Distribution Transformer Guide
Distribution Transformer Division Athens, Georgia Jefferson City, Missouri June, 1979 Revised March, 2002
ISO 9001 CERTIFIED
Foreword The purpose of this guide is to assemble fundamental information concerning common ratings, connections, and applications of distribution transformers. The information presented is a summary of these fundamentals and is intended as a reference for those who deal occasionally with distribution transformer applications. This guide does not purport to cover all aspects of selection and application; if questions arise or further details are required, contact ABB Inc.
1
Index I. General
Page A. Application ................................................................................ 4 B. Physical Description ................................................................. 4 C. Protection and Accessories ...................................................... 11
II. Performance A. B. C. D. E. F. G. H. I. J. K.
Designation of Winding Voltage Ratings ................................. Polarity ...................................................................................... Terminal Designations .............................................................. Short Circuit Ratings ................................................................ Sound Level Ratings ................................................................ Tolerance Definitions ................................................................ Impedance Calculations ........................................................... Efficiency Calculations ............................................................. Regulation Calculations ........................................................... Performance Example .............................................................. Secondary Fault Current—120/240 Volt Systems ...................
17 20 21 22 22 23 23 24 24 26 27
III. Three-Phase Transformers and Banks A. Application Considerations ...................................................... 34 B. Summary of Common Connections ......................................... 41 C. Common Three-Phase Banks Using Single-Phase Transformers ........................................................................... 48
IV. Loading A. B. C. D.
Paralleling ................................................................................. Delta-Delta Bank Loading ........................................................ Overloading .............................................................................. Single-Phase and Three-Phase Loading of Symmetrical and Unsymmetrical Transformer Banks .......................................... E. Dedicated Motor Loads ............................................................
51 51 52 53 66
V. Voltage Unbalance A. B. C. D.
Effects of Voltage Unbalance ................................................... Voltage Unbalance Definitions ................................................. Causes of Voltage Unbalance ................................................. Voltage Unbalance With Three-Phase Loading ...................... 1. Delta-Delta and Floating Wye-Delta Banks ........................ 2. Open-Delta Banks ...............................................................
71 71 73 73 74 75
VI. Reference Data Solid and Concentric Stranded Aluminum and Copper Conductors .................................................................................... Temperature Correction Factors for Resistance of Aluminum Conductors .................................................................................... Logarithm Tables ........................................................................... Nominal direct-Current Resistance, Ohms per 1000 Feet, at 20°C and 25°C of Solid and Concentric Stranded Conductors .... Natural Functions of Angles .......................................................... Typical Isokeraunic Map ................................................................ Selected SI Equivalents ................................................................
2
80 81 83 85 86 87 88
I. General
Page
A. Application ................................................................................ B. Physical Description ................................................................. 1. Pole Mounted ...................................................................... 2. Pad Mounted ....................................................................... C. Protection and Accessories ...................................................... 1. General ................................................................................ 2. Types of Accessories and Transformer Protection Packages — Pole Mounted ................................................ 3. Types of Accessories and Protection — Pad Mounted ......
4 4 4 6 11 11 11 13
3
I. General
A. Application ABB single-phase and three-phase, oil-filled, pole- and padmounted distribution transformers are specifically designed for servicing residential distribution loads; they are also suitable for light commercial loads, and industrial lighting and diversified power applications. The transformers described herein are designed for the application conditions normally encountered on electric utility power distribution systems. As such they are suitable for use under the “usual service conditions” described in ANSI C57.12.00 General Requirements for Liquid-Immersed Distribution, Power and Regulating Transformers. All other conditions are considered “unusual service” and should be avoided unless specific ABB Division approval is obtained.
B. Physical Description 1. Pole Mounted • Meets Industry Standard ANSI C57.12.20 • 0.5 - 1000kVA • 65° C temperature rise • Insulation levels: Rated Insulation Voltage Ranges Class 480- 600 1.2 2160- 2400 5.0 4160- 4800 8.7 7200-12470 1 15.0 13200-14400 18.0 19920-22900 2 25.0 -34400 34.5 1 Optional 125 kV BIL 12000 volts available 2 Optional 125 kV BIL 19920 volts available
4
Basic Impulse Level (kV) 30 60 75 95 125 150 200
Type CSP
Type S
kVA
High Low Voltage Voltage Pole Mounted (Single-Phase) 0.5 2400 through 120/240 1.5 34,400 volts 240/480 3 5 10 15 25 371/2 50 75 100 167 250 333 500 667 750 833 1000
Three-Phase
Jumbo
Pole Mounted (Three-Phrase) 15 2400 to 208/120 30 13,800T 240x480T 45 480/277 75 1121/2 150 225 330 500
JUMBO ® Step-Down Transformer. The ABB JUMBO single-phase step-down transformer is especially useful during utility system voltage conversions when it is desirable to convert a portion of a substation or a feeder to a higher voltage and still be able to supply the remaining customers at the existing voltage. The JUMBO ® is also available for industrial and commercial applications or for resale customers. 5
2. Pad Mounted A single-phase, single service, low profile distribution padmount transformer available in loop or radial feed. Designed to aesthetically, safely and economically provide underground electrical service to single loads, particularly, rural residences, farms and ranches.
Micro-Pak, 10-50 kVA A single-phase, multi-service, full-line, low profile padmount transformer designed for loop feed or radial feed on a grounded wye underground distribution system. The Mini-Pak can be furnished in a complete line of ratings and in a wide range of configurations to fully meet the reliability, safety and operating requirements of any distribution system.
Mini-Pak, 10-167 kVA The Maxi-Pak is designed specifically for those customers requiring straightup feed (Type I) rather than cross feed (Type II). The additional height of the Maxi-Pak allows installation of air load break switching in this low-profile design.
Maxi-Pak, 10-250 kVA
6
ABB single-phase padmounted Distribution Transformers meet the following Industry Standards: ANSI C57.12.00 - IEEE Standard General Requirements for Liquid Immersed…Transformers ANSI C57.12.25 - Pad-Mounted…Single-Phase Distribution Transformers with Separable Insulated High-Voltage Connectors… ANSI C57.12.28 - Pad-Mounted Equipment - Enclosure Integrity or ANSI C57.12.29 - …Pad-Mounted Equipment - Enclosure Integrity for Coastal Environments ANSI C57.12.70 - …Terminal Markings and Connections ANSI C57.12.80 - IEEE Standard Terminology… ANSI C57.12.90 - IEEE Standard Test Code… NEMA Tr-1 - Transformer Standards IEEE 386 - …Separable Insulated Connectors ABB recommends the use of ANSI C57.91 - IEEE Guide for Loading…for the establishment of proper distribution transformer loading practices. Ratings @65° Rise kVA: 10,25,371/2, 50, 75, 100, 167, 250 1 HV: 4160GY/2400 through 34500GY/19920V 3 BIL: 60, 75, 95, 125, 150 kV LV: 240/120, 480/240, 277 V, 120/240 3, 240/480 2 1 Maxi only 2 Available only on micros with cable lead secondary 3 Mini and Maxi only (micros available thru 24940GY/14400)
Standard Features: 1. Equipped with two universal high voltage bushing wells for loop feed. (Only one bushing well is provided for radial feed.) 2. A removable flip-top hood and heavy-duty 3/8 '', stainless steel hinge pins provide safe and durable service. 3. A recessed locking assembly with padlock provisions and a pentahead locking bolt is standard for tamper resistant operation. A hex-head locking bolt is available. 4. All tanks are constructed of heavy gauge steel. Tank seams are welded and each unit is pressure tested and inspected for leaks prior to shipment. In addition, all single phase transformers are supplied with: a. b. c. d. e.
/ ''-11 stainless steel lifting bosses Oil level/fill plug Oil drain plug Self-actuating pressure relief device Two ground bosses, 1/ 2''-13 NC tapped hole 7/16'' deep.
5 8
5. The front sill latches with the flip-top hood, is attached on the side of the tank, and is removable. 6. The high voltage universal bushing wells are externally clamped and removable. A parking stand between the bushing wells is provided for attachment of bushing accessories. 7. Externally clamped low voltage epoxy bushings. 8. Tamper-resistant design that exceeds ANSI C57.12.28. 9. NEMA safety labels per NEMA Publication 260-1982.
7
Minimum/Maximum Design Dimensions 1
Min. Max.
Micro-Pak A B C D 24 24 30.25 14 24 24 35.50 16
Mini-Pak A B C D 24 32 30.25 14 42 44 46.00 20
Maxi-Pak A B C D 32 32 30.25 14 42 44 46.00 20
1 Actual dimensions will vary according to voltage, loss evaluation,
and accessories.
Dimensions are in Inches
“C”+6 CABLE OPENING
5.0 “B”+6
5.0
Recommended Pad Dimensions
Optional Accessories 1. Overcurrent Protection a. An internal primary protective link to remove the transformer from the system in the event of an internal fault. b. A secondary breaker provides protection against secondary overloads and short circuits. c. An oil-immersed bayonet-type fuse link to remove the transformer from the system in case of an internal fault (fault sensing) or secondary short overload (overload sensing). This fuse is a drawout design and is supplied in series with an isolation link. A drip plate is provided to prevent oil from dripping onto the bushing or elbow. d. A current limiting fuse mounted in a dry well loadbreak canister.2 • The high interrupting rating of the CL fuse permits its use on systems where the available fault current exceeds the rating of normal expulsion fuses. e. A partial range current limiting fuse mounted under oil within the transformer tank.2 • An expulsion fuse is supplied in series with the partial range CL fuse. • Available at 95, 125, and 150 kV BIL. 2. Switching a. Externally operated tap changer. b. Externally operated dual voltage switch.2 c. Externally operated loadbreak oil rotary (LBOR) switch.2 d. EFD CL fused air loadbreak switch available for either radial or loop feed.3
2 Not available on Micro 3 Maxi only
8
3. Primary Connection a. Universal bushing wells (standard) and loadbreak inserts. b. Integral (one piece) loadbreak bushings. 4. Secondary Connections a. Copper studs with rotatable spade type bushings. • Four-hole, NEMA type, tinplated copper alloy spade. • Four-hole, in line, tin-plated copper alloy spade. b. Cable lead secondary. 4 5. Corrosion Resistance a. ANSI C57.12.29 Full 400 Series Stainless Steel b. Partial Stainless Steel • Mini-Skirt™ and Sill • Sill Only • Sill and Hood • Mini-Skirt™, Sill, and Hood 6. Miscellaneous a. Cleats for anchoring sill to pad. b. Polypad mounting base. 4 4 Micro only
9
The ABB MTR is an oil-filled, three phase, commercial padmounted distribution transformer specifically designed for servicing such underground distribution loads as shopping centers, schools, institutions and industrial plants. It is available both live front and dead front construction, for radial or loop feed applications, or without taps. Industry Standards ABB three-phase MTR units meet the following industry standards: The ABB MTR Padmounted Transformer Three-Phase 45-2500 kVA
ANSI C57.12.00 - IEEE Standard General Requirements for Liquid Immersed… Transformers ANSI C57.12.22 - Pad-Mounted… Three-Phase Distribution Transformers with High Voltage Bushings ANSI C57.12.26 - Pad-Mounted…ThreePhase Distribution Transformers… With Separable Insulated High-Voltage Connectors ANSI C57.12.28 - …Pad-Mounted Equipment - Enclosure Integrity
or ANSI C57.12.29 - …Pad-Mounted Equipment - Enclosure Integrity for Coastal Environments ANSI C57.12.70 - Terminal Markings and Connections… ANSI C57.12.80 - IEEE Standard Terminology… ANSI C57.12.90 - IEEE Standard Test Code… NEMA Tr-1 - Transformer Standards IEEE 386 - Separable Insulated Connectors ABB Recommends the use of ANSI C57.91 - IEEE Guide for Loading…for the establishment of proper distribution transformer loading practices. Ratings • 45 through 2500 kVA • 65°C average winding rise over 30°C average ambient. • Low voltages: 1 208Y/120, 216Y/125, 460Y/265, 480Y/277, 480d, 240d and 240d with 120 volt mid-tap in one phase. • High voltages: 4160 Grd Y/2400 through 34,500 Grd Y/19,920 for Grounded Wye systems; 2400 through 34,500 for Delta systems; various dual high voltages. • Taps: All voltages are available with or without taps. • Insulation classes: 35 kV (200 kV BIL) and below. 1 208Y/120, 216Y/125, 240d not available above 1500kVA. 10
C. Protection and Accessories 1. General The distribution transformer functions as an integral part of the distribution system and consideration must be given to proper protection of the transformer from system disturbances. In addition, it is normal practice to apply overcurrent protection on the primary side of the transformer so that a faulted transformer is isolated from the primary system. Protection from excessive voltage transients and severe overcurrents should be provided. Protection considerations include: (1) Protective devices must be rated for the conditions anticipated. (2) When the transformer(s) is provided with overcurrent devices — coordination with system devices should be achieved to allow proper fault isolation. Caution: Operation of a primary protective device may indicate a faulted transformer. Re-energizing should be performed from a remote location unless the cause of device operation is positively identified and corrected. To do otherwise presents a hazard to life and property in the event of violent transformer failure. 2. Types of Accessories and Transformer Protection Packages— Pole Mounted ABB offers four basic transformer types: S, SP, CP and CSP ®. Together they represent a wide range of protective capabilities to meet nearly every application.
• Conventional “S” Transformers This type transformer contains no protective equipment. Therefore, lightning, fault and overload protection for these transformers must be provided by the purchaser.
• Surge-Protecting “SP” Transformers The “SP” transformers include transformer-mounted lightning arresters and internally-mounted high voltage protective links, but omit the internally-mounted low voltage circuit breaker.
• Current-Protecting “CP” Transformers The “CP” transformers are equipped with the internallymounted low-voltage circuit breaker and high voltage protective links, but omit the lightning arresters.
11
Types of Accessories and Transformer Protection Packages— Pole Mounted (Continued) • Self-Protecting “CSP ® ” Transformers In a “CSP” transformer the arrester protects the transformer from over-voltage caused by lightning and/or high voltage switching surges. The protective link operates to remove a defective transformer from service if an internal failure occurs, thereby protecting the system. The breaker provides the transformer a degree of protection from overloads and short circuits on the secondary side of the transformer. This type transformer offers the most protection of all protected transformers except for a “CSP” with a current limiting fuse. a. CL Fuses Two basic types of current limiting fuses exist—partial range and general purpose (full range). The partial range fuse requires a protective link applied in series while the general purpose fuse does not. The partial range fuse is available on pole-type transformers (bushing mounted) and padmounted transformers (internally mounted). The general purpose fuse is only available on padmounted transformers. b. The Distribution Surge Arrester protects the transformer (and other electrical equipment) from dangerous overvoltages, whether caused by lightning surges, switching surges or other transients. The Type LV Surgemaster™ valve type arrester has one or more arc gap assemblies connected in series with one or more current limiting “valve” blocks. Under high voltage surge conditions, the resistance in the blocks drops, providing a low-resistance path to ground. Once the surge has passed, however, the block resistance rises again, restricting the flow of current. The gaps will then interrupt this low-magnitude current flow, restoring the arrestor to an insulator. The LVBB Surgemaster™ valve type arrester is a big block (heavy duty) design which is capable of discharging a 100 KA surge. The big block arrester operates the same way as the LV with additional protection capability. The HMX gapless metal oxide arrester is a heavy duty design utilizing the non-linearity of a metal oxide resistor to provide protection levels equivalent to gapped silicon carbide arresters. The metal oxide distribution arrester offers the benefits of reduced complexity, improved reliability and improved performance characteristics. The LV, LVBB Surgemaster™ and HMX distribution arresters are available for either pole, crossarm or transformer mounting.
12
c. The Secondary Circuit Breaker provides the transformer with a degree of protection from secondary overloads and short circuits. It is mounted under oil, usually on the core/ coil assembly, connected between the coil’s secondary leads and the secondary bushings. The breaker is calibrated to trip when its bimetal reaches a predetermined temperature. An additional instantaneous magnetic trip element which responds to high fault currents is available on some breakers. 3. Types of Accessories and Protection—Padmounted For system and transformer protection from surge currents, short circuits and overloads, ABB offers a number of devices including a protective link, distribution surge arrester, secondary circuit breaker and current limiting fuse. a.
