33 -316.67 1400 Difference 644.93 296.67 940.72 Table 4.8-10: Maximum, minimum and peak to peak values of synchronous torque for VZIM2, at the slip ‫ ݏ‬ൌ ͳ, the comparison of the calculated values by KLASYS and Flux2D Maximum torque Minimum torque Peak to peak value (N.m) (N.m) (N.m) Flux2D 313.32 244.56 68.76 KLASYS 474.995 341.665 133.33 Difference 161.675 70.105 64.57 Table 4.8-11: Maximum, minimum and peak to peak values of synchronous torque for VZIM2, at the slip ‫ ݏ‬ൌ ͳǤͲ͹ͳ, the comparison of the calculated values by KLASYS and Flux2D Maximum torque Minimum torque Peak to peak value (N.m) (N.m) (N.m) Flux2D 574.2 470.88 103.32 KLASYS 624.995 291.665 333.33 Difference 50.795 179.215 230.01

- 60 -

Table 4.8-12: Maximum, minimum and peak to peak values of synchronous torque for VZIM3, at the slip ‫ ݏ‬ൌ ͲǤͺͷ͹, the comparison of the calculated values by KLASYS and Flux2D Maximum torque Minimum torque Peak to peak value (N.m) (N.m) (N.m) Flux2D 370 -42.44 412.44 KLASYS 907.13 -464.29 1371.42 Difference 537.13 421.85 958.98 Table 4.8-13: Maximum, minimum and peak to peak values of synchronous torque for VZIM3, at the slip ‫ ݏ‬ൌ ͳ, the comparison of the calculated values by KLASYS and Flux2D Maximum torque Minimum torque Peak to peak value (N.m) (N.m) (N.m) Flux2D 300.88 185.68 114.32 KLASYS 358.925 198.215 160.71 Difference 58.045 12.535 46.39 Table 4.8-14: Maximum, minimum and peak to peak values of synchronous torque for VZIM3, at the slip ‫ ݏ‬ൌ ͳǤͲ͹ͳ, the comparison of the calculated values by KLASYS and Flux2D Maximum torque Minimum torque Peak to peak value (N.m) (N.m) (N.m) Flux2D 458.32 351.72 106.6 KLASYS 353.57 246.43 107.14 Difference 104.75 105.29 0.54

4.9 Power losses in different parts of the motor In order to determine the efficiency of the motor, it is necessary to calculate the losses in different parts of the motor, these loss components are: -stator and rotor ohmic losses -friction and windage losses -iron losses (mainly in stator iron) -additional no-load losses such as tooth pulsation and surface losses -additional load losses such as stator and rotor eddy current losses in conductors. In simulation by Flux2D the mechanical losses are neglected. Tables I, II and III in Appendix III show the iron losses which include eddy-current losses and hysteresis losses in addition to the stator and rotor ohmic losses, which have been calculated by Flux2D and compared with the results of KLASYS tool. In Flux2D regarding the iron losses, the volume density of the instantaneous power loss ‫݌‬ሺ‫ݐ‬ሻ ൌ ܲሺ‫ݐ‬ሻΤܸ , is written as [5]: ଷ

ଶ

ଶ ݀ ଶ ݀‫ܤ‬ ݀‫ܤ‬ ଶ ‫݌‬ሺ‫ݐ‬ሻ ൌ ݇௛ ‫ܤ‬௠ ݂ ൅ ߪ ൭ ሺ‫ݐ‬ሻ൱ ൅ ݇௘ ൭ ሺ‫ݐ‬ሻ൱ ሺ ଷ ሻሺͶǤͻ െ ͳሻ ͳʹ ݀‫ݐ‬ ݀‫ݐ‬

which is composed of losses by hysteresis, classical losses and losses in excess, where - 61 -

-݇௛ is the coefficient by hysteresis -݇௘ is the coefficient of losses in excess -ߪ is the conductivity of the material -d is the thickness of the lamination -‫ܤ‬௠ is the peak value of the magnetic flux density ଷ

Within the frame of the flux computation, ଶ

ଶ ݀‫ܤ‬ ݀ଶ ݀‫ܤ‬ ଶ ‫݌‬ሺ‫ݐ‬ሻ ൌ ൦݇௛ ‫ܤ‬௠ ݂ ൅ ߪ ൭ ሺ‫ݐ‬ሻ൱ ൅ ݇௘ ൭ ሺ‫ݐ‬ሻ൱ ൪ ή ݇௙௘ ሺ ଷ ሻሺͶǤͻ െ ʹሻ ݀‫ݐ‬ ͳʹ ݀‫ݐ‬

-݇௙௘ is the stack fill factor (close to 1), this coefficient considers the electrical insulation of the lamination of the magnetic core.

In addition, in steady state AC magnetic application the volume density of the average power ‫݌‬ሺ‫ݐ‬ሻ ൌ ܲሺ‫ݐ‬ሻΤܸ is: ଷ ߨ ଶ ߪ݀ ଶ ଶ ሺ‫ܤ‬௠ ݂ሻଶ ൅ ݇௘ ሺ‫ܤ‬௠ ݂ሻଶ ή ͺǤ͸͹ሺ ଷ ሻሺͶǤͻ െ ͵ሻ ݂ ൅ ‫݌‬ሺ‫ݐ‬ሻ ൌ ݇௛ ‫ܤ‬௠ ͸ where: -݇௛ is the coefficient by hysteresis -݇௘ is the coefficient of losses in excess -ߪ is the conductivity of the material -d is the thickness of the lamination -f is frequency -‫ܤ‬௠ is the peak value of the magnetic flux density

Coefficients݇௛ , ߪ and݇௘ are necessary to calculate the iron losses with Flux2D, according to the data sheet of the M270-50A [7] which is the material of the stator and rotor lamination and with consideration of the ‫ܤ‬௠ equal to 1.5T and with solving ሺͶǤͻ െ ͵ሻ, in three different frequencies 50Hz, 100Hz and 200Hz the values of݇௛ , ߪ and݇௘ are calculated.

The values of the specific total losses ሺܲ௦ ሻ in frequencies 50Hz, 100Hz and 200Hz, according to the data sheet of the M270-50A [7] are given in Table 4.9-1. In addition the density of the iron sheet M270-50A is ߩ ൌ ͹ͺͲͲሺ

୏୥

୫య

ሻ and the power loss is ‫݌‬ሺ‫ݐ‬ሻ ൌ ܲ௦ ή ߩ, the calculated

values of the power losses for the frequencies 50Hz, 100Hz and 200Hz and are shown in Table 4.9-1.

Table 4.9-1: Values of specific total losses according to the data sheet of the M270-50A [7], and calculated power losses, in frequencies 50Hz, 100Hz and 200Hz 50Hz 100Hz 200Hz ‫܅‬ ࡼ࢙ ሺ ൗ‫܏ܓ‬ሻ 19.45 ʹǤͷ͵ ͸Ǥ͹ ࢖ሺ࢚ሻሺ‫܅‬ൗ ૜ ሻ ͳͻ͹͵Ͷ ͷʹʹ͸Ͳ ͳͷͳ͹ͳͲ ‫ܕ‬ - 62 -

after the calculations the following results are obtained: ݇௛ ൌ ͻ͹Ǥʹͷ͸ (Ws/( ଶ ଷ )) ݇௘ ൌ ͲǤ͹ʹ͸͸ (1/(Ohm.m)), ߪ ൌ ʹͲ͵Ͷ͵͹ͷǤͲ͵͵(W/(ଷ ή ሺΤሻଵǤହ)) When these values are available with the help of the Flux2D iron losses are calculated. 4.9.1 Losses of VZIM at different speeds In this section the losses of VZIM1, VZIM2 and VZIM3 in three different speeds which have been calculated by Flux2D and KLASYS are compared. Here the components of stator and rotor iron and copper losses have been considered. Tables I, II and III in Appendix III show the losses of VZIM1, VZIM2 and VZIM3. Apparently there are differences between losses calculated by KLASYS and Flux2D, at 22kW nominal output power operation. In VZIM2 the stator copper losses are around 14%, the rotor copper losses about 30% and stator iron losses about 11%, calculated by Flux2D are higher than values calculated by KLASYS. In VZIM3 the stator copper losses are around 18%, the rotor copper losses about 30% and stator iron losses about 21%, calculated by Flux2D are higher than values calculated by KLASYS. Fig.4.9.1-1 shows the comparison between losses of VZIM1 at rated speed equal to 1390.9 rpm calculated with Flux2D and KLASYS, Fig.4.9.1-2 shows the comparison between losses of VZIM2 at the rated speed equal to 1455.8 rpm calculated with Flux2D and KLASYS and Fig.4.9.1-3 shows the comparison between losses of VZIM3 at the rated speed equal to 1474.65 rpm calculated with Flux2D and KLASYS.

Fig.4.9.1-1: Comparison between losses of VZIM1 at speed equal to 1390.9 rpm calculated with Flux2D and KLASYS

- 63 -



- 64 -

Chapter 5: Thermal analysis 5.1 Preface Calculating the temperature rise in electrical machines which occurs due to electrical and mechanical losses, is one of the important steps in design of electrical machines. According to Arrhenius’ law, velocity of chemical decomposition of materials increases exponentially with temperature and for solid insulation materials Montsinger’s rule is valid, that the insulation life span L decreases by 50% with increase of temperature ߴ by ͳͲ. Due to the high sensitivity of insulation materials to over-temperature, thermal classes for different types of insulation materials are defined, which gives the maximum permissible temperature limit in hot spot of insulation. For example in class F maximum temperature rise οߴሺሻ is 105 for the machines in the power range of ܲ୒ ൑ ͷ [1]. Cooling system of electrical machines influences their thermal utilization and with a high efficient cooling system, power per mass of electrical machines can be raised. There are different possibilities to propel the coolant flow in electrical machines. In case of motors without fan, cooling is done due to the natural convection and heat radiation. With shaft mounted fan motors the speed of air flow depends on the velocity of motor. The external fan is another way for cooling the electrical machine. According to the second fundamental law of thermodynamics, the natural heat flow is only possible from a hot to a cold region. Basic principles for heat transfer are -Conduction, by heat conducting materials -Convection, by moving coolants like air or water -Radiation, which does not need any medium for heat transfer. Calculating the temperature in electrical machines may be done either by numerical methods or equivalent circuits. Calculation by numerical method in this project has been done by ANSYS tool.

5.2 Temperature rise calculation by thermal equivalent circuits To have an estimation of temperature rise in the stator winding simplified thermal networks are used.

- 65 -

In a simplified thermal network only copper and iron losses in the stator and thermal resistances between copper and iron due to the slot insulation, heat convection from winding overhang to air and heat convection from the stator iron to air are considered. Stator copper losses in the winding (ܲ஼௨ ) and stator iron losses in the stator iron stack (ܲி௘ ) are loss sources. Fig.5.2-1 shows the simplified thermal equivalent network for an induction motor.

Fig.5.2-1: The simplified thermal equivalent network for an induction motor

Heat resistances are, ܴ௧௛ଵ between stator iron and ambient cooling air, given by convection, according to [1] ܴ௧௛ଵ ൌ ͳΤሺߙ‫ ீܣ‬ሻ where ‫ ீܣ‬is the surface of the stator iron housing and ߙ is the heat transfer coefficient which describes the cooling effect of flowing coolant. ܴ௧௛ଶ between slot conductor copper and iron stack, mainly determined by heat resistance of the slot insulation, according to [1] ܴ௧௛ଶ ൌ ݀ Τ൫ߣ௜௦ ‫ܣ‬ொ ൯, where ‫ܣ‬ொ is the slot surface, ߣ௜௦ is the thermal conductivity of the slot insulation and ݀ is the thickness of the slot insulation. ܴ௧௛ଷ andܴ௧௛ସ are thermal resistances between winding overhang and surrounding air given by convection. Due to special structure of the VZIM, as this motor is totally enclosed, at one side of the winding overhang, the heat transfer coefficient is considered as ߙ ൌ ͳͷ for natural convection and heat radiation and at the other side, the rotating centrifugal mechanism leads to a better cooling due to the convection. Therefore in order to calculate the heat transfer coefficient, with try and error and comparison with measured values, ߙ is estimated for the 5.5kW VZIM (ߙ ൌ ͷͲȀሺʹ ሻ). For the 22kW VZIM due to similar structure with the 5.5kW VZIM, a proportional ratio of ߙ was considered, the calculations are presented in Appendix IV and the values are shown in Table 5.2-1.

- 66 -

Table 5.2-1: Heat transfer coefficient for one side of the winding overhang close to the centrifugal mechanism Winding overhang VZIM2 22kW VZIM3 22kW

Heat transfer coefficient ࢻ ‫܅‬Ȁሺ‫ܕ‬૛ ۹ሻ

63.32

62.86

οߴ஼௨ and οߴி௘ are unknown temperature rises, which οߴ ൌ ߴ െ ߴ௔௠௕ is the temperature difference between motor local temperature and ambient temperature of surrounding air. In this calculation ambient temperature is considered to be 20Ԩ. With consideration of the steady state temperature rise only two algebraic linear equations have to be solved, οߴ஼௨ οߴ஼௨ οߴ஼௨ െ οߴி௘ ൅ ൅ ൌ ܲ஼௨ǡ௦ ሺͷǤʹ െ ͳሻ ܴ௧௛ଶ ܴ௧௛ଷ ܴ௧௛ସ οߴி௘ οߴ஼௨ െ οߴி௘ െ ൌ ܲி௘ǡ௦ ሺͷǤʹ െ ʹሻ ܴ௧௛ଵ ܴ௧௛ଶ As VZIM2 and VZIM3 are active in rated operating speed of the motor, thermal calculations of VZIM2 and VZIM3 are done in an operating speed which the average of the output power produced by VZIM2 and VZIM3 in this speed is equal to 22kW.Therefore as it is depicted in Fig.5.2-2, the average of output power of VZIM2 and VZIM3 versus slip is calculated and at a speed of 1467.36 rpm the average power is equal to 22kW. The losses of VZIM2 and VZIM3 in this speed (1467.36 rpm) have been calculated and are shown in Table 5.2-2. Based on those losses the temperature rise οߴ஼௨ and οߴி௘ for VZIM2 and VZIM3 are calculated in the following sections.

Power-Slip characteristic 80000

Output power/W

60000 40000 20000

VZIM2 power

0

VZIM3 power

-20000

0

0.5

1

1.5

2

2.5

Average power

-40000 -60000

Slip

Fig.5.2-2: Calculated output power-slip characteristics of 22kW VZIM2 and VZIM3 and the average of the powers

- 67 -

ࡼ࡯࢛ǡ࢙ /W 492.073 1395.6 943.83

ࡼ࡯࢛ǡ࢘ //W 466.2 760.52 613.36

ࡼࡲࢋǡ࢙ /W 192.6 159.64 162.62

ࡼࡲࢋǡ࢘ /W 31.36 33.8 32.58

ࡼ࢕࢛࢚ /W 16894.2 27374.75 22134.47

Table 5.2-2: Losses in different parts of the VZIM2 and VZIM3 in the speed equal to 1467.36 rpm

VZIM2 VZIM3 Average

In calculation of the average output power in Fig.5.2-2, it is assumed that the equivalent output power of the VZIM motor can be estimated using super-position method. More accurate calculations based on an equivalent circuit model [6] show that the super-position method overestimates the equivalent torque and output power of the VZIM motor, especially at lower speeds. But it is quite accurate at the rated operating region of the motor, as shown in Fig.5.2-3. As here, only the calculation of the output power in the operating region of the VZIM motor is required, hence more simple method of super-positioning is used.

Fig.5.2-3: Calculated output power of the VZIM motor with the supper-position and equivalent circuit methods

- 68 -

5.2-1 Calculation of οࣖ࡯࢛ and οࣖࡲࢋ for VZIM2 (22kW) The materials used in construction of VZIM2 and the related thermal conductivities are shown in Table 5.2.1-1. Table 5.2.1-1: Thermal conductivities of the materials used in VZIM2 thermal model Material Air Iron stack Insulation 0.031 40 0.2 Thermal conductivity,ࣅ࢚ࢎ W/(m.K)

Copper 380

After determination of materials, calculation of thermal resistances is done as following. ܴ௧௛ଵ ൌ ͳΤሺߙ‫ ீܣ‬ሻ is the heat resistance between the stator iron stack and the ambient cooling air, ‫ ீܣ‬is the surface of the stator iron housing and ߙ is the heat transfer coefficient, ‫ ீܣ‬is equal to 203472ή ͳͲି଺ ଶ , and ߙ according to [1] is calculated by ߙ ൌ ͳͷ‫ ݒ‬ଶΤଷ (ߙ in Ȁሺଶ ሻ, ‫ ݒ‬in Τ) for moving air over bare metallic hot surface. The speed of wind flow over the stator body, based on an empirical rule [9], is equal to 70 percent of the linear speed of the top point of the fan blade, therefore ʹߨ݊ ͹Ͳ ʹߨ ή ͳͶ͸͹Ǥ͵͸ ͹Ͳ ܴ ൌ ή ͳ͸͸Ǥʹͷ ή ͳͲିଷ ή ൌ ͳ͹Ǥͺ͹ Ǥ ‫ ݒ‬ൌ ͸Ͳ ͳͲͲ ͸Ͳ ͳͲͲ R is the radius of the fan blade and n is the rotational speed of the VZIM2 with 22kW output power. Consequently heat transfer coefficient can be calculated as: ߙ ൌ ͳͷ‫ ݒ‬ଶΤଷ ൌ ͳͷ ή ͳ͹Ǥͺ͹ଶΤଷ ൌ ͳͲʹǤͷʹȀሺʹ ሻ, ܴ௧௛ଵ ൌ ͳΤሺߙ‫ ீܣ‬ሻ ൌ ͳΤሺͳͲʹǤͷʹ ή ʹͲ͵Ͷ͹ʹ ‫ ଺ିͲͳ כ‬ሻ ൌ ͲǤͲͶ͹ͻ͵

Ǥ

ܴ௧௛ଶ ൌ ݀ Τ൫ߣ௜௦ ‫ܣ‬ொ ൯ where ‫ܣ‬ொ is slot surface, ߣ௜௦ is thermal conductivity of slot insulation and ݀ is the thickness of the slot insulation. ܴ௧௛ଶ ൌ ݀ Τ൫ߣ௜௦ ‫ܣ‬ொ ൯ ൌ ͲǤʹ ή ͳͲିଷ ΤሺͲǤʹ ή ͳ͵ͻͲͲǤͶ͵͸ ή ͳͲି଺ ሻ ൌ ͲǤͲ͹ͳͻͶ Ȁ ܴ௧௛ଷ ൌ ͳΤሺߙ‫ ீܣ‬ሻ ൌ ͳΤሺͳͷ ή ͻʹͻͶͶ ‫ ଺ିͲͳ כ‬ሻ ൌ ͲǤ͹ͳȀ ܴ௧௛ସ ൌ ͳΤሺߙ‫ ீܣ‬ሻ ൌ ͳΤሺ͸͵Ǥ͵ʹ ή ͻʹͻͶͶ ‫ ଺ିͲͳ כ‬ሻ ൌ ͲǤͳ͹Ȁ ܲ஼௨ǡ௦ ൌ ͶͻʹǤͲ͹ ܲி௘ǡ௦ ൌ ͳͻʹǤ͸ According to equations (5.2-1) and (5.2-2) we get: οߴி௘ ൌ ͳͻǤ͸͸ οߴ஼௨ ൌ ͵ͷǤ͵ͳǤ The maximum temperature rise for class F of insulations is 105. Here the value is much lower than the maximum allowable temperature rise; therefore insulation is in the safe side. The reason for this low temperature rise is the fact that the middle cage (VZIM2) is working under rated load (16.89kW instead of 22kW) at the rated speed of 1467.36 rpm. The final temperature rise will be the average value of both VZIM2 and VZIM3.

