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Distribution
Transformer
Guide
Distribution Transformer Division
Athens, Georgia
Jefferson City, Missouri
June, 1979
Revised March, 2002
ISO 9001 CERTIFIED
1
ForewordThe purpose of this guide is to assemble fundamental informationconcerning common ratings, connections, and applications of distri-bution transformers. The information presented is a summary of thesefundamentals and is intended as a reference for those who dealoccasionally with distribution transformer applications. This guide doesnot purport to cover all aspects of selection and application; if ques-tions arise or further details are required, contact ABB Inc.
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IndexI. General Page
A. Application ................................................................................ 4B. Physical Description ................................................................. 4C. Protection and Accessories ...................................................... 11
II. PerformanceA. Designation of Winding Voltage Ratings ................................. 17B. Polarity ...................................................................................... 20C. Terminal Designations .............................................................. 21D. Short Circuit Ratings ................................................................ 22E. Sound Level Ratings ................................................................ 22F. Tolerance Definitions ................................................................ 23G. Impedance Calculations ........................................................... 23H. Efficiency Calculations ............................................................. 24I. Regulation Calculations ........................................................... 24J. Performance Example.............................................................. 26K. Secondary Fault Current—120/240 Volt Systems................... 27
III. Three-Phase Transformers and BanksA. Application Considerations ...................................................... 34B. Summary of Common Connections ......................................... 41C. Common Three-Phase Banks Using Single-Phase
Transformers ........................................................................... 48
IV. LoadingA. Paralleling ................................................................................. 51B. Delta-Delta Bank Loading ........................................................ 51C. Overloading .............................................................................. 52D. Single-Phase and Three-Phase Loading of Symmetrical and
Unsymmetrical Transformer Banks .......................................... 53E. Dedicated Motor Loads ............................................................ 66
V. Voltage UnbalanceA. Effects of Voltage Unbalance ................................................... 71B. Voltage Unbalance Definitions ................................................. 71C. Causes of Voltage Unbalance ................................................. 73D. Voltage Unbalance With Three-Phase Loading ...................... 73
1. Delta-Delta and Floating Wye-Delta Banks ........................ 742. Open-Delta Banks ............................................................... 75
VI. Reference DataSolid and Concentric Stranded Aluminum and CopperConductors .................................................................................... 80Temperature Correction Factors for Resistance of AluminumConductors .................................................................................... 81Logarithm Tables ........................................................................... 83Nominal direct-Current Resistance, Ohms per 1000 Feet, at20°C and 25°C of Solid and Concentric Stranded Conductors .... 85Natural Functions of Angles .......................................................... 86Typical Isokeraunic Map ................................................................ 87Selected SI Equivalents ................................................................ 88
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I. General Page
A. Application ................................................................................ 4B. Physical Description ................................................................. 4
1. Pole Mounted ...................................................................... 42. Pad Mounted ....................................................................... 6
C. Protection and Accessories ...................................................... 111. General ................................................................................ 112. Types of Accessories and Transformer Protection
Packages — Pole Mounted ................................................ 113. Types of Accessories and Protection — Pad Mounted ...... 13
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I. General
�����A. Application
ABB single-phase and three-phase, oil-filled, pole- and pad-mounted distribution transformers are specifically designed forservicing residential distribution loads; they are also suitable forlight commercial loads, and industrial lighting and diversified powerapplications.
The transformers described herein are designed for the applica-tion conditions normally encountered on electric utility powerdistribution systems. As such they are suitable for use under the“usual service conditions” described in ANSI C57.12.00 GeneralRequirements for Liquid-Immersed Distribution, Power andRegulating Transformers. All other conditions are considered“unusual service” and should be avoided unless specific ABBDivision 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 Basic ImpulseVoltage Ranges Class Level (kV)
480- 600 1.2 302160- 2400 5.0 604160- 4800 8.7 757200-12470 1 15.0 95
13200-14400 18.0 12519920-229002 25.0 150
-34400 34.5 200
1 Optional 125 kV BIL 12000 volts available2 Optional 125 kV BIL 19920 volts available
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Type CSP Type S
kVA High LowVoltage Voltage
Pole Mounted (Single-Phase)
0.5 2400 through 120/2401.5 34,400 volts 240/48035101525371/2
50751001672503335006677508331000
Pole Mounted (Three-Phrase)
15 2400 to 208/12030 13,800T 240x480T45 480/277751121/2
150225330
Three-Phase 500
JUMBO® Step-Down Transformer. The ABB JUMBOsingle-phase step-down transformer is especially use-ful during utility system voltage conversions when it isdesirable to convert a portion of a substation or afeeder to a higher voltage and still be able to supplythe remaining customers at the existing voltage. TheJUMBO® is also available for industrial and commer-cial applications or for resale customers.Jumbo
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2. Pad Mounted
A single-phase, single service, low pro-file distribution padmount transformeravailable in loop or radial feed.
Designed to aesthetically, safely andeconomically provide undergroundelectrical service to single loads,particularly, rural residences, farms andranches.
Micro-Pak, 10-50 kVA
A single-phase, multi-service, full-line,low profile padmount transformerdesigned for loop feed or radial feedon a grounded wye underground dis-tribution system.
The Mini-Pak can be furnished in acomplete line of ratings and in a widerange of configurations to fully meet thereliability, safety and operating require-ments of any distribution system.
Mini-Pak, 10-167 kVA
The Maxi-Pak is designed specificallyfor those customers requiring straight-up feed (Type I) rather than cross feed(Type II). The additional height of theMaxi-Pak allows installation of air loadbreak switching in this low-profiledesign.
Maxi-Pak, 10-250 kVA
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ABB single-phase padmounted Distribution Transformers meet thefollowing Industry Standards:
ANSI C57.12.00 - IEEE Standard General Requirements for LiquidImmersed…Transformers
ANSI C57.12.25 - Pad-Mounted…Single-Phase DistributionTransformers with Separable InsulatedHigh-Voltage Connectors…
ANSI C57.12.28 - Pad-Mounted Equipment - Enclosure Integrityor
ANSI C57.12.29 - …Pad-Mounted Equipment - Enclosure Integrityfor Coastal Environments
ANSI C57.12.70 - …Terminal Markings and ConnectionsANSI C57.12.80 - IEEE Standard Terminology…ANSI C57.12.90 - IEEE Standard Test Code…NEMA Tr-1 - Transformer StandardsIEEE 386 - …Separable Insulated Connectors
ABB recommends the use of ANSI C57.91 - IEEE Guide for Loading…forthe establishment of proper distribution transformer loading practices.
Ratings @65° RisekVA: 10,25,371/2, 50, 75, 100, 167, 2501
HV: 4160GY/2400 through 34500GY/19920V3
BIL: 60, 75, 95, 125, 150 kVLV: 240/120, 480/240, 277 V, 120/2403, 240/4802
1 Maxi only2 Available only on micros with cable lead secondary3 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 hingepins provide safe and durable service.
3. A recessed locking assembly with padlock provisions and apentahead 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 arewelded and each unit is pressure tested and inspected for leaksprior to shipment. In addition, all single phase transformers are sup-plied with:
a. 5/8''-11 stainless steel lifting bossesb. Oil level/fill plugc. Oil drain plugd. Self-actuating pressure relief devicee. Two ground bosses, 1/2''-13 NC tapped hole 7/16'' deep.
5. The front sill latches with the flip-top hood, is attached on the sideof the tank, and is removable.
6. The high voltage universal bushing wells are externally clampedand removable. A parking stand between the bushing wells is pro-vided 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.
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Minimum/Maximum Design Dimensions 11111
Micro-Pak Mini-Pak Maxi-Pak
A B C D A B C D A B C D
Min. 24 24 30.25 14 24 32 30.25 14 32 32 30.25 14Max. 24 24 35.50 16 42 44 46.00 20 42 44 46.00 20
1 Actual dimensions will vary according to voltage, loss evaluation,and accessories.
Optional Accessories1. Overcurrent Protection
a. An internal primary protectivelink to remove the transformerfrom the system in the event ofan internal fault.
b. A secondary breaker providesprotection against secondaryoverloads and short circuits.
Dimensions are in Inches c. An oil-immersed bayonet-typefuse link to remove the trans-former from the system in caseof an internal fault (fault sensing)or secondary short overload(overload sensing). This fuse isa drawout design and is suppliedin series with an isolation link. Adrip plate is provided to preventoil from dripping onto the bush-ing or elbow.
d. A current limiting fuse mountedin a dry well loadbreak canister.2
5.0
5.0
CABLE OPENING
“B”+6
“C”+6
• The high interrupting rating ofthe CL fuse permits its use onsystems where the availablefault current exceeds the rat-ing of normal expulsion fuses.
e. A partial range current limitingfuse mounted under oil withinthe transformer tank.2
• An expulsion fuse is suppliedin series with the partial rangeCL fuse.
Recommended Pad • Available at 95, 125, and 150 Dimensions kV BIL.
2. Switchinga. Externally operated tap changer.b. Externally operated dual voltage
switch.2
c. Externally operated loadbreakoil rotary (LBOR) switch.2
d. EFD CL fused air loadbreakswitch available for either radialor loop feed.3
2 Not available on Micro3 Maxi only
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3. Primary Connectiona. Universal bushing wells (stan-
dard) and loadbreak inserts.b. Integral (one piece) loadbreak
bushings.
4. Secondary Connectionsa. Copper studs with rotatable
spade type bushings.
• Four-hole, NEMA type, tin-plated copper alloy spade.
• Four-hole, in line, tin-platedcopper alloy spade.
b. Cable lead secondary.4
5. Corrosion Resistancea. ANSI C57.12.29 Full 400 Series
Stainless Steelb. Partial Stainless Steel
• Mini-Skirt™ and Sill
• Sill Only
• Sill and Hood
• Mini-Skirt™, Sill, and Hood
6. Miscellaneousa. Cleats for anchoring sill to pad.b. Polypad mounting base.4
4 Micro only
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The ABB MTR is an oil-filled, threephase, commercial padmounted distribu-tion transformer specifically designed forservicing such underground distributionloads as shopping centers, schools,institutions and industrial plants. It isavailable both live front and dead frontconstruction, for radial or loop feedapplications, or without taps.
Industry StandardsABB three-phase MTR units meet thefollowing industry standards:
The ABB MTR Padmounted ANSI C57.12.00 - IEEE Standard General
Transformer Three-Phase Requirements for Liquid Immersed…
45-2500 kVA Transformers
ANSI C57.12.22 - Pad-Mounted…Three-Phase Distribution Transformers
with High Voltage BushingsANSI C57.12.26 - Pad-Mounted…Three-
Phase Distribution Transformers…
With Separable Insulated High-VoltageConnectors
ANSI C57.12.28 - …Pad-Mounted Equipment- Enclosure Integrity
orANSI C57.12.29 - …Pad-Mounted
Equipment - Enclosure Integrityfor Coastal Environments
ANSI C57.12.70 - Terminal Markingsand Connections…
ANSI C57.12.80 - IEEE StandardTerminology…
ANSI C57.12.90 - IEEE Standard TestCode…
NEMA Tr-1 - Transformer StandardsIEEE 386 - Separable Insulated
Connectors
ABB Recommends the use of ANSI C57.91- IEEE Guide for Loading…for the estab-lishment of proper distribution transformerloading practices.