The Protective Fuse Link is an internal, oil-immersed, expulsion type fuse consisting of a fiber tube supporting and surrounding the fuse element usually made of copper and EVERDUR ®. The link is sized to operate only in the event of a winding failure, isolating the transformer from the primary system. Interrupting rating is 3500 amperes at 7.2kV.
Protective Fuse Link b. The Bayonet-Type Fuse Cartridge contains an oil-immersed expulsion type fuse with an interrupting rating of 3800 amperes at 8.3 kV. It is a hookstick-operable, drawout loadbreak design available through 19.9kV1. Two types of fuse links are available—overload-sensing and fault-sensing—and an internal isolation link is supplied in series for additional safety. The fault-sensing link is sized to operate only in the event of a transformer failure; the overload-sensing link is sized for additional protection from secondary system faults or prolonged heavy overload conditions. Standard Ratings: Voltage Class 8.3 kV 15.5 kV 23.0 kV
Interrupting Amps (RMS) 3800 2000 600
L.B. Amps At .8 PF 135 135 45
Bayonet-Type Fuse
13
c.
Current Limiting Fuses are available through 15 kV in either the EFD air loadbreak switch or in a drawout, loadbreak dry fuse well. Some applications may require parallel current limiting fuses to obtain sufficient full-load or inrush current ratings. A mechanical interlock with a loadbreak oil switch (LBOR) is recommended when using parallel drawout, loadbreak, dry well fuses. Some of the higher kVA designs may require current fuse ratings that are not available—contact Division. Partial range, internal, block-mounted current limiting fuses, which are applied in series with an internal protective link, are also available through 23 kV.
Loadbreak, Drywell Current Limiting Fuse Canister
d.
14
The EFD is an loadbreak air switch available for radial feed applications. Switching, flexibility and safety are made possible by a compact, “dead front” type construction that enables the switch to be externally-mounted on the tank in the terminal compartment. A sealed, silver sand current limiting fuse is normally provided to the switch’s transformer connecting pole. High voltage cables are connected to the switch contacts by means of solderless, clamp-type connectors capable of accepting cable sizes ranging from #6 to #4/0.
EFD Switch Ratings Continuous current ................ 200 A Loadbreak ............................. 200 A Close-in ................................. 5,000 A Momentary ............................ 10,000 A e.
The LBOR (Loadbreak Oil Rotary) switch is gang-operated and available for either radial or loop feed switching. The stacked deck rotary switch has a unique, springloaded cam-operated kicker system which provides quick make and break action to the contacts. LBOR Ratings: BIL Maximum Voltage (L-L) (L-Grd) Continuous and Interrupting Current Momentary and Making Current (RMS Sym./Assym.)
95 kV
125 kV
150 kV
15.5 kV 8.9 kV
27 kV 38 kV 15.5 kV 21.9 kV
300 A 1
200 A
300 A
12 kA/ 19.2 kA
12 kA/ 19.2 kA
10 kA/ 16 kA
1 200 A 3c rating also available.
LBOR Switch f.
The Tap Changer and Series Multiple Switch. Both are oil-immersed, externally-operated, and are designed for de-energized operation only.
Tap Changer Operating Handle
15
II. Performance
Page
A. Designation of winding voltage ratings .................................... B. Polarity ...................................................................................... C. Terminal designations .............................................................. 1. Pad-Mounted ....................................................................... 2. Pole-Mounted ...................................................................... (a) 1c pole-mounted ........................................................... (b) 3c pole-mounted ........................................................... D. Short circuit ratings .................................................................. E. Sound level ratings ................................................................... F. Tolerance definitions ................................................................ G. Impedance calculations ........................................................... H. Efficiency calculations .............................................................. I. Regulation calculations ............................................................ J. Performance example .............................................................. K. Secondary fault current—120/240 volt systems ......................
16
17 20 21 21 21 21 22 22 22 23 23 24 24 26 27
II. Performance
A. Designation of Winding Voltage Ratings (from ANSI C57.12.00)
1. Single-Phase Symbol
Example
E
12000
Typical Diagram
Usage E shall indicate a winding of E volts which is suitable for g connection on an E volt system. E/E1 Y
2400/4160Y
Usage E/E 1Y shall indicate a winding of E volts which is suitable for g connection on an E volt system or for Y connection on an E 1 volt system. E 1 GrdY/E
12 470GrdY/7200
Usage E 1 GrdY/E shall indicate a winding of E volts with reduced insulation at the neutral end. The neutral end may be connected directly to the tank for Y or for single-phase operation on an E 1 volt system, provided the neutral end of the winding is effectively grounded. E/2E
120/240
Usage E/2E shall indicate a winding, the sections of which can be connected in parallel for operation at E volts, or which can be connected in series for operation at 2E volts, or connected in series with a center terminal for three-wire operation at 2E volts between the extreme terminals and E volts between the center terminal and each of the extreme terminals. 2E/E
240/120
Usage 2E/E shall indicate a winding for 2E volts, two-wire full kVA between extreme terminals, or for 2E/E volts three-wire service with 1 /2 kVA available only, from midpoint to each extreme terminal. V x V1
240 x 480
Usage V x V1 shall indicate a winding for parallel or series operation only but not suitable for three-wire service.
Notes: (1) E = line-to-neutral voltage of a “Y” winding, or line-to-line voltage of a delta winding. (2) E 1 = CF3F E
17
2. Three-Phase Symbol
Example
E
2400
Typical Diagram
Usage E shall indicate a winding which is permanently g connected for operation on an E volt system. E1Y
4160Y
Usage E1 Y shall indicate a winding which is permanently Y connected without a neutral brought out (isolated) for operation on an E 1 volt system. E1Y/E
4160Y/2400
Usage E1Y/E shall indicate a winding which is permanently Y connected with a fully insulated neutral brought out for operation on an E1 volt system, with E volts available from line to neutral. E/E1Y
2400/4160Y
Usage E/E1Y shall indicate a winding which may be g connected for operation on an E volt system, or may be Y connected without a neutral brought out (isolated) for operation on an E 1 volt system. E/E1Y/E
2400/4160Y/2400
Usage E/E 1Y/E shall indicate a winding which may be g connected for operation on an E volt system or may be Y connected with a fully insulated neutral brought out for operation on an E 1 volt system with E volts available from line to neutral. E1 GrdY/E
12470GrdY/7200
Usage E1 GrdY/E shall indicate a winding with reduced insulation and permanently Y connected, with a neutral brought out and effectively grounded for operation on an E1 volt system with E volts available from line to neutral.
Notes: (1) E = line-to-neutral voltage of a “Y” winding, or line-to-line voltage of a delta winding. (2) E1 = CF3F E
18
2. Three-Phase (continued) Symbol
Example
E/E1 GrdY/E
7200/12470GrdY/7200
Typical Diagram
Usage E/E1 GrdY/E shall indicate a winding, having reduced insulation, which may be g connected for operation on an E volt system or may be connected Y with a neutral brought out and effectively grounded for operation on an E 1 volt system with E volts available from line to neutral. V x V1
7200 x 14 400
Usage V x V 1 shall indicate a winding, the sections of which may be connected in parallel to obtain one of the voltage ratings (as defined above) of V, or may be connected in series to obtain one of the voltage ratings (as defined above) of V1 . Windings are permanently g or Y connected.
Notes: (1) E = line-to-neutral voltage of a “Y” winding, or line-to-line voltage of a delta winding. (2) E1 = CF3F E
19
B. Polarity The lead polarity (or polarity) of a transformer is a designation of the relative instantaneous directions of currents in its leads. Primary and secondary leads are said to have the same polarity when at a given instant the current enters the primary lead in question and leaves the secondary lead in question in the same direction as though the two leads formed a continuous circuit. The lead polarity of a single-phase transformer may be either additive or subtractive. If one pair of adjacent leads from the two windings in question is connected together and a small voltage is applied to one of the windings, then the connection behaves as an auto transformer with the secondary voltage adding to or subtracting from the primary voltage. The polarity determination is as follows: a. The lead polarity is additive if the voltage across the other two leads of the windings in question is greater than that of the higher voltage winding alone. b. The lead polarity is subtractive if the voltage across the other two leads of the windings in question is less than that of the higher voltage winding alone.
Additive E3 > E1 Subtrative E3 < E1
By industry standards, single-phase distribution transformers 200 kVA and smaller, having high voltage windings rated 8660 volts or less have additive polarity. All other single-phase transformers have subtractive polarity. The polarity of a three-phase transformer is fixed by the internal connections between phases as well as by the relative locations of leads; it is usually designated by means of a vector diagram showing the angular displacements of windings and a sketch showing the marking of leads. The vectors of the vector diagrams represent induced voltages, and the recognized counterclockwise direction of rotation of the vectors is used. The vector representing the voltage of a given winding is drawn parallel to that representing the corresponding voltage of any other winding having the same phase.
20
C. Terminal Designations 1. Pad Mounted The terminal designations for pad-mounted distribution transformers are clearly marked at the terminals of both the high and low voltage. 2. Pole Mounted For pole mounted distribution transformers, the terminal designations follows:
(a) Single-Phase Pole Mounted Connection
Additive Polarity
Subtractive Polarity
E/2E with three external low-voltage terminals Series or Three-Wire
Parallel E/2E with four external low-voltage terminals Series or Three-Wire
Parallel E
Note: The H1 terminal for either additive or subtractive polarity is located on the lefthand side when facing the low-voltage terminals.
21
Terminal Designations (Continued) (b) Three-Phase Pole Mounted
All three-phase pole mounted distribution transformers have terminal designations as shown above regardless of internal connection. Neutral terminals (HV and/or LV) will exist as required by the winding connection and will be noted on the transformer nameplate.
D. Short-Circuit Ratings (ANSI C57.12.00)
The short-circuit ratings for distribution transformers are set by industry standards. The maximum magnitude required for units with secondary voltages rated less than 600 V is as follows:
1 c kVA
3 c kVA
Rating (times normal)
5-25 37.5-100 167-500
15-75 112.5-300 500 750-2500
40 35 25 1/ZT*
Two winding distribution transformers with secondary voltages rated above 600 volts are required to withstand short-circuits limited only by the transformer’s impedance. The duration of the short-circuit current is determined by 500 kVA and Below ________________ 1250 t= l2
where:
750-2500 kVA ___________ t = 1.0
t = duration (seconds) I = symmetrical short-circuit current (per unit) *1/ZT = The short circuit current will be limited by the transformer impedance only. ZT is transformer per unit impedance.
E. Sound Level Ratings (NEMA TR-1)
The sound level ratings for distribution transformers are set by industry standards. The maximum sound level (A weighted response curve) is: kVA Rating 0-50 51-100 101-300 301-500 -750 -1000 -1500 -2000 -2500
22
Sound Level (dB) 48 51 55 56 57 58 60 61 62
F. Tolerance Definitions (ANSI C57.12.00-1987) 1. Impedance (two-winding transformers) The impedance of a two-winding transformer with impedance voltage larger than 2.5% shall have a tolerance of ± 7.5% of the specified value; and the tolerance for those with impedance voltage 2.5% or less shall have a tolerance of ± 10% of the specified value. Differences of impedance between two duplicate two-winding transformers when two or more units of a given rating are produced by one manufacturer at the same time shall not exceed 7.5% of the specified value. Transformers shall be considered suitable for operation in parallel if impedances come within the limitations of the foregoing paragraphs, provided turns ratios and other controlling characteristics are suitable for such operation. (See Paralleling) 2. Losses The total losses of a transformer are the sum of the excitation losses and the load losses at rated load (with winding temperature at 85°C). Unless otherwise specified, the losses represented by a test of a transformer, or transformers, on a given order, shall not exceed the specified losses by more than the percentages below:
No. of Units On One Order
Basis of Determination
1 2 or more 2 or more
1 unit each unit average of all units
No Load Losses (Percent)
Total Losses (Percent)
10 10 0
6 6 0
G. Impedance Calculations Transformer impedance is shown on the transformer nameplate (Note: transformer impedance, reactance and resistance are typically given in percent or per unit). If the transformer load losses are known, the impedance may be separated into its reactive and resistive components. Z R X kVA Cu R=
impedance (percent) resistance (percent) reactance (percent) transformer kVA rating load loss at rated load at 85°C (watts) Cu
10 kVA
X = CFF Z2FFFF – R2
23
H. Efficiency Calculations The efficiency of a transformer is defined as the ratio of the output power to the input power. It can be calculated at any load and power factor if the transformer losses are known. E L kVA Cu Fe a E=
-
efficiency (percent) load (per unit) transformer kVA rating load loss at rated load at 85°C (watts) no load (excitation) loss (watts) power factor angle
L.kVA.cosa.10 5 percent (L.kVA.cosa.10 3) + Fe + L2.Cu
At rated load and unity power factor E=
kVA.105 kVA.10 3 + Fe + .Cu
percent
Regardless of the load power factor angle, it can be shown that the per unit load which results in maximum efficiency is: L (maximum efficiency) =
I. Regulation Calculations The voltage regulation of a distribution transformer is the change in output voltage which occurs when the load is reduced from rated value to zero with the primary terminal voltage maintained constant. The regulation can be calculated from the equations below or by the nomograph which follows: R X REGa -
resistance(percent) reactance (percent) percent voltage regulation power factor angle (positive for inductive load)
1 REG = [R2 + X2 + 200. (X .sina + R.cosa) + 10,000] /2 – 100
For unity power factor 1 REG = [R2 + X2 + 200R + 10,000] /2 – 100
24
Regulation Chart
Place straight edge at percent resistance, scale one, and at percent reactance, scale nine. Read the percent regulation at different power factors as given by scales two to eight inclusive
25
J. Performance Example Example Transformer Ratings (Typical) Single-phase kVA High voltage Low voltage No load (excitation) loss Total loss at rated load Impedance
25 7200v 120/240v 104 watts 419 watts 1.6%
For the above transformer determine the following: (1) Nominal reactance (2) Minimum impedance (3) Minimum efficiency at rated load (4) Expected efficiency at 50% load (5) Expected regulation Assume an inductive power factor (cosa = 0.85) 1. Nominal Reactance Cu (419–104) R = ______ = _______ = 1.26% 10.kVA 10 .25 2 – 1.262 = 0.99% X = CFF FFFFFFFF Z2FFFF – R 2 = CFF 1.6
2. Minimum Impedance Minimum Z = (1 – 0.10) . (Nominal Z) = 1.44% 3. Minimum Efficiency at Rated Load Maximum total loss = 1.06 . 419 = 444 watts E=
=
kVA . cosa . 105 . kVA cosa . 10 3 + (Fe + Cu) (25) . (.85) . 105 = 98.0% (25) . (.85) . 10 3 + (444)
4. Expected Efficiency at 50% Load L=
0.5
L . kVA . cosa . 105 E= . L kVA . cosa . 10 3 + Fe + L2 Cu =
(0.5) . (25) . (.85) . 10 5 = 98.3% . . (0.5) (25) (.85) . 10 3 + 104 + (0.5)2 . (419 – 104)
5. Expected Regulation 1 REG = [R2 + X2 + 200. (X.sina + R. cosa) + 10,000] /2 – 100 1 = [1.262 + 0.992 + 200.(0.99.0.53 +1.26.0.85) + 10,000] /2 –100
= 1.596
26
K. Secondary Fault Currents — 120/240 Volt Systems Service to individual residences in the United States most always is single-phase three-wire operating at 120 volts from phase-to-neutral, and 240 volts from phase-to-phase. In order to select service entrance equipment with adequate interrupting rating, or to coordinate over-current protective devices in the transformer-secondary systems, the available currents for a bolted fault (short circuit) must be known. This section presents equations and data which can be used to calculate the available currents for both phase-tophase (240 volt) and phase-to-neutral (120 volt) faults. The equations for calculating these currents are quite simple and can be easily evaluated with a handheld pocket calculator Fault Current Equations Figure K.1 gives, for convenient reference, the equations necessary for calculating the available currents for both 240 volt and 120 volt bolted faults, and defines the terms appearing in the equations. Before explaining the use of the equations, the assumptions used in arriving at these are discussed.