- 69 -

5.2-2 Calculation of οࣖ࡯࢛ and οࣖࡲࢋ for VZIM3 (22kW) The materials used in construction of VZIM3 and the related thermal conductivities are shown in Table 5.2.2-1. Table 5.2.2-1: Thermal conductivities of the materials used in VZIM3 thermal model Material Air Iron stack Insulation Thermal 0.031 40 0.2 conductivity,ࣅ࢚ࢎ W/(m.K)

Copper 380

Calculations are similar to calculations for VZIM2 in the previous section. ܴ௧௛ଵ ൌ ͳΤሺߙ‫ ீܣ‬ሻ is the heat resistances between the stator iron stack and the ambient cooling air, ‫ ீܣ‬is the surface of the stator iron housing and ߙ is the heat transfer coefficient, ‫ ீܣ‬is equal to 203472ή ͳͲି଺ ଶ , and ߙ according to [1] can be calculated by ߙ ൌ ͳͷ‫ ݒ‬ଶΤଷ (ߙ in Ȁሺଶ ሻ, ‫ ݒ‬in Τ) for moving air over bare metallic hot surface. The speed of wind flow over the stator body, based on an empirical rule [9], is equal to 70 percent of the linear speed of the top point of the fan blade, therefore ͹Ͳ ʹߨ݊ ͹Ͳ ʹߨ ή ͳͶ͸͹Ǥ͵͸ ܴ ൌ ή ͳ͸͸Ǥʹͷ ή ͳͲିଷ ή ൌ ͳ͹Ǥͺ͹ Ǥ ‫ ݒ‬ൌ ͸Ͳ ͳͲͲ ͸Ͳ ͳͲͲ R is the radius of the fan blade and n is the rotational speed of the VZIM2 with 22kW output power. Consequently heat transfer coefficient can be calculated: ߙ ൌ ͳͷ‫ ݒ‬ଶΤଷ ൌ ͳͷ ή ͳ͹Ǥͺ͹ଶΤଷ ൌ ͳͲʹǤͷʹȀሺʹ ሻ, ܴ௧௛ଵ ൌ ͳΤሺߙ‫ ீܣ‬ሻ ൌ ͳΤሺͳͲʹǤͷʹ ή ʹͲ͵Ͷ͹ʹ ‫ ଺ିͲͳ כ‬ሻ ൌ ͲǤͲͶ͹ͻ͵ȀǤ ܴ௧௛ଶ ൌ ݀Τ൫ߣ௜௦ ‫ܣ‬ொ ൯ where ‫ܣ‬ொ is slot surface, ߣ௜௦ is thermal conductivity of slot insulation and ݀ is the thickness of slot insulation. ܴ௧௛ଶ ൌ ݀Τ൫ߣ௜௦ ‫ܣ‬ொ ൯ ൌ ͲǤʹ ή ͳͲିଷ ΤሺͲǤʹ ή ͳ͵ͻͲͲǤͶ͵͸ ή ͳͲି଺ ሻ ൌ ͲǤͲ͹ͳͻͶ Ȁ ܴ௧௛ଷ ൌ ͳΤሺߙ‫ ீܣ‬ሻ ൌ ͳΤሺͳͷ ή ͻʹͻͶͶ ‫ ଺ିͲͳ כ‬ሻ ൌ ͲǤ͹ͳȀ ܴ௧௛ସ ൌ ͳΤሺߙ‫ ீܣ‬ሻ ൌ ͳΤሺ͸ʹǤͺ͸ ή ͻʹͻͶͶ ‫ ଺ିͲͳ כ‬ሻ ൌ ͲǤͳ͹Ȁ ܲ஼௨ǡ௦ ൌ ͳ͵ͻͷǤ͸ ܲி௘ǡ௦ ൌ ͳͷͻǤ͸Ͷ According to equations (5.2-1) and (5.2-2) we get: οߴி௘ ൌ ͶͲǤͺͳ οߴ஼௨ ൌ ͻͲǤͷ͹Ǥ The calculated temperature rise in the winding οߴ஼௨ is lower than the temperature rise limit 105 for class F insulations used in this motor. Therefore the insulation operates in the safe side, although the motor operates at overload condition (27.37kW instead of 22kW). Table 5.2.2-2 shows the temperature rise in the stator winding and iron of VZIM2 and VZIM3 calculated based on simplified thermal equivalent circuit and the average values.

- 70 -

Table 5.2.2-2: Temperature rise in the stator winding of VZIM2 and VZIM3 calculated based on simplified thermal equivalent circuit in the nominal operating speed equal to 1467.36 rpm

οࣖ࡯࢛ οࣖࡲࢋ

VZIM2 ͵ͷǤ͵ͳ ͳͻǤ͸͸

VZIM3 ͻͲǤͷ͹ ͶͲǤͺͳ

Average value ͸ʹǤͻͶ ͵ͲǤʹ͵ͷ

The average value of the temperature rise of VZIM2 and VZIM3, which is equal to ͸ʹǤͻͶ, is considered as the temperature rise of the 22kW VZIM motor, which is well below the temperature rise limit of the used insulation materials (Thermal class F). 5.3 Numerical calculation of temperature rise In ANSYS as a finite element tool, first of all the geometry of the model has to be prepared, assigning the materials and meshing the areas are next steps. The loss density is used as heat source, therefore based on the different losses of the motor, the loss densities for different parts of the motor are calculated using the formula ‫ ݌‬ൌ ܲΤܸ , where P is the calculated losses in each part, and V is the volume in which the losses occur. Electromagnetic analysis of 22kW VZIM is done with Flux2D in previous chapters and just 2D model of the VZIM motors was prepared but in this section for thermal analysis 3D models of VZIM2 and VZIM3 are built to consider the thermal effect of the windingoverhangs and end-rings. As usual, the number of the stator teeth and rotor teeth are not equal, hence a symmetry can be considered if for the simplification, the number of rotor teeth is assumed to be 36 instead of the 28 which is the real number of rotor teeth. With this simplification simulation of a half of the stator and rotor slots is enough to do the thermal analysis of the VZIM motor. In order to take into account the extra losses which appear due to the simplification in this new model, the rotor copper losses are multiplied by the factor compensate the extra losses added to the simulating model.

ଶ଼ ଷ଺

to

In this section, first of all the numerical calculation of the temperature rise based on the finite element method with ANSYS tool for the 5.5kW VZIM is done and compared with measured values, because the 5.5kW VZIM was already manufactured and tested. Afterwards the numerical calculation of temperature rise in 22kW VZIM has been carried out using ANAYS tool. In three dimensional models considered for thermal analysis of 5.5kW and 22kW VZIM, the convection heat transfer at the stator outer surface, the outer surface of the winding overhang and the end rings are considered as the boundary conditions of the thermal model. For simplification the heat transfer due to heat conduction, through the motor shaft is neglected. Clearly due to the thermal conduction between the motor shaft and rotor, in case of modeling the motor shaft, the calculated rotor temperature would be lower than values are presented with the simplified model in following sections.

- 71 -

5.3.1 Thermal analysis of 5.5 kW VZIM2 First a 2D geometry of 5.5kW VZIM2 is prepared and after assigning the material and meshing the geometry, by extruding the 2D model, the 3D model is constructed. In Fig.5.3.1-1 the 2D model and meshing of VZIM2 are depicted, the 3D model and meshing are depicted in Fig.5.3.1-2 and Fig.5.3.1-3.

Fig.5.3.1-1: 2D model of the 5.5kW VZIM2 and meshing, by ANSYS

Fig.5.3.1-2: 3D model of the 5.5kW VZIM2, by ANSYS

- 72 -

Fig.5.3.1-3: 3D model meshing of the 5.5kW VZIM2, by ANSYS

Loss densities in all volumes of VZIM2 have been calculated and assigned to the simulation model before solving the thermal model. Table 5.3.1-1 shows the thermal loss densities in VZIM2 5.5kW. Table 5.3.1-1: Loss densities in different parts of the 5.5kW VZIM2 Stator Winding Rotor bars winding overhang 0.000453318 0.00023955 0.00082892 Loss density W/‫ܕܕ‬૜

Stator iron

Rotor iron

0.000045299

0.000000566

The air flow in three areas including, the stator outer surface, winding overhang and end-rings surfaces are considered and the heat transfer coefficients due to the convection are calculated for those areas. - The heat transfer coefficient for the stator outer surface is calculated as below: ߙ ൌ ͳͷ‫ ݒ‬ଶΤଷ ൌ ͳͷ ή ͳ͵ǤͲͶͶଶΤଷ ൌ ͺ͵ǤͳͳȀሺʹ ሻ ͹Ͳ ʹߨ݊ ͹Ͳ ʹߨ ή ͳͶͷͷǤʹ ‫ ݒ‬ൌ ܴ ൌ ή ͳʹʹǤ͵ͷ ή ͳͲିଷ ή ൌ ͳ͵ǤͲͶͶ Ǥ ͳͲͲ ͸Ͳ ͳͲͲ ͸Ͳ

-The heat transfer coefficients for winding overhangs are considered as below: For one side of the motor the heat transfer coefficient over winding overhang isߙ ൌ ͳͷȀሺʹ ሻ for natural convection and heat radiation, but for the other side of the motor, because of the rotating centrifugal mechanism, the heat transfer coefficient for winding overhang was drawn with try and error, considering the value of the maximum temperature rise of the winding overhang, already measured. By choosing the heat transfer coefficient ߙ ൌ ͷͲȀሺʹ ሻ, the calculated and measured values match well together, hence consideration of ߙ ൌ ͷͲȀሺʹ ሻis acceptable. - 73 -

-The heat transfer coefficient for the end-ring surface is considered as below: As in area near the end ring air is nearly not moving, therefore ߙ ൌ ͳͷȀሺʹ ሻ is considered. Table 5.3.1-2 shows the heat transfer coefficient at different boundary conditions of the 5.5kW VZIM2. Table 5.3.1-2: Heat transfer coefficient at different boundary conditions of the 5.5kW VZIM2

ࢻሺ‫܅‬Ȁሺ‫ܕ‬૛ ۹ሻሻ

Stator outer surface

End ring

Winding overhang far from centrifugal mechanism

83.11

15

15

Winding overhang close to centrifugal mechanism 50

Fig.5.3.1-4 shows the thermal solution of the 5.5kW VZIM2, temperature changes from 84.5Ԩ on the stator body to 147.89Ԩ inside the rotor. For thermal class F of insulations the maximum allowable temperature rise is 105. For VZIM2 the maximum temperature rise in the stator winding is 72.28 as it has been shown in Fig.5.3.1-5, and assures that insulation lies in a safe side. The measured value of the maximum overhang temperature rise is 67.7, which is 6.3% lower than the finite element calculated value. The ambient temperature is considered to be 20Ԩ.

Fig.5.3.1-4: The thermal solution of the 5.5kW VZIM2 under nominal operation, by ANSYS

- 74 -

Fig.5.3.1-5: The calculated temperature in the winding overhang of VZIM2, by ANSYS

5.3.2 Thermal analysis of 5.5kW VZIM3 Similar to 5.5kW VZIM2, first a 2D geometry of 5.5kW VZIM3 is prepared and after assigning the material and meshing the geometry, by extruding the 2D model, the 3D model is constructed. In Fig.5.3.2-1 2D model and meshing of 5.55kW VZIM3 are depicted, also in Fig.5.3.2-2 and Fig.5.3.2-3 the 3D model and meshing are depicted.

Fig.5.3.2-1: 2D model of the 5.5kW VZIM3 and meshing, by ANSYS

- 75 -

Fig.5.3.2-2: 3D model of the 5.5kW VZIM3, by ANSYS

Fig.5.3.2-3: 3D model meshing of the 5.5kW VZIM3, by ANSYS

Loss densities in all volumes of VZIM3 have been calculated and assigned to the simulation model before solving the thermal model. Table 5.3.2-1 shows the thermal loss densities in 5.5kW VZIM3. Table 5.3.2-1: Loss densities in different parts of the 5.5kW VZIM3 Stator Winding Rotor bars winding overhang 0.00047015 0.00024844 0.0003707 Loss density ૜ W/‫ܕܕ‬

Stator iron

Rotor iron

0.00004544

0.000000461

The air flow in three areas including the stator outer surface, winding overhang and end-rings surfaces are considered and the heat transfer coefficient due to the convection are calculated for those areas. - 76 -

- The heat transfer coefficient for the stator outer surface is calculated as below: ߙ ൌ ͳͷ‫ ݒ‬ଶΤଷ ൌ ͳͷ ή ͳ͵Ǥͳ͹ଶΤଷ ൌ ͺ͵Ǥ͸ͷȀሺʹ ሻ ʹߨ݊ ͹Ͳ ʹߨ ή ͳͶ͹Ͳ ͹Ͳ ܴ ൌ ή ͳʹʹǤ͵ͷ ή ͳͲିଷ ή ൌ ͳ͵Ǥͳ͹ Ǥ ‫ ݒ‬ൌ ͸Ͳ ͳͲͲ ͸Ͳ ͳͲͲ

- The heat transfer coefficients for winding overhangs are considered as below: For one side of the motor the heat transfer coefficient over winding overhang isߙ ൌ ͳͷȀሺʹ ሻ, for the other side of the motor, because of the rotating centrifugal mechanism, the heat transfer coefficient for winding overhang was drawn with try and error, considering the value of the maximum temperature rise of the winding overhang was already measured, by consideration of heat transfer coefficient ߙ ൌ ͷͲȀሺʹ ሻ the calculated values and measured values match well together, hence consideration of ߙ ൌ ͷͲȀሺʹ ሻ is acceptable.

-The heat transfer coefficient for the end-ring surface is considered as below: As in this area air is nearly not moving, therefore ߙ ൌ ͳͷȀሺʹ ሻ is considered.

Table 5.3.2-2 shows the heat transfer coefficient at different boundary conditions of the 5.5kW VZIM3. Table 5.3.2-2: Heat transfer coefficient at different boundary conditions of the 5.5kW VZIM3



End ring


83.65

15

15

Winding overhang close to centrifugal mechanism 50

Fig.5.3.2-4 shows the thermal solution of the VZIM3, temperature changes from 80.17Ԩ on the stator body to 121.969Ԩ inside the rotor. For thermal class F of insulations the maximum allowable temperature is 105. For VZIM3 the maximum temperature rise in stator winding overhang is 63.35 as it has been shown in Fig.5.3.2-5, and assures that insulation lies in a safe side. The measured value of the maximum overhang temperature rise is 67.7, which is 6.42% higher than the calculated value by finite element method. The ambient temperature is considered to be 20Ԩ.

- 77 -

Fig.5.3.2-4: The thermal solution of the 5.5kW VZIM3 under nominal operation, by ANSYS

Fig.5.3.2-5: The calculated temperature in winding overhang of 5.5kW VZIM3, by ANSYS

Table 5.3.2-3 shows the temperature rise in stator slot of 5.5kW VZIM2 and VZIM3 stator slot calculated by ANSYS. Table 5.3.2-3: Temperature rise in stator slot of 5.5kW VZIM2 and VZIM3 calculated by ANSYS

οࣖ࡯࢛

VZIM2 ͹ʹǤʹͺ

VZIM3 ͸͵Ǥ͵ͷ

Average value ͸͹Ǥͺͳ

The average value of the temperature rise of VZIM2 and VZIM3 which is equal to ͸͹Ǥͺͳ, is considered as the temperature rise of the 5.5kW VZIM motor. The measured value of the maximum overhang temperature is ͸͹Ǥ͹ which obviously shows a very good fitting between the measured value and calculated values of temperature rise by ANSYS tool. - 78 -

5.3.3 Thermal analysis of 22kW VZIM2 First a 2D geometry of 22kW VZIM2 is prepared and after assigning the material and meshing the geometry, by extruding the 2D model, the 3D model is constructed. In Fig.5.3.31 2D model and meshing of it are depicted, also in Fig.5.3.3-2 and Fig.5.3.3-3 the 3D model and its meshing are depicted.

Fig.5.3.3-1: 2D model of the 22kW VZIM2 and meshing, by ANSYS

Fig.5.3.3-2: 3D model of the 22kW VZIM2, by ANSYS

- 79 -

Fig.5.3.3-3: 3D model meshing of the 22kW VZIM2, by ANSYS

Loss densities in all volumes of VZIM2 have been calculated and assigned to the simulation model before solving the thermal model. Table 5.3.3-1 shows the thermal loss densities in 22kW VZIM2. Table 5.3.3-1: Loss densities in different parts of the 22kW VZIM2 Stator Winding Rotor bars winding overhang Loss density 0.000169 0.0001074 0.0007152 W/‫ܕܕ‬૜

Stator iron

Rotor iron

0.00002834

0.000007678

The air flow in three areas including the stator outer surface, winding over hang and end-rings surfaces are considered and the heat transfer coefficient due to the convection are calculated for those areas. -The heat transfer coefficient for the stator outer surface is calculated as below: ߙ ൌ ͳͷ‫ ݒ‬ଶΤଷ ൌ ͳͷ ή ͳ͹Ǥͺ͹ଶΤଷ ൌ ͳͲʹǤͷʹȀሺʹ ሻ ͹Ͳ ʹߨ݊ ͹Ͳ ʹߨ ή ͳͶ͸͹Ǥ͵͸ ‫ ݒ‬ൌ ܴ ൌ ή ͳ͸͸Ǥʹͷ ή ͳͲିଷ ή ൌ ͳ͹Ǥͺ͹ ͸Ͳ ͳͲͲ ͸Ͳ ͳͲͲ -The heat transfer coefficients for winding overhangs are considered as below: At one side, the heat transfer coefficient for winding overhang isߙ ൌ ͳͷȀሺʹ ሻ, for the other side of the motor due to the rotating centrifugal mechanism, as already was discussedߙ ൌ ͸͵Ǥ͵ʹȀሺʹ ሻ. -The heat transfer coefficient for the end-ring surface is considered as below: As in this area air is nearly not moving, therefore ߙ ൌ ͳͷȀሺʹ ሻ is considered.

Table 5.3.3-2 shows the heat transfer coefficient at different boundary conditions of the 22kW VZIM2.