Ratings
• 45 through 2500 kVA
• 65°C average winding rise over 30°Caverage ambient.
• Low voltages: 1208Y/120, 216Y/125,460Y/265, 480Y/277, 480d, 240d and240d with 120 volt mid-tap in one phase.
• High voltages: 4160 Grd Y/2400through 34,500 Grd Y/19,920 forGrounded Wye systems; 2400 through34,500 for Delta systems; various dualhigh voltages.
• Taps: All voltages are available with orwithout taps.
• Insulation classes: 35 kV (200 kV BIL)and below.
1 208Y/120, 216Y/125, 240d not avail-able above 1500kVA.
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�����C. Protection and Accessories
1. General
The distribution transformer functions as an integral part of thedistribution system and consideration must be given to properprotection of the transformer from system disturbances. Inaddition, it is normal practice to apply overcurrent protectionon the primary side of the transformer so that a faulted trans-former is isolated from the primary system. Protection fromexcessive voltage transients and severe overcurrents shouldbe provided. Protection considerations include:
(1) Protective devices must be rated for the conditionsanticipated.
(2) When the transformer(s) is provided with overcurrentdevices — coordination with system devices should beachieved to allow proper fault isolation.
Caution: Operation of a primary protective device may indi-cate a faulted transformer. Re-energizing should be performedfrom a remote location unless the cause of device operation ispositively identified and corrected. To do otherwise presents ahazard to life and property in the event of violent transformerfailure.
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 capabili-ties to meet nearly every application.
• Conventional “S” TransformersThis type transformer contains no protective equipment.Therefore, lightning, fault and overload protection for thesetransformers must be provided by the purchaser.
• Surge-Protecting “SP” TransformersThe “SP” transformers include transformer-mounted lightningarresters and internally-mounted high voltage protective links,but omit the internally-mounted low voltage circuit breaker.
• Current-Protecting “CP” TransformersThe “CP” transformers are equipped with the internally-mounted low-voltage circuit breaker and high voltageprotective links, but omit the lightning arresters.
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Types of Accessories and Transformer Protection Packages—Pole Mounted (Continued)
• Self-Protecting “CSP®” TransformersIn a “CSP” transformer the arrester protects the transformerfrom over-voltage caused by lightning and/or high voltageswitching surges. The protective link operates to remove adefective transformer from service if an internal failure occurs,thereby protecting the system. The breaker provides the trans-former a degree of protection from overloads and short circuitson the secondary side of the transformer. This type trans-former offers the most protection of all protected transformersexcept for a “CSP” with a current limiting fuse.
a. CL FusesTwo basic types of current limiting fuses exist—partialrange and general purpose (full range). The partial rangefuse requires a protective link applied in series while thegeneral purpose fuse does not. The partial range fuse isavailable on pole-type transformers (bushing mounted)and padmounted transformers (internally mounted). Thegeneral purpose fuse is only available on padmountedtransformers.
b. The Distribution Surge Arrester protects the transformer(and other electrical equipment) from dangerous over-voltages, whether caused by lightning surges, switchingsurges or other transients.
The Type LV Surgemaster™ valve type arrester has oneor more arc gap assemblies connected in series with oneor more current limiting “valve” blocks. Under high volt-age surge conditions, the resistance in the blocks drops,providing a low-resistance path to ground. Once the surgehas passed, however, the block resistance rises again,restricting the flow of current. The gaps will then interruptthis low-magnitude current flow, restoring the arrestor toan insulator.
The LVBB Surgemaster™ valve type arrester is a bigblock (heavy duty) design which is capable of discharg-ing a 100 KA surge. The big block arrester operates thesame way as the LV with additional protection capability.
The HMX gapless metal oxide arrester is a heavy dutydesign utilizing the non-linearity of a metal oxide resistorto provide protection levels equivalent to gapped siliconcarbide arresters. The metal oxide distribution arresteroffers the benefits of reduced complexity, improved reli-ability and improved performance characteristics.
The LV, LVBB Surgemaster™ and HMX distributionarresters are available for either pole, crossarm or trans-former mounting.
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c. The Secondary Circuit Breaker provides the transformerwith a degree of protection from secondary overloads andshort circuits. It is mounted under oil, usually on the core/coil assembly, connected between the coil’s secondaryleads and the secondary bushings. The breaker is cali-brated to trip when its bimetal reaches a predeterminedtemperature. An additional instantaneous magnetic tripelement which responds to high fault currents is availableon 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 devicesincluding a protective link, distribution surge arrester, second-ary circuit breaker and current limiting fuse.
a. The Protective Fuse Link is an internal, oil-immersed,expulsion type fuse consisting of a fiber tube supportingand surrounding the fuse element usually made of copperand EVERDUR ®. The link is sized to operate only in theevent of a winding failure, isolating the transformer fromthe primary system. Interrupting rating is 3500 amperesat 7.2kV.
Protective Fuse Link
b. The Bayonet-Type Fuse Cartridge contains an oil-im-mersed expulsion type fuse with an interrupting rating of3800 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-sensingand fault-sensing—and an internal isolation link is sup-plied in series for additional safety. The fault-sensing linkis sized to operate only in the event of a transformer fail-ure; the overload-sensing link is sized for additional pro-tection from secondary system faults or prolonged heavyoverload conditions.
Standard Ratings:
Voltage Interrupting L.B. AmpsClass Amps (RMS) At .8 PF
8.3 kV 3800 13515.5 kV 2000 13523.0 kV 600 45
Bayonet-Type Fuse
14
c. Current Limiting Fuses are available through 15 kV ineither the EFD air loadbreak switch or in a drawout,loadbreak dry fuse well.
Some applications may require parallel current limitingfuses 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 mayrequire current fuse ratings that are not available—contactDivision.
Partial range, internal, block-mounted current limitingfuses, which are applied in series with an internal protec-tive link, are also available through 23 kV.
Loadbreak, Drywell Current Limiting Fuse Canister
d. The EFD is an loadbreak air switch available for radialfeed applications. Switching, flexibility and safety aremade possible by a compact, “dead front” type construc-tion that enables the switch to be externally-mounted onthe tank in the terminal compartment. A sealed, silver sandcurrent limiting fuse is normally provided to the switch’stransformer connecting pole. High voltage cables are con-nected to the switch contacts by means of solderless,clamp-type connectors capable of accepting cable sizesranging from #6 to #4/0.
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EFD Switch Ratings
Continuous current ................ 200 ALoadbreak ............................. 200 AClose-in ................................. 5,000 AMomentary ............................ 10,000 A
e. The LBOR (Loadbreak Oil Rotary) switch is gang-operatedand available for either radial or loop feed switching. Thestacked deck rotary switch has a unique, springloadedcam-operated kicker system which provides quick makeand break action to the contacts.
LBOR Ratings:
BIL 95 kV 125 kV 150 kV
Maximum Voltage(L-L) 15.5 kV 27 kV 38 kV(L-Grd) 8.9 kV 15.5 kV 21.9 kV
Continuous andInterrupting Current 300 A 1 200 A 300 A
Momentary andMaking Current 12 kA/ 12 kA/ 10 kA/(RMS Sym./Assym.) 19.2 kA 19.2 kA 16 kA
1 200 A 3c rating also available.
LBOR Switch
f. The Tap Changer and Series Multiple Switch. Both areoil-immersed, externally-operated, and are designed forde-energized operation only.
Tap Changer Operating Handle
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I I. Performance Page
A. Designation of winding voltage ratings .................................... 17B. Polarity ...................................................................................... 20C. Terminal designations .............................................................. 21
1. Pad-Mounted ....................................................................... 212. Pole-Mounted ...................................................................... 21
(a) 1c pole-mounted ........................................................... 21(b) 3c pole-mounted ........................................................... 22
D. Short circuit ratings .................................................................. 22E. Sound level ratings................................................................... 22F. Tolerance definitions ................................................................ 23G. Impedance calculations ........................................................... 23H. Efficiency calculations .............................................................. 24I. Regulation calculations ............................................................ 24J. Performance example .............................................................. 26K. Secondary fault current—120/240 volt systems...................... 27
17
I I. Performance
�����A. Designation of Winding Voltage Ratings(from ANSI C57.12.00)
1. Single-Phase
Symbol Example Typical Diagram
E 12000
Usage E shall indicate a winding of E volts which is suitable for g
connection on an E volt system.
E/E1Y 2400/4160Y
Usage E/E1Y shall indicate a winding of E volts which is suitable forg connection on an E volt system or for Y connection on an E1 voltsystem.
E1GrdY/E 12 470GrdY/7200
Usage E1GrdY/E shall indicate a winding of E volts with reducedinsulation at the neutral end. The neutral end may be connected directlyto the tank for Y or for single-phase operation on an E1 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 beconnected in parallel for operation at E volts, or which can be con-nected in series for operation at 2E volts, or connected in series witha center terminal for three-wire operation at 2E volts between theextreme terminals and E volts between the center terminal and eachof the extreme terminals.
2E/E 240/120
Usage 2E/E shall indicate a winding for 2E volts, two-wire full kVAbetween extreme terminals, or for 2E/E volts three-wire service with1/2 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 operationonly 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 deltawinding.
(2) E1 = CFF3 E
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2. Three-Phase
Symbol Example Typical Diagram
E 2400
Usage E shall indicate a winding which is permanently g connectedfor operation on an E volt system.
E1Y 4160Y
Usage E1Y shall indicate a winding which is permanently Y connectedwithout a neutral brought out (isolated) for operation on an E1 voltsystem.
E1Y/E 4160Y/2400
Usage E1Y/E shall indicate a winding which is permanently Y con-nected with a fully insulated neutral brought out for operation on anE1 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 foroperation on an E volt system, or may be Y connected without a neutralbrought out (isolated) for operation on an E1 volt system.
E/E1Y/E 2400/4160Y/2400
Usage E/E1Y/E shall indicate a winding which may be g connectedfor operation on an E volt system or may be Y connected with a fullyinsulated neutral brought out for operation on an E1 volt system withE volts available from line to neutral.