Figure K.1 27
The impedance of the primary system supplying the distribution transformer is very small in comparison to that of the distribution transformer and secondary circuit up to the point of fault. The effect of this assumption is to make the calculated values of current for a bolted fault in the secondary system slightly higher than those which result when the effect of primary impedance is included. Increasing the “stiffness” of the primary system, reducing the kVA size of the transformer, or increasing the secondary circuit length to the fault point reduces the difference between the approximate and more exact calculated values of bolted fault current. In contrast, the difference between the approximate and more exact values will be greater for “weak” primary systems, large distribution transformers, and short secondary circuits. For most cases where the calculations are made to determine available fault current at the service entrance for sizing equipment, or to determine maximum currents at which overcurrent protective devices must coordinate, the difference resulting from the assumption is negligible. However, for those cases where the calculated current using methods neglecting primary impedance is slightly higher than the interrupting rating of a fuse or breaker in the secondary system, or where the calculated current is slightly above the value at which overcurrent protective device coordination can be achieved, then including the effect of primary system impedance may show that a “problem” does not exist. Calculations including the effects of primary system impedance are not contained in this guide. Reference to Figure K.1 shows that the expressions for calculating the available current for the 240 volt and 120 volt bolted faults are different. While the 240 volt fault current can be calculated from a knowledge of the “full winding” impedance of the transformer, the calculation of the 120 volt fault current requires a knowledge of the transformer “half winding” impedance. As the relationship between transformer “half winding” and “full winding” impedance is not fixed and can vary from design to design, the most typical relationship for present day designs was used in arriving at the equation for 120 volt fault current. Letting RT + jXT be the “full winding” impedance in percent on nameplate kVA rating looking into the primary winding, the “half winding” impedance in percent on nameplate kVA can be approximated by 1.5 R T + j2.0 XT. Also notice from Figure K.1 that the equations do not include the effect of any metering impedances which may be present in the circuit, or any “fault” impedance. Including these impedances will further reduce the calculated values of fault current.
28
The steps to follow when using the equations in Figure K.1 to calculate the bolted fault currents are as follows: 1. Calculate the transformer resistance in ohms at secondary terminals X1-X3 (RT in Figure K.1). This requires that the transformer total losses at full load in watts, and no load losses in watts be known (WTOT and WNL respectively in Figure K.1). 2. Calculate the transformer leakage impedance in ohms at secondary terminals X1-X3 (Z T in Figure K.1). This requires that the transformer nameplate impedance in percent (Z%) be known. 3. Calculate the transformer leakage reactance in ohms at secondary terminals X1-X3 (X T in Figure K.1). 4. Determine the resistance of the secondary circuit in ohms per 1000 feet for a 240 volt fault (RS). Also determine the reactance of the secondary circuit in ohms per 1000 feet for a 240 volt fault (X S). Typical values for RS and XS in ohms per 1000 feet are given in Tables 1 and 2 for circuits using aluminum phase conductors under the header “240 V FAULTS”. The values in Table 1 are for triplex cable, and those in Table 2 are for rack mounted conductors. From these tables notice that the resistance values are the same, but the reactance values are greater with the rack mounted conductors. This is due to the larger spacing. 5. Calculate the available current for a 240 volt bolted fault (I 240) using the equation in Figure K.1 and the values calculated in steps 1 through 4. 6. Determine the resistance (R S1) and reactance (X S1) of the secondary circuit in ohms per 1000 feet for a 120 volt fault. typical values for RS1 and X S1 in ohms per 1000 feet are given in Tables 1 and 2 for circuits using aluminum conductors under the header “120 V FAULTS”. In both tables, the values listed are for circuits using a reduced size neutral conductor. If a full size neutral conductor is used, then the impedance values given under the header “240 V FAULTS” should also be used for the calculation of the 120 volt fault currents. 7. Calculate the available current for a 120 volt bolted fault (I 120) using the equation in Figure K.1 and the values calculated in steps 1 through 3 and step 6.
29
Example Calculations The use of the equations in Figure K.1 is illustrated with the following: A 50 kVA transformer with total losses at full load of 759 watts, and no load losses of 204 watts has an impedance of 1.75 percent. A service entrance circuit which is 80 feet in length using 3/0 aluminum triplex with reduced neutral is connected directly to the transformer terminals. What is the available current for both a 240 and 120 volt bolted fault at the end of the service? From the statement of the problem: WTOT = 759 watts WNL = 204 watts kVA = 50
Z = 1.75 percent L = 80 feet
The calculations proceed following the steps outlined. 1. R T = 0.0576 2. Z T = 0.576
759 – 204 = 0.012787 ohms 502
1.75 = 0.02016 ohms 50
3. X T = CFFFFFFFFFFFFFFFFF .020162 – .0127872 = 0.015586 ohms 4. From Table 1, the resistive and reactive components of the impedance for a 240 volt fault with 3/0 aluminum triplex cable are: R S = 0.211 ohms per 1000 feet XS = 0.0589 ohms per 1000 feet 5. Placing these values of R T, X T, RS, X S, and L into the equation for I 240 in Figure K.1 gives: I 240 = 6676.6 amperes rms symmetrical Note that the large number of significant digits included in these calculations is not to suggest that they are accurate to the last digit, but to aid those who want to check their own calculations. 6. From Table 1, the resistive and reactive components of the impedance for a 120 volt fault with 3/0 aluminum triplex cable (reduced neutral) are: R S1 = 0.273 ohms per 1000 feet X S1 = 0.0604 ohms per 1000 feet 7. Placing the above values into the equation for I 120 in Figure K.1 gives: I120 = 4071.1 amperes rms symmetrical For this example notice that at a distance of 80 feet from the transformer, the available current for the 120 volt bolted fault is considerably less than that for a 240 volt fault. However, from the equation for I 240 and I120 in Figure K.1, notice that for a fault at the transformer secondary terminals (L = 0.0 feet), the available current for a bolted 120 volt fault is greater than that for a 240 volt fault. Thus at some distance L from the transformer, I 240 and I 120 would be equal, and at distances greater than this, the available current for a 240 volt fault will be higher.
30
Figure K.2 is a plot of the available current for both the 120 and 240 volt bolted faults vs. the distance from the transformer terminals to the fault point in feet. The curves are for transformer sizes of 50, 75, and 100 kVA supplying a secondary circuit made with 3/0 aluminum triplex with reduced neutral. From these curves notice that:
Figure K.2 (a) The available current for both the 120 and 240 volt faults is rapidly reduced as the fault is moved away from the transformer, even for the rather large 3/0 aluminum service conductor. (b) With the 3/0 aluminum service conductor, the available current for a 120 volt fault is less than that of a 240 volt fault at distances greater than about 10 feet from the 50, 75, or 100 kVA transformer. For most all single-phase services rated 200 amperes or less, the available current at the service entrance for the 120 volt fault is less than that of the 240 volt fault. (c) As the distance from the transformer to the fault location becomes large, the available current for both the 120 and 240 volt faults becomes independent of the transformer size, especially for the 120 volt fault.
31
32
Aluminum Neutral Cond. ____________________ Size No. of (AWG or MCM) Strands ___________ ______ 4 7 3 7 2 7 1 19 1/0 19 2/0 19 3/0 19 4/0 19 300 37
120 Volt Faults ____________ RS1 XS1 (j/1000 Ft.) (j/1000 Ft.) ________ ________ .691 .0652 .547 .0659 .435 .0628 .345 .0629 .273 .0604 .217 .0588 .177 .0583 .134 .0570 .095 .0547 240 Volt Faults ____________ RS XS (j/1000 Ft.) (j/1000 Ft.) ________ ________ .534 .0633 .424 .0659 .335 .0616 .266 .0596 .211 .0589 .167 .0576 .142 .0574 .102 .0558 .072 .0530
Aluminum Phase Cond. ___________________ Size No. of (AWG or MCM) Strands ___________ ______ 2 7 1 19 1/0 19 2/0 19 3/0 19 4/0 19 250 37 350 37 500 37
Aluminum Neutral Cond. ____________________ Size No. of (AWG or MCM) Strands ___________ ______ 4 7 3 7 2 7 1 19 1/0 19 2/0 19 3/0 19 4/0 19 300 37
120 Volt Faults ____________ RS1 XS1 (j/1000 Ft.) (j/1000 Ft.) ________ ________ .691 .223 .547 .217 .435 .211 .345 .205 .273 .199 .217 .193 .177 .189 .134 .182 .095 .174 240 Volt Faults ____________ RS XS (j/1000 Ft.) (j/1000 Ft.) ________ ________ .534 .217 .424 .212 .335 .204 .266 .199 .211 .193 .167 .188 .142 .184 .102 .176 .072 .168
Table 2. Typical Impedances for 120/240 Volt Circuits With Rack Mounted Conductors
Aluminum Phase Cond. ___________________ Size No. of (AWG or MCM) Strands ___________ ______ 2 7 1 19 1/0 19 2/0 19 3/0 19 4/0 19 250 37 350 37 500 37
Table 1. Typical Impedances for 120/240 Volt Circuits With Triplex Cable
Notes: (1) Resistance values based on a conductor temperature of 25°C. (2) Reactance values based on secondary rack with 12 inch spacing between conductors with neutral in top position and phase conductors in the two lower positions. Resistance and reactance values given for 120 volt fault assume fault is to phase conductor in middle position in rack. (3) For secondary circuits with full size neutral, use resistance and reactance values given for 240 volt fault for both 120 and 240 volt faults.
Notes: (1) Resistance values based on a conductor temperature of 25°C. (2) Reactance based on following: (a) 600 volt insulation with all 3 insulated conductors in contact. (b) For 120 volt (Phase-to-Neutral Fault), all current returns in the neutral conductor with no current returning in the earth. (3) Insulation thickness is 0.062 inch for #4 to #2, 0.078 inch for #1 to 4/0, and .094 inch for 250 to 500 MCM. (4) For secondary circuits with full size neutral, use resistance and reactance values given for 240 volt fault for both 120 and 240 volt faults.
III. Three-Phase Transformers and Banks
Page
A. Application Considerations ...................................................... 1. Types of distribution systems ............................................. a. Primary (source) systems .............................................. b. Secondary (service) systems ......................................... 2. Angular displacement (phase shift) .................................... 3. Neutral grounding ............................................................... a. Primary neutral grounding ............................................. b. Secondary neutral grounding ......................................... 4. Ferroresonance ................................................................... a. Primary winding connections which can result in Ferroresonance .......................................................... b. Primary winding connections which can prevent or minimize the possibility of Ferroresonance ................... B. Summary of Common Connections ......................................... 1. Delta-delta ........................................................................... 2. Delta-wye ............................................................................ 3. Wye-delta ............................................................................ 4. Wye-wye .............................................................................. 5. Grounded wye-wye ............................................................. 6. T-T (O degree angular displacement) ................................ 7. T-T (30 degree angular displacement) ............................... 8. Open Wye-Open Delta ........................................................ 9. Open Delta-Open Delta ...................................................... C. Common Three-Phase Banks Using Single-Phase Transformers .....................................................
34 34 34 34 35 35 35 35 36 37 38 41 41 42 43 44 45 45 46 46 47 48
33
III. Three-Phase Transformers and Banks This section presents many important factors to be considered when selecting the connections used for both three phase transformers and three-phase banks of the single-phase transformers applied in threephase distribution systems. A summary of commonly encountered connections is provided. In addition, connection diagrams using singlephase transformers for three-phase banks are shown.
A. Application Considerations 1. Types of distribution systems A three-phase distribution transformer, or a bank of singlephase distribution transformers should be thought of as a system component which connects the primary to the secondary system. Since it is a system component, proper application and determination of permissible connections requires an understanding of the characteristics of both the primary system which will supply the transformer, and the secondary system which will be supplied by the transformer. a.
Primary (Source) Systems Distribution systems are either effectively grounded, impedance grounded, or ungrounded. Most electric utility distribution systems in this country are three-phase 4-wire multi-grounded neutral systems which are effectively grounded. (An effectively grounded system is one where at any point in the system the ratio of zero-sequence reactance to positive-sequence reactance is less than three, and the ration of zero-sequence resistance to positivesequence reactance is less than one.) With a 4-wire effectively grounded neutral system, the primary windings of the distribution transformers can be connected from either phase to phase or phase to neutral. This permits usage of the following connections: Delta, open delta, grounded wye, open wye, floating wye, and T. Whether the neutral point of wye connected primary windings should or should not be connected to the system neutral depends upon the connections used for the secondary windings. Although they are not commonly used by electric utilities for distribution, impedance grounded or ungrounded systems are frequently found in industrial plants. These systems provide no path to carry neutral load current. Thus, distribution transformers applied must be connected phase to phase using either delta, open delta, floating wye, or T connected windings.
b.
Secondary (Service) Systems Secondary systems supplied from distribution transformers and operating at 600 volts or less usually are either 3-wire ungrounded, or 4-wire grounded. To supply a 3-wire ungrounded (delta) system, the transformer secondary winding may be connected in delta, open delta, floating wye, or T. Loads which require both single-phase 3-wire 120/240 volt service and three-phase 240 volt service can be supplied by a 4-wire service consisting of transformers with secondary windings connected delta or open delta
34
with a center tap ground on one leg of the delta. In the 4-wire grounded (wye) system, the transformer secondary windings must have a neutral point which can be grounded. The 4-wire grounded secondary service can be supplied by either the wye connection or the T connection with the neutral point grounded. 2. Angular Displacement (Phase Shift) For standard three-phase connections the phase-to-neutral voltage on the primary side either leads that on the secondary side by 30° or is in phase with the phase-to-neutral voltage on the secondary side. The delta delta and wye wye connections produce no phase shift. The delta wye and wye delta connections produce the 30° phase shift. The T-T transformer can be designed to exhibit either a 30° or a 0° phase shift. When paralleling three-phase transformers or banks, the phase shift of each must be the same. In addition, the 30° phase shift has an effect on the coordination of overcurrent protective devices located on the primary and secondary sides of the transformer. For unsymmetrical faults the line currents do not transform in proportion to the voltage ratings. Of particular importance is a line-to-line fault on the transformer secondary. For the connections which have a 30° phase shift, this fault produces a fault current in one primary phase which is 1.15 times the secondary fault current on a per unit basis. This additional 15% must be considered to achieve selective coordination. 3. NeutraI Grounding Some transformer connections or winding connections (wye or T) have a neutral point on either the primary windings, secondary windings, or both, which can be grounded. That is, the neutral point of the primary windings can be connected to the multi-grounded neutral conductor of the primary system, or the neutral point of the secondary windings can be grounded to establish a 4-wire grounded wye system. Whether the neutral point of windings should or should not be grounded depends on factors discussed below. a.
Primary Neutral Grounding For the primary neutral point to be grounded, the primary source must be a 4-wire multi-grounded neutral system. In addition, it is generally undesirable that a distribution bank act as a ground source for the primary system. To prevent creation of a grounding bank, a primary wye should only be grounded if the secondary is also connected in wye and a T primary should never be grounded. Note however that the open wye connection must be grounded at the neutral point to function properly.
b.
Secondary Neutral Grounding To supply phase to neutral connected load on the secondary, a low impedance ground source must be established. This can be achieved by grounding the neutral of a secondary wye connection provided that the primary is connected either delta or wye grounded supplied by a 4-wire multi-grounded neutral (effectively grounded) source. The neutral of a secondary T connection may also be grounded. In addition, a delta or open delta winding may be grounded at any one point. 35
4. Ferroresonance Ferroresonance is a non-linear resonance which can occur during open conductor (single-phase) conditions in the distribution system. When ferroresonance occurs, it is characterized by high overvoltages whose waveform contains appreciable harmonics. The transformers involved in the ferroresonant circuit may emit unusual noises which frequently are described as rattling, rumbling, or whining sounds. These are considerably different than those which emanate from the transformer when energized at rated frequency and voltage. Overvoltages of five times normal and higher have been measured during ferroresonant oscillations in test circuits. Some causes of open conductor conditions which may result in ferroresonance are: (1) the operation of single-pole overcurrent protective devices such as fuses or single-pole reclosers, (2) normal switching operations with single-pole devices such as distribution cutouts to energize or de-energize a transformer, and (3) failure to connect jumpers. Whether ferroresonance will occur during open conductor conditions depends to a great extent upon the connections used for the primary windings in a distribution transformer bank or in a three-phase distribution transformer. Under normal conditions where all three primary phases to the transformer bank are energized through a continuous path from the source, ferroresonance will not occur for any of the connections used for the primary windings. But when an open conductor condition occurs, the non-linear inductance of a transformer or transformer bank, with certain connections, can be placed in series with system capacitance. If the capacitance lies within a specified range, ferroresonance may result. However, with other transformer connections, ferroresonance will not occur during open conductor conditions because the non-linear inductances cannot be inserted in series with system capacitances.