- 80 -

Table 5.3.3-2: Heat transfer coefficient at different boundary conditions of the 22kW VZIM2



End ring


102.52

15

15

Winding overhang close to centrifugal mechanism 63.32

Fig.5.3.3-4 shows the thermal solution of the 22kW VZIM2, temperature changes from 59.7929Ԩ on the stator body to 150.95Ԩ inside the rotor. For thermal class F of insulations the maximum allowable temperature rise is 105. For VZIM2 the maximum temperature rise in the stator winding overhang is 42.73 as it has been shown in Fig.5.3.3-5, and assures that insulation lies in a safe side. The calculated value of the stator winding temperature rise is 35.31 which is 17.3% lower than finite element calculated value.

Fig.5.3.3-4: The thermal solution of the 22kW VZIM2 under nominal operation, by ANSYS

- 81 -

Fig.5.3.3-5: The calculated temperature in the winding overhang of 22kW VZIM2, by ANSYS

5.3.4 Thermal analysis of 22kW VZIM3 Similar to 22kW VZIM2, first a 2D geometry of 22kW VZIM3 is prepared and after assigning the material and meshing the geometry, by extruding the 2D model, the 3D model is constructed. In Fig.5.3.4-1 2D model and meshing of 22kW VZIM3 are depicted, also in Fig.5.3.4-2 and Fig.5.3.4-3 the 3D model and meshing are depicted.

Fig.5.3.4-1: 2D model of the 22kW VZIM3 and meshing, by ANSYS

- 82 -

Fig.5.3.4-2: 3D model of the 22kW VZIM3, by ANSYS

Fig.5.3.4-3: 3D model meshing of the 22kW VZIM3, by ANSYS

Loss densities in all volumes of VZIM3 have been calculated and assigned to the simulation model before solving the thermal model. Table 5.3.4-1 shows the thermal loss densities in 22kW VZIM3. Table 5.3.4-1: Loss densities in different parts of the 22kW VZIM3 Stator Winding Rotor bars winding overhang Loss density 0.00048 0.000304 0.000606 W/‫ܕܕ‬૜

- 83 -

Stator iron

Rotor iron

0.00002349

0.00000934

The air flow in three areas including the stator outer surface, winding over hang and end-rings surfaces are considered and the heat transfer coefficient due to the convection are calculated for those areas. -The heat transfer coefficient for the stator outer surface is calculated as below: ߙ ൌ ͳͷ‫ ݒ‬ଶΤଷ ൌ ͳͷ ή ͳ͹Ǥͺ͹ଶΤଷ ൌ ͳͲʹǤͷʹȀሺʹ ሻ ʹߨ݊ ͹Ͳ ʹߨ ή ͳͶ͸͹Ǥ͵͸ ͹Ͳ ܴ ൌ ή ͳ͸͸Ǥʹͷ ή ͳͲିଷ ή ൌ ͳ͹Ǥͺ͹ ‫ ݒ‬ൌ ͸Ͳ ͳͲͲ ͸Ͳ ͳͲͲ

-The heat transfer coefficients for winding overhangs are considered as below: At one side, the heat transfer coefficient for winding overhang isߙ ൌ ͳͷȀሺʹ ሻ, for the other side of the motor due to the rotating centrifugal mechanism, as already was discussedߙ ൌ ͸ʹǤͺ͸Ȁሺʹ ሻ. - The heat transfer coefficient for the end-ring surface is considered as below: As in this area air is nearly not moving, therefore ߙ ൌ ͳͷȀሺʹ ሻ is considered.

Table 5.3.4-2 shows the heat transfer coefficient at different boundary conditions of the 22kW VZIM3. Table 5.3.4-2: Heat transfer coefficient at different boundary conditions of the 22kW VZIM3



End ring


102.52

15

15

Winding overhang close to centrifugal mechanism 62.86

Fig.5.3.4-4 shows the thermal solution of the 22kW VZIM3, temperature changes from 98.7877Ԩ on the stator body to 224.19Ԩ inside the rotor. For thermal class F of insulations the maximum allowable temperature rise is 105. For VZIM3 the temperature in stator winding overhang is 85.72 as it has been shown in Fig.5.3.4-5, and assures that insulation lies in a safe side. The calculated value of the stator winding temperature rise is 90.57 which is 5.5% higher than calculated value by finite element method.

- 84 -

Fig.5.3.4-4: The thermal solution of the 22kW VZIM3 under nominal operation, by ANSYS

Fig.5.3.4-5: The calculated temperature in the winding overhang of 22kW VZIM3, by ANSYS

Table 5.3.4-3 shows the temperature rise in stator slot of 22kW VZIM2 and VZIM3 stator slot calculated by ANSYS. Table 5.3.4-3: Temperature rise in stator slot of 22kW VZIM2 and VZIM3 calculated by ANSYS

οࣖ࡯࢛

VZIM2 ͶʹǤ͹͵

VZIM3 ͺͷǤ͹ʹ

- 85 -

Average value ͸ͶǤʹʹ

The average value of the temperature rise of VZIM2 and VZIM3 (͸ͶǤʹʹ) is considered as the temperature rise of the 22kW VZIM motor which is well below the maximum temperature rise for the insulation class F (ͳͲͷ). The mean value of the calculated values of the maximum overhang temperature with simplified equivalent thermal circuit is͸ʹǤͻͶ, which shows a 1.99% difference between finite element method calculations and the calculations based on the simplified thermal circuits. In order to summarize the thermal calculations of 5.5kW and 22kW VZIM, the values of the stator maximum winding temperature rise and the rotor maximum temperature rise are shown in Table 5.3.4-4. The thermal analyses of 5.5kW VZIM2 and VZIM3 motors are done in the operation point of the motor where the output power of each motor is 5.5kW. But the thermal analyses of 22kW VZIM2 and VZIM3 motors are done in an operation point of the motor, where the average output power of 22kW motors (VZIM2 and VZIM3) is 22kW. Therefore according to the previous description and calculation in section 5.2, as it is shown in Table 5.2-2, the 22kW VZIM2 is analyzed in an operation point, that its output power is 16.89kW and the 22kW VZIM3 is analyzed in an operation point, where its output power is 27.37kW. Therefore the 22kW VZIM3 motor with the output power 27.37kW is clearly overloaded, therefore the temperature rise of the rotor and stator of 22kW VZIM3 is relatively high. As previously mentioned, the average temperature rise of VZIM2 and VZIM3 in the rated operating point of the VZIM is important and shows the temperature rise of the real motor. Table 5.3.4-4: Temperature rise in stator slot of 22kW VZIM2 and VZIM3 calculated by ANSYS

Stator max. winding temperature rise Rotor max. temperature rise

5.5kW VZIM2

5.5kW VZIM3

Average value of 5.5kW VZIM

22kW VZIM2

22kW VZIM3

Average value of 22kW VZIM

͹ʹǤʹͺ

͸͵Ǥ͵ͷ

͸͹Ǥͺͳ

ͶʹǤ͹͵

ͺͷǤ͹ʹ

͸ͶǤʹʹ

ͳͶ͹Ǥͺͻͷ

ͳʹͳǤͻ͸ͻ

ͳ͵ͶǤͻ͵ʹ

- 86 -

ͳ͵ͲǤͻͷ

ʹͲͶǤͳͻ

ͳ͸͹Ǥͷ͹

Chapter 6: Conclusion Electromagnetic and thermal analyses of Variable Impedance Induction Motor (VZIM) have been done for a 4 pole 22kW motor. The electromagnetic analyses show that the expected goals of the high starting torque and high efficiency at the rated operating point of the VZIM motor are fulfilled. Electromagnetic analysis of the 22kW VZIM motor is carried out with consideration of three independent motors VZIM1, VZIM2 and VZIM3, using the finite element software Flux2D and the results are compared with the analytical calculations performed by KLASYS tool. In the steady state AC analysis the torque-slip, stator phase current-slip, normal air-gap flux density and flux lines in the electromagnetically active parts of the motors are drawn. The results of the analytical and finite element method, are compared which shows a good match between them. Moreover in time stepping analysis torque-slip, fundamental stator phase current-slip, input power-slip, efficiency-slip and power factor-slip characteristics are drawn and compared. Comparison of torque-slip characteristics in steady state AC and time stepping analysis show that the KLASYS tool always predicts higher values than Flux2D. The efficiency-slip, phase current-slip, input power-slip and power factor-slip characteristics of the VZIM motor in time stepping and steady state AC analysis, calculated by Flux2D and KLASYS match very well in all slips. Thermal calculations have been done for VZIM with the help of the ANSYS tool which is a finite element method tool. Comparing the results obtained by ANSYS and simplified thermal circuits of the 22kW VZIM motor shows a low difference about 1.99%, and the results assure that the insulation of the stator winding always stays in a safe side, and the temperature never exceeds the critical values. In addition, the average value of the temperature rise of VZIM2 and VZIM3 (͸ͶǤʹʹሻ calculated by ANSYS, is considered as the temperature rise of the 22kW VZIM motor which is well below the maximum temperature rise for the Thermal Class F (ͳͲͷ).

- 87 -

Bibliography [1] Binder, A. ,‘‘CAD and system dynamics of electrical machines,’’ Lecture script, TU Darmstadt, 2010. [2] Binder, A. ,‘‘Motor development for electrical drive systems, ’’ Lecture script, TU Darmstadt, 2010. [3] Binder, A. ,‘‘Electrical Machines and Drives, ’’ Lecture script, TU Darmstadt, 2009. [4] Binder, A. ; Funieru, B. ‚‘‘Design of Electrical Machines and Actuators with Numerical Field Calculations, ’’ Seminary notes, TU Darmstadt, 2010. [5] Flux2D Application‚‘‘Induction machine tutorial calculations in Flux2D,’’ Cedrat corporation, 2011. [6] Gholizad, H. ; Binder, A. ,“Analytical Modeling of Variable Impedance Induction Motors’’, in proc. IEEE IEMDC, 15-18 May 2011, Niagara Falls, Canada, pp. 1504-1509. [7] Data sheet, Power Core M270-50A, Elektroband NO / NGO electrical steel. [8] Xyptras, J. ; Hatziathanassiou, V. ,“Thermal analysis of an electrical machine taking into the account the iron losses and the deep bar effect,’’ IEEE Transactions on Energy Conversion vol. 14, no. 4, December 1999, pp. 996-1003. [9] Huai, Y. ; Melnik, R. ; Thogersen, P. ,“Computational analysis of temperature rise phenomena in induction motors’’, Applied Thermal engineering vol. 23, no. 7, May 2003, pp. 779-795.

- 88 -

8. Appendix

8.1 Appendix I: Calculation of magnetizing main and stray inductances Magnetizing main inductance ‫ܮ‬௛ǡஶ for infinite iron permeability is only determined by air gap. According to [1] the magnetizing main inductance ‫ܮ‬௛ǡஶ is calculated as: ʹ ή ͵ ʹͶͲ ή ͳͲିଷ ή ͳ͵͵ǤͶͷ ή ͳͲିଷ ʹ݉ ݈߬௣ ଶ ଶ ି଻ ଶ ଶ ൌ Ͷߨ ή ͳͲ ή ͹ʹ ή ͲǤͻ͸ ή ଶ ‫ܮ‬௛ǡஶ ൌ ߤ଴ ܰ ݇௪ ଶ ߨ ߨ ‫ߜ݌‬ ʹ ή ͲǤͶͷ ή ͳͲିଷ ൌ ͲǤͳʹͻͻͷ ߨ ቀ͸ቁ ݇௪ ൌ ߨ ൌ ͲǤͻ͸ ͵ ቀ ቁ ͳͺ ݀௦௜ ߨ ൌ ͳ͵͵ǤͶͷ ൉ ͳͲିଷ ߬௣ ൌ ʹ‫݌‬ Stator harmonic stray inductance is calculated as: ‫ܮ‬ఙǡ௢௦ ൌ ߪ௦ǡ௢ ‫ܮ‬௦௛ ൌ ͲǤͲͳʹ͸ ή ͳʹͻͻǤͷ ή ͳͲିସ ൌ ͳ͸Ǥ͵͹͵͹ ή ͳͲିସ ‫ ܪ‬ൌ ͳ͸Ǥ͵͹͵͹ where stator harmonic stray coefficient is calculated: ଶ ͷ൉ͻ൅ͳ ߨ ଶ ͷ‫ݍ‬௦ଶ ൅ ͳ ߨ ሻ ൉ െͳൌቀ ቁ ൉ െ ͳ ൌ ͲǤͲͳʹ͸Ǥ ߪ௦ǡ௢ ൌ ሺ ଶ ͵ ൉ ͲǤͻ͸ ͸൉ͻ ͵݇௪௦ǡଵ ͸‫ݍ‬௦ where rotor stray harmonic inductance is calculated as: ‫ܮ‬ᇱఙǡ௢௥ ൌ ߪ௥ǡ௢ ‫ܮ‬௛ ൌ ͲǤͲͳ͸ ή ͳʹͻͻǤͷ ή ͳͲିସ ൌ ʹͲǤ͹ͻʹ ή ͳͲିସ Rotor harmonic stray coefficient is calculated: ͳ ͳ െͳൌ ߪ௥ǡ௢ ൌ ଶ െ ͳ ൌ ͲǤͲͳ͸ ሺ‫ ݌‬൉ ߨΤܳ௥ ሻ ଶ ሺʹ ൉ ߨΤʹͺሻ ሻ ሺ ൬ ൰ ‫ ݌‬൉ ߨΤܳ௥ ʹ ൉ ߨΤʹͺ Inductance of stator winding overhang is calculated as below: ͹ʹଶ ʹ ή ʹ ൉ ʹʹ͸Ǥ͸ ή ͳͲିଷ ൉ ͲǤʹͺ͹ ֜ ‫ܮ‬௦ǡఙ௕ ൌ ͲǤͶʹͶ ‫ܮ‬௦ǡఙ௕ ൌ ߤ଴ ܰ௦ଶ ߣ௕ ݈௕ ൌ Ͷߨ ή ͳͲି଻ ή ‫݌‬ ʹ ߣ௕ ൌ ͲǤ͵͹ ൉ ܿ ൉ ൤ͳ ൅ ͲǤͻ ൉

݀௪ ൌ ඨ

݈௕ െ ͳͲ ή ݀௪ ʹʹ͸Ǥ͸ െ ͳͲ ή ͳͷǤͻ ൨ ൌ ͲǤ͵͹ ൉ ͲǤ͸͹ ൉ ൤ͳ ൅ ͲǤͻ ൉ ൨ ൌ ͲǤʹͺ͹ ʹʹ͸Ǥ͸ ൅ ͳͲ ή ͳͷǤͻ ݈௕ ൅ ͳͲ ή ݀௪

ଶ ܰ ܽ௜ ݀௖௨ ͻ ൉ ͳ ൉ ͳʹ ௖ ൌඨ ൌ ͳͷǤͻ ݇௙ ͲǤͶʹ͸

End-ring stray inductance of the rotor cage is calculated: ߣ௕ ൌ ͲǤͳʹ,݈௕ ൌ ߬௣

ͳ ଶ ʹ ʹ ି଻ ‫ܮ‬௥ǡఙ௕ ൌ ߣ ݈ ൌ Ͷߨ ή ͳͲ ൉ ൬ ൰ ή ൉ ͲǤͳʹ ൉ ͳ͵͵Ǥͷ ൉ ͳͲିଷ ൌ ͷǤͲ͵ ൉ ͳͲିଽ ʹ ʹ ‫ ݌‬௕௕ Slot stray inductance of stator winding is calculated: ߤ଴ ܰ௦ଶ

‫ܮ‬௦ఙொ ൌ ߤ଴ ܰ௦ଶ

ଶ

௣Ǥ௤ೞ

ߣொ௦ ݈௘ ൌ Ͷߨ ή ͳͲି଻ ൉ ሺ͹ʹሻଶ ൉

ଶ

ଶήଷ

ή ͳǤͷͺ͸ ή ʹͶͲ ή ͳͲିଷ ൌ ͺǤʹ͸ͳ ή ͳͲିସ

- 89 -

݄ଵ ݄ସ ͳ͹ǤͶͲͺ ൉ ͳͲିଷ ͲǤ͹͵͹ ൉ ͳͲିଷ ߣொ௦ ൌ ൅ ͲǤ͸ͺͷ ൅ ൌ ൅ ͲǤ͸ͺͷ ൅ ൌ ͳǤͷͺ͹ ͵ܾொ௔௩ ‫ݏ‬ொ ͵ ൉ ͺǤ͹Ͷ ൉ ͳͲିଷ ͵Ǥͳ ൉ ͳͲିଷ ܾ௜ ൅ ܾ௔ ͹Ǥʹͳʹ ൅ ͳͲǤʹ͹͹ ൌ ή ͳͲିଷ ൌ ͺǤ͹ͶͶͷ ή ͳͲିଷ Ǥ ܾொ௔௩ ൌ ʹ ʹ Rotor slot stray inductance of VZIM1 (Round bar) calculation: Rotor slot stray inductance of VZIM1 (round bar) is calculated according to [1] ‫ܮ‬ఙǡ௕௔௥ ൌ ߤ଴ Ǥ ߣொ௥ Ǥ ݈௘ ൌ Ͷߨ ή ͳͲି଻ ൉ ͲǤͺͶ͵͵ ൉ ʹͶͲ ή ͳͲିଷ ൌ ͲǤʹͷʹ ή ͳͲି଺ ݇௅ǡ௥௢௨௡ௗ ൌ ෥ͳ ݀ ݄ସ ͳͲǤʹ ή ͳͲିଷ ͳ ൉ ͳͲିଷ ߣொ௥ ൌ ቆ ൅ ͲǤͶ͹ቇ ൉ ݇௅ǡ௥௢௨௡ௗ ൅ ൌቆ ൅ ͲǤͶ͹ቇ ൉ ݇ ൅ ௅ǡ௥௢௨௡ௗ ͳͷ‫ݏ‬ொ ‫ݏ‬ொ ͳͷ ൉ ͶǤͷ ή ͳͲିଷ ͶǤͷ ൉ ͳͲିଷ ൌ ͲǤͺͶ͵͵Ǥ Rotor slot stray inductance of VZIM2 (Deep bar) calculation: Rotor slot stray inductance of VZIM2 (Deep bar) is calculated according to [1] ‫ܮ‬ఙǡ௕௔௥ ൌ ߤ଴ Ǥ ߣொ௥ Ǥ ݈௘ ൌ Ͷߨ ή ͳͲି଻ ൉ ͲǤͷͶʹͺͺ ൉ ʹͶͲ ή ͳͲିଷ ൌ ͲǤͳ͸͵͸ ή ͳͲି଺ where:

ߦ ൌ ݄ଵ ඥߨ ή ݂௥ ή ߤ଴ ή ˍ௖௨ ݂௥ ൌ ‫ ݏ‬ή ݂௦ ൌ ͳ ή ͷͲ ൌ ͷͲ (At standstill)