E1GrdY/E 12470GrdY/7200
Usage E1GrdY/E shall indicate a winding with reduced insulation andpermanently Y connected, with a neutral brought out and effectivelygrounded for operation on an E1 volt system with E volts availablefrom 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 = CFF3 E
19
2. Three-Phase (continued)
Symbol Example Typical Diagram
E/E1GrdY/E 7200/12470GrdY/7200
Usage E/E1GrdY/E shall indicate a winding, having reduced insula-tion, which may be g connected for operation on an E volt system ormay be connected Y with a neutral brought out and effectivelygrounded for operation on an E1 volt system with E volts availablefrom line to neutral.
V x V1 7200 x 14 400
Usage V x V1 shall indicate a winding, the sections of which may beconnected in parallel to obtain one of the voltage ratings (as definedabove) of V, or may be connected in series to obtain one of the volt-age ratings (as defined above) of V1. Windings are permanently g orY connected.
Notes:(1) E = line-to-neutral voltage of a “Y” winding, or line-to-line voltage of a delta
winding.
(2) E1 = CFF3 E
20
�����B. Polarity
The lead polarity (or polarity) of a transformer is a designation ofthe relative instantaneous directions of currents in its leads. Primaryand secondary leads are said to have the same polarity when at agiven instant the current enters the primary lead in question andleaves the secondary lead in question in the same direction asthough the two leads formed a continuous circuit. The lead polar-ity of a single-phase transformer may be either additive orsubtractive. If one pair of adjacent leads from the two windings inquestion is connected together and a small voltage is applied toone of the windings, then the connection behaves as an auto trans-former with the secondary voltage adding to or subtracting fromthe primary voltage. The polarity determination is as follows:
a. The lead polarity is additive if the voltage across the other twoleads of the windings in question is greater than that of thehigher voltage winding alone.
b. The lead polarity is subtractive if the voltage across the othertwo leads of the windings in question is less than that of thehigher voltage winding alone.
AdditiveE3 > E1
SubtrativeE3 < E1
By industry standards, single-phase distribution transformers 200kVA and smaller, having high voltage windings rated 8660 volts orless have additive polarity. All other single-phase transformers havesubtractive polarity.
The polarity of a three-phase transformer is fixed by the internalconnections between phases as well as by the relative locationsof leads; it is usually designated by means of a vector diagramshowing the angular displacements of windings and a sketchshowing the marking of leads. The vectors of the vector diagramsrepresent induced voltages, and the recognized counterclockwisedirection of rotation of the vectors is used. The vector representingthe voltage of a given winding is drawn parallel to that represent-ing the corresponding voltage of any other winding having thesame phase.
21
�����C. Terminal Designations
1. Pad MountedThe terminal designations for pad-mounted distribution trans-formers are clearly marked at the terminals of both the highand low voltage.
2. Pole MountedFor pole mounted distribution transformers, the terminaldesignations follows:
(a) Single-Phase Pole Mounted
Connection Additive Polarity Subtractive Polarity
E/2Ewith threeexternallow-voltageterminals
Series orThree-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 left-hand side when facing the low-voltage terminals.
22
Terminal Designations (Continued)
(b) Three-Phase Pole Mounted
All three-phase pole mounted distribution transformers have terminaldesignations as shown above regardless of internal connection.
Neutral terminals (HV and/or LV) will exist as required by the windingconnection 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 industrystandards. The maximum magnitude required for units with secondaryvoltages rated less than 600 V is as follows:
1 c kVA 3 c kVA Rating (times normal)
5-25 15-75 4037.5-100 112.5-300 35
167-500 500 25
750-2500 1/ZT*
Two winding distribution transformers with secondary voltages ratedabove 600 volts are required to withstand short-circuits limited only bythe transformer’s impedance.
The duration of the short-circuit current is determined by
500 kVA and Below 750-2500 kVA________________ ___________
t =1250 t = 1.0
l 2
where: t = duration (seconds)I = symmetrical short-circuit current (per unit)
*1/ZT = The short circuit current will be limited by the trans-former impedance only. ZT is transformer per unitimpedance.
�����E. Sound Level Ratings (NEMA TR-1)
The sound level ratings for distribution transformers are set by industrystandards. The maximum sound level (A weighted response curve) is:
kVA Rating Sound Level (dB)0-50 48
51-100 51101-300 55301-500 56
-750 57-1000 58-1500 60-2000 61-2500 62
23
�����F. Tolerance Definitions (ANSI C57.12.00-1987)
1. Impedance (two-winding transformers)
The impedance of a two-winding transformer with impedancevoltage larger than 2.5% shall have a tolerance of ± 7.5% ofthe specified value; and the tolerance for those with impedancevoltage 2.5% or less shall have a tolerance of ± 10% of thespecified value.
Differences of impedance between two duplicate two-windingtransformers when two or more units of a given rating are pro-duced by one manufacturer at the same time shall not exceed7.5% of the specified value.
Transformers shall be considered suitable for operation inparallel if impedances come within the limitations of the fore-going paragraphs, provided turns ratios and other controllingcharacteristics are suitable for such operation. (See Paralleling)
2. Losses
The total losses of a transformer are the sum of the excitationlosses and the load losses at rated load (with winding tem-perature at 85°C).
Unless otherwise specified, the losses represented by a test ofa transformer, or transformers, on a given order, shall not exceedthe specified losses by more than the percentages below:
No Load TotalNo. of Units Basis of Losses Losses
On One Order Determination (Percent) (Percent)
1 1 unit 10 62 or more each unit 10 62 or more average of 0 0
all units
�����G. Impedance Calculations
Transformer impedance is shown on the transformer nameplate(Note: transformer impedance, reactance and resistance are typi-cally given in percent or per unit). If the transformer load lossesare known, the impedance may be separated into its reactive andresistive components.
Z - impedance (percent)R - resistance (percent)X - reactance (percent)
kVA - transformer kVA ratingCu - load loss at rated load at 85°C (watts)
R =Cu
10 kVA
X = CFFFFFFZ2 – R2
24
�����H. Efficiency Calculations
The efficiency of a transformer is defined as the ratio of the outputpower to the input power. It can be calculated at any load andpower factor if the transformer losses are known.
E - efficiency (percent)L - load (per unit)
kVA - transformer kVA ratingCu - load loss at rated load at 85°C (watts)Fe - no load (excitation) loss (watts)a - power factor angle
E =L.kVA.cosa.105
percent(L.kVA.cosa.10 3) + Fe + L2.Cu
At rated load and unity power factor
E =kVA.105
percentkVA.10 3 + Fe +.Cu
Regardless of the load power factor angle, it can be shown thatthe per unit load which results in maximum efficiency is:
L (maximum efficiency) =
�����I. Regulation Calculations
The voltage regulation of a distribution transformer is the changein output voltage which occurs when the load is reduced fromrated value to zero with the primary terminal voltage maintainedconstant. The regulation can be calculated from the equationsbelow or by the nomograph which follows:
R - resistance(percent)X - reactance (percent)
REG- percent voltage regulationa - power factor angle (positive for inductive load)
REG = [R2 + X2 + 200. (X.sina + R.cosa) + 10,000]1/2 – 100
For unity power factor
REG = [R2 + X2 + 200R + 10,000]1/2 – 100
25
Regulation Chart
Place straight edge at percent resistance, scale one, and at percentreactance, scale nine. Read the percent regulation at different powerfactors as given by scales two to eight inclusive
26
�����J. Performance Example
Example Transformer Ratings (Typical)
Single-phase kVA 25High voltage 7200vLow voltage 120/240vNo load (excitation) loss 104 wattsTotal loss at rated load 419 wattsImpedance 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
R =Cu
=(419–104)
= 1.26%______ _______
10.kVA 10.25
X = CFFFFFF = CFFFFFFFFFF = 0.99%Z2 – R2 1.62 – 1.262
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
E =L . kVA . cosa . 105
L . kVA . cosa . 103 + Fe + L2Cu
=(0.5)
. (25)
. (.85)
. 105
= 98.3%(0.5)
. (25)
. (.85)
. 103 + 104 + (0.5)2
. (419 – 104)
5. Expected Regulation
REG = [R2 + X2 + 200. (X.sina + R.cosa) + 10,000]1/2 – 100
= [1.262 + 0.992 + 200.(0.99.0.53 + 1.26.0.85) + 10,000]1/2–100
= 1.596
27
�����K. Secondary Fault Currents — 120/240 Volt Systems
Service to individual residences in the United States most always issingle-phase three-wire operating at 120 volts from phase-to-neutral,and 240 volts from phase-to-phase. In order to select serviceentrance equipment with adequate interrupting rating, or to coor-dinate over-current protective devices in the transformer-secondarysystems, the available currents for a bolted fault (short circuit)must be known. This section presents equations and data whichcan be used to calculate the available currents for both phase-to-phase (240 volt) and phase-to-neutral (120 volt) faults. Theequations for calculating these currents are quite simple and canbe easily evaluated with a handheld pocket calculator
Fault Current EquationsFigure K.1 gives, for convenient reference, the equationsnecessary for calculating the available currents for both 240 voltand 120 volt bolted faults, and defines the terms appearing in theequations. Before explaining the use of the equations, the as-sumptions used in arriving at these are discussed.
Figure K.1
28
The impedance of the primary system supplying the distributiontransformer is very small in comparison to that of the distribution trans-former and secondary circuit up to the point of fault. The effect of thisassumption is to make the calculated values of current for a boltedfault in the secondary system slightly higher than those which resultwhen the effect of primary impedance is included. Increasing the“stiffness” of the primary system, reducing the kVA size of the trans-former, or increasing the secondary circuit length to the fault pointreduces the difference between the approximate and more exact cal-culated values of bolted fault current. In contrast, the difference betweenthe approximate and more exact values will be greater for “weak” primarysystems, large distribution transformers, and short secondary circuits.For most cases where the calculations are made to determine avail-able fault current at the service entrance for sizing equipment, or todetermine maximum currents at which overcurrent protective devicesmust coordinate, the difference resulting from the assumption is negli-gible. However, for those cases where the calculated current usingmethods neglecting primary impedance is slightly higher than theinterrupting rating of a fuse or breaker in the secondary system, orwhere the calculated current is slightly above the value at whichovercurrent protective device coordination can be achieved, thenincluding the effect of primary system impedance may show that a“problem” does not exist. Calculations including the effects of primarysystem impedance are not contained in this guide.
Reference to Figure K.1 shows that the expressions for calculating theavailable current for the 240 volt and 120 volt bolted faults are different.While the 240 volt fault current can be calculated from a knowledge ofthe “full winding” impedance of the transformer, the calculation of the120 volt fault current requires a knowledge of the transformer “halfwinding” impedance. As the relationship between transformer “halfwinding” and “full winding” impedance is not fixed and can vary fromdesign to design, the most typical relationship for present day designswas used in arriving at the equation for 120 volt fault current. LettingRT + jXT be the “full winding” impedance in percent on nameplate kVArating looking into the primary winding, the “half winding” impedance inpercent on nameplate kVA can be approximated by 1.5 RT + j2.0 XT.