Figure 4.1: Cable-fed transformer with single-pole switching devices located at the junction between the overhead and underground circuits. Figure 4.1 illustrates a frequently encountered system condition which produces ferroresonance. An unloaded three-phase pad mounted transformer with delta connected primary windings is supplied from an open-wire line through a cable circuit. At the riser or transition pole, the cable circuit is connected to the open wire line using distribution cutouts. Notice that during the switching operation (open conductor condition) where only 36
the switch in phase A is closed as illustrated, the non-linear inductances of the transformer windings between phases A and B, and phases A and C, are placed in series with the cable capacitance on the open phases. This makes a series L-C circuit where the L is non-linear, and if the parameters are in the proper range, ferroresonance will occur. a.
Primary Winding Connections Which Can Result in Ferroresonance
Theoretically, ferroresonance can occur during open conductor conditions in either one or two phases if the primary windings of the distribution transformers are connected in delta, open delta, floating wye, or tee. Whether it does or does not occur with these “ungrounded” connections for the primary windings depends upon the amount of capacitance between the open conductor and transformer, the transformer internal capacitances, the transformer size, the system voltage, and the amount of load connected to the secondary terminals of the transformer, or the amount of load on the primary circuit between the open conductor and transformer. Studies have shown that ferroresonance is more likely to occur with cable circuits (due to higher capacitance) than open-wire lines, with small transformers, at higher primary voltage levels (more likely at 35 kV than 4 kV voltage level), and with unloaded transformers. Industry experience has shown that in overhead distribution systems operating at 15 kV and below, overvoltages and ferroresonance usually do not occur during open conductor conditions, even when the ungrounded primary winding connections are used for transformers. Ferroresonance became an important concern in the utility industry with the advent of underground distribution and the use of 25 and 35 kV class voltages. In higher voltage (25 kV and 35 kV) overhead systems, overvoltages and ferroresonance have occurred when single-pole switching is performed at the terminals of small transformer banks with their primary windings connected in floating wye or delta. This is due to the internal capacitances of the transformers. Figure 4.2 summarizes in a qualitative fashion the probability of ferroresonance occurring in 15, 25, and 35 kV class overhead systems when the switching is performed at the terminals of small banks made from single-phase units. The probability of ferroresonance and the associated overvoltages is very high if the circuit between the location of the open conductor and the transformer is made from shielded cable and operates at voltage levels in either the 15, 25, or 35 kV class. This is because the capacitance per unit length of a cable circuit is in the range of 50 times that of open wire lines. A system illustrating this situation is shown in Figure 4.1. Because of the high probability of ferroresonance in underground systems using conventional single-pole switching devices, many system operators will not use the ungrounded primary winding connections in cable-fed transformers. If, however the transformer primary windings are ungrounded, as with the delta, open delta, wye, and tee connections, and the circuit between the transformer and possible location of an open conductor (single-phase) condition is made from cable, the possibility of ferroresonance can be minimized with the following measures. 37
(1) Application of only three-pole gang operated switches and fault interrupters. This minimizes the possibility of having single-phase conditions. (2) Location of the single-pole switches and overcurrent protective devices only at the transformer terminals. (3) Connection of resistive load to the secondary terminals of the transformer during remote single-pole switching. Although these measures can be very effective, many operators of underground systems consider them unacceptable for either economical, operational, or technical reasons. Instead, they prefer to use transformer connections which have either a zero or very low probability of ferroresonance during open conductor conditions at a location remote from the transformer.
Figure 4.2: Probability of ferroresonance in overhead systems when switching is performed at the terminals of small transformer banks made from single-phase units. b.
Primary Winding Connections Which Prevent Or Minimize Possibility of Ferroresonance
When the primary windings of single-phase distribution transformers used in a bank are connected in open wye or grounded wye, or if a three-phase unit with the grounded wye primary windings employs triplex construction, ferroresonance will not occur during most open conductor conditions in the primary system. This is true for both overhead and underground systems operating up through 35 kV. But if either a floating wye or delta connected shunt capacitor bank is installed on the primary line between the transformer bank and location of the open conductor, ferroresonance may occur. However, the use of these connections for capacitor banks is very uncommon in distribution systems operating in the 15 kV class and above. If there is a very long length of open wire line between 38
the location of the open conductor and transformer bank with grounded wye or open wye primary windings, and no other load is connected to the line beyond the open point, ferroresonance can occur because of the phase-to-phase capacitance of the open wire line. The probability of such conditions existing, even in 25 and 35 kV rural distribution systems, is very remote. Thus, for practical purposes, ferroresonance will not occur when the grounded wye or open wye connections are used for the primary windings with single-phase units, or a three-phase unit with triplex construction. The probability of ferroresonance is zero when the switching is performed at the terminals of transformer banks in overhead systems with the grounded wye or open wye connected primaries at all voltages as illustrated in Figure 4.2. When the grounded wye-grounded wye or grounded wyefloating wye connections are used in a transformer constructed on a four- or five-legged core, overvoltages and ferroresonance may occur during open conductor conditions at a remote point when cable circuits are involved. Test data shows that crest voltages as high as 2.35 per unit are possible, but usually they are considerably less than this. In contrast, overvoltages of 5 per unit and higher are possible when the transformer has the ungrounded primary winding connections. Furthermore, the length of primary cable circuit which can be used with transformers with four- or five-legged core and grounded-wye primary is in the range of 50 times that possible when the ungrounded primary connections are used when the voltage on the open phase is limited to 1.25 per unit. Although the use of triplex construction essentially eliminates the possibility of ferroresonance in cable-fed three-phase transformers with the grounded wye primary, such construction generally makes the transformer larger, heavier, and more costly than conventional four- or five-legged core units. Most system operators, based on the good experience and performance they have had with the grounded wye primaries on fourand five-legged cores, have not been able to justify the added cost for triplex construction. If it is necessary to further minimize the possibility of ferroresonance when the grounded wye primary is used on a fouror five-legged core, the measures listed below can be employed: (1) Application of only three-pole gang operated switches and fault interrupters. This minimizes the possibility of having single phase conditions. (2) Location of single-pole switches and overcurrent protective devices only at the transformer terminals. (3) Connection of resistive load to the secondary terminals of the transformer during remote single-pole switching. The preceding discussion of ferroresonance is both very brief and very qualitative in content. As it may be necessary to quantify certain aspects of ferroresonance, such as determining the maximum length of cable circuit which can be used between a switch and transformer if voltage is to be limited to a specified value, the reader is referred to the many references which exist on the subject. A few are listed below. 39
References 1. Schmid, R. L. “An Analysis and Results of Ferroresonance”. Transmission and Distribution, pp. 114-117, Oct. 1969. 2. Kratz, E. F., Manning, L. W., and M. Maxwell. “Ferroresonance in Series Capacitor-Distribution Transformer Applications.” AIEE Transaction (Power Apparatus & Systems), vol. 78, pp. 438-449, August 1959. 3. Young, F. S., Schmid, R. L., and P. I. Fergestad. “A Laboratory Investigation of Ferroresonance in Cable Connected Transformers,” IEEE Transactions on Power Apparatus and Systems, vol. PAS-87, pp.1240-1249, May 1968. 4. Crann, L. B., and R. B. Flickinger, “Overvoltages on 14.4/24.9 kV Rural Distribution Systems.” AIEE Transactions (Power Apparatus and Systems), vol. 73, pp. 1208-1212, Oct. 1954. 5. Smith, D. R., Swanson, S. R., and J. D. Borst. “Overvoltages With Remotely-Switched Cable-Fed Grounded Wye-Wye Transformers.” IEEE Transactions on Power Apparatus and Systems, vol. PAS-94, pp. 1843-1853, Sept./Oct. 1975.
40
B. Summary of Common Connections
DELTA-DELTA Connection Phasor Diagram:
Angular Displacement (Degrees): 0 Source: Suitable for both ungrounded and effectively grounded sources. Service: Suitable for 3-wire service or for 4-wire service with a midtap ground. Notes: 1. With one unit out of service, a bank of single-phase units can be reconnected as an open delta, open delta bank. With one of three identical units out of service, the rating of the bank when supplying only three-phase load is about 57.7 percent of the bank rating when all three units are in service. 2. Caution: Each unit in a bank of single-phase units must be connected for the same voltage ratio, otherwise high circulating currents can occur. Prior to completing a closed delta secondary connection, the voltage between the two transformers closing the delta should be checked to verify the voltage ratios and connections. 3. Impedance mismatch among units of a single-phase bank will require derating of the bank. 4. Single-phase units having a secondary breaker should not be used for a bank providing 4-wire (mid-tap ground) delta service. 5. Frequently installed with mid-tap ground on one leg when supplying combination three-phase and single-phase load where the three-phase load is much larger than single-phase load. 6. Single-phase transformers with primary windings rated E volts usually are used for this bank.
41
DELTA-WYE Connection Phasor Diagram:
Angular Displacement (Degrees): 30 Source: Suitable for both ungrounded and effectively grounded sources. Service: Suitable for 3-wire service or for 4-wire grounded service with a XO grounded. Notes: 1. With XO grounded, the bank acts as a ground source for the secondary system. 2. Fundamental and harmonic frequency zero-sequence currents in the secondary lines supplied by the transformer do not flow in the primary lines. Instead these zero-sequence currents circulate in the closed delta primary windings. 3. When supplied from effectively grounded primary system, ground relay for primary system does not see load unbalances and ground faults in the secondary system. 4. Single-phase transformers with primary windings rated E volts usually are used for this bank.
42
WYE-DELTA Connection Phasor Diagram:
Angular Displacement (Degrees): 30 Source: Suitable for both ungrounded and effectively grounded sources. Service: Suitable for 3-wire service or for 4-wire delta service with a mid-tap ground. Notes: 1. Neutral point of primary windings with unbalanced and/or singlephase secondary load is locked at ground potential if each unit in bank has same impedance. Even with different units in the bank, neutral point of primary windings is essentially locked at ground potential. 2. Grounding the primary neutral of this connection would create a ground source for the primary system. This could subject the transformer to severe overloading during a primary system disturbance, or load unbalance. 3. With one unit out of service, a bank of single-phase units can be reconnected as an open wye—open delta bank provided that the source is 4-wire effectively grounded. With one of three identical units out of service, the rating of the bank when supplying only three-phase load is about 57.7 percent of the bank rating when all three units are in service. 4. Single-phase units with secondary breakers should not be used whether there is or is not a center tap ground on one leg. Opening of breaker in one leg causes severe voltage unbalance and wave form distortion. 5. Frequently installed with mid-tap ground on one leg when supplying combination three-phase and single-phase load where the three-phase load is much larger than the single-phase load. 6. When used in 25 and 35 kV three-phase 4-wire primary systems, ferroresonance can occur when energizing or de-energizing the bank using single pole switches located at the primary terminals. With smaller kVA transformers in the bank, the probability of ferroresonance is higher. 7. Single-phase transformers rated E/E1, Y volts usually are used for this bank (E1 = CFF 3 E).
43
WYE-WYE Connection Phasor Diagram:
Angular Displacement (Degrees): 0 Source: Suitable for both ungrounded and effectively grounded sources. Service: Suitable for 3-wire service only, even if XO is grounded. Notes: 1. This connection is incapable of furnishing a stabilized neutral and its use may result in phase-to-neutral overvoltage (neutral shift) as a result of unbalanced phase-to-neutral load. 2. When supplied from effectively grounded source and made from single-phase units, very high third harmonic voltage (of the order of 50%) appears between neutral point of primary windings and ground (tank). 3. When supplied from ungrounded source and made from singlephase units, third harmonic voltages appear from neutral point of primary windings and ground, and from primary lines to ground. Division of total third harmonic voltage (of order of 50%) depends upon capacitances of primary lines and transformers. 4. If a three-phase unit is built on a three-legged core, the neutral point of primary windings is practically locked at ground potential.
44
GROUNDED WYE-WYE Connection Phasor Diagram:
Angular Displacement (Degrees): 0 Source: Suitable for a 4-wire effectively grounded source only. Service: Suitable for 3-wire service or for 4-wire grounded service with XO grounded. Notes: 1. Three-phase transformers with this connection may experience stray flux tank heating during certain external system unbalances unless the core configuration utilized provides a return path for the flux. 2. Fundamental and harmonic frequency zero-sequence currents in the secondary lines supplied by the transformer also flow in the primary lines (and primary neutral conductor). 3. Ground relay for the primary system may see load unbalances and ground faults in the secondary system. This must be considered when coordinating overcurrent protective devices. 4. Three-phase transformers with the neutral points of the high voltage and low voltage windings connected together internally and brought out through an HOXO bushing should not be operated with the HOXO bushing ungrounded (floating). To do so can create very high voltages in the secondary systems. T-T Connection Phasor Diagram:
Angular Displacement (Degrees): 0 Source: Suitable for both ungrounded and effectively grounded sources. Service: Suitable for 3-wire service or for 4-wire service with XO grounded. Can also supply 4-wire delta service. Notes: 1. Because of winding voltages required, this connection is generally only available as a three-phase transformer. 2. Neutral point of primary windings, if available, should not be grounded unless it is desired that the transformer serve as a grounding bank.
45
T-T Connection Phasor Diagram:
Angular Displacement (Degrees): 30 Source: Suitable for both ungrounded and effectively grounded sources. Service: Suitable for 3-wire service or for 4-wire service with XO grounded. Can also supply 4-wire delta service. Notes: 1. Because of winding voltages required, this connection is generally only available as a three-phase transformer. 2. Neutral point of primary windings, if available, should not be grounded unless it is desired that the transformer serve as a grounding bank. OPEN WYE-OPEN DELTA Connection Phasor Diagram:
Angular Displacement (Degrees): 30 Source: Suitable for a 4-wire effectively grounded source only. Service: Suitable for 3-wire service or for 4-wire delta service with a mid-tap ground. Notes: 1. When two units of the same kVA rating are used to supply only a balanced three-phase load, the combined rating of the two units must be 115 percent of the three-phase load if the load on each transformer is not to exceed nameplate rating. 2. Single-phase units with secondary breaker can be used, even with a mid-tap ground on one leg. However, with the secondary breaker open in only the grounded leg, high voltages due to capacitive coupling may appear from each terminal to ground of the transformer in the other leg. Sufficient phase-to-neutral connected load will limit these voltages. 3. Can be connected to either a three-phase or V phase primary line. 4. Frequently installed with one large and one small transformer to supply a combination of single-phase and three-phase load where single-phase load is much larger than the three-phase load. 5. With ungrounded secondary windings (3-wire service), voltage to ground from one or more secondary phases can be greater than secondary phase-to-phase voltage due to unbalances in the capacitance network. With sufficient length of secondary circuit or connected load, phase-to-ground voltage for each phase will approach in magnitude the phase-to-phase voltage divided by CFF 3. 46
6. When primary terminals H1 and H2 are supplied from the same system phase, the open circuit phase to phase voltage from secondary terminal X1 to X3 is two (2) times normal phase to phase voltage. OPEN DELTA-OPEN DELTA Connection Phasor Diagram:
Angular Displacement (Degrees): 0 Source: Suitable for both ungrounded and effectively grounded sources. Service: Suitable for 3-wire service or for 4-wire delta service with a mid-tap ground. Notes: 1. When two units of the same kVA rating are used to supply only a balanced three-phase load, the combined rating of the two units must be 115 percent of the three-phase load if the load on each transformer is not to exceed nameplate rating. 2. Single-phase units with a secondary breaker can be used, even with a mid-tap ground on one leg. However, with the secondary breaker open in only the grounded leg, high voltages due to capacitive coupling may appear from each terminal to ground of the transformer in the other leg. Sufficient phase-to-neutral connected load will limit these voltages. 3. Can be connected to only a three-phase primary line. 4. Frequently installed with one large and one small transformer to supply a combination of single-phase and three-phase load where single-phase load is much larger than the three-phase load. 5. With ungrounded secondary windings (3-wire service), voltage to ground from one or more secondary phases can be greater than secondary phase-to-phase voltage due to unbalances in the capacitance network. With sufficient length of secondary circuit or connected load, phase to ground voltage for each phase will approach in magnitude the phase-to-phase voltage divided by CFF 3.