ߦ ൌ ͺǤʹ ή ͳͲିଷ ξߨ ή ͷͲ ή Ͷ ή ߨ ή ͳͲି଻ ή ͷǤ͹ ή ͳͲ଻ ൌ ͲǤͺ͸ͻ͵ ͵ ሺͳǤ͹͵ͺ͹ሻ െ ሺͳǤ͹͵ͺ͹ሻ ͵ ሺʹߦሻ െ ሺʹߦሻ ൌ ൌ ͲǤͻͺͷͺ ݇௅ ൌ ͳǤ͹͵ͺ͹ ሺͳǤ͹͵ͺ͹ሻ െ ሺͳǤ͹͵ͺ͹ሻ ʹߦ ሺʹߦሻ െ ሺʹߦሻ ݄ଵ ݄ସ ͺǤʹ ͳ ߣொ௥ ൌ ൉ ݇௅ ൅ ൌ ൉ ͲǤͻͺͷͺ ൅ ൌ ͲǤͷͶʹͺͺǤ ͵ ή ͻǤʹ Ͷ ή ͳͲିଷ ͵ܾொ ‫ݏ‬ொ

Rotor slot stray inductance of VZIM3 (Wedge bar) calculation: Rotor slot stray inductance of VZIM2 (Wedge bar) is calculated according to [1] ‫ܮ‬ఙǡ௕௔௥ ൌ ߤ଴ Ǥ ߣொ௥ Ǥ ݈௘ ൌ Ͷߨ ή ͳͲି଻ ൉ ͲǤͻ͵ͷͺ ൉ ʹͶͲ ή ͳͲିଷ ൌ ͲǤʹͺʹ ή ͳͲି଺ ݄ଵ ݄ସ ͳ͹Ǥ͹ ͳ ߣொ௥ ൌ ൉ ݇௅ǡ௪௘ௗ௚௘ ൅ ൌ ൉ ͲǤ͹͵ ൅ ൌ ͲǤͻ͵ͷͺ ‫ݏ‬ொ ͵ ή ͻǤͺͺʹͺ ʹ ͵ܾ௘ where:

ߦ ൌ ݄ଵ ඥߨ ή ݂௥ ή ߤ଴ ή ˍ௖௨ ݂௥ ൌ ‫ ݏ‬ή ݂௦ ൌ ͳ ή ͷͲ ൌ ͷͲ (At standstill)

ߦ ൌ ͳ͹Ǥ͹ ή ͳͲିଷ ξߨ ή ͷͲ ή Ͷ ή ߨ ή ͳͲି଻ ή ͷǤ͹ ή ͳͲ଻ ൌ ͳǤͺ͹͸ͷǤ

- 90 -

8.2 Appendix II: a) Calculation of slips, where a synchronous harmonic torque of VZIM1, VZIM2, VZIM3 happens In order to derive the slips that synchronous harmonics of the VZIM occur, according to [2], Stator ordinal numbers are derived as, ‫ ݒ‬ൌ ͳ ൅ ʹ݉௦ ݃ ൌ ͳ ൅ ͸݃ ൌ ͳǡ െͷǡ͹ǡ െͳͳǡͳ͵ǡ െͳ͹ǡͳͻǡ െʹ͵ǡʹͷǡ െʹͻǡ͵ͳ Rotor ordinal numbers are derived as, ʹͺ ܳ௥ ߤ ൌ ͳ ൅ ݃ ൌ ͳ ൅ ݃ ൌ ͳǡ െͳ͵ǡͳͷǡ െʹ͹ǡʹͻǡ െͶͳ ʹ ‫݌‬ Condition for slip where synchronous harmonic occurs is derived as, ‫ ݒ‬ൌ ߤ ֜ ‫ כ ݏ‬ൌ ͳ ‫ ݒ‬ൌ ߤ ൌ ͳ ֜ ‫ כݏ‬ൌ ͳ ‫ݒ‬െͳ ‫ ݒ‬ൌ െߤ ֜ ‫ כ ݏ‬ൌ ‫ݒ‬൅ͳ ‫ ݒ‬െ ͳ ͳ͵ െ ͳ ‫ ݒ‬ൌ െߤ ൌ ͳ͵ ֜ ‫ כ ݏ‬ൌ ൌ ൌ ͲǤͺͷ͹ ‫ ݒ‬൅ ͳ ͳ͵ ൅ ͳ ‫ ݒ‬െ ͳ െʹͻ െ ͳ ൌ ൌ ͳǤͲ͹ͳ ‫ ݒ‬ൌ െߤ ൌ െʹͻ ֜ ‫ כ ݏ‬ൌ ‫ ݒ‬൅ ͳ െʹͻ ൅ ͳ After deriving the slips, the related speeds of synchronous harmonic torque is calculated as ݊௦௬௖ െ ݊ ‫ݏ‬ൌ ݊௦௬௡ ݊ ൌ ሺͳ െ ‫ݏ‬ሻ݊௦௬௡ ൌ ሺͳ െ ͲǤͺͷ͹ሻͳͷͲͲ ൌ ʹͳͶǤͶ‫݉݌ݎ‬ ݊ ൌ ሺͳ െ ‫ݏ‬ሻ݊௦௬௡ ൌ ሺͳ െ ͳǤͲ͹ͳሻͳͷͲͲ ൌ െͳͲ͸Ǥͷ‫݉݌ݎ‬ In Table I synchronous harmonic torque slips which are the same for VZIM1, VZIM2, and VZIM3 are shown. Table I: Synchronous harmonic torque slip Harmonic 13 -29

slip 0.857 1.071

Rotor speed(rpm) 214.5 -106.5

b) Calculation of the slips which asynchronous harmonic torque happens in VZIM1 Slips that asynchronous harmonic torques of the VZIM1 occur are calculated according to [2]. Stator ordinal numbers are derived as ‫ ݒ‬ൌ ͳ ൅ ʹ݉௦ ݃ ൌ ͳ ൅ ͸݃ ൌ ͳǡ െͷǡ͹ǡ െͳͳǡͳ͵ǡ െͳ͹ǡͳͻǡ െʹ͵ǡʹͷǡ െʹͻǡ͵ͳǡ െ͵ͷǡ͵͹Ǥ The asynchronous harmonic slips are derived as ͳ ‫ ݏ‬ൌ ͳ െ Ǥ ‫ݒ‬

- 91 -

Table II: Stator ordinal numbers and related asynchronous harmonic torque slips -11 13 -17 19 -23 25 -29 31 -35 37 ࢜ 1 -5 7 ࢙ 0 1.2 0.857 1.09 0.92 1.0588 0.947 1.043 0.96 1.034 0.96 1.028 0.072 Harmonic break down slips, are calculated for each asynchronous harmonic torque, according to [2] as ܴ௥ ǡ ‫ݏ‬௩௕ ൌ േ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ where ‫ܮ‬௥ఙ௩ is calculated as ‫ܮ‬௥ఙ௩ ൌ ሺͳΤߟ௩ଶ െ ͳሻ ή ‫ܮ‬௥௛௩ ൅ ‫ܮ‬௥ఙ . For all ordinal numbers the breakdown slips are calculated as following: For ‫ ݒ‬ൌ െͷ:

ܴ௥ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ߬௣ ݈ி௘ ʹ݉௥ ଶ ή ଶ ଶ ‫ܮ‬௥௛௩ ൌ ߤ଴ ܰ௥ଶ ݇௪௥ ή ߨ ή‫ ݒ‬ή‫ߜ ݌‬ ଶ ͳ ʹ ή ʹͺ ͳ͵͵Ǥͷ ή ͳͲିଷ ή ʹͶͲ ή ͳͲିଷ ି଻ ή ൌ ʹǤͷ͵ ൉ ͳͲି଺ ‫ܪ‬ ‫ܮ‬௥௛ǡିହ ൌ Ͷߨ ൉ ͳͲ ൬ ή ͳ൰ ή ଶ ߨ ή ʹͷ ή ʹ ͲǤͶͷ ή ͳͲିଷ ʹ ሺ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ሻ ሺെͷ ή ʹ ή ߨΤʹͺሻ ൌ ൌ ͲǤͺͲ͵ ߟ௩ୀିହ ൌ െͷ ή ʹ ή ߨΤʹͺ ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ‫ܮ‬௥ఙǡିହ ൌ ሺͳΤߟ௩ଶ െ ͳሻ ή ‫ܮ‬௥௛௩ ൅ ‫ܮ‬௥ఙ ൌ ሺͳΤሺͲǤͺͲ͵ሻଶ െ ͳሻ ή ʹǤͷ͵ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ൌ ʹǤ͸͸ ή ͳͲି଺ ‫ܪ‬ ܴ௥ ʹǤͳ͸ ൉ ͳͲିସ ൌൌ േ ൌ േͲǤͳ͵ ‫ݏ‬௩ୀିହǡ௕ ൌ േ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ʹ ή ߨ ή ͷͲ ή ሺʹǤͷ͵ ή ͳͲି଺ ൅ ʹǤ͸͸ ή ͳͲି଺ ሻ ͳ െ ‫ݏ‬௩ ‫ݏ‬௩ ൌ ͳ െ ‫ݒ‬ሺͳ െ ‫ݏ‬ሻ ֜ ‫ ݏ‬ൌ ͳ െ ‫ݒ‬ ͳ െ ͲǤͳ͵ ͳ െ ‫ݏ‬௩ ൌ ͳെ ൌ ͳǤͳ͹Ͷ ‫ ݏ‬ൌ ͳെ െͷ ‫ݒ‬ ͳ ൅ ͲǤͳ͵ ͳ െ ‫ݏ‬௩ ‫ ݏ‬ൌ ͳെ ൌ ͳെ ൌ ͳǤʹʹ͸ െͷ ‫ݒ‬ ‫ݏ‬௩௕ ൌ േ

For‫ ݒ‬ൌ ͹:

ܴ௥ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ߬௣ ݈ி௘ ʹ݉௥ ଶ ή ή ଶ ଶ ‫ܮ‬௥௛௩ ൌ ߤ଴ ܰ௥ଶ ݇௪௥ ߨ ή‫ ݒ‬ή‫ߜ ݌‬ ଶ ͳ ʹ ή ʹͺ ͳ͵͵Ǥͷ ή ͳͲିଷ ή ʹͶͲ ή ͳͲିଷ ି଻ ‫ܮ‬௥௛ǡ଻ ൌ Ͷߨ ൉ ͳͲ ൬ ή ͳ൰ ή ଶ ή ൌ ʹǤʹͻͷ ൉ ͳͲି଺ ‫ܪ‬ ͲǤͶͷ ή ͳͲିଷ ߨ ή Ͷͻ ή ʹ ʹ ሺ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ሻ ሺ͹ ή ʹ ή ߨΤʹͺሻ ൌ ൌ ͲǤ͸͵͸ ߟ௩ୀ଻ ൌ ͹ ή ʹ ή ߨΤʹͺ ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ‫ܮ‬௥ఙǡିହ ൌ ሺͳΤߟ௩ଶ െ ͳሻ ή ‫ܮ‬௥௛௩ ൅ ‫ܮ‬௥ఙ ൌ ሺͳΤሺͲǤ͸͵͸ሻଶ െ ͳሻ ή ʹǤʹͻͷ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ൌ ͶǤ͸Ͷ ή ͳͲି଺ ‫ܪ‬ ‫ݏ‬௩௕ ൌ േ

- 92 -

ʹǤ͹ͷ͸ ൉ ͳͲିସ ܴ௥ ൌൌ േ ൌ േͲǤͲͻͻ ʹ ή ߨ ή ͷͲ ή ሺͶǤ͸Ͷ ή ͳͲି଺ ൅ ʹǤʹͻͷ ή ͳͲି଺ ሻ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ͳ െ ‫ݏ‬௩ ‫ݏ‬௩ ൌ ͳ െ ‫ݒ‬ሺͳ െ ‫ݏ‬ሻ ֜ ‫ ݏ‬ൌ ͳ െ ‫ݒ‬ ͳ െ ͲǤͲͻͻ ͳ െ ‫ݏ‬௩ ൌ ͳെ ൌ ͲǤͺ͹ͳ ‫ ݏ‬ൌ ͳെ ‫ݒ‬ ͹ ͳ െ ‫ݏ‬௩ ͳ ൅ ͲǤͲͻͻ ൌ ͳെ ൌ ͲǤͺͶ͵ ‫ ݏ‬ൌ ͳെ ‫ݒ‬ ͹

‫ݏ‬௩ୀ଻ǡ௕ ൌ േ

For‫ ݒ‬ൌ െͳͳ:

ܴ௥ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ߬௣ ݈ி௘ ʹ݉௥ ଶ ή ଶ ଶ ή ‫ܮ‬௥௛௩ ൌ ߤ଴ ܰ௥ଶ ݇௪௥ ߨ ή‫ ݒ‬ή‫ߜ ݌‬ ଶ ͳ ʹ ή ʹͺ ͳ͵͵Ǥͷ ή ͳͲିଷ ή ʹͶͲ ή ͳͲିଷ ି଻ ‫ܮ‬௥௛ǡ଻ ൌ Ͷߨ ൉ ͳͲ ൬ ή ͳ൰ ή ଶ ή ൌ ͲǤͷʹͶ ൉ ͳͲି଺ ‫ܪ‬ ߨ ή ͳʹͳ ή ʹ ͲǤͶͷ ή ͳͲିଷ ʹ ሺ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ሻ ሺͳͳ ή ʹ ή ߨΤʹͺሻ ൌ ൌ ͲǤʹͷʹ ߟ௩ୀ଻ ൌ ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ͳͳ ή ʹ ή ߨΤʹͺ ‫ܮ‬௥ఙǡିହ ൌ ሺͳΤߟ௩ଶ െ ͳሻ ή ‫ܮ‬௥௛௩ ൅ ‫ܮ‬௥ఙ ൌ ሺͳΤሺͲǤʹͷʹሻଶ െ ͳሻ ή ͲǤͷʹͶ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ൌ ͺǤͻͻ ή ͳͲି଺ ‫ܪ‬ ܴ௥ ʹǤͳ͸ ൉ ͳͲିସ ‫ݏ‬௩ୀିଵଵǡ௕ ൌ േ ൌൌ േ ൌ േͲǤͲ͹ʹ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ʹ ή ߨ ή ͷͲ ή ሺͲǤͷʹͶ ή ͳͲି଺ ൅ ͺǤͻͻ ή ͳͲି଺ ሻ ͳ െ ‫ݏ‬௩ ‫ݏ‬௩ ൌ ͳ െ ‫ݒ‬ሺͳ െ ‫ݏ‬ሻ ֜ ‫ ݏ‬ൌ ͳ െ ‫ݒ‬ ͳ െ ‫ݏ‬௩ ͳ െ ͲǤͲ͹ʹ ‫ ݏ‬ൌ ͳെ ൌ ͳെ ൌ ͳǤͲͺͶ ‫ݒ‬ െͳͳ ͳ ൅ ͲǤͲ͹ʹ ͳ െ ‫ݏ‬௩ ൌ ͳെ ൌ ͳǤͲͻ͹ ‫ ݏ‬ൌ ͳെ െͳͳ ‫ݒ‬ ‫ݏ‬௩ୀିଵଵǡ௕ ൌ േ

For‫ ݒ‬ൌ ͳ͵:

ܴ௥ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ߬௣ ݈ி௘ ʹ݉௥ ଶ ‫ܮ‬௥௛௩ ൌ ߤ଴ ܰ௥ଶ ݇௪௥ ή ଶ ଶ ή ߨ ή‫ ݒ‬ή‫ߜ ݌‬ ଶ ʹ ή ʹͺ ͳ͵͵Ǥͷ ή ͳͲିଷ ή ʹͶͲ ή ͳͲିଷ ͳ ି଻ ή ൌ ͲǤ͵͹ͷ ൉ ͳͲି଺ ‫ܪ‬ ‫ܮ‬௥௛ǡଵଷ ൌ Ͷߨ ൉ ͳͲ ൬ ή ͳ൰ ή ଶ ߨ ή ͳ͸ͻ ή ʹ ͲǤͶͷ ή ͳͲିଷ ʹ ሺ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ሻ ሺͳ͵ ή ʹ ή ߨΤʹͺሻ ߟ௩ୀଵଷ ൌ ൌ ൌ ͲǤͲ͹͸ ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ͳ͵ ή ʹ ή ߨΤʹͺ ‫ܮ‬௥ఙǡଵଷ ൌ ሺͳΤߟ௩ଶ െ ͳሻ ή ‫ܮ‬௥௛௩ ൅ ‫ܮ‬௥ఙ ൌ ሺͳΤሺͲǤͲ͹͸ሻଶ െ ͳሻ ή ͲǤ͵͹ͷ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ൌ ͸Ǥͷͺͳ ή ͳͲିହ ‫ܪ‬ ܴ௥ ʹǤͳ͸ ൉ ͳͲିସ ‫ݏ‬௩ୀଵଷǡ௕ ൌ േ ൌൌ േ ൌ േͲǤͲͳ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ʹ ή ߨ ή ͷͲ ή ሺ͸Ǥͷͺͳ ή ͳͲି଺ ൅ ͲǤ͵͹ͷ ή ͳͲି଺ ሻ ͳ െ ‫ݏ‬௩ ‫ݏ‬௩ ൌ ͳ െ ‫ݒ‬ሺͳ െ ‫ݏ‬ሻ ֜ ‫ ݏ‬ൌ ͳ െ ‫ݒ‬ ‫ݏ‬௩௕ ൌ േ

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ͳ െ ‫ݏ‬௩ ͳ െ ͲǤͲͳ ൌ ͳെ ൌ ͲǤͻʹ͵ ‫ݒ‬ ͳ͵ ͳ ൅ ͲǤͲͳ ͳ െ ‫ݏ‬௩ ൌ ͳെ ൌ ͲǤͻʹ͵ ‫ ݏ‬ൌ ͳെ ͳ͵ ‫ݒ‬

‫ ݏ‬ൌ ͳെ

For‫ ݒ‬ൌ െͳ͹:

ܴ௥ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ߬௣ ݈ி௘ ʹ݉௥ ଶ ‫ܮ‬௥௛௩ ൌ ߤ଴ ܰ௥ଶ ݇௪௥ ή ଶ ଶ ή ߨ ή‫ ݒ‬ή‫ߜ ݌‬ ଶ ͳ ʹ ή ʹͺ ͳ͵͵Ǥͷ ή ͳͲିଷ ή ʹͶͲ ή ͳͲିଷ ή ൌ ͲǤʹͳͻ ൉ ͳͲି଺ ‫ܪ‬ ‫ܮ‬௥௛ǡିଵ଻ ൌ Ͷߨ ൉ ͳͲି଻ ൬ ή ͳ൰ ή ଶ ʹ ߨ ή ʹͺͻ ή ʹ ͲǤͶͷ ή ͳͲିଷ ሺ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ሻ ሺͳ͹ ή ʹ ή ߨΤʹͺሻ ൌ ൌ െͲǤͳ͸͵ ߟ௩ୀିଵ଻ ൌ ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ͳ͹ ή ʹ ή ߨΤʹͺ ‫ܮ‬௥ఙǡିଵ଻ ൌ ሺͳΤߟ௩ଶ െ ͳሻ ή ‫ܮ‬௥௛௩ ൅ ‫ܮ‬௥ఙ ൌ ሺͳΤሺͲǤͳ͸͵ሻଶ െ ͳሻ ή ͲǤʹͳͻ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ൌ ͻǤʹͻ ή ͳͲି଺ ‫ܪ‬ ܴ௥ ʹǤͳ͸ ൉ ͳͲିସ ‫ݏ‬௩ୀିଵ଻ǡ௕ ൌ േ ൌൌ േ ൌ േͲǤͲ͹ʹ ʹ ή ߨ ή ͷͲ ή ሺͻǤʹͻ ή ͳͲି଺ ൅ ͲǤʹͳͻ ή ͳͲି଺ ሻ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ͳ െ ‫ݏ‬௩ ‫ݏ‬௩ ൌ ͳ െ ‫ݒ‬ሺͳ െ ‫ݏ‬ሻ ֜ ‫ ݏ‬ൌ ͳ െ ‫ݒ‬ ͳ െ ͲǤͲ͹ʹ ͳ െ ‫ݏ‬௩ ‫ ݏ‬ൌ ͳെ ൌ ͳെ ൌ ͳǤͲͷͶ െͳ͹ ‫ݒ‬ ͳ െ ‫ݏ‬௩ ͳ ൅ ͲǤͲ͹ʹ ‫ ݏ‬ൌ ͳെ ൌ ͳെ ൌ ͳǤͲ͸͵ െͳ͹ ‫ݒ‬ ‫ݏ‬௩௕ ൌ േ