Also notice from Figure K.1 that the equations do not include the effectof any metering impedances which may be present in the circuit, orany “fault” impedance. Including these impedances will further reducethe calculated values of fault current.
29
The steps to follow when using the equations in Figure K.1 to calculatethe bolted fault currents are as follows:
1. Calculate the transformer resistance in ohms at secondary termi-nals X1-X3 (RT in Figure K.1). This requires that the transformertotal losses at full load in watts, and no load losses in watts beknown (WTOT and WNL respectively in Figure K.1).
2. Calculate the transformer leakage impedance in ohms at second-ary terminals X1-X3 (ZT in Figure K.1). This requires that thetransformer nameplate impedance in percent (Z%) be known.
3. Calculate the transformer leakage reactance in ohms at secondaryterminals X1-X3 (XT in Figure K.1).
4. Determine the resistance of the secondary circuit in ohms per1000 feet for a 240 volt fault (RS). Also determine the reactance ofthe secondary circuit in ohms per 1000 feet for a 240 volt fault (XS).Typical values for RS and XS in ohms per 1000 feet are given inTables 1 and 2 for circuits using aluminum phase conductors underthe header “240 V FAULTS”. The values in Table 1 are for triplexcable, and those in Table 2 are for rack mounted conductors. Fromthese tables notice that the resistance values are the same, but thereactance 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 (I240) usingthe equation in Figure K.1 and the values calculated in steps1 through 4.
6. Determine the resistance (RS1) and reactance (XS1) of the second-ary circuit in ohms per 1000 feet for a 120 volt fault. typical valuesfor RS1 and XS1 in ohms per 1000 feet are given in Tables 1 and 2for circuits using aluminum conductors under the header “120 VFAULTS”. In both tables, the values listed are for circuits using areduced size neutral conductor. If a full size neutral conductor isused, then the impedance values given under the header “240 VFAULTS” should also be used for the calculation of the 120 voltfault currents.
7. Calculate the available current for a 120 volt bolted fault (I120) usingthe equation in Figure K.1 and the values calculated in steps 1through 3 and step 6.
30
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 noload losses of 204 watts has an impedance of 1.75 percent. A serviceentrance circuit which is 80 feet in length using 3/0 aluminum triplexwith reduced neutral is connected directly to the transformer terminals.What is the available current for both a 240 and 120 volt bolted fault atthe end of the service? From the statement of the problem:
WTOT = 759 watts Z = 1.75 percentWNL = 204 watts L = 80 feetkVA = 50
The calculations proceed following the steps outlined.
1. RT = 0.0576759 – 204
= 0.012787 ohms502
2. ZT
= 0.5761.75
= 0.02016 ohms50
3. XT = CFFFFFFFFFFFFFFFFF = 0.015586 ohms.020162 – .0127872
4. From Table 1, the resistive and reactive components of the imped-ance for a 240 volt fault with 3/0 aluminum triplex cable are:
RS = 0.211 ohms per 1000 feet
XS = 0.0589 ohms per 1000 feet
5. Placing these values of RT, X T, RS, X S, and L into the equation forI240 in Figure K.1 gives:
I240 = 6676.6 amperes rms symmetrical
Note that the large number of significant digits included in thesecalculations 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 imped-ance for a 120 volt fault with 3/0 aluminum triplex cable (reducedneutral) are:
RS1 = 0.273 ohms per 1000 feet
XS1 = 0.0604 ohms per 1000 feet
7. Placing the above values into the equation for I120 in Figure K.1gives:
I120 = 4071.1 amperes rms symmetrical
For this example notice that at a distance of 80 feet from the trans-former, the available current for the 120 volt bolted fault is considerablyless than that for a 240 volt fault. However, from the equation for I240
and I120 in Figure K.1, notice that for a fault at the transformer second-ary terminals (L = 0.0 feet), the available current for a bolted 120 voltfault is greater than that for a 240 volt fault. Thus at some distance Lfrom the transformer, I240 and I120 would be equal, and at distancesgreater than this, the available current for a 240 volt fault will be higher.
31
Figure K.2 is a plot of the available current for both the 120 and 240volt bolted faults vs. the distance from the transformer terminals to thefault point in feet. The curves are for transformer sizes of 50, 75, and100 kVA supplying a secondary circuit made with 3/0 aluminum triplexwith reduced neutral. From these curves notice that:
Figure K.2
(a) The available current for both the 120 and 240 volt faults israpidly 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 currentfor a 120 volt fault is less than that of a 240 volt fault at distancesgreater than about 10 feet from the 50, 75, or 100 kVA trans-former. For most all single-phase services rated 200 amperesor less, the available current at the service entrance for the 120volt fault is less than that of the 240 volt fault.
(c) As the distance from the transformer to the fault location be-comes large, the available current for both the 120 and 240 voltfaults becomes independent of the transformer size, especiallyfor the 120 volt fault.
32
No
tes
:(1
)Re
sis
tan
ce
va
lue
s b
ase
d o
n a
co
nd
ucto
r te
mp
era
ture
of 25
°C.
(2)R
ea
cta
nce
ba
se
d o
n f
ollo
win
g:
(a)
60
0 v
olt in
su
latio
n w
ith
all
3 in
su
late
d c
on
du
cto
rs in
co
nta
ct.
(b)
Fo
r 1
20
vo
lt (
Ph
ase
-to
-Ne
utr
al
Fa
ult),
all
cu
rre
nt
retu
rns i
n t
he
ne
utr
al
co
nd
ucto
r w
ith
no
cu
rre
nt
retu
rnin
g in
th
e e
art
h.
(3) I
nsu
latio
n t
hic
kn
ess i
s 0
.06
2 i
nch
fo
r #
4 t
o #
2,
0.0
78
inch
fo
r #
1 t
o 4
/0,
an
d .
09
4 in
ch
fo
r 2
50
to
50
0 M
CM
.(4
)Fo
r se
co
nd
ary
circu
its w
ith
fu
ll siz
e n
eu
tra
l, u
se
re
sis
-ta
nce
an
d r
ea
cta
nce
va
lue
s g
ive
n f
or
24
0 v
olt f
au
lt f
or
bo
th 1
20
an
d 2
40
vo
lt f
au
lts.
No
tes
:(1
)Re
sis
tan
ce
va
lue
s b
ase
d o
n a
co
nd
ucto
r te
mp
era
ture
of 25
°C.
(2)R
eacta
nce v
alu
es b
ased o
n s
econdary
rack w
ith 1
2in
ch
sp
acin
g b
etw
ee
n c
on
du
cto
rs w
ith
ne
utr
al
in t
op
po
si-
tio
n a
nd
ph
ase
co
nd
ucto
rs in
th
e t
wo
lo
we
r p
ositio
ns.
Resis
tance a
nd reacta
nce v
alu
es g
iven for 120 v
olt fault
assu
me
fa
ult is t
o p
ha
se
co
nd
ucto
r in
mid
dle
po
sitio
nin
ra
ck.
(3)F
or
se
co
nd
ary
circu
its w
ith
fu
ll siz
e n
eu
tra
l, u
se
re
sis
-ta
nce
an
d r
ea
cta
nce
va
lue
s g
ive
n f
or
24
0 v
olt f
au
lt f
or
bo
th 1
20
an
d 2
40
vo
lt f
au
lts.
Tab
le 1
.Typ
ical
Imp
ed
an
ce
s f
or
12
0/2
40
Vo
lt C
irc
uit
s W
ith
Tri
ple
x C
ab
le
Alu
min
um
Ph
ase
Co
nd
. A
lum
inu
m N
eu
tra
l C
on
d.
12
0 V
olt F
au
lts
24
0 V
olt F
au
lts
___________________
____________________
____________
____________
Siz
eN
o. of
Siz
eN
o. of
RS
1X
S1
RS
XS
(AW
G o
r M
CM
)S
trands
(AW
G o
r M
CM
)S
trands
(j/1
00
0 F
t.)
(j/1
00
0 F
t.)
(j/1
00
0 F
t.)
(j/1
00
0 F
t.)
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
27
47
.69
1.0
65
2.5
34
.06
33
11
93
7.5
47
.06
59
.42
4.0
65
9
1/0
19
27
.43
5.0
62
8.3
35
.06
16
2/0
19
11
9.3
45
.06
29
.26
6.0
59
63
/01
91/0
19
.27
3.0
60
4.2
11
.05
89
4/0
19
2/0
19
.21
7.0
58
8.1
67
.05
76
25
03
73
/01
9.1
77
.05
83
.14
2.0
57
43
50
37
4/0
19
.13
4.0
57
0.1
02
.05
58
50
03
73
00
37
.09
5.0
54
7.0
72
.05
30
Ta
ble
2.
Typ
ical
Imp
ed
an
ce
s f
or
12
0/2
40
Vo
lt C
irc
uit
s W
ith
Ra
ck
Mo
un
ted
Co
nd
uc
tors
Alu
min
um
Ph
ase
Co
nd
. A
lum
inu
m N
eu
tra
l C
on
d.
12
0 V
olt F
au
lts
24
0 V
olt F
au
lts
___________________
____________________
____________
____________
Siz
eN
o. of
Siz
eN
o. of
RS
1X
S1
RS
XS
(AW
G o
r M
CM
)S
trands
(AW
G o
r M
CM
)S
trands
(j/1
00
0 F
t.)
(j/1
00
0 F
t.)
(j/1
00
0 F
t.)
(j/1
00
0 F
t.)
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
27
47
.69
1.2
23
.53
4.2
17
11
93
7.5
47
.21
7.4
24
.21
2
1/0
19
27
.43
5.2
11
.33
5.2
04
2/0
19
11
9.3
45
.20
5.2
66
.19
93
/01
91/0
19
.27
3.1
99
.211
.19
34
/01
92/0
19
.21
7.1
93
.16
7.1
88
25
03
73
/01
9.1
77
.18
9.1
42
.18
43
50
37
4/0
19
.13
4.1
82
.10
2.1
76
50
03
73
00
37
.09
5.1
74
.07
2.1
68
33
I I I. Three-Phase Transformersand Banks Page
A. Application Considerations ...................................................... 341. Types of distribution systems ............................................. 34
a. Primary (source) systems .............................................. 34b. Secondary (service) systems ......................................... 34
2. Angular displacement (phase shift) .................................... 353. Neutral grounding ............................................................... 35
a. Primary neutral grounding ............................................. 35b. Secondary neutral grounding ......................................... 35
4. Ferroresonance ................................................................... 36a. Primary winding connections which can result
in Ferroresonance .......................................................... 37b. Primary winding connections which can prevent or
minimize the possibility of Ferroresonance ................... 38B. Summary of Common Connections ......................................... 41
1. Delta-delta ........................................................................... 412. Delta-wye ............................................................................ 423. Wye-delta ............................................................................ 434. Wye-wye.............................................................................. 445. Grounded wye-wye ............................................................. 456. T-T (O degree angular displacement) ................................ 457. T-T (30 degree angular displacement) ............................... 468. Open Wye-Open Delta ........................................................ 469. Open Delta-Open Delta ...................................................... 47
C. Common Three-Phase Banks UsingSingle-Phase Transformers ..................................................... 48
34
II I. Three-Phase Transformers and BanksThis section presents many important factors to be considered whenselecting the connections used for both three phase transformers andthree-phase banks of the single-phase transformers applied in three-phase distribution systems. A summary of commonly encounteredconnections is provided. In addition, connection diagrams using single-phase transformers for three-phase banks are shown.