47
C. Common Three-Phase Banks Using Single-Phase Transformers
Phase Relation Diagram
HV Connection Diagrams
48
Angular Displacement
Polarity
Connection Diagrams
LV Connection Diagrams
* Represents opposite end of winding from X1; may be X2, X3, or X4 depending upon the low voltage rating (2, 3, or 4 bushing). 49
IV. Loading
Page
A. Paralleling ................................................................................. B. Delta-delta bank loading ........................................................... C. Overloading ............................................................................... D. Single-phase and three-phase loading of symmetrical and unsymmetrical transformer banks .................................... E. Dedicated motor loads ..............................................................
50
51 51 52 53 66
IV. Loading
A. Paralleling Transformers or transformer banks may be connected in parallel to increase capacity by connecting terminals of like designation together provided that the frequency and voltage (including tap setting) ratings are the same. In addition, three-phase transformers or banks must have the same phase shift. Mismatched impedance between the parallel units or banks requires a derating because the load does not then divide in proportion to the kVA ratings. This derating can be approximated as follows: K 1 - Capacity of the unit or bank with the larger percent impedance K 2 - Capacity of the unit or bank with the smaller percent impedance Z 1 - Impedance of unit or bank 1 Z 2 - Impedance of unit or bank 2 Derating factor = e
Z2 . K 1 + K 2 f / (K 1 + K 2) Z1
Example: 25 and 50 kVA single-phase transformers with 1.6 and 2.0 percent impedance respectively. Derating factor = e
1.6 50 + 25 f / (50 + 25) = 0.87 2.0
Parallel rating = 0.87 (50 + 25) = 65 kVA
B. Delta-Delta Bank Loading Unequal turns ratios (voltage rating and tap setting) in delta-delta connected transformer banks can cause large circulating currents within the deltas. Thus, a requirement for such banks is equal turns ratios for all units. Similarly, an impedance imbalance can cause a small circulating current which makes it necessary to derate the bank. For units of equal capacity with one odd impedance, the derating for balanced loading is approximated in the following table: Ratio of odd unit impedance to impedance of other two units _________________________ 1.6 1.5 1.4 1.3 1.2 1.1 10 .9 .8 .7
Derating Factor _______ 0.91 0.93 0.94 0.95 0.97 0.98 1.00 .97 .93 .90 51
C. Overloading The overloading of distribution transformers is a complex subject requiring knowledge of load characteristics, transformer parameters and environmental conditions to be accomplished without damaging the transformer. ABB distribution transformers are fit for planned overloading providing that such overloading is in accord with the ANSI Loading Guide (C57.91). The table below shows the approximate peak overload capability for a typical distribution transformer for normal life expectancy (these values are extracted from C57.91). The table applies to a 30°C ambient; the load capability at other ambients (0–50°C) can be estimated by (1) decreasing the load capability by 1.5% for each degree C that the ambient exceeds 30°C or (2) increasing the load capability by 1.0% for each degree C that the ambient is below 30°C. Peak Loading Capability For Normal Life Expectancy (Per Unit) Peak Load Duration (Hours) _____ 1 2 4 8 24
52
Equivalent Continuous Preload (per unit) 0.50 0.75 0.90 ________________________________ 2.12 1.79 1.50 1.28 1.08
1.96 1.68 1.44 1.25 1.07
1.82 1.57 1.36 1.21 1.07
D. Single-Phase and Three-Phase Loading of Symmetrical and Unsymmetrical Transformer Banks Single-phase distribution transformers can be connected in banks to supply a combination of single-phase and three-phase load. The transformer bank supplying the combination load may be either symmetrical or unsymmetrical. A symmetrical bank is one consisting of three identical single-phase transformers. Most often the primary and secondary windings are connected in either wye or delta. An unsymmetrical bank is one containing only two singlephase transformers, or a bank with three single-phase transformers where all three transformers are not the same. (Note: for loading considerations [not ferro, tank heating], a three-phase transformer can be considered to be a bank consisting of three identical singlephase transformers.) The “4-wire delta” system is a common type used to supply a combination of single-phase and three-phase load. These systems are supplied from a transformer bank with the secondary windings connected in either delta or open delta with a center tap ground on one leg of the delta. The service to the three-phase load is 3-wire at 240 volts, and the service to the single-phase load is 3-wire at 120/240 volts. Transformer banks with their secondary windings connected in grounded wye can also be used to supply a combination of single-phase and three-phase load. In a 4-wire grounded wye system supplied from such a bank, the singlephase load may be connected from either phase-to-neutral or phase-to-phase. When known single-phase and three-phase loads are to be fed from a transformer bank, frequently it is necessary to know the load which will be supplied by each transformer so that it may be properly sized. Equations for calculating the load supplied by each single-phase transformer in the bank are given in Figures D.1 through D.8. The basis and assumptions used in deriving these equations are discussed in the following. Basis For Loading Equations A cursory look at the loading equations in Figures D.1 to D.8 shows that they can be easily evaluated numerically using a hand-held pocket calculator. In order to arrive at these relatively simple equations, it is necessary that certain assumptions be made concerning the characteristics of the three-phase and single-phase loads; and both the primary and secondary systems. The three-phase load is assumed to be a constant “current sink”' which draws only balanced (positive-sequence) currents. Losses in the secondary conductor between the transformer terminals and both single-phase and three-phase loads are negligible such that the phase voltages at the load and transformer are the same. The single-phase load supplied from a delta or open delta secondary is balanced between the two phase wires and neutral wire such that current does not flow in the neutral. Furthermore, the voltages impressed on the primary windings of the transformer are of a magnitude and angle which results in balanced output voltages from the transformer secondary terminals. Although these conditions rarely exist in practice, they are the assumptions traditionally used in the industry, although not always stated, to arrive at these simplified loading equations. 53
If it is desired to make more exact calculations for the kVA load supplied by each transformer in the bank, providing sufficient information is available for representing the load and system, then the methods originally developed by Neupauer 1,2, or by Seematter and Richards3 may be used. However, these methods do not result in simple expressions similar to those given in Figures D.1 to D.8, but require the use of a digital computer for implementation. Load Equations For Symmetrical and Unsymmetrical Transformer Banks Use of the simplified loading equations is discussed in the following sections for the more common symmetrical and unsymmetrical transformer connections. Examples are given to demonstrate the use of these equations. Open Wye—Open Delta Bank With the open wye-open delta transformer bank, the single-phase load may be connected to either the lagging phase as shown at the top of Figure D.1, or to the leading phase as shown at the top of Figure D.2. The transformer across which the single-phase load is connected is sometimes referred to as the “lighting leg” and the other transformer is referred to as the “power leg.” These are designated as L and P respectively in Figures D.1 and D.2. Equations for calculating the load in kVA supplied by the lighting leg transformer (kVA L) and that supplied by the power leg transformer (kVA P) are given in Figures D.1 and D.2. Furthermore, the symbols used in these equations are defined in the Figures. The use of the equations is illustrated with the following example. Open WYE—Open DELTA (Lagging)
Figure D 1: Load equations for the open wye-open delta bank with the single-phase load connected to the lagging phase. 54
An open wye-open delta bank supplies a single-phase load of 70 kVA at 0.95 lagging power factor, and a three-phase load of 30 kVA at 0.8 lagging power factor. The power factor angles (a3 and a1) are the arc cosine of the power factors. Thus: a3 = arc cos (.8) = 36.87° a1 = arc cos (.95) = 18.19° The load in kVA supplied by the lighting leg and power leg transformers for both the leading and lagging connection will be determined. First, consider the lagging connection in Figure D.1. The numerical values for the symbols in Figure D.1 are as follows:
Next, consider the leading connection shown in Figure D.2. The expression for the load in kVA supplied by the power leg transformer is the same as for the lagging connection. The expression for the load supplied by the lighting leg transformer is identical to that for the lagging connection except for the argument of the cosine term. Evaluation of the expression for kVA L in Figure D.2 shows that the lighting leg transformer supplies 82.47 kVA with the leading connection. For most combination loads, the power factor of the three-phase load is less than that of the single-phase load. Thus a3 – a1 is positive in sign, and m is a positive number. For expected values of m, the magnitude of the argument of the cosine term in the expressions for kVA L will be greater for the leading connection, and thus the cosine of the argument will be less. Consequently, when the leading connection is used the kVA load supplied by the lighting leg transformer usually is less than for the lagging connection.
55
Open WYE—Open DELTA (Leading)
Figure D.2: Load equations for the open wye-open delta bank with the single-phase load connected to the leading phase. Open DELTA—Open DELTA Bank (Leading or Lagging) The equations for calculating the load in kVA supplied by the lighting leg and power leg transformers in the open delta-open delta bank are the same as for the open wye-open delta bank. Thus the equations in Figures D.1 are used for the lagging connection, and those in Figure D.2 are used for the leading connection of the open delta-open delta bank. Figure D.3 is a loading curve chart for the open delta-open delta (leading) connection. • Transformer output limited to 100% of rated • Upper number — required kVA of power leg • Lower number — required kVA of lighting leg
Figure D.3 56
Floating WYE—DELTA
Figure D.4: Load equations for the floating wye-delta connected transformer bank. Floating WYE—DELTA Bank Equations for calculating the load in kVA supplied by each transformer in the floating wye-delta bank are given at the top of Figure D.4. Notice in these equations that a double subscript is used to specify the phases to which each transformer is connected, and the single-phase load is connected from phases b-to-c. Since the primary windings of the transformers in the bank are connected in floating wye, the single-phase load division is independent of transformer characteristics and zerosequence current cannot circulate in the secondary delta. Because of this and the assumptions concerning the characteristics of the threephase load, the equations in Figure D.4 for determining the load supplied by each transformer are independent of transformer impedance. Use of the equations is demonstrated with the following example. A floating wye-delta bank is to supply a three-phase load of 100 kVA at 0.8 power factor lagging, and a single-phase load of 50 kVA at 0.95 power factor lagging. What is the smallest size transformer which can be used in each leg if the load supplied by each transformer is not to exceed nameplate rating? From the specified power factors: a3 = arc cosine (.8) = 36.87° a1 = arc cosine (.95) = 18.19°
57
Evaluating the equations in Figure D.4 with K 3 equal to 100, K 1 equal to 50, and m equal to 18.68 degrees results in the following: KVA ab = 40.09 KVA bc = 65.78 KVA ca = 47.15 Thus the transformers connected between a and b, and between a and c should be 50 kVA units. The one between b and c should be a 75 kVA unit. Figure D.5 is a loading curve chart for the floating wye-delta connection. • Transformer output limited to 100% of rated • Upper number – required kVA of power leg • Lower number – required kVA of lighting leg
Figure D.5
58
DELTA - DELTA
Figure D.6: Load equations for the delta-delta connected bank with identical transformers in each power leg, and a different unit in the lighting leg. DELTA—DELTA Bank The equations for calculating the load in kVA supplied by each transformer in a delta-delta bank are given at the top of Figure D.6. The assumptions used in deriving these equations are the same as previously outlined, plus it is assumed that the impedance of the transformers between a and b, and between a and c are identical. These two units are sometimes referred to as the “power leg” transformers, and their 59
leakage impedance is designated as Z P. Across the transformer connected between b and c is the single-phase load. This unit is referred to as the “lighting leg” transformer and its impedance is designated at Z L. Although the equations in Figure D.6 may seem rather complicated, their evaluation is quite simple as illustrated by the following example: A delta-delta bank containing a 50 kVA unit in each power leg and a 75 kV unit in the lighting leg supplies a 100 kVA three-phase load (0.8 power factor lagging) and a 50 kVA single-phase load (0.95 power factor lagging). The impedance of each transformer in percent is: Z P = 1.1 + j1.3
% on 50 kVA base
Z L = 1.0 + j1.5
% on 75 kVA base
From the previous examples where the three-phase and single-phase power factors also were 0.8 and 0.95 respectively: m = 18.68 degrees Notice that the three equations at the top of Figure D.6 for calculating the load supplied by each transformer contain the terms M 1, M 2, M 3, M 4, b 2, b 3, b 4. These are real numbers which are a function of Z P and Z L. The M’s are the magnitude of the impedance functions as shown in the figure, and the b ’s are the angles in degrees for the impedance functions. The equations for calculating the M’s and b ’s are also given in the Figure. To calculate these, first put Z L on the same kVA base as Z P. Z L = (1.0+ j1.5)
50 = .6667 + j1.0 % ON 50 kVA 75
Placing the values of Z L and Z P into the equations yields the following: M1 = M2 = M3 = M4 =
2.702 .9802 .7835 .9199
b 2 = -113.47° b3 = 4.84° b 4 = 115.12°
Placing these values plus the values of K 3, K 1, and m into the loading equations gives the following for the load in kVA supplied by each transformer. kVA ab = 33.66 kVA bc = 73.04 kVA ca = 43.15
60
If the delta-delta bank is made from three transformers with the same leg impedance (on a common kVA base, or in actual ohms), then the loading equations reduce to the simpler form shown in Figure D.7. Notice that these are the same equations as used for the floating wye-delta bank in Figure D.4. If the delta-delta bank is supplying only three-phase load, but one of the units has a different impedance, the loading equations in Figure D.6 reduce to the relatively simple form shown in Figure D.8 if all transformers have the same impedance angle. When the impedance of each transformer is the same, the load supplied by each is 1/3 of the total three-phase load. Figure D.11 is a plot of the per unit load supplied by each transformer as a function of the ratio of Z L to Z P where the impedances are on a common base. One per unit load is the load carried by the transformer when all three have the same impedance. From this plot, notice that reasonable difference in impedances do not produce large unbalances in loading. Thus, although it is desirable that each transformer in a delta-delta bank supplying a three-phase load have the same impedances, this is not an absolute necessity. However, in contrast, these transformers must have the same voltage rating and tap settings as discussed in Section III.B.1. DELTA-DELTA with Equal Leg Impedances
Figure D.7: Load equations for the delta-delta connected bank with identical transformers in each leg.
61
DELTA-DELTA, Three-Phase Load, Same Impedance Angle for Z P and Z L
Figure D.8: Load equations for the delta-delta connected bank supplying only a three-phase load. Equations apply only when impedance angle of all transformers are the same. Grounded WYE-DELTA Bank The equations for calculating the load supplied by each transformer in a grounded wye-delta bank are the same as those for a delta-delta bank as given in Figure D.6. However, for reasons previously discussed in Section III, this connection is not recommended to supply distribution loads.
62
DELTA-Grounded WYE Bank With the delta-grounded wye bank, the single-phase load on the secondary may be connected from either phase-to-neutral as shown in Figure D.9 or from phase-to-phase as shown in Figure D.10. The equations for calculating the load supplied by each transformer are given at the top of each figure. The terms appearing in each equation are the same as used in the equations for the other connections for which examples have been given. DELTA-WYE, Phase-to-Neutral Single-Phase Load
Figure D.9: Load equations for the delta-wye connected bank when the single-phase load is connected from phaseto-neutral.
63
DELTA-WYE, Phase-to-Phase Single-Phase Load
Figure D.10: Load equations for the delta-wye connected bank when the single-phase load is connected phase-tophase. Grounded WYE-Grounded WYE Bank The grounded wye-grounded wye bank also is used to supply a combination of three-phase and single-phase load. Single-phase secondary load may be connected either phase-to-neutral, or phase-to-phase. The load supplied by each transformer in the grounded wye-grounded wye bank can be calculated using the equations for the delta-grounded wye bank in Figures D.9 and D.10.
64
Figure D.11: Per unit load supplied by each transformer in a deltadelta bank to a three-phase load when one of the transformers (connected between phases b and c with impedance Z L) has a different impedance. The transformers between phases a and b, and between phases c and a have impedance Z P as shown in Figure D.6. References 1. J. C. Neupauer. “Unbalanced Open-Wye Open-Delta Transformer Banks.” A.l.E.E. Transactions PAS., Vol. 75, pt. III, pp. 570-572, August 1956. 2. Neupauer, J. C., and C. L. Smith. “Motor-Starting Lamp Flicker on Open-Delta Transformer Banks.” A.l.E.E. Transactions PAS., Vol. 77, pt. III, pp. 1568-1576, February 1959. 3. Seematter, S. C., and E. F. Richard. “Computer Analysis of 3-Phase Induction Motor Operation on Rural Open Delta Distribution Systems.” I.E.E.E. Trans. on Industry Applications, Vol. 1A-12, No.5, pp. 479-485, Sept./Oct. 1976.