For‫ ݒ‬ൌ ͳͻ:

ܴ௥ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ߬௣ ݈ி௘ ʹ݉௥ ଶ ή ଶ ଶ ή ‫ܮ‬௥௛௩ ൌ ߤ଴ ܰ௥ଶ ݇௪௥ ߨ ή‫ ݒ‬ή‫ߜ ݌‬ ଶ ͳ ʹ ή ʹͺ ͳ͵͵Ǥͷ ή ͳͲିଷ ή ʹͶͲ ή ͳͲିଷ ‫ܮ‬௥௛ǡଵଽ ൌ Ͷߨ ൉ ͳͲି଻ ൬ ή ͳ൰ ή ଶ ή ൌ ͲǤͳ͹ͷ ൉ ͳͲି଺ ‫ܪ‬ ͲǤͶͷ ή ͳͲିଷ ߨ ή ͵͸ͳ ή ʹ ʹ ሺ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ሻ ሺͳͻ ή ʹ ή ߨΤʹͺሻ ൌ ൌ െͲǤʹͳͳ ߟ௩ୀଵଽ ൌ ͳͻ ή ʹ ή ߨΤʹͺ ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ‫ܮ‬௥ఙǡଵଽ ൌ ሺͳΤߟ௩ଶ െ ͳሻ ή ‫ܮ‬௥௛௩ ൅ ‫ܮ‬௥ఙ ൌ ሺͳΤሺͲǤʹͳͳሻଶ െ ͳሻ ή ͲǤͳ͹ͷ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ൌ ͷǤͲʹͷ ή ͳͲି଺ ‫ܪ‬ ܴ௥ ʹǤͳ͸ ൉ ͳͲିସ ‫ݏ‬௩ୀଵଽǡ௕ ൌ േ ൌൌ േ ൌ േͲǤͳ͵ʹ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ʹ ή ߨ ή ͷͲ ή ሺͲǤͳ͹ͷ ή ͳͲି଺ ൅ ͷǤͲʹͷ ή ͳͲି଺ ሻ ͳ െ ‫ݏ‬௩ ‫ݏ‬௩ ൌ ͳ െ ‫ݒ‬ሺͳ െ ‫ݏ‬ሻ ֜ ‫ ݏ‬ൌ ͳ െ ‫ݒ‬ ͳ െ ‫ݏ‬௩ ͳ െ ͲǤͳ͵ʹ ‫ ݏ‬ൌ ͳെ ൌ ͳെ ൌ ͲǤͻͷͶ ‫ݒ‬ ͳͻ ͳ ൅ ͲǤͳ͵ʹ ͳ െ ‫ݏ‬௩ ൌ ͳെ ൌ ͲǤͻͶ ‫ ݏ‬ൌ ͳെ ‫ݒ‬ ͳͻ ‫ݏ‬௩௕ ൌ േ

- 94 -

For‫ ݒ‬ൌ െʹ͵:

ܴ௥ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ߬௣ ݈ி௘ ʹ݉௥ ଶ ή ଶ ଶ ή ‫ܮ‬௥௛௩ ൌ ߤ଴ ܰ௥ଶ ݇௪௥ ߨ ή‫ ݒ‬ή‫ߜ ݌‬ ଶ ͳ ʹ ή ʹͺ ͳ͵͵Ǥͷ ή ͳͲିଷ ή ʹͶͲ ή ͳͲିଷ ି଻ ‫ܮ‬௥௛ǡିଶଷ ൌ Ͷߨ ൉ ͳͲ ൬ ή ͳ൰ ή ଶ ή ൌ ͲǤͳʹ ൉ ͳͲି଺ ‫ܪ‬ ߨ ή ͷʹͻ ή ʹ ͲǤͶͷ ή ͳͲିଷ ʹ ሺ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ሻ ሺʹ͵ ή ʹ ή ߨΤʹͺሻ ൌ ൌ െͲǤͳ͹Ͷ ߟ௩ୀିଶଷ ൌ ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ʹ͵ ή ʹ ή ߨΤʹͺ ‫ܮ‬௥ఙǡିଶଷ ൌ ሺͳΤߟ௩ଶ െ ͳሻ ή ‫ܮ‬௥௛௩ ൅ ‫ܮ‬௥ఙ ൌ ሺͳΤሺͲǤͳ͹Ͷሻଶ െ ͳሻ ή ͲǤͳ͹ͷ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ൌ ͷǤͳͳ ή ͳͲି଺ ‫ܪ‬ ܴ௥ ʹǤͳ͸ ൉ ͳͲିସ ൌൌ േ ൌ േͲǤͳ͵ͳͷ ‫ݏ‬௩ୀିଶଷǡ௕ ൌ േ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ʹ ή ߨ ή ͷͲ ή ሺͲǤͳʹ ή ͳͲି଺ ൅ ͷǤͳͳ ή ͳͲି଺ ሻ ͳ െ ‫ݏ‬௩ ‫ݏ‬௩ ൌ ͳ െ ‫ݒ‬ሺͳ െ ‫ݏ‬ሻ ֜ ‫ ݏ‬ൌ ͳ െ ‫ݒ‬ ͳ െ ͲǤͳ͵ͳͷ ͳ െ ‫ݏ‬௩ ൌ ͳെ ൌ ͳǤͲ͵͹ ‫ ݏ‬ൌ ͳെ െʹ͵ ‫ݒ‬ ͳ ൅ ͲǤͳ͵ͳͷ ͳ െ ‫ݏ‬௩ ൌ ͳെ ൌ ͳǤͲͶͻ ‫ ݏ‬ൌ ͳെ െʹ͵ ‫ݒ‬ ‫ݏ‬௩௕ ൌ േ

For‫ ݒ‬ൌ ʹͷ:

ܴ௥ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ߬௣ ݈ி௘ ʹ݉௥ ଶ ή ଶ ଶ ή ‫ܮ‬௥௛௩ ൌ ߤ଴ ܰ௥ଶ ݇௪௥ ߨ ή‫ ݒ‬ή‫ߜ ݌‬ ଶ ͳ ʹ ή ʹͺ ͳ͵͵Ǥͷ ή ͳͲିଷ ή ʹͶͲ ή ͳͲିଷ ି଻ ‫ܮ‬௥௛ǡଶହ ൌ Ͷߨ ൉ ͳͲ ൬ ή ͳ൰ ή ଶ ή ൌ ͲǤͳͲͳͷ ൉ ͳͲି଺ ‫ܪ‬ ߨ ή ͸ʹͷ ή ʹ ͲǤͶͷ ή ͳͲିଷ ʹ ሺ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ሻ ሺʹͷ ή ʹ ή ߨΤʹͺሻ ൌ ൌ െͲǤͳͳͳ ߟ௩ୀଶହ ൌ ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ʹͷ ή ʹ ή ߨΤʹͺ ‫ܮ‬௥ఙǡଶହ ൌ ሺͳΤߟ௩ଶ െ ͳሻ ή ‫ܮ‬௥௛௩ ൅ ‫ܮ‬௥ఙ ൌ ሺͳΤሺͲǤͳͳͳሻଶ െ ͳሻ ή ͲǤͳ͹ͷ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ൌ ͻǤͶ ή ͳͲି଺ ‫ܪ‬ ܴ௥ ʹǤͳ͸ ൉ ͳͲିସ ‫ݏ‬௩ୀଶହǡ௕ ൌ േ ൌൌ േ ൌ േͲǤͲ͹ʹ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ʹ ή ߨ ή ͷͲ ή ሺͲǤͳͲͳͷ ή ͳͲି଺ ൅ ͻǤͶ ή ͳͲି଺ ሻ ͳ െ ‫ݏ‬௩ ‫ݏ‬௩ ൌ ͳ െ ‫ݒ‬ሺͳ െ ‫ݏ‬ሻ ֜ ‫ ݏ‬ൌ ͳ െ ‫ݒ‬ ͳ െ ‫ݏ‬௩ ͳ െ ͲǤͲ͹ʹ ‫ ݏ‬ൌ ͳെ ൌ ͳെ ൌ ͲǤͻ͸ʹ ʹͷ ‫ݒ‬ ͳ ൅ ͲǤͲ͹ʹ ͳ െ ‫ݏ‬௩ ൌ ͳെ ൌ ͲǤͻͷͺ ‫ ݏ‬ൌ ͳെ ʹͷ ‫ݒ‬

‫ݏ‬௩௕ ൌ േ

For‫ ݒ‬ൌ െʹͻ: ‫ݏ‬௩௕ ൌ േ

ܴ௥ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ - 95 -

߬௣ ݈ி௘ ʹ݉௥ ή ଶ ή‫ ݒ‬ή‫ߜ ݌‬ ଶ ͳ ʹ ή ʹͺ ͳ͵͵Ǥͷ ή ͳͲିଷ ή ʹͶͲ ή ͳͲିଷ ‫ܮ‬௥௛ǡିଶଽ ൌ Ͷߨ ൉ ͳͲି଻ ൬ ή ͳ൰ ή ଶ ή ൌ ͲǤ͹ͷͶ ൉ ͳͲି଻ ‫ܪ‬ ʹ ͲǤͶͷ ή ͳͲିଷ ߨ ή ͺͶͳ ή ʹ ሺ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ሻ ሺʹͻ ή ʹ ή ߨΤʹͺሻ ൌ ൌ ͲǤͲ͵Ͷʹ ߟ௩ୀିଶଽ ൌ ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ʹͻ ή ʹ ή ߨΤʹͺ ‫ܮ‬௥ఙǡିଶଽ ൌ ሺͳΤߟ௩ଶ െ ͳሻ ή ‫ܮ‬௥௛௩ ൅ ‫ܮ‬௥ఙ ൌ ሺͳΤሺͲǤͲ͵Ͷʹሻଶ െ ͳሻ ή ͲǤ͹ͷͶ ή ͳͲି଻ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ൌ ͸Ǥͷ͸ ή ͳͲିହ ‫ܪ‬ ܴ௥ ʹǤͳ͸ ൉ ͳͲିସ ൌൌ േ ൌ േͲǤͲͳ ‫ݏ‬௩ୀିଶଽǡ௕ ൌ േ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ʹ ή ߨ ή ͷͲ ή ሺ͸Ǥͷ͹͵ ή ͳͲିହ ሻ ͳ െ ‫ݏ‬௩ ‫ݏ‬௩ ൌ ͳ െ ‫ݒ‬ሺͳ െ ‫ݏ‬ሻ ֜ ‫ ݏ‬ൌ ͳ െ ‫ݒ‬ ͳ െ ‫ݏ‬௩ ͳ െ ͲǤͲͳ ‫ ݏ‬ൌ ͳെ ൌ ͳെ ൌ ͳǤͲ͵Ͷ ‫ݒ‬ െʹͻ ͳ ൅ ͲǤͲͳ ͳ െ ‫ݏ‬௩ ൌ ͳെ ൌ ͳǤͲ͵Ͷͺ ‫ ݏ‬ൌ ͳെ െʹͻ ‫ݒ‬ ଶ ή ‫ܮ‬௥௛௩ ൌ ߤ଴ ܰ௥ଶ ݇௪௥

ߨଶ

All slips, in which asynchronous harmonic torques happen, and the related slips of the breakdown harmonic torques, in addition to selected slips for time stepping calculation of the VZIM1, are shown in Table V. c) Calculation of the slips, where an Asynchronous harmonic torque happens in VZIM2 Similar to VZIM1 all the calculations have been done for VZIM2. Stator ordinal numbers are derived as ‫ ݒ‬ൌ ͳ ൅ ʹ݉௦ ݃ ൌ ͳ ൅ ͸݃ ൌ ͳǡ െͷǡ͹ǡ െͳͳǡͳ͵ǡ െͳ͹ǡͳͻǡ െʹ͵ǡʹͷǡ െʹͻǡ͵ͳǡ െ͵ͷǡ͵͹Ǥ The asynchronous harmonic slips are derived as ͳ ‫ ݏ‬ൌ ͳ െ Ǥ ‫ݒ‬ Table III: Stator ordinal numbers and related asynchronous harmonic torque slips -11 13 -17 19 -23 25 -29 31 -35 37 ‫ ݒ‬1 -5 7 ‫ ݏ‬0 1.2 0.857 1.09 0.92 1.0588 0.947 1.043 0.96 1.034 0.96 1.028 0.972 Harmonic break down slips are calculated for each asynchronous harmonic torque, according to [2] as ܴ௥ ǡ ‫ݏ‬௩௕ ൌ േ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ where ‫ܮ‬௥ఙ௩ is calculated as ‫ܮ‬௥ఙ௩ ൌ ሺͳΤߟ௩ଶ െ ͳሻ ή ‫ܮ‬௥௛௩ ൅ ‫ܮ‬௥ఙ , For all ordinal numbers the breakdown slips are calculated as following: ‫ ݒ‬ൌ െͷ: ‫ݏ‬௩௕ ൌ േ

ܴ௥ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ - 96 -

߬௣ ݈ி௘ ʹ݉௥ ή ଶ ή‫ ݒ‬ή‫ߜ ݌‬ ଶ ͳ ʹ ή ʹͺ ͳ͵͵Ǥͷ ή ͳͲିଷ ή ʹͶͲ ή ͳͲିଷ ‫ܮ‬௥௛ǡିହ ൌ Ͷߨ ൉ ͳͲି଻ ൬ ή ͳ൰ ή ଶ ή ൌ ʹǤͷ͵ ൉ ͳͲି଺ ‫ܪ‬ ʹ ͲǤͶͷ ή ͳͲିଷ ߨ ή ʹͷ ή ʹ ሺ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ሻ ሺെͷ ή ʹ ή ߨΤʹͺሻ ൌ ൌ ͲǤͺͲ͵ ߟ௩ୀିହ ൌ ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ െͷ ή ʹ ή ߨΤʹͺ ‫ܮ‬௥ఙǡିହ ൌ ሺͳΤߟ௩ଶ െ ͳሻ ή ‫ܮ‬௥௛௩ ൅ ‫ܮ‬௥ఙ ൌ ሺͳΤሺͲǤͺͲ͵ሻଶ െ ͳሻ ή ʹǤͷ͵ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ൌ ʹǤ͸͸ ή ͳͲି଺ ‫ܪ‬ ܴ௥ ͻǤʹͳ ൉ ͳͲିହ ൌൌ േ ൌ േͲǤͲͷ͸ͷ ‫ݏ‬௩ୀିହǡ௕ ൌ േ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ʹ ή ߨ ή ͷͲ ή ሺʹǤͷ͵ ή ͳͲି଺ ൅ ʹǤ͸͸ ή ͳͲି଺ ሻ ͳ െ ‫ݏ‬௩ ‫ݏ‬௩ ൌ ͳ െ ‫ݒ‬ሺͳ െ ‫ݏ‬ሻ ֜ ‫ ݏ‬ൌ ͳ െ ‫ݒ‬ ͳ െ ͲǤͲͷ͸ͷ ͳ െ ‫ݏ‬௩ ൌ ͳെ ൌ ͳǤͳͺͺ͹ ‫ ݏ‬ൌ ͳെ െͷ ‫ݒ‬ ͳ ൅ ͲǤͲͷ͸ͷ ͳ െ ‫ݏ‬௩ ൌ ͳെ ൌ ͳǤʹͳͳ ‫ ݏ‬ൌ ͳെ െͷ ‫ݒ‬ ଶ ή ‫ܮ‬௥௛௩ ൌ ߤ଴ ܰ௥ଶ ݇௪௥

ߨଶ

‫ ݒ‬ൌ ͹:

ܴ௥ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ߬௣ ݈ி௘ ʹ݉௥ ଶ ή ‫ܮ‬௥௛௩ ൌ ߤ଴ ܰ௥ଶ ݇௪௥ ή ଶ ଶ ߨ ή‫ ݒ‬ή‫ߜ ݌‬ ଶ ͳ ʹ ή ʹͺ ͳ͵͵Ǥͷ ή ͳͲିଷ ή ʹͶͲ ή ͳͲିଷ ‫ܮ‬௥௛ǡ଻ ൌ Ͷߨ ൉ ͳͲି଻ ൬ ή ͳ൰ ή ଶ ή ൌ ʹǤʹͻͷ ൉ ͳͲି଺ ‫ܪ‬ ʹ ߨ ή Ͷͻ ή ʹ ͲǤͶͷ ή ͳͲିଷ ሺ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ሻ ሺ͹ ή ʹ ή ߨΤʹͺሻ ߟ௩ୀ଻ ൌ ൌ ൌ ͲǤ͸͵͸ ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ͹ ή ʹ ή ߨΤʹͺ ‫ܮ‬௥ఙǡିହ ൌ ሺͳΤߟ௩ଶ െ ͳሻ ή ‫ܮ‬௥௛௩ ൅ ‫ܮ‬௥ఙ ൌ ሺͳΤሺͲǤ͸͵͸ሻଶ െ ͳሻ ή ʹǤʹͻͷ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ൌ ͶǤ͸Ͷ ή ͳͲି଺ ‫ܪ‬ ܴ௥ ͻǤʹͳ ൉ ͳͲିହ ൌൌ േ ൌ േͲǤͲͶ͵͸ ‫ݏ‬௩ୀ଻ǡ௕ ൌ േ ʹ ή ߨ ή ͷͲ ή ሺͶǤ͸Ͷ ή ͳͲି଺ ൅ ʹǤʹͻͷ ή ͳͲି଺ ሻ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ͳ െ ‫ݏ‬௩ ‫ݏ‬௩ ൌ ͳ െ ‫ݒ‬ሺͳ െ ‫ݏ‬ሻ ֜ ‫ ݏ‬ൌ ͳ െ ‫ݒ‬ ͳ െ ‫ݏ‬௩ ͳ െ ͲǤͲͶ͵͸ ‫ ݏ‬ൌ ͳെ ൌ ͳെ ൌ ͲǤͺ͸͵͵ ‫ݒ‬ ͹ ͳ ൅ ͲǤͲͶ͵͸ ͳ െ ‫ݏ‬௩ ൌ ͳെ ൌ ͲǤͺͷ ‫ ݏ‬ൌ ͳെ ͹ ‫ݒ‬ ‫ݏ‬௩௕ ൌ േ

‫ ݒ‬ൌ െͳͳ:

ܴ௥ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ߬௣ ݈ி௘ ʹ݉௥ ଶ ‫ܮ‬௥௛௩ ൌ ߤ଴ ܰ௥ଶ ݇௪௥ ή ଶ ଶ ή ߨ ή‫ ݒ‬ή‫ߜ ݌‬ ଶ ͳ ʹ ή ʹͺ ͳ͵͵Ǥͷ ή ͳͲିଷ ή ʹͶͲ ή ͳͲିଷ ି଻ ή ൌ ͲǤͷʹͶ ൉ ͳͲି଺ ‫ܪ‬ ‫ܮ‬௥௛ǡ଻ ൌ Ͷߨ ൉ ͳͲ ൬ ή ͳ൰ ή ଶ ͲǤͶͷ ή ͳͲିଷ ʹ ߨ ή ͳʹͳ ή ʹ

‫ݏ‬௩ୀିଵଵǡ௕ ൌ േ

- 97 -

ሺ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ሻ ሺͳͳ ή ʹ ή ߨΤʹͺሻ ൌ ൌ ͲǤʹͷʹ ͳͳ ή ʹ ή ߨΤʹͺ ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ‫ܮ‬௥ఙǡିହ ൌ ሺͳΤߟ௩ଶ െ ͳሻ ή ‫ܮ‬௥௛௩ ൅ ‫ܮ‬௥ఙ ൌ ሺͳΤሺͲǤʹͷʹሻଶ െ ͳሻ ή ͲǤͷʹͶ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ൌ ͺǤͻͻ ή ͳͲି଺ ‫ܪ‬ ܴ௥ ʹǤͳ͸ ൉ ͳͲିସ ൌൌ േ ൌ േͲǤͲ͵ʹ ‫ݏ‬௩ୀିଵଵǡ௕ ൌ േ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ʹ ή ߨ ή ͷͲ ή ሺͲǤͷʹͶ ή ͳͲି଺ ൅ ͺǤͻͻ ή ͳͲି଺ ሻ ͳ െ ‫ݏ‬௩ ‫ݏ‬௩ ൌ ͳ െ ‫ݒ‬ሺͳ െ ‫ݏ‬ሻ ֜ ‫ ݏ‬ൌ ͳ െ ‫ݒ‬ ͳ െ ‫ݏ‬௩ ͳ െ ͲǤͲ͵ʹ ‫ ݏ‬ൌ ͳെ ൌ ͳെ ൌ ͳǤͲͺͺ ‫ݒ‬ െͳͳ ͳ െ ‫ݏ‬௩ ͳ ൅ ͲǤͲ͵ʹ ‫ ݏ‬ൌ ͳെ ൌ ͳെ ൌ ͳǤͲͻ͵ ‫ݒ‬ െͳͳ ߟ௩ୀ଻ ൌ

‫ ݒ‬ൌ ͳ͵:

ܴ௥ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ߬௣ ݈ி௘ ʹ݉௥ ଶ ή ଶ ଶ ή ‫ܮ‬௥௛௩ ൌ ߤ଴ ܰ௥ଶ ݇௪௥ ߨ ή‫ ݒ‬ή‫ߜ ݌‬ ଶ ͳ ʹ ή ʹͺ ͳ͵͵Ǥͷ ή ͳͲିଷ ή ʹͶͲ ή ͳͲିଷ ή ൌ ͲǤ͵͹ͷ ൉ ͳͲି଺ ‫ܪ‬ ‫ܮ‬௥௛ǡଵଷ ൌ Ͷߨ ൉ ͳͲି଻ ൬ ή ͳ൰ ή ଶ ߨ ή ͳ͸ͻ ή ʹ ͲǤͶͷ ή ͳͲିଷ ʹ ሺ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ሻ ሺͳ͵ ή ʹ ή ߨΤʹͺሻ ߟ௩ୀଵଷ ൌ ൌ ൌ ͲǤͲ͹͸ ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ͳ͵ ή ʹ ή ߨΤʹͺ ‫ܮ‬௥ఙǡଵଷ ൌ ሺͳΤߟ௩ଶ െ ͳሻ ή ‫ܮ‬௥௛௩ ൅ ‫ܮ‬௥ఙ ൌ ሺͳΤሺͲǤͲ͹͸ሻଶ െ ͳሻ ή ͲǤ͵͹ͷ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ൌ ͸Ǥͷͺͳ ή ͳͲିହ ‫ܪ‬ ܴ௥ ͻǤʹͳ ൉ ͳͲିହ ‫ݏ‬௩ୀଵଷǡ௕ ൌ േ ൌൌ േ ൌ േͲǤͲͲͶͶ͵ʹ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ʹ ή ߨ ή ͷͲ ή ሺ͸Ǥͷͺͳ ή ͳͲି଺ ൅ ͲǤ͵͹ͷ ή ͳͲି଺ ሻ ͳ െ ‫ݏ‬௩ ‫ݏ‬௩ ൌ ͳ െ ‫ݒ‬ሺͳ െ ‫ݏ‬ሻ ֜ ‫ ݏ‬ൌ ͳ െ ‫ݒ‬ ͳ െ ͲǤͲͲͶͶ͵ʹ ͳ െ ‫ݏ‬௩ ൌ ͳെ ൌ ͲǤͻʹ͵Ͷ ‫ ݏ‬ൌ ͳെ ͳ͵ ‫ݒ‬ ͳ െ ‫ݏ‬௩ ͳ ൅ ͲǤͲͲͶͶ͵ʹ ‫ ݏ‬ൌ ͳെ ൌ ͳെ ൌ ͲǤͻʹʹ͹ ͳ͵ ‫ݒ‬ ‫ݏ‬௩௕ ൌ േ

‫ ݒ‬ൌ െͳ͹:

ܴ௥ ሺ‫ܮ‬ ߱௦ ௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ߬௣ ݈ி௘ ʹ݉௥ ଶ ή ଶ ଶ ‫ܮ‬௥௛௩ ൌ ߤ଴ ܰ௥ଶ ݇௪௥ ή ߨ ή‫ ݒ‬ή‫ߜ ݌‬ ଶ ͳ ʹ ή ʹͺ ͳ͵͵Ǥͷ ή ͳͲିଷ ή ʹͶͲ ή ͳͲିଷ ‫ܮ‬௥௛ǡିଵ଻ ൌ Ͷߨ ൉ ͳͲି଻ ൬ ή ͳ൰ ή ଶ ή ൌ ͲǤʹͳͻ ൉ ͳͲି଺ ‫ܪ‬ ͲǤͶͷ ή ͳͲିଷ ʹ ߨ ή ʹͺͻ ή ʹ ሺ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ሻ ሺͳ͹ ή ʹ ή ߨΤʹͺሻ ൌ ൌ െͲǤͳ͸͵ ߟ௩ୀିଵ଻ ൌ ͳ͹ ή ʹ ή ߨΤʹͺ ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ‫ܮ‬௥ఙǡିଵ଻ ൌ ሺͳΤߟ௩ଶ െ ͳሻ ή ‫ܮ‬௥௛௩ ൅ ‫ܮ‬௥ఙ ൌ ሺͳΤሺͲǤͳ͸͵ሻଶ െ ͳሻ ή ͲǤʹͳͻ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ൌ ͻǤʹͻ ή ͳͲି଺ ‫ܪ‬

‫ݏ‬௩௕ ൌ േ

- 98 -

ܴ௥ ͻǤʹͳ ൉ ͳͲିହ ‫ݏ‬௩ୀିଵ଻ǡ௕ ൌ േ ൌൌ േ ൌ േͲǤͲ͵Ͳͺ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ʹ ή ߨ ή ͷͲ ή ሺͻǤʹͻ ή ͳͲି଺ ൅ ͲǤʹͳͻ ή ͳͲି଺ ሻ ͳ െ ‫ݏ‬௩ ‫ݏ‬௩ ൌ ͳ െ ‫ݒ‬ሺͳ െ ‫ݏ‬ሻ ֜ ‫ ݏ‬ൌ ͳ െ ‫ݒ‬ ͳ െ ‫ݏ‬௩ ͳ െ ͲǤͲ͵Ͳͺ ൌ ͳെ ൌ ͳǤͲͷ͹ ‫ ݏ‬ൌ ͳെ െͳ͹ ‫ݒ‬ ͳ ൅ ͲǤͲ͵Ͳͺ ͳ െ ‫ݏ‬௩ ൌ ͳെ ൌ ͳǤͲ͸Ͳ͸ ‫ ݏ‬ൌ ͳെ െͳ͹ ‫ݒ‬ ‫ ݒ‬ൌ ͳͻ:

ܴ௥ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ߬௣ ݈ி௘ ʹ݉௥ ଶ ή ଶ ଶ ή ‫ܮ‬௥௛௩ ൌ ߤ଴ ܰ௥ଶ ݇௪௥ ߨ ή‫ ݒ‬ή‫ߜ ݌‬ ଶ ͳ ʹ ή ʹͺ ͳ͵͵Ǥͷ ή ͳͲିଷ ή ʹͶͲ ή ͳͲିଷ ି଻ ‫ܮ‬௥௛ǡଵଽ ൌ Ͷߨ ൉ ͳͲ ൬ ή ͳ൰ ή ଶ ή ൌ ͲǤͳ͹ͷ ൉ ͳͲି଺ ‫ܪ‬ ͲǤͶͷ ή ͳͲିଷ ߨ ή ͵͸ͳ ή ʹ ʹ ሺ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ሻ ሺͳͻ ή ʹ ή ߨΤʹͺሻ ߟ௩ୀଵଽ ൌ ൌ ൌ െͲǤʹͳͳ ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ͳͻ ή ʹ ή ߨΤʹͺ ‫ܮ‬௥ఙǡଵଽ ൌ ሺͳΤߟ௩ଶ െ ͳሻ ή ‫ܮ‬௥௛௩ ൅ ‫ܮ‬௥ఙ ൌ ሺͳΤሺͲǤʹͳͳሻଶ െ ͳሻ ή ͲǤͳ͹ͷ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ൌ ͷǤͲʹͷ ή ͳͲି଺ ‫ܪ‬ ܴ௥ ͻǤʹͳ ൉ ͳͲିହ ൌൌ േ ൌ േͲǤͲͷ͸Ͷ ‫ݏ‬௩ୀଵଽǡ௕ ൌ േ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ʹ ή ߨ ή ͷͲ ή ሺͲǤͳ͹ͷ ή ͳͲି଺ ൅ ͷǤͲʹͷ ή ͳͲି଺ ሻ ͳ െ ‫ݏ‬௩ ‫ݏ‬௩ ൌ ͳ െ ‫ݒ‬ሺͳ െ ‫ݏ‬ሻ ֜ ‫ ݏ‬ൌ ͳ െ ‫ݒ‬ ͳ െ ͲǤͲͷ͸Ͷ ͳ െ ‫ݏ‬௩ ൌ ͳെ ൌ ͲǤͻͷ ‫ ݏ‬ൌ ͳെ ͳͻ ‫ݒ‬ ͳ െ ‫ݏ‬௩ ͳ ൅ ͲǤͲͷ͸Ͷ ‫ ݏ‬ൌ ͳെ ൌ ͳെ ൌ ͲǤͻͶͶ ‫ݒ‬ ͳͻ ‫ݏ‬௩௕ ൌ േ

‫ ݒ‬ൌ െʹ͵:

ܴ௥ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ߬௣ ݈ி௘ ʹ݉௥ ଶ ή ଶ ଶ ‫ܮ‬௥௛௩ ൌ ߤ଴ ܰ௥ଶ ݇௪௥ ή ߨ ή‫ ݒ‬ή‫ߜ ݌‬ ଶ ͳ ʹ ή ʹͺ ͳ͵͵Ǥͷ ή ͳͲିଷ ή ʹͶͲ ή ͳͲିଷ ି଻ ‫ܮ‬௥௛ǡିଶଷ ൌ Ͷߨ ൉ ͳͲ ൬ ή ͳ൰ ή ଶ ή ൌ ͲǤͳʹ ൉ ͳͲି଺ ‫ܪ‬ ͲǤͶͷ ή ͳͲିଷ ߨ ή ͷʹͻ ή ʹ ʹ ሺ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ሻ ሺʹ͵ ή ʹ ή ߨΤʹͺሻ ൌ ൌ െͲǤͳ͹Ͷ ߟ௩ୀିଶଷ ൌ ʹ͵ ή ʹ ή ߨΤʹͺ ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ‫ܮ‬௥ఙǡିଶଷ ൌ ሺͳΤߟ௩ଶ െ ͳሻ ή ‫ܮ‬௥௛௩ ൅ ‫ܮ‬௥ఙ ൌ ሺͳΤሺͲǤͳ͹Ͷሻଶ െ ͳሻ ή ͲǤͳ͹ͷ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ൌ ͷǤͳͳ ή ͳͲି଺ ‫ܪ‬ ܴ௥ ͻǤʹͳ ൉ ͳͲିହ ‫ݏ‬௩ୀିଶଷǡ௕ ൌ േ ൌൌ േ ൌ േͲǤͲͷ͸ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ʹ ή ߨ ή ͷͲ ή ሺͲǤͳʹ ή ͳͲି଺ ൅ ͷǤͳͳ ή ͳͲି଺ ሻ ͳ െ ‫ݏ‬௩ ‫ݏ‬௩ ൌ ͳ െ ‫ݒ‬ሺͳ െ ‫ݏ‬ሻ ֜ ‫ ݏ‬ൌ ͳ െ ‫ݒ‬

‫ݏ‬௩௕ ൌ േ

- 99 -

ͳ െ ͲǤͲǤͲͷ͸ ͳ െ ‫ݏ‬௩ ൌ ͳെ ൌ ͳǤͲͶͳ െʹ͵ ‫ݒ‬ ͳ ൅ ͲǤͲͷ͸ ͳ െ ‫ݏ‬௩ ൌ ͳെ ൌ ͳǤͲͶͷ ‫ ݏ‬ൌ ͳെ െʹ͵ ‫ݒ‬ ‫ ݏ‬ൌ ͳെ

‫ ݒ‬ൌ ʹͷ:

ܴ௥ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ߬௣ ݈ி௘ ʹ݉௥ ଶ ή ଶ ଶ ή ‫ܮ‬௥௛௩ ൌ ߤ଴ ܰ௥ଶ ݇௪௥ ߨ ή‫ ݒ‬ή‫ߜ ݌‬ ଶ ʹ ή ʹͺ ͳ͵͵Ǥͷ ή ͳͲିଷ ή ʹͶͲ ή ͳͲିଷ ͳ ή ൌ ͲǤͳͲͳͷ ൉ ͳͲି଺ ‫ܪ‬ ‫ܮ‬௥௛ǡଶହ ൌ Ͷߨ ൉ ͳͲି଻ ൬ ή ͳ൰ ή ଶ ߨ ή ͸ʹͷ ή ʹ ͲǤͶͷ ή ͳͲିଷ ʹ ሺ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ሻ ሺʹͷ ή ʹ ή ߨΤʹͺሻ ߟ௩ୀଶହ ൌ ൌ ൌ െͲǤͳͳͳ ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ʹͷ ή ʹ ή ߨΤʹͺ ‫ܮ‬௥ఙǡଶହ ൌ ሺͳΤߟ௩ଶ െ ͳሻ ή ‫ܮ‬௥௛௩ ൅ ‫ܮ‬௥ఙ ൌ ሺͳΤሺͲǤͳͳͳሻଶ െ ͳሻ ή ͲǤͳ͹ͷ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ൌ ͻǤͶ ή ͳͲି଺ ‫ܪ‬ ܴ௥ ͻǤʹͳ ൉ ͳͲିହ ൌൌ േ ൌ േͲǤͲ͵Ͳͺ ‫ݏ‬௩ୀଶହǡ௕ ൌ േ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ʹ ή ߨ ή ͷͲ ή ሺͲǤͳͲͳͷ ή ͳͲି଺ ൅ ͻǤͶ ή ͳͲି଺ ሻ ͳ െ ‫ݏ‬௩ ‫ݏ‬௩ ൌ ͳ െ ‫ݒ‬ሺͳ െ ‫ݏ‬ሻ ֜ ‫ ݏ‬ൌ ͳ െ ‫ݒ‬ ͳ െ ‫ݏ‬௩ ͳ െ ͲǤͲ͵Ͳͺ ‫ ݏ‬ൌ ͳെ ൌ ͳെ ൌ ͲǤͻ͸ͳ ʹͷ ‫ݒ‬ ͳ ൅ ͲǤͲ͵Ͳͺ ͳ െ ‫ݏ‬௩ ൌ ͳെ ൌ ͲǤͻͷͺ͹ ‫ ݏ‬ൌ ͳെ ʹͷ ‫ݒ‬ ‫ݏ‬௩௕ ൌ േ

‫ ݒ‬ൌ െʹͻ:

ܴ௥ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ߬௣ ݈ி௘ ʹ݉௥ ଶ ‫ܮ‬௥௛௩ ൌ ߤ଴ ܰ௥ଶ ݇௪௥ ή ଶ ଶ ή ߨ ή‫ ݒ‬ή‫ߜ ݌‬ ଶ ͳ ʹ ή ʹͺ ͳ͵͵Ǥͷ ή ͳͲିଷ ή ʹͶͲ ή ͳͲିଷ ‫ܮ‬௥௛ǡିଶଽ ൌ Ͷߨ ൉ ͳͲି଻ ൬ ή ͳ൰ ή ଶ ή ൌ ͲǤ͹ͷͶ ൉ ͳͲି଻ ‫ܪ‬ ߨ ή ͺͶͳ ή ʹ ͲǤͶͷ ή ͳͲିଷ ʹ ሺ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ሻ ሺʹͻ ή ʹ ή ߨΤʹͺሻ ߟ௩ୀିଶଽ ൌ ൌ ൌ ͲǤͲ͵Ͷʹ ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ʹͻ ή ʹ ή ߨΤʹͺ ‫ܮ‬௥ఙǡିଶଽ ൌ ሺͳΤߟ௩ଶ െ ͳሻ ή ‫ܮ‬௥௛௩ ൅ ‫ܮ‬௥ఙ ൌ ሺͳΤሺͲǤͲ͵Ͷʹሻଶ െ ͳሻ ή ͲǤ͹ͷͶ ή ͳͲି଻ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ൌ ͸Ǥͷ͸ ή ͳͲିହ ‫ܪ‬ ͻǤʹͳ ൉ ͳͲିହ ܴ௥ ൌൌ േ ൌ േͲǤͲͲͶͶ͸ ‫ݏ‬௩ୀିଶଽǡ௕ ൌ േ ʹ ή ߨ ή ͷͲ ή ሺ͸Ǥͷ͹͵ ή ͳͲିହ ሻ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ͳ െ ‫ݏ‬௩ ‫ݏ‬௩ ൌ ͳ െ ‫ݒ‬ሺͳ െ ‫ݏ‬ሻ ֜ ‫ ݏ‬ൌ ͳ െ ‫ݒ‬ ͳ െ ‫ݏ‬௩ ͳ െ ͲǤͲͲͶͶ͸ ‫ ݏ‬ൌ ͳെ ൌ ͳെ ൌ ͳǤͲ͵Ͷ͵ʹ െʹͻ ‫ݒ‬ ͳ െ ‫ݏ‬௩ ͳ ൅ ͲǤͲͲͶͶ͸ ‫ ݏ‬ൌ ͳെ ൌ ͳെ ൌ ͳǤͲ͵Ͷ͸͵ ‫ݒ‬ െʹͻ ‫ݏ‬௩௕ ൌ േ