�����A. Application Considerations
1. Types of distribution systems
A three-phase distribution transformer, or a bank of single-phase distribution transformers should be thought of as asystem component which connects the primary to the second-ary system. Since it is a system component, proper applicationand determination of permissible connections requires anunderstanding of the characteristics of both the primary sys-tem which will supply the transformer, and the secondarysystem which will be supplied by the transformer.
a. Primary (Source) Systems
Distribution systems are either effectively grounded,impedance grounded, or ungrounded. Most electric utilitydistribution systems in this country are three-phase 4-wiremulti-grounded neutral systems which are effectivelygrounded. (An effectively grounded system is one whereat any point in the system the ratio of zero-sequencereactance to positive-sequence reactance is less than three,and the ration of zero-sequence resistance to positive-sequence reactance is less than one.) With a 4-wireeffectively grounded neutral system, the primary windingsof the distribution transformers can be connected from eitherphase to phase or phase to neutral. This permits usage ofthe following connections: Delta, open delta, grounded wye,open wye, floating wye, and T. Whether the neutral point ofwye connected primary windings should or should not beconnected to the system neutral depends upon the con-nections used for the secondary windings.
Although they are not commonly used by electric utilitiesfor distribution, impedance grounded or ungroundedsystems are frequently found in industrial plants. Thesesystems provide no path to carry neutral load current. Thus,distribution transformers applied must be connected phaseto phase using either delta, open delta, floating wye, or Tconnected windings.
b. Secondary (Service) Systems
Secondary systems supplied from distribution transformersand operating at 600 volts or less usually are either 3-wireungrounded, or 4-wire grounded. To supply a 3-wireungrounded (delta) system, the transformer secondarywinding may be connected in delta, open delta, floatingwye, or T.
Loads which require both single-phase 3-wire 120/240volt service and three-phase 240 volt service can besupplied by a 4-wire service consisting of transformerswith secondary windings connected delta or open delta
35
with a center tap ground on one leg of the delta. In the4-wire grounded (wye) system, the transformer second-ary windings must have a neutral point which can begrounded. The 4-wire grounded secondary service canbe supplied by either the wye connection or the T con-nection with the neutral point grounded.
2. Angular Displacement (Phase Shift)
For standard three-phase connections the phase-to-neutralvoltage on the primary side either leads that on the secondaryside by 30° or is in phase with the phase-to-neutral voltage onthe secondary side. The delta delta and wye wye connectionsproduce no phase shift. The delta wye and wye delta connec-tions produce the 30° phase shift. The T-T transformer can bedesigned to exhibit either a 30° or a 0° phase shift.
When paralleling three-phase transformers or banks, the phaseshift of each must be the same. In addition, the 30° phase shifthas an effect on the coordination of overcurrent protective deviceslocated on the primary and secondary sides of the transformer.For unsymmetrical faults the line currents do not transform inproportion to the voltage ratings. Of particular importance is aline-to-line fault on the transformer secondary. For the connec-tions which have a 30° phase shift, this fault produces a faultcurrent in one primary phase which is 1.15 times the secondaryfault current on a per unit basis. This additional 15% must beconsidered to achieve selective coordination.
3. NeutraI Grounding
Some transformer connections or winding connections (wyeor 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 tothe multi-grounded neutral conductor of the primary system,or the neutral point of the secondary windings can be groundedto establish a 4-wire grounded wye system. Whether the neutralpoint of windings should or should not be grounded dependson factors discussed below.
a. Primary Neutral Grounding
For the primary neutral point to be grounded, the primarysource must be a 4-wire multi-grounded neutral system.In addition, it is generally undesirable that a distributionbank act as a ground source for the primary system. Toprevent creation of a grounding bank, a primary wyeshould only be grounded if the secondary is also con-nected in wye and a T primary should never be grounded.Note however that the open wye connection must begrounded at the neutral point to function properly.
b. Secondary Neutral Grounding
To supply phase to neutral connected load on the sec-ondary, a low impedance ground source must beestablished. This can be achieved by grounding the neutralof a secondary wye connection provided that the primaryis connected either delta or wye grounded supplied by a4-wire multi-grounded neutral (effectively grounded)source. The neutral of a secondary T connection may alsobe grounded. In addition, a delta or open delta windingmay be grounded at any one point.
36
4. Ferroresonance
Ferroresonance is a non-linear resonance which can occurduring open conductor (single-phase) conditions in the distri-bution system. When ferroresonance occurs, it is characterizedby high overvoltages whose waveform contains appreciableharmonics. The transformers involved in the ferroresonantcircuit may emit unusual noises which frequently are describedas rattling, rumbling, or whining sounds. These are considerablydifferent than those which emanate from the transformer whenenergized at rated frequency and voltage. Overvoltages offive times normal and higher have been measured duringferroresonant oscillations in test circuits. Some causes of openconductor conditions which may result in ferroresonance are:(1) the operation of single-pole overcurrent protective devicessuch as fuses or single-pole reclosers, (2) normal switchingoperations with single-pole devices such as distribution cut-outs to energize or de-energize a transformer, and (3) failureto connect jumpers.
Whether ferroresonance will occur during open conductor con-ditions depends to a great extent upon the connections used forthe primary windings in a distribution transformer bank or in athree-phase distribution transformer. Under normal conditionswhere all three primary phases to the transformer bank areenergized through a continuous path from the source, ferro-resonance will not occur for any of the connections used for theprimary 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 systemcapacitance. If the capacitance lies within a specified range,ferroresonance may result. However, with other transformerconnections, ferroresonance will not occur during open conduc-tor conditions because the non-linear inductances cannot beinserted in series with system capacitances.
Figure 4.1: Cable-fed transformer with single-pole switchingdevices located at the junction between the overheadand underground circuits.
Figure 4.1 illustrates a frequently encountered system condi-tion which produces ferroresonance. An unloaded three-phasepad mounted transformer with delta connected primary wind-ings is supplied from an open-wire line through a cable circuit.At the riser or transition pole, the cable circuit is connected tothe open wire line using distribution cutouts. Notice that duringthe switching operation (open conductor condition) where only
37
the switch in phase A is closed as illustrated, the non-linearinductances of the transformer windings between phases Aand B, and phases A and C, are placed in series with the cablecapacitance on the open phases. This makes a series L-Ccircuit where the L is non-linear, and if the parameters are inthe proper range, ferroresonance will occur.
a. Primary Winding Connections Which Can Result inFerroresonance
Theoretically, ferroresonance can occur during open conductorconditions in either one or two phases if the primary windings ofthe 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 dependsupon the amount of capacitance between the open conductorand transformer, the transformer internal capacitances, thetransformer size, the system voltage, and the amount of loadconnected to the secondary terminals of the transformer, or theamount of load on the primary circuit between the open conduc-tor and transformer. Studies have shown that ferroresonance ismore likely to occur with cable circuits (due to higher capaci-tance) than open-wire lines, with small transformers, at higherprimary voltage levels (more likely at 35 kV than 4 kV voltagelevel), and with unloaded transformers.
Industry experience has shown that in overhead distributionsystems operating at 15 kV and below, overvoltages andferroresonance usually do not occur during open conductor con-ditions, even when the ungrounded primary winding connectionsare used for transformers. Ferroresonance became an impor-tant concern in the utility industry with the advent of undergrounddistribution and the use of 25 and 35 kV class voltages.
In higher voltage (25 kV and 35 kV) overhead systems, over-voltages and ferroresonance have occurred when single-poleswitching is performed at the terminals of small transformer bankswith 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 offerroresonance occurring in 15, 25, and 35 kV class overheadsystems when the switching is performed at the terminals ofsmall banks made from single-phase units.
The probability of ferroresonance and the associated overvolt-ages is very high if the circuit between the location of the openconductor and the transformer is made from shielded cable andoperates at voltage levels in either the 15, 25, or 35 kV class.This is because the capacitance per unit length of a cable circuitis in the range of 50 times that of open wire lines. A systemillustrating this situation is shown in Figure 4.1. Because of thehigh probability of ferroresonance in underground systems usingconventional single-pole switching devices, many systemoperators will not use the ungrounded primary winding connec-tions in cable-fed transformers.
If, however the transformer primary windings are ungrounded,as with the delta, open delta, wye, and tee connections, and thecircuit between the transformer and possible location of an openconductor (single-phase) condition is made from cable, thepossibility of ferroresonance can be minimized with the follow-ing measures.
38
(1) Application of only three-pole gang operated switches andfault interrupters. This minimizes the possibility of havingsingle-phase conditions.
(2) Location of the single-pole switches and overcurrent pro-tective devices only at the transformer terminals.
(3) Connection of resistive load to the secondary terminalsof the transformer during remote single-pole switching.
Although these measures can be very effective, many opera-tors of underground systems consider them unacceptable foreither economical, operational, or technical reasons. Instead,they prefer to use transformer connections which have eithera zero or very low probability of ferroresonance during openconductor conditions at a location remote from the transformer.
Figure 4.2: Probability of ferroresonance in overhead systemswhen switching is performed at the terminals of smalltransformer banks made from single-phase units.
b. Primary Winding Connections Which Prevent OrMinimize Possibility of Ferroresonance
When the primary windings of single-phase distribution trans-formers used in a bank are connected in open wye or groundedwye, or if a three-phase unit with the grounded wye primarywindings employs triplex construction, ferroresonance will notoccur during most open conductor conditions in the primarysystem. This is true for both overhead and undergroundsystems operating up through 35 kV. But if either a floatingwye or delta connected shunt capacitor bank is installed onthe primary line between the transformer bank and location ofthe open conductor, ferroresonance may occur. However, theuse of these connections for capacitor banks is very uncom-mon in distribution systems operating in the 15 kV class andabove. If there is a very long length of open wire line between
39
the location of the open conductor and transformer bank withgrounded wye or open wye primary windings, and no otherload is connected to the line beyond the open point, ferro-resonance can occur because of the phase-to-phasecapacitance of the open wire line. The probability of such con-ditions existing, even in 25 and 35 kV rural distribution systems,is very remote. Thus, for practical purposes, ferroresonancewill not occur when the grounded wye or open wye connec-tions 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 switchingis performed at the terminals of transformer banks in over-head systems with the grounded wye or open wye connectedprimaries at all voltages as illustrated in Figure 4.2.