65
E. Dedicated Motor Loads Many different types of motors are used today and are an important consideration in sizing a transformer to supply power to a given load. For most transformers supplying individual and multiunit residences, the motor load can be ignored because it represents a small percentage of the total load connected to the transformer, and the motors are only started on an infrequent basis. When applying transformers to commercial or industrial loads, the motors that are to be served can present a major limiting factor on what size transformer is necessary to serve the load. Motor Starting Load The major consideration in sizing transformers for Motor Application is limiting the starting current so that it will not shorten the life of the transformer due to thermal or mechanical damage from the starting pulse. Extensive data has been gathered on pulse duty on power transformers and the conclusion was that if the current pulses per hour exceed 4.25 f n = e lp
4
where n = number of starts per hour Ip = the pulse current in per unit of transformer rated current
then the transformer will fail prematurely due to the repeated mechanical strains placed on the coil. (See Curve 1.)
66
67
Number of Current Pulses Per Hour
Curve 1
Maximum Allowable per Unit Pulse
Dedicated Transformers (One motor is the entire load on the transformer) When a transformer is dedicated to supplying the power to only one motor then the problem of sizing the transformer can be solved very methodically. On squirrel cage induction motors Nema Standards call for a starting code letter which corresponds to the kVA per horsepower required to start the motor; a table giving this relationship is shown on Curve 2. Curve 2 is based on the locked rotor code letters but it can be used for any motor by selecting the curve that corresponds to locked rotor kVA/HP of the motor that the transformer is being sized for. The procedure to size the transformer proceeds as follows: 1. If the starting kVA or starting code letter is unknown, calculate the motors locked rotor kVA (kVA’s = CF3F x V R x IS x 10-3) Where IS = starting current at rated voltage V R = rated phase-to-phase voltage of motor 2. Determine the number of starts per hour planned for the motor under normal operating conditions. 3. On Curve 2 find the curve letter that corresponds to the locked rotor kVA/HP of the motor. Enter Curve 2 on the abscissa at the correct starts per hour for the motor application. 4. Move up to the intersection of the starts/hour and the correct locked rotor code letter curve and read the kVA of transformer required per horsepower of motor. 5. MuItiply the kVA/HP found in “Step 4” by the rated HP of the motor and that is the smallest transformer that should be used in that application. Sizing the transformer with this procedure is conservative since it assumes that the voltage maintained at the motor terminals during starting is motor rated voltage. 6. Most motors require 60–80% of rated voltage at the terminal under locked rotor conditions to successfully start. After the transformer has been sized so that it can withstand the starting pulse due to the motor, the voltage regulation of the system must be checked to determine if the voltage is adequate under locked rotor condition to start the motor.
68
69
Locked-Rotor kVA per Hp
0 -3.15 3.15 -3.55 3.55 -4.0 4.0 -4.5 4.5 -5.0 5.0 -5.6 5.6 -6.3 6.3 -7.1 7.1 -8.0 8.0 -9.0 9.0 -10.0 10.0 -11.2 11.2 -12.5 12.5 -14.0 14.0 -16.0 16.0 -18.0 18.0 -20.0 20.0 -22.4 22.4 and up
Code Letter
A B C D E F G H J K L M N P R S T U V
NEMA StartingCode Letter
Curve 2
V. Voltage Unbalance
Page
A. Effects of Voltage Unbalance ................................................... B. Voltage Unbalance Definitions.................................................. C. Causes of Voltage Unbalance .................................................. D. Voltage Unbalance With Three-Phase Loading ....................... 1. Delta-Delta and Floating Wye-Delta Banks ......................... 2. Open Delta Banks ................................................................
70
71 71 73 73 74 75
V. Voltage Unbalance
A. Effects of Voltage Unbalance Voltage unbalance in secondary distribution systems affects the performance of induction motors, with motor derating required when voltage unbalance exceeds 1.0 percent. Figure 5.1, extracted from NEMA MG1-14.34, dated June 1980, gives the derating factor for fractional and integral-horsepower induction motors. The performance of semi-conductor rectifier circuits also can be affected by voltage unbalance, with proposed revisions to ANSI C34.2 indicating the application is unusual if either the negativeor zero-sequence component of voltage exceeds 5 percent of the positive-sequence component. This level of unbalance usually is not present in utility distribution systems. However, some voltage unbalance will be present in any type of low-voltage system, whether it be 4-wire wye, 3-wire delta, or 4-wire delta. The subject of voltage unbalance in secondary distribution system is a very complex matter, due to the many system and transformer parameters which affect unbalance, and does not allow a detailed discussion in this Guide. Considered in the following sections are several special cases where the transformer bank is supplying just a balanced three-phase load through a symmetrical secondary circuit, with each transformer in the bank having the same kVA rating, although not necessarily the same impedance. Furthermore, the primary system voltages are assumed to be balanced. For these special cases, the following points are noteworthy. 1. With either the floating wye-delta or delta-delta bank supplying three-phase load, it is not necessary from a voltage unbalance standpoint that low impedance transformers be used. For these banks, voltage unbalance is caused by impedance differences between the single-phase transformers in the bank. Reasonable differences in impedances are tolerable and will not cause objectionable voltage unbalance. 2. With either the open wye-open delta or open delta-open delta banks supplying just three-phase load, voltage unbalance is caused by the dissymmetries of the transformer bank (due to use of only two transformers), regardless of transformer impedance magnitudes. Voltage unbalance with the open delta bank can be significantly higher than that with a closed delta bank supplying the same load.
B. Voltage Unbalance Definitions Three different definitions are employed to quantify voltage unbalance. The one used depends upon the task being performed, for example, calculating unbalance from measured quantities or developing equations for unbalance. When quantifying voltage unbalance in 3-wire and 4-wire delta circuits under unfaulted conditions, all three definitions give nearly the same result.
71
In NEMA induction motor standards, percent voltage unbalance is defined as: Percent Maximum Voltage Deviation from Voltage Average Phase Voltage = x 100 Unbalance Average Phase Voltage When phase-to-phase voltages are measured at an actual installation, voltage unbalance is easily calculated with this definition, the one frequently preferred by individuals not familiar with symmetrical components. For example, with phase-to-phase voltages of 235, 230 and 222 volts, average voltage is 229 volts, maximum deviation from average is 7.0 volts, and percent voltage unbalance is 3.06 percent. Some engineers have advocated that percent voltage unbalance be defined as 100 times the ratio of the magnitude of the negativesequence voltage to the magnitude of the positive-sequence voltage. When an analysis is performed with symmetrical components to obtain sequence quantities, it is expedient to calculate voltage unbalance with this definition as this eliminates the need to calculate the phase-to-phase voltages required with the NEMA definition. In a three-phase system where zero-sequence voltages are not present, the ratio of the magnitude of the negativesequence voltage to the magnitude of the positive-sequence voltage also can be found from the following equation.
In this equation a, b, and c are the magnitudes of the three line-toline voltages (or line-to-ground voltages when zero-sequence is not present) and: S=
a+b+c 2
For example, with phase-to-phase voltages of 235, 230, and 222 volts, the equation shows that the ratio of the magnitude of V2 to V1 is 3.30 percent. In comparison, the percent voltage unbalance for these voltages using the NEMA definition is 3.06 percent. A third definition for percent voltage unbalance is 100 times the per unit negative-sequence voltage, where the per unit value is the actual value in volts divided by the system nominal voltage. The voltage unbalance calculated in this fashion does not differ significantly from 100 times the ratio of V2 to V1, as V1 is close to 1.0 per unit under loading conditions. In the curves in section V-D, voltage unbalance is quantified in terms of negative-sequence voltage.
72
C. Causes of Voltage Unbalance Voltage unbalance in secondary distribution systems is caused by dissymmetries in either the primary system, distribution transformer bank, secondary circuit, or loading on the transformer bank. Transformer bank symmetry is defined in Section IV-D of this guide. A symmetrical secondary circuit is one which has identical conductors in each phase with the conductors arranged such that the mutual impedances between its sequence networks are zero. An unsymmetrical circuit is one where the same size conductor is not in each phase, or the same size conductor is used but the spacings are such that the sequence mutual impedances are not zero. With a symmetrical primary system, the open circuit voltages at the transformer are perfectly balanced. Whenever a symmetrical transformer bank supplies a perfectly balanced three-phase load (one where there is no coupling between the sequence networks) through a symmetrical secondary circuit and the voltages at the primary terminals of the bank are balanced, the voltages at the secondary terminals also will be balanced. That is, a negative- or zero-sequence component will not be present in the secondary phase-to-neutral voltages. However, when a symmetrical transformer bank supplies a perfectly balanced three-phase load and a single-phase load, the secondary voltages will be unbalanced regardless of the symmetry of the secondary circuit. Furthermore, when an unsymmetrical transformer bank supplies only a balanced three-phase load, or both a balanced three-phase load and a single-phase load, the secondary voltages will be unbalanced irrespective of the secondary circuit symmetry. The main parameters which can affect voltage unbalance in secondary systems are transformer bank connection, transformer impedance, primary system impedance, secondary circuit characteristics, three-phase and single-phase load magnitudes, load power factors, and primary system voltage unbalance. Any one of these parameters can have a significant effect on voltage unbalance in secondary systems, with the only exception being primary system impedance which usually has a minor effect. This is because the impedance of the primary system typically is much smaller than that of the distribution transformers. A complete discussion on the effect of all of these parameters on voltage unbalance is beyond the scope of this guide. Illustrated in the following is the effect of distribution transformer impedance on voltage unbalance for the situation where the unsymmetrical transformer bank supplies only balanced three-phase load through a symmetrical secondary circuit, is supplied from an infinite bus primary system, and is made from transformers of equal kVA rating.
D. Voltage Unbalance With Three-Phase Loading Figures 5.2, 5.3, and 5.5 show the effect of transformer impedance on the maximum negative-sequence voltage in percent which could appear in the secondary system with balanced three-phase nominal loading on the bank. They apply respectively to the deltadelta, floating wye-delta, and open delta transformer banks. For a given bank, each transformer has the same kVA rating. Nominal loading occurs when the positive-sequence current in the secondary windings equals winding rated current. Since, in general,
73
a negative-sequence component of current is present under nominal loading conditions, the actual winding currents may be somewhat greater than or less than winding rated current. The curves apply to banks supplying balanced three-phase loads. For these loads, the positive- and negative-sequence impedances are either the same or different, but there is no coupling between the sequence networks representing the load. Loads with these characteristics are three-phase induction motors, or impedances of equal magnitude and angle connected in either wye or delta. For the three-phase induction motor, the negative-sequence impedance is less than the motor’s positive-sequence impedance, whereas these impedances are equal for loads made from either wye or delta connected impedances. Furthermore, the curves are plotted assuming the primary system voltages are of equal magnitude and 120 electrical degrees displaced from each other. Also, the impedance angles of the transformers in a bank were assumed equal, with the impedance magnitudes being the same or different. It is emphasized that the curves give the maximum, or greatest upper bound on negative-sequence voltage at the load in the secondary with nominal loading (three-phase) on the bank. Actual unbalance can be considerably less, depending upon the relationship between the transformer bank impedances, secondary circuit impedances, and the three-phase loads negative sequence impedance. 1. Delta-Delta and Floating Wye Delta Banks The curves of Figure 5.2 and 5.3 respectively show on the ordinate the maximum negative-sequence voltage in percent in the secondary system with balanced three-phase load supplied from a delta-delta bank and floating wye-delta bank. The curves apply to banks with transformers of the same kVA and voltage ratings. Two transformers in the bank have the same leakage impedance, designated as Z P, and the third unit’s impedance is Z L. Given on the abscissa is the ratio of Z L to Z P. The curves show that if all three transformers in the bank have equal impedance, regardless of impedance magnitude, the bank will not produce voltage unbalance with only balanced threephase load. Also with balanced three-phase nominal loading, the maximum negative-sequence voltage (voltage unbalance) will not exceed 0.6 percent, as long as the ratio of Z L to Z P is between 0.5 to 1.5, and Z P is 3.0 percent or less on nameplate rating. Thus, considering voltage unbalance when serving balanced three-phase load, it is not necessary that all units have the same impedance or low impedance in closed delta banks. If one unit fails in a bank made from three “old” units of “low” impedance, it could, in most cases, be replaced by a “new” unit with a higher impedance without creating objectionable unbalance. It is unduly restrictive from a voltage unbalance standpoint to require the same impedance for all units in the delta-delta and floating wye-delta bank supplying a three-phase load. Reasonable impedance differences are tolerable. For an example of the use of the curves in Figure 5.2, consider the following situation. A utility had some 1500 kVA delta-delta banks made from 500 kVA units with nameplate impedances of 2.2 percent. The utility decided to order several spare units and requested units with 2.2 percent impedance. When the supplier
74
quoted units with an impedance of 1.9 percent, the utility objected because it was thought that voltage unbalance problems would be created by the “low impedance” units. If two units in the bank have an impedance of 2.2 percent (Z P) and one has an impedance of 1.9 percent (Z L), the curves of Figure 5.2 show that the maximum negative-sequence voltage possible at nominal load would not exceed 0.13 percent. If two units have an impedance of 1.9 percent (Z P) and one unit has an impedance of 2.2 percent (X L), the maximum negative-sequence voltage possible at nominal load would not exceed 0.1 percent. For this situation, the impedance differences will not cause significant voltage unbalance. However, when one unit in the delta-delta bank has a different impedance, it may be necessary to derate the bank for thermal reasons. Section IV-B of this guide presents derating factors for the delta-delta bank made from 3 units of equal capacity supplying perfectly balanced three-phase load where one unit has a different impedance. A perfectly balanced three-phase load, for purpose of derating is defined as one which draws only positivesequence current. Figure 5.4 is a plot of derating factor as a function of the ratio of Z L to Z P for the delta-delta bank. 2. Open-Delta Banks The curves of Figure 5.5 show the maximum negative-sequence voltage (voltage unbalance) in percent which could appear in the secondary with balanced three-phase load supplied from either the open wye-open delta or open delta-open delta bank. The curves apply to banks made from two single-phase units with the same kVA and voltage ratings. The impedance of one unit is Z P and that of the other is Z L. For the open delta bank, nominal load is that which makes the positive-sequence current in each secondary winding equal to rated current of the winding. Practically, nominal load is when the positive-sequence kVA of the balanced three-phase load equals 1.732 times the kVA rating of one transformer. When both transformers in the open delta bank have the same impedance, the upper bound on negative-sequence voltage in percent at nominal load is the impedance in percent divided by the square root of 3. A comparison of Figure 5.5 with either Figure 5.2 or 5.3 shows that the maximum negative-sequence voltage at nominal load is much greater for the open delta bank than for the closed delta banks. Even when both transformers have the same impedance (Z L = Z P), the maximum negative-sequence voltage at nominal load can be appreciably above 1.0 percent. Also, for any ratio of Z L to Z P, the impedance of the transformers in the open delta bank must be low if the negative-sequence voltage (voltage unbalance) is to be limited to less than 1.0 percent. For example, consider an open delta bank made from units with 3 percent impedance. This impedance is typical in units purchased today by some users. From Figure 5.5, the voltage unbalance at nominal load due to the open delta bank is 1.73 percent. This would be the total unbalance (upper bound) at the load if the primary system voltages were perfectly balanced. Recognizing that the voltage unbalance of the primary system could be in the 1 to 2 percent range, a worst case upper bound on the voltage unbalance at the load is in the range of 2.7 to 3.7 percent.