- 100 -

All slips, in which asynchronous harmonic torques happen, and the related slips of the breakdown harmonic torques, in addition to selected slips for time stepping calculation of the VZIM2 are shown in Table VI. d) Calculation of the slips where an Asynchronous harmonic torque happens in VZIM3 Similar to VZIM1and VZIM2 all the calculations have been done for VZIM3. Stator ordinal numbers are derived as ‫ ݒ‬ൌ ͳ ൅ ʹ݉௦ ݃ ൌ ͳ ൅ ͸݃ ൌ ͳǡ െͷǡ͹ǡ െͳͳǡͳ͵ǡ െͳ͹ǡͳͻǡ െʹ͵ǡʹͷǡ െʹͻǡ͵ͳǡ െ͵ͷǡ͵͹Ǥ The asynchronous harmonic slips are derived as, ͳ ‫ ݏ‬ൌ ͳ െ Ǥ ‫ݒ‬ Table IV: Stator ordinal numbers and related asynchronous harmonic torque slips -11 13 -17 19 -23 25 -29 31 -35 37 ‫ ݒ‬1 -5 7 ‫ ݏ‬0 1.2 0.857 1.09 0.92 1.0588 0.947 1.043 0.96 1.034 0.96 1.028 0.972 Harmonic break down slips are calculated for each asynchronous harmonic torque, according to [2] as ܴ௥ ǡ ‫ݏ‬௩௕ ൌ േ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ where ‫ܮ‬௥ఙ௩ is calculated as ‫ܮ‬௥ఙ௩ ൌ ሺͳΤߟ௩ଶ െ ͳሻ ή ‫ܮ‬௥௛௩ ൅ ‫ܮ‬௥ఙ . For all ordinal numbers the breakdown slips are calculated as following: ‫ ݒ‬ൌ െͷ:

ܴ௥ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ߬௣ ݈ி௘ ʹ݉௥ ଶ ή ଶ ଶ ή ‫ܮ‬௥௛௩ ൌ ߤ଴ ܰ௥ଶ ݇௪௥ ߨ ή‫ ݒ‬ή‫ߜ ݌‬ ଶ ͳ ʹ ή ʹͺ ͳ͵͵Ǥͷ ή ͳͲିଷ ή ʹͶͲ ή ͳͲିଷ ି଻ ‫ܮ‬௥௛ǡିହ ൌ Ͷߨ ൉ ͳͲ ൬ ή ͳ൰ ή ଶ ή ൌ ʹǤͷ͵ ൉ ͳͲି଺ ‫ܪ‬ ͲǤͶͷ ή ͳͲିଷ ߨ ή ʹͷ ή ʹ ʹ ሺ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ሻ ሺെͷ ή ʹ ή ߨΤʹͺሻ ൌ ൌ ͲǤͺͲ͵ ߟ௩ୀିହ ൌ ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ െͷ ή ʹ ή ߨΤʹͺ ‫ܮ‬௥ఙǡିହ ൌ ሺͳΤߟ௩ଶ െ ͳሻ ή ‫ܮ‬௥௛௩ ൅ ‫ܮ‬௥ఙ ൌ ሺͳΤሺͲǤͺͲ͵ሻଶ െ ͳሻ ή ʹǤͷ͵ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ൌ ʹǤ͸͸ ή ͳͲି଺ ‫ܪ‬ ܴ௥ ͵ǤͺͶ ൉ ͳͲିହ ‫ݏ‬௩ୀିହǡ௕ ൌ േ ൌൌ േ ൌ േͲǤͲʹ͵ͷ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ʹ ή ߨ ή ͷͲ ή ሺʹǤͷ͵ ή ͳͲି଺ ൅ ʹǤ͸͸ ή ͳͲି଺ ሻ ͳ െ ‫ݏ‬௩ ‫ݏ‬௩ ൌ ͳ െ ‫ݒ‬ሺͳ െ ‫ݏ‬ሻ ֜ ‫ ݏ‬ൌ ͳ െ ‫ݒ‬ ͳ െ ‫ݏ‬௩ ͳ െ ͲǤͲʹ͵ͷ ‫ ݏ‬ൌ ͳെ ൌ ͳെ ൌ ͳǤͳͻͷ ‫ݒ‬ െͷ ͳ ൅ ͲǤͲʹ͵ͷ ͳ െ ‫ݏ‬௩ ൌ ͳെ ൌ ͳǤʹͲͻ ‫ ݏ‬ൌ ͳെ ‫ݒ‬ െͷ ‫ݏ‬௩௕ ൌ േ

- 101 -

‫ ݒ‬ൌ ͹:

ܴ௥ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ߬௣ ݈ி௘ ʹ݉௥ ଶ ή ଶ ଶ ή ‫ܮ‬௥௛௩ ൌ ߤ଴ ܰ௥ଶ ݇௪௥ ߨ ή‫ ݒ‬ή‫ߜ ݌‬ ଶ ͳ ʹ ή ʹͺ ͳ͵͵Ǥͷ ή ͳͲିଷ ή ʹͶͲ ή ͳͲିଷ ି଻ ‫ܮ‬௥௛ǡ଻ ൌ Ͷߨ ൉ ͳͲ ൬ ή ͳ൰ ή ଶ ή ൌ ʹǤʹͻͷ ൉ ͳͲି଺ ‫ܪ‬ ߨ ή Ͷͻ ή ʹ ͲǤͶͷ ή ͳͲିଷ ʹ ሺ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ሻ ሺ͹ ή ʹ ή ߨΤʹͺሻ ൌ ൌ ͲǤ͸͵͸ ߟ௩ୀ଻ ൌ ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ͹ ή ʹ ή ߨΤʹͺ ‫ܮ‬௥ఙǡିହ ൌ ሺͳΤߟ௩ଶ െ ͳሻ ή ‫ܮ‬௥௛௩ ൅ ‫ܮ‬௥ఙ ൌ ሺͳΤሺͲǤ͸͵͸ሻଶ െ ͳሻ ή ʹǤʹͻͷ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ൌ ͶǤ͸Ͷ ή ͳͲି଺ ‫ܪ‬ ܴ௥ ͵ǤͺͶ ൉ ͳͲିହ ൌൌ േ ൌ േͲǤͲͳ͹͹ ‫ݏ‬௩ୀ଻ǡ௕ ൌ േ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ʹ ή ߨ ή ͷͲ ή ሺͶǤ͸Ͷ ή ͳͲି଺ ൅ ʹǤʹͻͷ ή ͳͲି଺ ሻ ͳ െ ‫ݏ‬௩ ‫ݏ‬௩ ൌ ͳ െ ‫ݒ‬ሺͳ െ ‫ݏ‬ሻ ֜ ‫ ݏ‬ൌ ͳ െ ‫ݒ‬ ͳ െ ͲǤͲͳ͹͹ ͳ െ ‫ݏ‬௩ ൌ ͳെ ൌ ͲǤͺͷͻ ‫ ݏ‬ൌ ͳെ ͹ ‫ݒ‬ ͳ െ ‫ݏ‬௩ ͳ ൅ ͲǤͲͳ͹͹ ‫ ݏ‬ൌ ͳെ ൌ ͳെ ൌ ͲǤͺͷͷ ͹ ‫ݒ‬ ‫ݏ‬௩௕ ൌ േ

‫ ݒ‬ൌ െͳͳ:

ܴ௥ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ߬௣ ݈ி௘ ʹ݉௥ ଶ ‫ܮ‬௥௛௩ ൌ ߤ଴ ܰ௥ଶ ݇௪௥ ή ଶ ଶ ή ߨ ή‫ ݒ‬ή‫ߜ ݌‬ ଶ ͳ ʹ ή ʹͺ ͳ͵͵Ǥͷ ή ͳͲିଷ ή ʹͶͲ ή ͳͲିଷ ି଻ ‫ܮ‬௥௛ǡ଻ ൌ Ͷߨ ൉ ͳͲ ൬ ή ͳ൰ ή ଶ ή ൌ ͲǤͷʹͶ ൉ ͳͲି଺ ‫ܪ‬ ʹ ߨ ή ͳʹͳ ή ʹ ͲǤͶͷ ή ͳͲିଷ ሺ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ሻ ሺͳͳ ή ʹ ή ߨΤʹͺሻ ൌ ൌ ͲǤʹͷʹ ߟ௩ୀ଻ ൌ ͳͳ ή ʹ ή ߨΤʹͺ ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ‫ܮ‬௥ఙǡିହ ൌ ሺͳΤߟ௩ଶ െ ͳሻ ή ‫ܮ‬௥௛௩ ൅ ‫ܮ‬௥ఙ ൌ ሺͳΤሺͲǤʹͷʹሻଶ െ ͳሻ ή ͲǤͷʹͶ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ൌ ͺǤͻͻ ή ͳͲି଺ ‫ܪ‬ ܴ௥ ͵ǤͺͶ ൉ ͳͲିହ ൌൌ േ ൌ േͲǤͲͳʹͻ ‫ݏ‬௩ୀିଵଵǡ௕ ൌ േ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ʹ ή ߨ ή ͷͲ ή ሺͲǤͷʹͶ ή ͳͲି଺ ൅ ͺǤͻͻ ή ͳͲି଺ ሻ ͳ െ ‫ݏ‬௩ ‫ݏ‬௩ ൌ ͳ െ ‫ݒ‬ሺͳ െ ‫ݏ‬ሻ ֜ ‫ ݏ‬ൌ ͳ െ ‫ݒ‬ ͳ െ ‫ݏ‬௩ ͳ െ ͲǤͲͳʹͻ ‫ ݏ‬ൌ ͳെ ൌ ͳെ ൌ ͳǤͲͺͻ െͳͳ ‫ݒ‬ ͳ ൅ ͲǤͲͳʹͻ ͳ െ ‫ݏ‬௩ ൌ ͳെ ൌ ͳǤͲͻͳ ‫ ݏ‬ൌ ͳെ െͳͳ ‫ݒ‬ ‫ݏ‬௩ୀିଵଵǡ௕ ൌ േ

‫ ݒ‬ൌ ͳ͵: ‫ݏ‬௩௕ ൌ േ

ܴ௥ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ - 102 -

߬௣ ݈ி௘ ʹ݉௥ ή ଶ ή‫ ݒ‬ή‫ߜ ݌‬ ଶ ͳ ʹ ή ʹͺ ͳ͵͵Ǥͷ ή ͳͲିଷ ή ʹͶͲ ή ͳͲିଷ ‫ܮ‬௥௛ǡଵଷ ൌ Ͷߨ ൉ ͳͲି଻ ൬ ή ͳ൰ ή ଶ ή ൌ ͲǤ͵͹ͷ ൉ ͳͲି଺ ‫ܪ‬ ʹ ͲǤͶͷ ή ͳͲିଷ ߨ ή ͳ͸ͻ ή ʹ ሺ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ሻ ሺͳ͵ ή ʹ ή ߨΤʹͺሻ ൌ ൌ ͲǤͲ͹͸ ߟ௩ୀଵଷ ൌ ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ͳ͵ ή ʹ ή ߨΤʹͺ ‫ܮ‬௥ఙǡଵଷ ൌ ሺͳΤߟ௩ଶ െ ͳሻ ή ‫ܮ‬௥௛௩ ൅ ‫ܮ‬௥ఙ ൌ ሺͳΤሺͲǤͲ͹͸ሻଶ െ ͳሻ ή ͲǤ͵͹ͷ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ൌ ͸Ǥͷͺͳ ή ͳͲିହ ‫ܪ‬ ܴ௥ ͵ǤͺͶ ൉ ͳͲିହ ൌൌ േ ൌ േͲǤͲͲͳͺ͸ ‫ݏ‬௩ୀଵଷǡ௕ ൌ േ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ʹ ή ߨ ή ͷͲ ή ሺ͸Ǥͷͺͳ ή ͳͲି଺ ൅ ͲǤ͵͹ͷ ή ͳͲି଺ ሻ ͳ െ ‫ݏ‬௩ ‫ݏ‬௩ ൌ ͳ െ ‫ݒ‬ሺͳ െ ‫ݏ‬ሻ ֜ ‫ ݏ‬ൌ ͳ െ ‫ݒ‬ ͳ െ ͲǤͲͲͳͺ͸ ͳ െ ‫ݏ‬௩ ൌ ͳെ ൌ ͲǤͻʹ͵ʹ ‫ ݏ‬ൌ ͳെ ͳ͵ ‫ݒ‬ ͳ െ ‫ݏ‬௩ ͳ ൅ ͲǤͲͲͳͺ͸ ‫ ݏ‬ൌ ͳെ ൌ ͳെ ൌ ͲǤͻͳ͸ͺ ͳ͵ ‫ݒ‬ ଶ ή ‫ܮ‬௥௛௩ ൌ ߤ଴ ܰ௥ଶ ݇௪௥

ߨଶ

‫ ݒ‬ൌ െͳ͹:

ܴ௥ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ߬௣ ݈ி௘ ʹ݉௥ ଶ ‫ܮ‬௥௛௩ ൌ ߤ଴ ܰ௥ଶ ݇௪௥ ή ଶ ଶ ή ߨ ή‫ ݒ‬ή‫ߜ ݌‬ ଶ ͳ ʹ ή ʹͺ ͳ͵͵Ǥͷ ή ͳͲିଷ ή ʹͶͲ ή ͳͲିଷ ή ൌ ͲǤʹͳͻ ൉ ͳͲି଺ ‫ܪ‬ ‫ܮ‬௥௛ǡିଵ଻ ൌ Ͷߨ ൉ ͳͲି଻ ൬ ή ͳ൰ ή ଶ ߨ ή ʹͺͻ ή ʹ ͲǤͶͷ ή ͳͲିଷ ʹ ሺ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ሻ ሺͳ͹ ή ʹ ή ߨΤʹͺሻ ߟ௩ୀିଵ଻ ൌ ൌ ൌ െͲǤͳ͸͵ ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ͳ͹ ή ʹ ή ߨΤʹͺ ‫ܮ‬௥ఙǡିଵ଻ ൌ ሺͳΤߟ௩ଶ െ ͳሻ ή ‫ܮ‬௥௛௩ ൅ ‫ܮ‬௥ఙ ൌ ሺͳΤሺͲǤͳ͸͵ሻଶ െ ͳሻ ή ͲǤʹͳͻ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ൌ ͻǤʹͻ ή ͳͲି଺ ‫ܪ‬ ͵ǤͺͶ ൉ ͳͲିହ ܴ௥ ൌൌ േ ൌ േͲǤͲͳʹͻ ‫ݏ‬௩ୀିଵ଻ǡ௕ ൌ േ ʹ ή ߨ ή ͷͲ ή ሺͻǤʹͻ ή ͳͲି଺ ൅ ͲǤʹͳͻ ή ͳͲି଺ ሻ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ͳ െ ‫ݏ‬௩ ‫ݏ‬௩ ൌ ͳ െ ‫ݒ‬ሺͳ െ ‫ݏ‬ሻ ֜ ‫ ݏ‬ൌ ͳ െ ‫ݒ‬ ͳ െ ͲǤͲͳʹͻ ͳ െ ‫ݏ‬௩ ൌ ͳെ ൌ ͳǤͲͷͺ ‫ ݏ‬ൌ ͳെ െͳ͹ ‫ݒ‬ ͳ െ ‫ݏ‬௩ ͳ ൅ ͲǤͲͳʹͻ ‫ ݏ‬ൌ ͳെ ൌ ͳെ ൌ ͳǤͲͷͻ͸ ‫ݒ‬ െͳ͹ ‫ݏ‬௩௕ ൌ േ

‫ ݒ‬ൌ ͳͻ:

ܴ௥ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ߬௣ ݈ி௘ ʹ݉௥ ଶ ή ‫ܮ‬௥௛௩ ൌ ߤ଴ ܰ௥ଶ ݇௪௥ ή ଶ ଶ ߨ ή‫ ݒ‬ή‫ߜ ݌‬ ଶ ͳ ʹ ή ʹͺ ͳ͵͵Ǥͷ ή ͳͲିଷ ή ʹͶͲ ή ͳͲିଷ ‫ܮ‬௥௛ǡଵଽ ൌ Ͷߨ ൉ ͳͲି଻ ൬ ή ͳ൰ ή ଶ ή ൌ ͲǤͳ͹ͷ ൉ ͳͲି଺ ‫ܪ‬ ʹ ͲǤͶͷ ή ͳͲିଷ ߨ ή ͵͸ͳ ή ʹ

‫ݏ‬௩௕ ൌ േ

- 103 -

ሺ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ሻ ሺͳͻ ή ʹ ή ߨΤʹͺሻ ൌ ൌ െͲǤʹͳͳ ͳͻ ή ʹ ή ߨΤʹͺ ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ‫ܮ‬௥ఙǡଵଽ ൌ ሺͳΤߟ௩ଶ െ ͳሻ ή ‫ܮ‬௥௛௩ ൅ ‫ܮ‬௥ఙ ൌ ሺͳΤሺͲǤʹͳͳሻଶ െ ͳሻ ή ͲǤͳ͹ͷ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ൌ ͷǤͲʹͷ ή ͳͲି଺ ‫ܪ‬ ܴ௥ ͵ǤͺͶ ൉ ͳͲିହ ‫ݏ‬௩ୀଵଽǡ௕ ൌ േ ൌൌ േ ൌ േͲǤͲͲʹ͵͹ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ʹ ή ߨ ή ͷͲ ή ሺͲǤͳ͹ͷ ή ͳͲି଺ ൅ ͷǤͲʹͷ ή ͳͲି଺ ሻ ͳ െ ‫ݏ‬௩ ‫ݏ‬௩ ൌ ͳ െ ‫ݒ‬ሺͳ െ ‫ݏ‬ሻ ֜ ‫ ݏ‬ൌ ͳ െ ‫ݒ‬ ͳ െ ‫ݏ‬௩ ͳ െ ͲǤͲͲʹ͵͹ ‫ ݏ‬ൌ ͳെ ൌ ͳെ ൌ ͲǤͻͶ͹Ͷ ͳͻ ‫ݒ‬ ͳ ൅ ͲǤͲͲʹ͵͹ ͳ െ ‫ݏ‬௩ ൌ ͳെ ൌ ͲǤͻͶ͸͸ ‫ ݏ‬ൌ ͳെ ͳͻ ‫ݒ‬ ߟ௩ୀଵଽ ൌ

‫ ݒ‬ൌ െʹ͵:

ܴ௥ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ߬௣ ݈ி௘ ʹ݉௥ ଶ ή ଶ ଶ ή ‫ܮ‬௥௛௩ ൌ ߤ଴ ܰ௥ଶ ݇௪௥ ߨ ή‫ ݒ‬ή‫ߜ ݌‬ ଶ ͳ ʹ ή ʹͺ ͳ͵͵Ǥͷ ή ͳͲିଷ ή ʹͶͲ ή ͳͲିଷ ή ൌ ͲǤͳʹ ൉ ͳͲି଺ ‫ܪ‬ ‫ܮ‬௥௛ǡିଶଷ ൌ Ͷߨ ൉ ͳͲି଻ ൬ ή ͳ൰ ή ଶ ߨ ή ͷʹͻ ή ʹ ͲǤͶͷ ή ͳͲିଷ ʹ ሺ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ሻ ሺʹ͵ ή ʹ ή ߨΤʹͺሻ ߟ௩ୀିଶଷ ൌ ൌ ൌ െͲǤͳ͹Ͷ ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ʹ͵ ή ʹ ή ߨΤʹͺ ‫ܮ‬௥ఙǡିଶଷ ൌ ሺͳΤߟ௩ଶ െ ͳሻ ή ‫ܮ‬௥௛௩ ൅ ‫ܮ‬௥ఙ ൌ ሺͳΤሺͲǤͳ͹Ͷሻଶ െ ͳሻ ή ͲǤͳ͹ͷ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ൌ ͷǤͳͳ ή ͳͲି଺ ‫ܪ‬ ܴ௥ ͵ǤͺͶ ൉ ͳͲିହ ‫ݏ‬௩ୀିଶଷǡ௕ ൌ േ ൌൌ േ ൌ േͲǤͲʹ͵ͷ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ʹ ή ߨ ή ͷͲ ή ሺͲǤͳʹ ή ͳͲି଺ ൅ ͷǤͳͳ ή ͳͲି଺ ሻ ͳ െ ‫ݏ‬௩ ‫ݏ‬௩ ൌ ͳ െ ‫ݒ‬ሺͳ െ ‫ݏ‬ሻ ֜ ‫ ݏ‬ൌ ͳ െ ‫ݒ‬ ͳ െ ͲǤͲʹ͵ͷ ͳ െ ‫ݏ‬௩ ൌ ͳെ ൌ ͳǤͲͶʹ ‫ ݏ‬ൌ ͳെ െʹ͵ ‫ݒ‬ ͳ െ ‫ݏ‬௩ ͳ ൅ ͲǤͲʹ͵ͷ ‫ ݏ‬ൌ ͳെ ൌ ͳെ ൌ ͳǤͲͶͶ െʹ͵ ‫ݒ‬ ‫ݏ‬௩௕ ൌ േ

‫ ݒ‬ൌ ʹͷ:

ܴ௥ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ߬௣ ݈ி௘ ʹ݉௥ ଶ ή ଶ ଶ ή ‫ܮ‬௥௛௩ ൌ ߤ଴ ܰ௥ଶ ݇௪௥ ߨ ή‫ ݒ‬ή‫ߜ ݌‬ ଶ ͳ ʹ ή ʹͺ ͳ͵͵Ǥͷ ή ͳͲିଷ ή ʹͶͲ ή ͳͲିଷ ‫ܮ‬௥௛ǡଶହ ൌ Ͷߨ ൉ ͳͲି଻ ൬ ή ͳ൰ ή ଶ ή ൌ ͲǤͳͲͳͷ ൉ ͳͲି଺ ‫ܪ‬ ͲǤͶͷ ή ͳͲିଷ ʹ ߨ ή ͸ʹͷ ή ʹ ሺ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ሻ ሺʹͷ ή ʹ ή ߨΤʹͺሻ ൌ ൌ െͲǤͳͳͳ ߟ௩ୀଶହ ൌ ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ʹͷ ή ʹ ή ߨΤʹͺ ‫ܮ‬௥ఙǡଶହ ൌ ሺͳΤߟ௩ଶ െ ͳሻ ή ‫ܮ‬௥௛௩ ൅ ‫ܮ‬௥ఙ ൌ ሺͳΤሺͲǤͳͳͳሻଶ െ ͳሻ ή ͲǤͳ͹ͷ ή ͳͲି଺ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ൌ ͻǤͶ ή ͳͲି଺ ‫ܪ‬

‫ݏ‬௩௕ ൌ േ

- 104 -

ܴ௥ ͵ǤͺͶ ൉ ͳͲିହ ‫ݏ‬௩ୀଶହǡ௕ ൌ േ ൌൌ േ ൌ േͲǤͲͳʹͻ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ʹ ή ߨ ή ͷͲ ή ሺͲǤͳͲͳͷ ή ͳͲି଺ ൅ ͻǤͶ ή ͳͲି଺ ሻ ͳ െ ‫ݏ‬௩ ‫ݏ‬௩ ൌ ͳ െ ‫ݒ‬ሺͳ െ ‫ݏ‬ሻ ֜ ‫ ݏ‬ൌ ͳ െ ‫ݒ‬ ͳ െ ‫ݏ‬௩ ͳ െ ͲǤͲͳʹͻ ൌ ͳെ ൌ ͲǤͻ͸Ͳͷ ‫ ݏ‬ൌ ͳെ ʹͷ ‫ݒ‬ ͳ ൅ ͲǤͲͳʹͻ ͳ െ ‫ݏ‬௩ ൌ ͳെ ൌ ͲǤͻͷͻͷ ‫ ݏ‬ൌ ͳെ ʹͷ ‫ݒ‬ ‫ ݒ‬ൌ െʹͻ:

ܴ௥ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ߬௣ ݈ி௘ ʹ݉௥ ଶ ή ଶ ଶ ή ‫ܮ‬௥௛௩ ൌ ߤ଴ ܰ௥ଶ ݇௪௥ ߨ ή‫ ݒ‬ή‫ߜ ݌‬ ଶ ͳ ʹ ή ʹͺ ͳ͵͵Ǥͷ ή ͳͲିଷ ή ʹͶͲ ή ͳͲିଷ ି଻ ‫ܮ‬௥௛ǡିଶଽ ൌ Ͷߨ ൉ ͳͲ ൬ ή ͳ൰ ή ଶ ή ൌ ͲǤ͹ͷͶ ൉ ͳͲି଻ ‫ܪ‬ ͲǤͶͷ ή ͳͲିଷ ߨ ή ͺͶͳ ή ʹ ʹ ሺ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ሻ ሺʹͻ ή ʹ ή ߨΤʹͺሻ ߟ௩ୀିଶଽ ൌ ൌ ൌ ͲǤͲ͵Ͷʹ ‫ ݒ‬ή ‫ ݌‬ή ߨΤܳ௥ ʹͻ ή ʹ ή ߨΤʹͺ ‫ܮ‬௥ఙǡିଶଽ ൌ ሺͳΤߟ௩ଶ െ ͳሻ ή ‫ܮ‬௥௛௩ ൅ ‫ܮ‬௥ఙ ൌ ሺͳΤሺͲǤͲ͵Ͷʹሻଶ െ ͳሻ ή ͲǤ͹ͷͶ ή ͳͲି଻ ൅ ͲǤͲͳʹ͹ ή ͳͲିସ ൌ ͸Ǥͷ͸ ή ͳͲିହ ‫ܪ‬ ܴ௥ ͵ǤͺͶ ൉ ͳͲିହ ‫ݏ‬௩ୀିଶଽǡ௕ ൌ േ ൌൌ േ ൌ േͲǤͲͲͳͺ͹ͷ ߱௦ ሺ‫ܮ‬௥ఙ௩ ൅ ‫ܮ‬௥௛௩ ሻ ʹ ή ߨ ή ͷͲ ή ሺ͸Ǥͷ͹͵ ή ͳͲିହ ሻ ͳ െ ‫ݏ‬௩ ‫ݏ‬௩ ൌ ͳ െ ‫ݒ‬ሺͳ െ ‫ݏ‬ሻ ֜ ‫ ݏ‬ൌ ͳ െ ‫ݒ‬ ͳ െ ͲǤͲͲͳͺ͹ͷ ͳ െ ‫ݏ‬௩ ൌ ͳെ ൌ ͳǤͲ͵ͶͶͳͺ ‫ ݏ‬ൌ ͳെ െʹͻ ‫ݒ‬ ͳ െ ‫ݏ‬௩ ͳ ൅ ͲǤͲͲͳͺ͹ͷ ‫ ݏ‬ൌ ͳെ ൌ ͳെ ൌ ͳǤͲ͵Ͷ͵ͺͳ ‫ݒ‬ െʹͻ ‫ݏ‬௩௕ ൌ േ

All slips, in which asynchronous harmonic torques happen, and the related slips of the breakdown harmonic torques, in addition to selected slips for time stepping calculation of the VZIM2, are shown in Table VII.

- 105 -

Table V: Slips, which are used to perform time-stepping analysis of VZIM1 Slip Speed(rpm) 0.00001 1500 0.01 1485 0.02 1470 0.0253 1462 0.03 1455 0.04 1440 0.05 1425 0.06 1410 0.08 1380 0.1 1350 0.12 1320 0.15 1275 0.2 1200 0.3 1050 0.45 825 0.65 525 0.843 235.5 0.857 214.5 0.871 193.5 0.917 124.5 0.92 120 0.923 115.5 0.94 90 0.947 79.5 0.954 69 0.958 63 0.96 60 0.962 57 0.987 19.5 1 0 1.013 -19.5 1.034 -51 1.0344 -51.6 1.0348 -52.2 1.037 -55.5 1.043 -64.5 1.054 -81 1.0588 -88.2 1.063 -94.5 1.071 -106.5 1.077 -115.5 1.084 -126 1.09 -135 1.096 -144 1.174 -261 1.2 -300 1.226 -339 1.33 -495 1.66 -990 2 -1500 - 106 -

Table VI: Slips, which are used to perform time-stepping analysis of VZIM2 Slip Speed(rpm) 0.00001 1500 0.01 1485 0.02 1470 0.0253 1462 0.03 1455 0.04 1440 0.05 1425 0.06 1410 0.08 1380 0.1 1350 0.12 1320 0.15 1275 0.2 1200 0.3 1050 0.45 825 0.65 525 0.85 225 0.857 214.5 0.8633 205.05 0.9166 125.1 0.92 120 0.9234 114.9 0.944 84 0.947 79.5 0.95 75 0.9587 61.95 0.96 60 0.961 58.5 0.995 7.5 1.0044 -6.6 1.013 -19.5 1.0344 -51 1.03432 -51.48 1.03463 -51.94 1.041 -61.5 1.043 -64.5 1.045 -73.5 1.057 -85.5 1.0588 -88.2 1.0606 -90.9 1.0643 -96.45 1.071 -106.5 1.077 -115.5 1.088 -132 1.09 -135 1.093 -139.5 1.1887 -283.05 1.211 -316.5 1.33 -495 1.66 -990 2 -1500 - 107 -

Table VII: Slips, which are used to perform time-stepping analysis of VZIM3 Slip Speed(rpm) 0.00001 1500 0.01 1485 0.02 1470 0.0253 1462 0.03 1455 0.04 1440 0.05 1425 0.06 1410 0.08 1380 0.1 1350 0.12 1320 0.15 1275 0.2 1200 0.3 1050 0.45 825 0.65 525 0.855 217.5 0.859 211.5 0.8633 205.05 0.9168 124.5 0.92 120 0.9232 115.2 0.9466 80.1 0.947 79.5 0.9474 78.9 0.9595 60.75 0.96 60 1.34 -51 1.03438 -51.57 1.0344 -51.6 1.034418 -51.62 1.042 -63 1.043 -64.5 1.044 -66 1.058 -87 1.0588 -88.2 1.0596 -89.4 1.071 -106.5 1.089 -133.5 1.09 -135 1.091 -136.5 1.191 -286.5 1.2 -300 1.209 -313.5 1.33 -495 1.66 -990 2 -1500

- 108 -

Cos(Fi) 0,967 0,915 0,872 0,769

0,8071 0,805

0,61 0,595

I1/A 37,39 37,4 15,49

16,36

200,99

208,6

290,97

306,6

126104,2

123100

115439,3

112400

8722,3

Pin/W 25053,7 23695,5 9368,97

305

288,06

207,2

203,17

12,84

I2'/A 37,1343 35,66 12,82

Method Speed/rpm I1/A Flux2D 35,91 Klasys 1455,8 32,38 Flux2D 31,202 Klasys 1462 28,43 Flux2D 284,04 Klasys 525 300,6 Flux2D 336,36 Klasys -1500 369,2

Pin/W 24188 20425,1 21100 17704,8

133000 147789,1

124000 139378,3

Cos(Fi) 0,97 0,916 0,97 0,904

0,67 0,714

0,53 0,546

367,1

333,577

298,8

287,5

26,32

I2'/A 34,31 30,44 30,61

Table II: Losses in VZIM2 calculated by Flux2D and KLASYS

Method Speed/rpm Flux2D Klasys 1390,9 Flux2D Klasys 1462 Flux2D Klasys 525 Flux2D Klasys -1500

Table I: Losses in VZIM1 calculated by Flux2D and KLASYS

300,3

236,36

548,1

318,76

105,1

Torque/Nm 144,52 121,3 126,04

403,6

355,52

543,4

443,72

51,55

Torque/Nm 148,468 140,7 55,64

- 109 -

-47168,6

-39413,28

30133,2

17190,558

16098,2

Pout/W 22021,051 18486,2 20101,057

-63392,4

-58127,68

29873,1

24219,762

7892

Pout/W 21605,23 20492,4 9175,239

8.3 Appendix III: Calculated losses of VZIM1, VZIM2 and VZIM3

-0,338

-0,317

0,204

0,1292

0,909

Efficiency 0,9104 0,905 0,95

-0,503

-0,472

0,208

0,2154

0,905

Efficiency 0,86 0,865 0,9793

124693,8

95580

55836,4

43031,48

213,3

92248,7

76571,83

61150,9

54603,35

546,9

93373

68382,892

56222,1

40938,52

430,1

LOSScu1/W LOSScu2/W 872,75 833,24 709,8 575,7 658,908 625,172

63644,6

57300,28

29440,1

27340,67

181,2

73,89

291,776

88,01

127,016

179,2

LOSSfes/W 200,08 177,9 201,852

58,8

319,1

78,02

136,288

119,7

13,05

131,08

7,976

38,72

0,615

LOSSfer/W 37,32 0,724 33,56

26,51

124,92

15,72

38,44

0,669

LOSScu1/W LOSScu2/W LOSSfes/W LOSSfer/W 946,17 1497,44 161,12 25,32 947 1645,6 116 2,034 162,39 197,76 126,52 12,52

Method Speed/rpm I1/A cos(Fi) Flux2D 36,62 0,9525 Klasys 1474,65 33,02 0,918 Flux2D 52,63 0,957 Klasys 1462 47,4 0,932 Flux2D 303,929 0,593 Klasys 525 318,6 0,584 Flux2D 342,61 0,522 Klasys -1500 360 0,507 316,5

337,75 358,1

126307,7

302,9

125000 124000

45,91

30381,1 128486,7

I2'/A 30,34 31,33 53,08

Pin/W 24166,01 20931,6 34900

Table III: Losses in VZIM3 calculated by Flux2D and KLASYS

-38838,9

247,3

231,04

377,9

208,88

180,4

Torque/Nm 143,132 125,4 202,44

- 110 -

-36273,28

20774,3

11277,938

27612

Pout/W 22040,7 19367,3 31749,14

-0,307

-0,292

0,162

0,092

0,909

Efficiency 0,912 0,925 0,909

87737,7

79443,875

68713,5

62517,93

1520,9

76732,1

63657,052

38776,4

30445,928

726,7

76,22

222,44

82,29

110,28

173,2

18,52

112,88

7,799

33,96

0,944

LOSScu1/W LOSScu2/W LOSSfes/W LOSSfer/W 907,605 487,816 140,124 23,756 738,1 338,2 178,2 0,607 1874,67 998,45 167,16 36,92

8.4 APPENDIX IV: Estimation of the heat transfer coefficient at the winding-overhang close to the centrifugal mechanism Although VZIM motor is totally enclosed, at one side of the winding overhang, heat transfer coefficient is considered as ߙ ൌ ͳͷȀሺʹ ሻ for the natural convection and heat radiation, but at the other side of the motor, the rotating centrifugal mechanism leads to a better cooling due to the forced convection. Therefore in order to calculate the heat transfer coefficient for this winding overhang close to the centrifugal mechanism, with try and error and comparison with measured values, ߙ is calculated for 5.5kW VZIM (ߙ ൌ ͷͲȀሺʹ ሻ ) and for 22kW VZIM due to the similar structure with 5.5kW VZIM, a proportional value of ߙ is considered. In the following, it is described how the value of heat transfer coefficient is derived. Measured values at the rated load thermal test for 5.5kW VZIM shows that the maximum temperature rise for the winding overhang is 67.7. According to this value a finite element model of 5.5kW VZIM is prepared using ANSYS, and the value of ߙ is adjusted, and finally with a heat transfer coefficient about ߙ ൌ ͷͲȀሺʹ ሻ the calculated values, based on the finite element method with ANSYS and measured values are matched together. For 5.5kW VZIM2 (nominal speed for 5.5kW VZIM2 is equal to 1455.2rpm) with consideration of the centrifugal mechanism as a standard fan, the heat transfer coefficient is equal to ߙ ൌ ͺ‫ ݒ‬ଷΤସ ൌ ͺሺͳ͵ǤͲͶͶሻଷΤସ ൌ ͷͶǤͻȀሺʹ ሻ, besides for 22kW VZIM2 motor (nominal speed for 22kW VZIM2 is equal to 1467.36 rpm) the heat transfer coefficient for winding overhang with considering the mechanical part of the motor, as a standard fan, is equal to ߙ ൌ ͺ‫ ݒ‬ଷΤସ ൌ ͺሺͳ͹Ǥͺ͹ሻଷΤସ ൌ ͸ͻǤͷ͵Ȁሺʹ ሻ. Therefore due to the similarity of the 2 types of the motor (5.5kW and 22kW) the applied value of ߙ for VZIM2 22kW is ൌ ͷͲΤͷͶǤͻ ή ͸ͻǤͷ͵ ൌ ͸͵Ǥ͵ʹȀሺʹ ሻ .

For 5.5kW VZIM3 (nominal speed for 5.5kW VZIM3 is equal to 1470rpm) with consideration of the centrifugal mechanism as a standard fan, the heat transfer coefficient is equal to ߙ ൌ ͺ‫ ݒ‬ଷΤସ ൌ ͺሺͳ͵Ǥͳ͹ሻଷΤସ ൌ ͷͷǤ͵Ȁሺʹ ሻ. Besides for 22kW VZIM3 motor (nominal speed for 22kW VZIM3 is equal to 1467.36rpm) the heat transfer coefficient for the winding overhang, with considering the mechanical part of the motor as a standard fan, is equal to ߙ ൌ ͺ‫ ݒ‬ଷΤସ ൌ ͺሺͳ͹Ǥͺ͹ሻଷΤସ ൌ ͸ͻǤͷ͵Ȁሺʹሻ. Therefore due to the similarity of the 2 types of the motor (5.5kW and 22kW) the applied value of ߙ for VZIM2 22kW is ߙ ൌ ͷͲΤͷͷǤ͵ ή ͸ͻǤͷ͵ ൌ ͸ʹǤͺ͸Ȁሺʹ ሻ.

- 111 -

Induction Motor Analysis

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