When the grounded wye-grounded wye or grounded wye-floating wye connections are used in a transformer constructedon a four- or five-legged core, overvoltages and ferroresonancemay occur during open conductor conditions at a remote pointwhen cable circuits are involved. Test data shows that crestvoltages as high as 2.35 per unit are possible, but usually theyare considerably less than this. In contrast, overvoltages of5 per unit and higher are possible when the transformer hasthe ungrounded primary winding connections. Furthermore, thelength of primary cable circuit which can be used with trans-formers with four- or five-legged core and grounded-wyeprimary is in the range of 50 times that possible when theungrounded primary connections are used when the voltageon the open phase is limited to 1.25 per unit.
Although the use of triplex construction essentially eliminatesthe possibility of ferroresonance in cable-fed three-phase trans-formers with the grounded wye primary, such constructiongenerally makes the transformer larger, heavier, and morecostly than conventional four- or five-legged core units. Mostsystem operators, based on the good experience and perfor-mance they have had with the grounded wye primaries on four-and five-legged cores, have not been able to justify the addedcost for triplex construction.
If it is necessary to further minimize the possibility of ferro-resonance when the grounded wye primary is used on a four-or five-legged core, the measures listed below can beemployed:
(1) Application of only three-pole gang operated switches andfault interrupters. This minimizes the possibility of havingsingle phase conditions.
(2) Location of single-pole switches and overcurrent protec-tive devices only at the transformer terminals.
(3) Connection of resistive load to the secondary terminalsof the transformer during remote single-pole switching.
The preceding discussion of ferroresonance is both very briefand very qualitative in content. As it may be necessary to quan-tify certain aspects of ferroresonance, such as determining themaximum length of cable circuit which can be used between aswitch and transformer if voltage is to be limited to a specifiedvalue, the reader is referred to the many references which existon the subject. A few are listed below.
40
References
1. Schmid, R. L. “An Analysis and Results of Ferroresonance”. Trans-mission and Distribution, pp. 114-117, Oct. 1969.
2. Kratz, E. F., Manning, L. W., and M. Maxwell. “Ferroresonance inSeries Capacitor-Distribution Transformer Applications.” AIEETransaction (Power Apparatus & Systems), vol. 78, pp. 438-449,August 1959.
3. Young, F. S., Schmid, R. L., and P. I. Fergestad. “A LaboratoryInvestigation of Ferroresonance in Cable Connected Transform-ers,” 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 kVRural Distribution Systems.” AIEE Transactions (Power Apparatusand Systems), vol. 73, pp. 1208-1212, Oct. 1954.
5. Smith, D. R., Swanson, S. R., and J. D. Borst. “Overvoltages WithRemotely-Switched Cable-Fed Grounded Wye-Wye Transformers.”IEEE Transactions on Power Apparatus and Systems, vol. PAS-94,pp. 1843-1853, Sept./Oct. 1975.
41
�����B. Summary of Common Connections
DELTA-DELTA Connection
PhasorDiagram:
Angular Displacement (Degrees): 0
Source: Suitable for both ungrounded and effectively groundedsources.
Service: Suitable for 3-wire service or for 4-wire service with a mid-tap 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 ofthree identical units out of service, the rating of the bank whensupplying only three-phase load is about 57.7 percent of thebank rating when all three units are in service.
2. Caution: Each unit in a bank of single-phase units must be con-nected for the same voltage ratio, otherwise high circulatingcurrents can occur. Prior to completing a closed delta second-ary connection, the voltage between the two transformers closingthe delta should be checked to verify the voltage ratios andconnections.
3. Impedance mismatch among units of a single-phase bank willrequire derating of the bank.
4. Single-phase units having a secondary breaker should not beused for a bank providing 4-wire (mid-tap ground) delta service.
5. Frequently installed with mid-tap ground on one leg when sup-plying combination three-phase and single-phase load wherethe three-phase load is much larger than single-phase load.
6. Single-phase transformers with primary windings rated E voltsusually are used for this bank.
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DELTA-WYE Connection
PhasorDiagram:
Angular Displacement (Degrees): 30
Source: Suitable for both ungrounded and effectively groundedsources.
Service: Suitable for 3-wire service or for 4-wire grounded service witha 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 currentsin the secondary lines supplied by the transformer do not flow inthe primary lines. Instead these zero-sequence currents circu-late in the closed delta primary windings.
3. When supplied from effectively grounded primary system, groundrelay for primary system does not see load unbalances andground faults in the secondary system.
4. Single-phase transformers with primary windings rated E voltsusually are used for this bank.
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WYE-DELTA Connection
PhasorDiagram:
Angular Displacement (Degrees): 30
Source: Suitable for both ungrounded and effectively groundedsources.
Service: Suitable for 3-wire service or for 4-wire delta service with amid-tap ground.
Notes:1. Neutral point of primary windings with unbalanced and/or single-
phase secondary load is locked at ground potential if each unitin bank has same impedance. Even with different units in thebank, neutral point of primary windings is essentially locked atground potential.
2. Grounding the primary neutral of this connection would create aground source for the primary system. This could subject thetransformer to severe overloading during a primary systemdisturbance, or load unbalance.
3. With one unit out of service, a bank of single-phase units can bereconnected as an open wye—open delta bank provided thatthe source is 4-wire effectively grounded. With one of three iden-tical units out of service, the rating of the bank when supplyingonly three-phase load is about 57.7 percent of the bank ratingwhen all three units are in service.
4. Single-phase units with secondary breakers should not be usedwhether there is or is not a center tap ground on one leg. Open-ing of breaker in one leg causes severe voltage unbalance andwave form distortion.
5. Frequently installed with mid-tap ground on one leg when sup-plying combination three-phase and single-phase load wherethe 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 thebank using single pole switches located at the primary termi-nals. With smaller kVA transformers in the bank, the probabilityof ferroresonance is higher.
7. Single-phase transformers rated E/E1, Y volts usually are usedfor this bank (E1 = CFF3 E).
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WYE-WYE Connection
PhasorDiagram:
Angular Displacement (Degrees): 0
Source: Suitable for both ungrounded and effectively groundedsources.
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 (neutralshift) as a result of unbalanced phase-to-neutral load.
2. When supplied from effectively grounded source and made fromsingle-phase units, very high third harmonic voltage (of the orderof 50%) appears between neutral point of primary windings andground (tank).
3. When supplied from ungrounded source and made from single-phase units, third harmonic voltages appear from neutral pointof primary windings and ground, and from primary lines to ground.Division of total third harmonic voltage (of order of 50%) dependsupon capacitances of primary lines and transformers.
4. If a three-phase unit is built on a three-legged core, the neutralpoint of primary windings is practically locked at ground potential.
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GROUNDED WYE-WYE Connection
PhasorDiagram:
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 withXO grounded.
Notes:1. Three-phase transformers with this connection may experience
stray flux tank heating during certain external system unbalancesunless the core configuration utilized provides a return path forthe flux.
2. Fundamental and harmonic frequency zero-sequence currentsin the secondary lines supplied by the transformer also flow inthe primary lines (and primary neutral conductor).
3. Ground relay for the primary system may see load unbalancesand ground faults in the secondary system. This must be con-sidered when coordinating overcurrent protective devices.
4. Three-phase transformers with the neutral points of the high volt-age and low voltage windings connected together internally andbrought out through an HOXO bushing should not be operatedwith the HOXO bushing ungrounded (floating).
To do so can create very high voltages in the secondary systems.
T-T Connection
PhasorDiagram:
Angular Displacement (Degrees): 0
Source: Suitable for both ungrounded and effectively groundedsources.
Service: Suitable for 3-wire service or for 4-wire service with XOgrounded. Can also supply 4-wire delta service.
Notes:1. Because of winding voltages required, this connection is gener-
ally only available as a three-phase transformer.
2. Neutral point of primary windings, if available, should not begrounded unless it is desired that the transformer serve as agrounding bank.
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T-T Connection
PhasorDiagram:
Angular Displacement (Degrees): 30
Source: Suitable for both ungrounded and effectively groundedsources.
Service: Suitable for 3-wire service or for 4-wire service with XOgrounded. Can also supply 4-wire delta service.
Notes:1. Because of winding voltages required, this connection is gener-
ally only available as a three-phase transformer.
2. Neutral point of primary windings, if available, should not begrounded unless it is desired that the transformer serve as agrounding bank.
OPEN WYE-OPEN DELTA Connection
PhasorDiagram:
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 amid-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 unitsmust be 115 percent of the three-phase load if the load on eachtransformer is not to exceed nameplate rating.
2. Single-phase units with secondary breaker can be used, evenwith a mid-tap ground on one leg. However, with the secondarybreaker open in only the grounded leg, high voltages due tocapacitive coupling may appear from each terminal to ground ofthe transformer in the other leg. Sufficient phase-to-neutral con-nected load will limit these voltages.
3. Can be connected to either a three-phase or V phase primaryline.
4. Frequently installed with one large and one small transformer tosupply a combination of single-phase and three-phase load wheresingle-phase load is much larger than the three-phase load.
5. With ungrounded secondary windings (3-wire service), voltageto ground from one or more secondary phases can be greaterthan secondary phase-to-phase voltage due to unbalances inthe capacitance network. With sufficient length of secondarycircuit or connected load, phase-to-ground voltage for each phasewill approach in magnitude the phase-to-phase voltage dividedby CFF3 .
47
6. When primary terminals H1 and H2 are supplied from the samesystem phase, the open circuit phase to phase voltage fromsecondary terminal X1 to X3 is two (2) times normal phase tophase voltage.
OPEN DELTA-OPEN DELTA Connection
PhasorDiagram:
Angular Displacement (Degrees): 0
Source: Suitable for both ungrounded and effectively groundedsources.
Service: Suitable for 3-wire service or for 4-wire delta service with amid-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 unitsmust be 115 percent of the three-phase load if the load on eachtransformer is not to exceed nameplate rating.
2. Single-phase units with a secondary breaker can be used, evenwith a mid-tap ground on one leg. However, with the secondarybreaker open in only the grounded leg, high voltages due tocapacitive coupling may appear from each terminal to ground ofthe transformer in the other leg. Sufficient phase-to-neutralconnected 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 tosupply a combination of single-phase and three-phase loadwhere single-phase load is much larger than the three-phaseload.
5. With ungrounded secondary windings (3-wire service), voltageto ground from one or more secondary phases can be greaterthan secondary phase-to-phase voltage due to unbalances inthe capacitance network. With sufficient length of secondary cir-cuit or connected load, phase to ground voltage for each phasewill approach in magnitude the phase-to-phase voltage dividedby CFF3 .