75
As another example, one utility was serving a 460-Volt load, predominately motors, from an open delta bank with two 1000 kVA, 5.8 percent impedance transformers, with a load of 1360 kVA or about 79 percent of bank rating. Measured voltage unbalance at the service entrance was 2.3 percent. In comparison, the upper bound on voltage unbalance with just three-phase load, assuming a balanced primary, is calculated as 5.8 x 0.79/ CFF 3 or 2.65 percent. When the open delta bank was replaced with three 500 kVA units with 4.8 percent impedance, the measured voltage unbalance at about the same loading was less than 0.5 percent, due mainly to primary system unbalance. Considering voltage unbalance, successful operation of the open delta transformer bank supplying balanced three-phase load frequently is enhanced by the use of low impedance distribution transformers. As transformer impedance decreases, the negativesequence voltage due to transformer bank dissymmetries decreases. When primary voltages are balanced the negativesequence voltage at the load due to the dissymmetries of the open delta bank will be reduced by either closing the bank (adding a third transformer), by using transformers with the same kVA rating but with lower impedance, or by using transformers of a higher kVA rating. Using transformers with a higher kVA rating (same percent impedance) corresponds to loading the bank to less than “nominal load.” When primary system voltages are unbalanced, the total voltage unbalance (negative-sequence voltage) at the balanced threephase load fed from the open delta bank is the vector sum of that due to the transformer bank dissymmetries and the voltage unbalance of the primary system. Decreasing the component of voltage unbalance due to the transformer bank by reducing transformer impedance may either increase or decrease the total negative-sequence voltage at the load. The effect of reducing impedance depends upon the relative magnitude of the primary system voltage unbalance and the voltage unbalance due to transformer bank dissymmetries, plus the angle between these sources of voltage unbalance. Under conditions where the two sources of voltage unbalance are in-phase, reducing transformer impedance reduces voltage unbalance at the secondary load.
Figure 5.1:
76
Fractional and integral-horsepower induction motor derating factor.
Figure 5.2:
Maximum negative-sequence voltage at nominal load in the secondary system with balanced three-phase load supplied from a delta-delta bank made from three transformers with the same kVA and voltage ratings.
Figure 5.3:
Maximum negative-sequence voltage at nominal load in the secondary system with balanced three-phase load supplied from a floating wyedelta bank made from three transformers with the same kVA and voltage ratings.
77
78
Figure 5.4:
Derating factor for a delta-delta bank made from three units of equal capacity (kVA rating) with the impedance of one unit being different from that of the other two. Derating factor is approximate as it assumes perfectly balanced threephase load drawing only positive-sequence current.
Figure 5.5:
Maximum negative-sequence voltage at nominal load in the secondary system with balanced three-phase load supplied from an open delta transformer bank made from two transformers with the same kVA and voltage ratings.
Vl. Reference Data
Page
Solid and Concentric Stranded Aluminum and Copper Conductors . Temperature Correction Factors for Resistance of Aluminum Conductors .................................................................................... Logarithm Tables ........................................................................... Nominal Direct-Current Resistance, Ohms per 1000 Feet, at 20°C and 25°C of Solid and Concentric Stranded Conductors .... Natural Functions of Angles .......................................................... TypicaI Isokeraunic Map ............................................................... Selected Sl Equivalents ................................................................
80 81 83 85 86 87 88
79
80
10380 16510 26240 41740
52620 66360 83690 105600 133100 167800 211600
………. ………. ………. ………. ………. ……….
………. ………. ………. ……….
………. ………. ………. ……….
10 8 6 4
3 2 1 1/0 2/0 3/0 4/0
250 300 350 400 450 500
600 750 800 1000
1250 1500 1750 2000
…….. …….. …….. ……..
…….. …….. …….. ……..
…….. …….. …….. …….. …….. ……..
229.4 257.6 289.3 324.9 364.8 409.6 466.0
101.9 128.5 162.0 204.3
32.0 40.3 50.8 64.1 80.8
.…… .…… .…… .……
.…… .…… .…… .……
.…… .…… .…… .…… .…… .……
48.43 61.07 77.00 97.13 122.5 154.4 194.7
9.56 15.20 24.15 38.41
0.942 1.49 2.38 3.78 6.01
.……. .……. .……. .…….
.……. .……. .……. .…….
.……. .……. .……. .……. .……. .…….
159.3 200.9 253.3 319.5 402.8 507.8 640.5
31.43 49.98 79.44 126.3
3.10 4.92 7.81 12.4 19.8
Solid __________________________________ Nominal Approximate Weight, Diameter, Pounds per 1000 Feet _____________________ Mils Aluminum Copper
•IPCEA Standard Publication S-66-524, NEMA WC 7-1971.
1020 1620 2580 4110 6530
20 18 16 14 12
Crosssectional Area, CM
91 91 127 127
61 61 61 61
37 37 37 37 37 37
7 7 19 19 19 19 19
7 7 7 7
7 7 7 7 7
117.2 128.4 117.4 125.5
99.2 110.9 114.5 128.0
82.2 90.0 97.3 104.0 110.3 116.2
86.7 97.4 66.4 74.5 83.7 94.0 105.5
38.5 48.6 61.2 77.2
12.1 15.2 19.2 24.2 30.5
1.289 1.412 1.526 1.632
0.893 0.998 1.031 1.152
0.575 0.630 0.681 0.728 0.772 0.813
0.260 0.292 0.332 0.373 0.418 0.470 0.528
0.116 0.146 0.184 0.232
0.036 0.046 0.058 0.073 0.092
1173 1408 1643 1877
563 704 751 939
235 282 329 376 422 469
. . . .
. . . .
. . . . . .
49.4 62.3 78.6 99.1 125 . 157 . 199 .
9.75 15.5 24.6 39.2
………. ………. ………. ………. 6.13
3859 4632 5403 6176
1850 2316 2469 3086
772 925 1080 1236 1390 1542
162 205 259 326 411 518 653
32.06 51.0 80.9 129
3.154 5.015 7.974 12.68 20.16
Class B ____________________________________________________________ Number Nominal Approximate Approximate Weight, of Diameter Outside Pounds per 1000 Feet ___________________ Wires of Each Diameter, Wire, Mils Inches Aluminum Copper
Solid and Concentric Stranded Aluminum and Copper Conductors*
Conductor Size, Awg or kcmil
127 127 169 169
91 91 91 91
61 61 61 61 61 61
19 19 37 37 37 37 37
19 19 19 19
…. …. …. 19 19
99.2 108.7 101.8 108.8
81.2 90.8 93.8 104.8
64.0 70.1 75.7 81.0 85.9 90.5
52.6 59.1 47.6 53.4 60.0 67.3 75.6
23.4 29.5 37.2 46.9
…… …… …… 14.7 18.5
Class C _____________________ Number Nominal of Diameter Wires of Each Wire, Mils
169 169 217 217
127 127 127 127
91 91 91 91 91 91
37 37 61 61 61 61 61
37 37 37 37
…. …. …. 37 37
86.0 94.2 89.8 96.0
68.7 76.8 79.4 88.7
52.4 57.4 62.0 66.3 70.3 74.1
37.7 42.4 37.0 41.6 46.7 52.4 58.9
16.7 21.1 26.6 33.6
…… …… …… 10.5 13.3
Class D ___________________ Number Nominal of Diameter Wires of Each Wire, Mils
Temperature Correction Factors for Resistance of Aluminum Conductors* Temperature, Degrees °C
Multiplying Factors for Reduction to 20°C 25°C
0 5 10 15
1.088 1.064 1.042 1.020
1.110 1.085 1.063 1.041
20 25 30 35
1.000 0.980 0.961 0.943
1.020 1.000 0.981 0.962
40 45 50 55
0.925 0.908 0.892 0.876
0.944 0.927 0.910 0.894
60 65 70 75
0.861 0.846 0.832 0.818
0.878 0.863 0.849 0.835
80 85 90
0.805 0.792 0.780
0.821 0.808 0.796
The correction factors given in this table are satisfactory for most applications. They are based upon aluminum having 61 percent conductivity and are derived from the formulae:
where
R1 = R 2
248 228 + T2
R3 = R 2
253 228 + T2
R1 – Resistance at 20°C R2 – Measured resistance at test temperature, T2 R3 – Resistance at 25°C
*IPCEA Publication S-66-524, NEMA WC 7-1971
81
Temperature Correction Factors for Resistance of Copper Conductors* Temperature, Degrees °C
Multiplying Factors for Reduction to 20°C 25°C
0 5 10 15
1.085 1.063 1.041 1.020
1.107 1.084 1.061 1.040
20 25 30 35
1.000 0.981 0.962 0.944
1.020 1.000 0.981 0.963
40 45 50 55
0.927 0.911 0.895 0.879
0.945 0.928 0.912 0.896
60 65 70 75
0.864 0.850 0.836 0.822
0.881 0.866 0.852 0.838
80 85 90
0.809 0.797 0.784
0.825 0.812 0.800
The correction factors given in this table are satisfactory for most applications. They are based upon copper having 100 percent conductivity and are derived from the formulae:
where
R1 = R 2
254.5 234.5 + T2
R3 = R 2
259.5 234.5 + T2
R1 – Resistance at 20°C R2 – Measured resistance at test temperature, T2 R3 – Resistance at 25°C
For more accurate determination of resistance, allowing for different conductivities, see “Copper Wire Tables,” National Bureau of Standards, Handbook No. 100.
*IPCEA Publications S-66-524, NEMA WE 7-1971
82
Logarithm Tables Four Place Mantissas for Common Logarithms N
0
1
2
3
4
5
6
7
8
9
1
2
10 11 12 13 14
0000 0414 0792 1139 1461
0043 0453 0828 1173 1492
0086 0492 0864 1206 1523
0128 0531 0899 1239 1553
0170 0569 0934 1271 1584
0212 0607 0969 1303 1614
0253 0645 1004 1335 1644
0294 0682 1038 1367 1673
0334 0719 1072 1399 1703
0374 0755 1106 1430 1732
*4* 4 3 3 3
8 8 7 6 6
15 16 17 18 19
1761 2041 2304 2553 2788
1790 2068 2330 2577 2810
1818 2095 2355 2601 2833
1847 2122 2380 2625 2856
1875 2148 2405 2648 2878
1903 2175 2430 2672 2900
1931 2201 2455 2695 2923
1959 2227 2480 2718 2945
1987 2253 2504 2742 2967
2014 2279 2529 2765 2989
*3* 3 2 2 2
20 21 22 23 24
3010 3222 3424 3617 3802
3032 3243 3444 3636 3820
3054 3263 3464 3655 3838
3075 3284 3483 3674 3856
3096 3304 3502 3692 3874
3118 3324 3522 3711 3892
3139 3345 3541 3729 3909
3160 3365 3560 3747 3927
3181 3385 3579 3766 3945
3201 3404 3598 3784 3962
25 26 27 28 29
3979 4150 4314 4472 4624
3997 4166 4330 4487 4639
4014 4183 4346 4502 4654
4031 4200 4362 4518 4669
4048 4216 4378 4533 4683
4065 4232 4393 4548 4698
4082 4249 4409 4564 4713
4099 4265 4425 4579 4728
4116 4281 4440 4594 4742
30 31 32 33 34
4771 4914 5051 5185 5315
4786 4928 5065 5198 5328
4800 4942 5079 5211 5340
4814 4955 5092 5224 5353
4829 4969 5101 5237 5366
4843 4983 5119 5250 5378
4857 4997 5132 5263 5391
4871 5011 5145 5276 5403
35 36 37 38 39
5441 5563 5682 5798 5911
5453 5575 5694 5809 5922
5465 5587 5705 5821 5933
5478 5599 5717 5832 5944
5490 5611 5729 5843 5955
5502 5623 5740 5855 5966
5514 5635 5752 5866 5977
40 41 42 43 44
6021 6128 6232 6335 6435
6031 6138 6243 6345 6444
6042 6149 6253 6355 6454
6053 6160 6263 6365 6464
6064 6170 6274 6375 6474
6075 6180 6284 6385 6484
45 46 47 48 49
6532 6628 6721 6812 6902
6542 6637 6730 6821 6911
6551 6646 6739 6830 6920
6561 6656 6749 6839 6928
6571 6665 6758 6848 6937
50 51 52 53 54
6990 7076 7160 7243 7324
6998 7084 7168 7251 7332
7007 7093 7177 7259 7340
7016 7101 7185 7267 7348
N
0
1
2
3
Proportional Parts 3 4 5 6 7 8 12 11 10 10 9
17 15 14 13 12
21 19 17 16 15
25 23 21 19 18
29 26 24 23 21
9
33 30 28 26 24
37 34 31 29 27
6 5 5 5 4
8 11 14 17 20 22 8 11 13 16 18 21 7 10 12 15 17 20 7 9 12 14 16 19 7 9 11 13 16 18
25 24 22 21 20
2 2 2 2 2
4 4 4 4 4
6 6 6 6 5
8 11 13 15 17 19 8 10 12 14 16 18 8 10 12 14 15 17 7 9 11 13 15 17 7 9 11 12 14 16
4133 4298 4456 4609 4757
2 2 2 2 1
3 3 3 3 3
5 5 5 5 4
7 7 6 6 6
9 10 12 14 15 8 10 11 13 15 8 9 11 13 14 8 9 11 12 14 7 9 10 12 13
4886 5024 5159 5289 5416
4900 5038 5172 5302 5428
1 1 1 1 1
3 3 3 3 3
4 4 4 4 4
6 6 5 5 5
7 7 7 6 6
9 10 11 13 8 10 11 12 8 9 11 12 8 9 10 12 8 9 10 11
5527 5647 5763 5877 5988
5539 5658 5775 5888 5999
5551 5670 5786 5899 6010
1 1 1 1 1
2 2 2 2 2
4 4 3 3 3
5 5 5 5 4
6 6 6 6 5
7 7 7 7 7
9 10 11 8 10 11 8 9 10 8 9 10 8 9 10
6085 6191 6294 6395 6493
6096 6201 6304 6405 6503
6107 6212 6314 6415 6513
6117 6222 6325 6425 6522
1 1 1 1 1
2 2 2 2 2
3 3 3 3 3
4 4 4 4 4
5 5 5 5 5
6 6 6 6 6
8 7 7 7 7
9 10 8 9 8 9 8 9 8 9
6580 6675 6767 6857 6946
6590 6684 6776 6866 6955
6599 6693 6785 6875 6964
6609 6702 6794 6884 6972
6618 6712 6803 6893 6981
1 1 1 1 1
2 2 2 2 2
3 3 3 3 3
4 4 4 4 4
5 5 5 4 4
6 6 5 5 5
7 7 6 6 6
8 7 7 7 7
9 8 8 8 8
7024 7110 7193 7275 7356
7033 7118 7202 7284 7364
7042 7126 7210 7292 7372
7050 7135 7218 7300 7380
7059 7143 7226 7308 7388
7067 7152 7235 7316 7396
1 1 1 1 1
2 2 2 2 2
3 3 2 2 2
3 3 3 3 3
4 4 4 4 4
5 5 5 5 5
6 6 6 6 6
7 7 7 6 6
8 8 7 7 7
4
5
6
7
8
9
1
2
3
4
5
6
7
8 9
* Interpolation in this.section of the table is inaccurate.
83
Logarithm Tables Four Place Mantissas for Common Logarithms (Continued) Proportional Parts 3 4 5 6 7 8
N
0
1
2
3
4
5
6
7
8
9
1
2
55 56 57 53 59
7404 7482 7559 7634 7709
7412 7490 7566 7642 7716
7419 7497 7574 7649 7723
7427 7505 7582 7657 7731
7435 7513 7589 7664 7738
7443 7520 7597 7672 7745
7451 7528 7604 7679 7752
7459 7536 7612 7686 7760
7466 7543 7619 7694 7767
7474 7551 7627 7701 7774
1 1 1 1 1
2 2 2 1 1
2 2 2 2 2
3 3 3 3 3
4 4 4 4 4
5 5 5 4 4
5 5 5 5 5
6 6 6 6 6
7 7 7 7 7
60 61 62 63 64
7782 7853 7924 7993 8062
7789 7860 7931 8000 8069
7796 7868 7938 8007 8075
7803 7875 7945 8014 8082
7810 7882 7952 8021 8089
7818 7889 7959 8028 8096
7825 7896 7966 8035 8102
7832 7903 7973 8041 8109
7839 7910 7980 8048 8116
7846 7917 7987 8055 8122
1 1 1 1 1
1 1 1 1 1
2 2 2 2 2
3 3 3 3 3
4 4 3 3 3
4 4 4 4 4
5 5 5 5 5
6 6 6 5 5
6 6 6 6 6
65 66 67 68 69
8129 8195 8261 8325 8388
8136 8202 8267 8331 8395
8142 8209 8274 8338 8401
8149 8215 8280 8344 8407
8156 8222 8287 8351 8414
8162 8228 8293 8357 8420
8169 8235 8299 8363 8426
8176 8241 8306 8370 8432
8182 8248 8312 8376 8439
8189 8254 8319 8382 8445
1 1 1 1 1
1 1 1 1 1
2 2 2 2 2
3 3 3 3 2
3 3 3 3 3
4 4 4 4 4
5 5 5 4 4
5 5 5 5 5
6 6 6 6 6
70 71 72 73 74
8451 8513 8573 8633 8692
8457 8519 8579 8639 8698
8463 8525 8585 8645 8704
8470 8531 8591 8651 8710
8476 8537 8597 8657 8716
8482 8543 8603 8663 8722
8488 8549 8609 8669 8727
8494 8555 8615 8675 8733
8500 8561 8621 8681 8739
8506 8567 8627 8686 8745
1 1 1 1 1
1 1 1 1 1
2 2 2 2 2
2 2 2 2 2
3 3 3 3 3
4 4 4 4 4
4 4 4 4 4
5 5 5 5 5
6 5 5 5 5
75 76 77 78 79
8751 8808 8865 8921 8976
8756 8814 8871 8927 8982
8762 8820 8876 8932 8987
8768 8825 8882 8938 8993
8774 8831 8887 8943 8998
8779 8837 8893 8949 9004
8785 8842 8899 8954 9009
8791 8848 8904 8960 9015
8797 8854 8910 8965 9020
8802 8859 8915 8971 9025
1 1 1 1 1
1 1 1 1 1
2 2 2 2 2
2 2 2 2 2
3 3 3 3 3
3 3 3 3 3
4 4 4 4 4
5 5 4 4 4
5 5 5 5 5
80 81 82 83 84
9031 9085 9138 9191 9243
9036 9090 9143 9196 9248
9042 9096 9149 9201 9253
9047 9101 9154 9206 9258
9053 9106 9159 9212 9263
9058 9112 9165 9217 9269
9063 9117 9170 9222 9274
9069 9122 9175 9227 927Si
9074 9128 9180 9232 9284
9079 9133 9186 9238 9289
1 1 1 1 1
1 1 1 1 1
2 2 2 2 2
2 2 2 2 2
3 3 3 3 3
3 3 3 3 3
4 4 4 4 4
4 4 4 4 4
5 5 5 5 5
85 86 87 83 89
9294 9345 9395 9445 9494
9299 9350 9400 9450 9499
9304 9355 9405 9455 9504
9309 9360 9410 9460 9509
9315 9365 9415 9465 9513
9320 9370 9420 9469 9518
9325 9375 9425 9474 9523
9330 9380 9430 9470 9528
9335 9385 9435 9484 9533
9340 9390 9440 9489 9538
1 1 0 0 0
1 1 1 1 1
2 2 1 1 1
2 2 2 2 2
3 3 2 2 2
3 3 3 3 3
4 4 3 3 3
4 4 4 4 4
5 5 4 4 4
90 91 92 93 94
9542 9590 9638 9685 9731
9547 9595 9643 9689 9736
9552 9600 9647 9894 9741
9557 9605 9652 9699 9745
9562 9609 9657 9703 9750
9566 9614 9661 9708 9754
9571 9619 9666 9713 9759
9576 9624 9671 9717 9763
9581 9628 9675 9722 9768
9586 9633 9680 9727 9773
0 0 0 0 0
1 1 1 1 1
1 1 1 1 1
2 2 2 2 2
2 2 2 2 2
3 3 3 3 3
3 3 3 3 3
4 4 4 4 4
4 4 4 4 4
95 96 97 98 99
9777 9823 9868 9912 9956
9782 9827 9872 9917 9961
9786 9832 9877 9921 9965
9791 9836 9881 9926 9969
9795 9841 9886 9930 9974
9800 9845 9890 9934 9978
9805 9850 9894 9939 9983
9809 9854 9899 9943 9987
9814 9859 9903 9948 9991
9818 9863 9908 9952 9996
0 0 0 0 0
1 1 1 1 1
1 1 1 1 1
2 2 2 2 2
2 2 2 2 2
3 3 3 3 3
3 3 3 3 3
4 4 4 4 3
4 4 4 4 4
N
0
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
84
9
85
0.323 0.256 0.203 0.161 0.128 0.101 0.0803
………. ………. ………. ………. ………. ……….
………. ………. ………. ……….
………. ………. ………. ……….
3 2 1 1/0 2/0 3/0 4/0
250 300 350 400 450 500
600 750 800 1000
1250 1500 1750 2000
………... ………... ………... ………...
………... ………... ………... ………...
………... ………... ………... ………... ………... ………...
0.1971 0.1563 0.1239 0.09825 0.07793 0.06182 0.04901
0.9988 0.6281 0.3952 0.2485
•IPCEA Standards Publication S-66-524, NEMA WC 7-1971.
1.64 1.03 0.648 0.407
10 8 6 4
………. ………. ………. ……….
………. ………. ………. ……….
………. ………. ………. ………. ………. ……….
0.330 0.261 0.207 0.164 0.130 0.103 0.0820
1.67 1.05 0.661 0.415
17.0 10.7 6.72 4.22 2.66
Aluminum
……….. ……….. ……….. ………..
……….. ……….. ……….. ………..
……….. ……….. ……….. ……….. ……….. ………..
0.2010 0.1594 0.1264 0.1002 0.07946 0.06303 0.04998
1.018 0.6404 0.4029 0.2534
10.3 6.51 4.10 2.57 1.62
Copper
0.0139 0.0116 0.00991 0.00867
0.0289 0.0231 0.0217 0.0173
0.0694 0.0578 0.0495 0.0434 0.0385 0.0347
0.329 0.261 0.207 0.164 0.130 0.103 0.0820
1.67 1.05 0.661 0.415
……… ……… ……… ……… 2.66
0.00846 0.00705 0.00605 0.00529
0.0176 0.0141 0.0132 0.0106
0.0423 0.0353 0.0302 0.0264 0.0235 0.0212
0.201 0.159 0.126 0.100 0.0795 0.0630 0.0500
1.02 0.641 0.403 0.253
10.4 6.53 4.10 2.57 1.62
Copper
0.0142 0.0118 0.0101 0.00885
0.0295 0.0236 0.0221 0.0177
0.0708 0.0590 0.0505 0.0442 0.0393 0.0354
0.336 0.266 0.211 0.168 0.133 0.105 0.0836
1.70 1.07 0.674 0.424
………… ………… ………… ………… 2.71
Aluminum
0.00863 0.00719 0.00616 0.00539
0.0180 0.0144 0.0135 0.0108
0.0431 0.0360 0.0308 0.0270 0.0240 0.0216
0.205 0.162 0.129 0.102 0.0811 0.0642 0.0509
1.04 0.654 0.410 0.259
10.6 6.66 4.18 2.62 1.65
Copper
Aluminum
10.1 6.39 4.02 2.52 1.59
Copper
Aluminum
16.6 10.5 6.58 4.14 2.60
20 18 16 14 12
Stranded Classes B, C and D _____________________________________________ °C 20 °C 25 ____________________ ____________________
Solid ______________________________________________ °C 20 °C 25 ____________________ ____________________
Aluminum and Annealed Uncoated Copper
………. ………. ………. ……….
………. ………. ………. ……….
………. ………. ………. ………. ………. ……….
0.203 0.161 0.128 0.101 0.0798 0.0633 0.0502
1.04 0.646 0.407 0.256
10.6 6.66 4.18 2.62 1.65
………. ………. ………. ……….
………. ………. ………. ……….
………. ………. ………. ………. ………. ……….
0.207 0.164 0.130 0.103 0.0814 0.0645 0.0512
1.06 0.659 0.415 0.261
10.8 6.79 4.26 2.68 1.68
0.00871 0.00726 0.00622 0.00544
0.0183 0.0145 0.0136 0.0109
0.0440 0.0367 0.0314 0.0272 0.0242 0.0218
0.209 0.166 0.131 0.104 0.0827 0.0656 0.0515
1.06 0.666 0.419 0.264
11.0 6.92 4.35 2.68 1.68
0.00888 0.00740 0.00634 0.00555
0.0187 0.0148 0.0139 0.0111
0.0449 0.0374 0.0320 0.0278 0.0247 0.0222
0.213 0.169 0.134 0.106 0.0843 0.0668 0.0525
1.08 0.679 0.427 0.269
11.2 7.05 4.44 2.73 1.72
Annealed Coated Copper _______________________________________ Solid Stranded Class B ___________________ ___________________ 25°C 20°C 25°C 20°C
Nominal Direct-Current Resistance, Ohms per 1000 Feet, at 20°C and 25°C of Solid and Concentric Stranded Conductors*
Conductor Size, Awg or kcmil
Natural Functions of Angles Deg.
Sine
Cosine
Tangent
Cotangent
Deg.
0 1 2
0.0000 0.0175 0.0349
1.0000 0.9998 0.9994
0.0000 0.0175 0.0349
oo 57.2900 28.6363
90 89 88
3 4 5
0.0523 0.0698 0.0872
0.9986 0.9976 0.9962
0.0524 0.0699 0.0875
19.0811 14.3007 11.4301
87 86 85
6 7 8
0.1045 0.1219 0.1392
0.9945 0.9925 0.9903
0.1051 0.1228 0.1405
9.5144 8.1443 7.1154
84 83 82
9 10 11
0.1564 0.1736 0.1908
0.9877 0.9848 0.9816
0.1584 0.1763 0.1944
6.3138 5.6713 5.1446
81 80 79
12 13 14
0.2079 0.2250 0.2419
0.9781 0.9744 0.9703
0.2126 0.2309 0.2493
4.7046 4.3315 4.0108
78 77 76
15 16 17
0.2588 0.2756 0.2924
0.9659 0.9613 0.9563
0.2679 0.2867 0.3057
3.7321 3.4874 3.2709
75 74 73
18 19 20
0.3090 0.3256 0.3420
0.9511 0.9455 0.9397
0.3249 0.3443 0.3640
3.0777 2.9042 2.7475
72 71 70
21 22 23
0.3584 0.3746 0.3907
0.9336 0.9272 0.9205
0.3839 0.4040 0.4245
2.6051 2.4751 2.3559
69 68 67
24 25 26
0.4067 0.4226 0.4384
0.9135 0.9063 0.8988
0.4452 0.4663 0.4877
2.2460 2.1445 2.0503
66 65 64
27 28 29
0.4540 0.4695 0.4848
0.8910 0.8829 0.8746
0.5095 0.5317 0.5543
1.9626 1.8807 1.8040
63 62 61
30 31 32
0.5000 0.5150 0.5299
0.8660 0.8572 0.8480
0.5774 0.6009 0.6249
1.7321 1.6643 1.6003
60 59 58
33 34 35
0.5446 0.5592 0.5736
0.8387 0.8290 0.8192
0.6494 0.6745 0.7002
1.5399 1.4826 1.4281
57 56 55
36 37 38
0.5878 0.6018 0.6157
0.8090 0.7986 0.7880
0.7265 0.7536 0.7813
1.3764 1.3270 1.2799
54 53 52
39 40 41
0.6293 0.6428 0.6561
0.7771 0.7660 0.7547
0.8098 0.8391 0.8693
1.2349 1.1918 1.1504
51 50 49
42 43 44
0.6691 0.6820 0.6947
0.7331 0.7314 0.7193
0.9004 0.9325 0.9657
1.1106 1.0724 1.0355
48 47 46
45
0.7071
0.7071
1.0000
1.0000
45
Deg.
Cosine
Sine
Cotangent
Tangent
Deg.
86
87
Typical Isokeraunic Map
Selected Sl Equivalents LENGTH 1 in 1 ft 1 yd 1 mile
TEMPERATURE = = = =
25.40 mm 0.3048 m 0.9144 m 1.609 km
= = = = =
506.7 um2 6.452 cm 2 0.0929 m2 4047 m2 2.590 km 2
= = = = = =
1 dm3 29.57 ml 3.785 litres 16.39 cm 3 28.32 dm3 0.7646 m3
AREA 1 cmil 1 in2 1 ft2 1 acre 1 mile2
VOLUME 1 litre 1 fl oz (US) 1 gal (US) 1 in3 1 ft3 1 yd 3
1°F (interval) = 5/9°C temp (°F) = (9/5) temp (°C) + 32 Ice . point at 1 atm = 32°F = 0°C = 273.15 K Triple-point of water = 0.01°C = 273.16 K
POWER 1 watt 1 Btu/hr 1 hp
= 1 joule/sec = 0.2931 W = 746.0 W
POWER FLUX 1 Btu/hr. ft2
= 3.152 W/m2
POWER DENSITY 1 Btu/hr.ft3
= 10.34 W/m3
SPECIFIC POWER 1 hp/lb
= 1.645 kW/kg
VELOCITY 1 in/min 1 ft/min 1 mile/hr
= 25.4 mm/min = 0.3048 m/min = 1.609 km/hr
STEFAN-BOLTZMANN CONST. = 56.7 nW/m2.K 4
o
THERMALCONDUCTIVITY MASS 1 oz (avdp) 1 lb 1 short ton
1 Btu/hr.ft F = 28.35 gram = 0.4536 kg = 0.9072 Mg
DENSITY 1 Ib/ft3 1 Ib/in3
= 1.731 W/m.K
THERMAL CONDUCTANCE 1 Btu/hr.ft2 F
= 5.678 W/m2.K
VISCOSITY, ABSOLUTE = 16.02 kg/m3 = 27.68 Mg/m3
1 poise 1 Ib/ft.sec 1 lbf.sec/ft2
= 0.1 Pa. 2 = 1.488 Pa.s = 47.88 Pa.s
FLOW RATE 1 gal/min 1 ft3/min
= 63.09 cm 3/s = 0.4719 dm3
VISCOSITY, KINEMATIC 1 stoke 1 ft 2/sec
= 10 -4m2/s = 0.0929 m2 /s
MASS VELOCITY 1 Ib/hr . ft2
= 0.08137 kg/s.m2
FORCE 1 newton 1 Ibf 1 kgf
2
= 1 kg.m/s = 4.448 N = 9.8066 N
PRESSURE
1 coulomb 1 volt 1 ohm 1 farad 1 henry
= 1 ampere second = 1 joule/coulomb = 1 volt/ampere = 1 coulomb/volt = 1 volt sec/ampere
MAGNETIC FLUX
1 pascal = 1 N/m2 1 mm Hg = 133.3 Pa 1 in H2O (60°F) = 248.8 Pa 1 in Hg = 3.377 kPa 1 psi = 6.895 kPa 1 kgf/cm2 = 98.07 kPa 1 bar = 100 kPa 1 atm = 101.3 kPa
ENERGY 1 joule 1 ft.Ibf 1 cal 1 Btu 1 kW.h 1 MWD
ELECTRICITY
1 weber 1 maxwell 1 kiloline
= 1 volt second = 10-8 Wb = 10-5 Wb
MAGNETIC INDUCTION, B 1 tesla 1 gamma 1 gauss
= 1 Wb/m2 = 10-9 T = 10-4 T
MAGNETOMOTIVE FORCE = = = = = =
1 N.m 1.356 J 4.187 J 1055 J 3.600 MJ 86.40 GJ
1 gilbert
= 0.7958 amp-turn
MAGNETIC FIELD STRENGTH, H 1 oersted
= 79.58 amp-turn/m
DIELECTRIC COEFFICIENT 1 farad/m
= 1 coul 2 /N.m2
SPECIFIC ENERGY = = = =
2.989 J/kg 2.326 kJ/kg 37.26 kJ/m 3 278.7 kJ/m 3
SPECIFIC HEAT: ENTROPY 1 Btu/lb.F = 4.187 kJ/kg.K Gas constant: R = 8.314 kJ/kg-mol.K
88
PERMITTIVITY CONST.
b
1 ft.Ibf/lb 1 Btu/lb 1 Btu/ft3 1 Btu/gal
O
= 8.8542 pF/m
PERMEABILITY CONST. uo c2 uoeo
= 4n x 10-7 henry/m =1
3A49299H01