48
�����C. Common Three-Phase Banks Using Single-PhaseTransformers
Phase Relation Diagram Angular Polarity Connection
Displacement Diagrams
HV Connection Diagrams
49
LV Connection Diagrams
* Represents opposite end of winding from X1; may be X2, X3, or X4depending upon the low voltage rating (2, 3, or 4 bushing).
50
IV. Loading Page
A. Paralleling ................................................................................. 51B. Delta-delta bank loading ........................................................... 51C. Overloading............................................................................... 52D. Single-phase and three-phase loading of symmetrical
and unsymmetrical transformer banks .................................... 53E. Dedicated motor loads .............................................................. 66
51
IV. Loading
�����A. Paralleling
Transformers or transformer banks may be connected in parallelto increase capacity by connecting terminals of like designationtogether provided that the frequency and voltage (including tapsetting) ratings are the same. In addition, three-phase transformersor banks must have the same phase shift.
Mismatched impedance between the parallel units or banksrequires a derating because the load does not then divide in pro-portion to the kVA ratings. This derating can be approximated asfollows:
K1 - Capacity of the unit or bank with the larger percentimpedance
K2 - Capacity of the unit or bank with the smaller percentimpedance
Z1 - Impedance of unit or bank 1Z2 - Impedance of unit or bank 2
Derating factor = e Z2 . K 1 + K 2 f / (K1 + K2)
Z1
Example: 25 and 50 kVA single-phase transformers with 1.6 and2.0 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-deltaconnected transformer banks can cause large circulating currentswithin the deltas. Thus, a requirement for such banks is equalturns ratios for all units.
Similarly, an impedance imbalance can cause a small circulatingcurrent which makes it necessary to derate the bank. For units ofequal capacity with one odd impedance, the derating for balancedloading is approximated in the following table:
Ratio of odd unit impedance Derating
to impedance of other two units Factor_________________________ _______
1.6 0.911.5 0.931.4 0.941.3 0.951.2 0.971.1 0.981 0 1.00.9 .97.8 .93.7 .90
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�����C. Overloading
The overloading of distribution transformers is a complex subjectrequiring knowledge of load characteristics, transformer param-eters and environmental conditions to be accomplished withoutdamaging the transformer. ABB distribution transformers are fitfor planned overloading providing that such overloading is in accordwith the ANSI Loading Guide (C57.91). The table below showsthe approximate peak overload capability for a typical distributiontransformer for normal life expectancy (these values are extractedfrom C57.91). The table applies to a 30°C ambient; the loadcapability at other ambients (0–50°C) can be estimated by (1)decreasing the load capability by 1.5% for each degree C that theambient exceeds 30°C or (2) increasing the load capability by1.0% for each degree C that the ambient is below 30°C.
Peak Loading Capability For Normal Life Expectancy (Per Unit)
Peak LoadDuration Equivalent Continuous Preload (per unit)(Hours) 0.50 0.75 0.90 _____ ________________________________
1 2.12 1.96 1.822 1.79 1.68 1.574 1.50 1.44 1.368 1.28 1.25 1.21
24 1.08 1.07 1.07
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�����D. Single-Phase and Three-Phase Loading of Symmetricaland Unsymmetrical Transformer Banks
Single-phase distribution transformers can be connected in banksto supply a combination of single-phase and three-phase load.The transformer bank supplying the combination load may be eithersymmetrical or unsymmetrical. A symmetrical bank is one con-sisting of three identical single-phase transformers. Most oftenthe primary and secondary windings are connected in either wyeor delta. An unsymmetrical bank is one containing only two single-phase transformers, or a bank with three single-phase transformerswhere all three transformers are not the same. (Note: for loadingconsiderations [not ferro, tank heating], a three-phase transformercan be considered to be a bank consisting of three identical single-phase transformers.)
The “4-wire delta” system is a common type used to supply acombination of single-phase and three-phase load. These sys-tems are supplied from a transformer bank with the secondarywindings connected in either delta or open delta with a center tapground on one leg of the delta. The service to the three-phaseload is 3-wire at 240 volts, and the service to the single-phaseload is 3-wire at 120/240 volts. Transformer banks with their sec-ondary windings connected in grounded wye can also be used tosupply a combination of single-phase and three-phase load. In a4-wire grounded wye system supplied from such a bank, the single-phase load may be connected from either phase-to-neutral orphase-to-phase.
When known single-phase and three-phase loads are to be fedfrom a transformer bank, frequently it is necessary to know theload which will be supplied by each transformer so that it may beproperly sized. Equations for calculating the load supplied by eachsingle-phase transformer in the bank are given in Figures D.1through D.8. The basis and assumptions used in deriving theseequations are discussed in the following.
Basis For Loading Equations
A cursory look at the loading equations in Figures D.1 to D.8 showsthat they can be easily evaluated numerically using a hand-heldpocket calculator. In order to arrive at these relatively simple equa-tions, it is necessary that certain assumptions be made concerningthe characteristics of the three-phase and single-phase loads; andboth the primary and secondary systems.
The three-phase load is assumed to be a constant “current sink”'which draws only balanced (positive-sequence) currents. Lossesin the secondary conductor between the transformer terminalsand both single-phase and three-phase loads are negligible suchthat the phase voltages at the load and transformer are the same.The single-phase load supplied from a delta or open delta sec-ondary is balanced between the two phase wires and neutral wiresuch that current does not flow in the neutral. Furthermore, thevoltages impressed on the primary windings of the transformerare of a magnitude and angle which results in balanced outputvoltages from the transformer secondary terminals. Although theseconditions rarely exist in practice, they are the assumptionstraditionally used in the industry, although not always stated, toarrive at these simplified loading equations.
54
If it is desired to make more exact calculations for the kVA load sup-plied by each transformer in the bank, providing sufficient informationis available for representing the load and system, then the methodsoriginally developed by Neupauer1,2, or by Seematter and Richards3
may be used. However, these methods do not result in simple expres-sions similar to those given in Figures D.1 to D.8, but require the use ofa digital computer for implementation.
Load Equations For Symmetrical and UnsymmetricalTransformer Banks
Use of the simplified loading equations is discussed in the followingsections for the more common symmetrical and unsymmetrical trans-former connections. Examples are given to demonstrate the use ofthese equations.
Open Wye—Open Delta Bank
With the open wye-open delta transformer bank, the single-phase loadmay be connected to either the lagging phase as shown at the top ofFigure 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 issometimes referred to as the “lighting leg” and the other transformer isreferred to as the “power leg.” These are designated as L and Prespectively in Figures D.1 and D.2. Equations for calculating the loadin kVA supplied by the lighting leg transformer (kVAL) and that suppliedby 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 theFigures. The use of the equations is illustrated with the followingexample.
Open WYE—Open DELTA (Lagging)
Figure D 1: Load equations for the open wye-open delta bank withthe single-phase load connected to the lagging phase.
55
An open wye-open delta bank supplies a single-phase load of 70 kVAat 0.95 lagging power factor, and a three-phase load of 30 kVA at 0.8lagging power factor. The power factor angles (a3 and a1) are the arccosine 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 transformersfor both the leading and lagging connection will be determined. First,consider the lagging connection in Figure D.1. The numerical valuesfor the symbols in Figure D.1 are as follows:
Next, consider the leading connection shown in Figure D.2. Theexpression for the load in kVA supplied by the power leg transformer isthe same as for the lagging connection. The expression for the loadsupplied by the lighting leg transformer is identical to that for the lag-ging connection except for the argument of the cosine term. Evaluationof the expression for kVAL in Figure D.2 shows that the lighting legtransformer supplies 82.47 kVA with the leading connection.
For most combination loads, the power factor of the three-phase loadis less than that of the single-phase load. Thus a3 – a1 is positive insign, and m is a positive number. For expected values of m, the magni-tude of the argument of the cosine term in the expressions for kVAL willbe greater for the leading connection, and thus the cosine of the argu-ment will be less. Consequently, when the leading connection is usedthe kVA load supplied by the lighting leg transformer usually is lessthan for the lagging connection.
56
Open WYE—Open DELTA (Leading)
Figure D.2: Load equations for the open wye-open delta bank withthe 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 legand power leg transformers in the open delta-open delta bank are thesame as for the open wye-open delta bank. Thus the equations in Fig-ures D.1 are used for the lagging connection, and those in Figure D.2are 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
57
Floating WYE—DELTA
Figure D.4: Load equations for the floating wye-delta connectedtransformer bank.
Floating WYE—DELTA Bank
Equations for calculating the load in kVA supplied by each transformerin the floating wye-delta bank are given at the top of Figure D.4. Noticein these equations that a double subscript is used to specify the phasesto which each transformer is connected, and the single-phase load isconnected from phases b-to-c. Since the primary windings of the trans-formers in the bank are connected in floating wye, the single-phaseload division is independent of transformer characteristics and zero-sequence current cannot circulate in the secondary delta. Because ofthis and the assumptions concerning the characteristics of the three-phase load, the equations in Figure D.4 for determining the load suppliedby each transformer are independent of transformer impedance. Useof the equations is demonstrated with the following example.
A floating wye-delta bank is to supply a three-phase load of 100 kVA at0.8 power factor lagging, and a single-phase load of 50 kVA at 0.95power factor lagging. What is the smallest size transformer which canbe used in each leg if the load supplied by each transformer is not toexceed nameplate rating? From the specified power factors:
a3 = arc cosine (.8) = 36.87°
a1 = arc cosine (.95) = 18.19°
58
Evaluating the equations in Figure D.4 with K3 equal to 100, K1 equalto 50, and m equal to 18.68 degrees results in the following:
KVAab = 40.09
KVAbc = 65.78
KVAca = 47.15
Thus the transformers connected between a and b, and between aand c should be 50 kVA units. The one between b and c should be a 75kVA 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
59
DELTA - DELTA
Figure D.6: Load equations for the delta-delta connected bankwith identical transformers in each power leg, and adifferent unit in the lighting leg.
DELTA—DELTA Bank
The equations for calculating the load in kVA supplied by each trans-former in a delta-delta bank are given at the top of Figure D.6. Theassumptions used in deriving these equations are the same as previ-ously outlined, plus it is assumed that the impedance of the transformersbetween a and b, and between a and c are identical. These two unitsare sometimes referred to as the “power leg” transformers, and their
60
leakage impedance is designated as ZP. Across the transformer con-nected between b and c is the single-phase load. This unit is referredto as the “lighting leg” transformer and its impedance is designated atZL. 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 a75 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 powerfactor lagging). The impedance of each transformer in percent is:
ZP = 1.1 + j1.3 % on 50 kVA base
ZL = 1.0 + j1.5 % on 75 kVA base
From the previous examples where the three-phase and single-phasepower 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 calculatingthe load supplied by each transformer contain the terms M 1, M 2, M3,M 4, b 2, b 3, b 4. These are real numbers which are a function of ZP andZL. The M’s are the magnitude of the impedance functions as shown inthe figure, and the b ’s are the angles in degrees for the impedancefunctions. The equations for calculating the M’s and b ’s are also givenin the Figure. To calculate these, first put ZL on the same kVA baseas ZP.
ZL = (1.0+ j1.5)50
= .6667 + j1.0 % ON 50 kVA75
Placing the values of ZL and ZP into the equations yields the following:
M 1 = 2.702M 2 = .9802 b 2 = -113.47°M 3 = .7835 b 3 = 4.84°M 4 = .9199 b 4 = 115.12°
Placing these values plus the values of K3, K 1, and m into the loadingequations gives the following for the load in kVA supplied by each trans-former.
kVAab = 33.66
kVAbc = 73.04
kVAca = 43.15
61
If the delta-delta bank is made from three transformers with the sameleg impedance (on a common kVA base, or in actual ohms), then theloading equations reduce to the simpler form shown in Figure D.7. Noticethat these are the same equations as used for the floating wye-deltabank in Figure D.4.
If the delta-delta bank is supplying only three-phase load, but one ofthe units has a different impedance, the loading equations in FigureD.6 reduce to the relatively simple form shown in Figure D.8 if all trans-formers have the same impedance angle. When the impedance of eachtransformer is the same, the load supplied by each is 1/3 of the totalthree-phase load. Figure D.11 is a plot of the per unit load supplied byeach transformer as a function of the ratio of ZL to ZP where the imped-ances are on a common base. One per unit load is the load carried bythe transformer when all three have the same impedance. From thisplot, notice that reasonable difference in impedances do not producelarge unbalances in loading. Thus, although it is desirable that eachtransformer in a delta-delta bank supplying a three-phase load havethe same impedances, this is not an absolute necessity. However, incontrast, these transformers must have the same voltage rating andtap settings as discussed in Section III.B.1.
DELTA-DELTA with Equal Leg Impedances
Figure D.7: Load equations for the delta-delta connected bankwith identical transformers in each leg.
62
DELTA-DELTA, Three-Phase Load, Same ImpedanceAngle for ZP and ZL
Figure D.8: Load equations for the delta-delta connected banksupplying only a three-phase load. Equations applyonly when impedance angle of all transformers arethe same.
Grounded WYE-DELTA Bank
The equations for calculating the load supplied by each transformer ina grounded wye-delta bank are the same as those for a delta-deltabank as given in Figure D.6. However, for reasons previously discussedin Section III, this connection is not recommended to supply distribu-tion loads.
63
DELTA-Grounded WYE Bank
With the delta-grounded wye bank, the single-phase load on the sec-ondary may be connected from either phase-to-neutral as shown inFigure D.9 or from phase-to-phase as shown in Figure D.10. The equa-tions for calculating the load supplied by each transformer are given atthe top of each figure. The terms appearing in each equation are thesame as used in the equations for the other connections for whichexamples have been given.
DELTA-WYE, Phase-to-Neutral Single-Phase Load
Figure D.9: Load equations for the delta-wye connected bankwhen the single-phase load is connected from phase-to-neutral.
64
DELTA-WYE, Phase-to-Phase Single-Phase Load
Figure D.10: Load equations for the delta-wye connected bankwhen the single-phase load is connected phase-to-phase.
Grounded WYE-Grounded WYE Bank
The grounded wye-grounded wye bank also is used to supply a combi-nation of three-phase and single-phase load. Single-phase secondaryload may be connected either phase-to-neutral, or phase-to-phase.The load supplied by each transformer in the grounded wye-groundedwye bank can be calculated using the equations for the delta-groundedwye bank in Figures D.9 and D.10.
65
Figure D.11: Per unit load supplied by each transformer in a delta-delta bank to a three-phase load when one of thetransformers (connected between phases b and c withimpedance ZL) has a different impedance. The trans-formers between phases a and b, and between phasesc and a have impedance ZP as shown in Figure D.6.
References
1. J. C. Neupauer. “Unbalanced Open-Wye Open-Delta TransformerBanks.” 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 onOpen-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-PhaseInduction Motor Operation on Rural Open Delta Distribution Sys-tems.” I.E.E.E. Trans. on Industry Applications, Vol. 1A-12, No.5,pp. 479-485, Sept./Oct. 1976.
66
�����E. Dedicated Motor Loads
Many different types of motors are used today and are an impor-tant consideration in sizing a transformer to supply power to agiven load. For most transformers supplying individual and multi-unit residences, the motor load can be ignored because itrepresents a small percentage of the total load connected to thetransformer, and the motors are only started on an infrequent basis.When applying transformers to commercial or industrial loads, themotors that are to be served can present a major limiting factor onwhat size transformer is necessary to serve the load.
Motor Starting Load
The major consideration in sizing transformers for Motor Applica-tion is limiting the starting current so that it will not shorten the lifeof the transformer due to thermal or mechanical damage from thestarting pulse. Extensive data has been gathered on pulse dutyon power transformers and the conclusion was that if the currentpulses per hour exceed
n = e 4.25 f4 where n = number of starts per hour
lp Ip = the pulse current in per unitof transformer rated current
then the transformer will fail prematurely due to the repeatedmechanical strains placed on the coil. (See Curve 1.)
67
Maximum Allowable per Unit Pulse
Cu
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1 Nu
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of
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Pu
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s P
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Ho
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68
Dedicated Transformers(One motor is the entire load on the transformer)
When a transformer is dedicated to supplying the power to only onemotor then the problem of sizing the transformer can be solved verymethodically. On squirrel cage induction motors Nema Standards callfor a starting code letter which corresponds to the kVA per horsepowerrequired to start the motor; a table giving this relationship is shown onCurve 2. Curve 2 is based on the locked rotor code letters but it can beused for any motor by selecting the curve that corresponds to lockedrotor 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 themotors locked rotor kVA (kVA’s = CFF3 x VR x IS x 10-3)
Where IS = starting current at rated voltage
VR = rated phase-to-phase voltage of motor
2. Determine the number of starts per hour planned for the motorunder normal operating conditions.
3. On Curve 2 find the curve letter that corresponds to the lockedrotor kVA/HP of the motor. Enter Curve 2 on the abscissa at thecorrect starts per hour for the motor application.
4. Move up to the intersection of the starts/hour and the correct lockedrotor code letter curve and read the kVA of transformer requiredper horsepower of motor.
5. MuItiply the kVA/HP found in “Step 4” by the rated HP of the motorand that is the smallest transformer that should be used in thatapplication. Sizing the transformer with this procedure is conser-vative since it assumes that the voltage maintained at the motorterminals during starting is motor rated voltage.
6. Most motors require 60–80% of rated voltage at the terminal underlocked rotor conditions to successfully start. After the transformerhas been sized so that it can withstand the starting pulse due tothe motor, the voltage regulation of the system must be checkedto determine if the voltage is adequate under locked rotor condi-tion to start the motor.
69
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70
V. Voltage Unbalance Page
A. Effects of Voltage Unbalance ................................................... 71B. Voltage Unbalance Definitions.................................................. 71C. Causes of Voltage Unbalance .................................................. 73D. Voltage Unbalance With Three-Phase Loading ....................... 73
1. Delta-Delta and Floating Wye-Delta Banks ......................... 742. Open Delta Banks ................................................................ 75
71
V. Voltage Unbalance
�����A. Effects of Voltage Unbalance
Voltage unbalance in secondary distribution systems affects theperformance of induction motors, with motor derating requiredwhen voltage unbalance exceeds 1.0 percent. Figure 5.1, extractedfrom NEMA MG1-14.34, dated June 1980, gives the derating factorfor fractional and integral-horsepower induction motors. Theperformance of semi-conductor rectifier circuits also can beaffected by voltage unbalance, with proposed revisions to ANSIC34.2 indicating the application is unusual if either the negative-or zero-sequence component of voltage exceeds 5 percent of thepositive-sequence component. This level of unbalance usually isnot present in utility distribution systems. However, some voltageunbalance 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 systemis a very complex matter, due to the many system and trans-former parameters which affect unbalance, and does not allow adetailed discussion in this Guide. Considered in the followingsections are several special cases where the transformer bankis supplying just a balanced three-phase load through a sym-metrical secondary circuit, with each transformer in the bankhaving the same kVA rating, although not necessarily the sameimpedance. Furthermore, the primary system voltages areassumed to be balanced. For these special cases, the followingpoints are noteworthy.
1. With either the floating wye-delta or delta-delta bank supply-ing three-phase load, it is not necessary from a voltageunbalance standpoint that low impedance transformers beused. For these banks, voltage unbalance is caused byimpedance differences between the single-phase transformersin the bank. Reasonable differences in impedances are toler-able and will not cause objectionable voltage unbalance.
2. With either the open wye-open delta or open delta-open deltabanks supplying just three-phase load, voltage unbalance iscaused by the dissymmetries of the transformer bank (due touse of only two transformers), regardless of transformerimpedance magnitudes. Voltage unbalance with the open deltabank can be significantly higher than that with a closed deltabank supplying the same load.
�����B. Voltage Unbalance Definitions
Three different definitions are employed to quantify voltageunbalance. The one used depends upon the task being performed,for example, calculating unbalance from measured quantities ordeveloping equations for unbalance. When quantifying voltageunbalance in 3-wire and 4-wire delta circuits under unfaultedconditions, all three definitions give nearly the same result.
72
In NEMA induction motor standards, percent voltage unbalanceis defined as:
Percent Maximum Voltage Deviation fromVoltage
=Average Phase Voltage
x 100Unbalance Average Phase Voltage
When phase-to-phase voltages are measured at an actual instal-lation, voltage unbalance is easily calculated with this definition,the one frequently preferred by individuals not familiar with sym-metrical components. For example, with phase-to-phase voltagesof 235, 230 and 222 volts, average voltage is 229 volts, maximumdeviation from average is 7.0 volts, and percent voltage unbalanceis 3.06 percent.
Some engineers have advocated that percent voltage unbalancebe defined as 100 times the ratio of the magnitude of the negative-sequence voltage to the magnitude of the positive-sequencevoltage. When an analysis is performed with symmetrical compo-nents to obtain sequence quantities, it is expedient to calculatevoltage unbalance with this definition as this eliminates the needto calculate the phase-to-phase voltages required with the NEMAdefinition. In a three-phase system where zero-sequence voltagesare not present, the ratio of the magnitude of the negative-sequence voltage to the magnitude of the positive-sequencevoltage also can be found from the following equation.
In this equation a, b, and c are the magnitudes of the three line-to-line voltages (or line-to-ground voltages when zero-sequence isnot present) and:
S =a + b + c
2
For example, with phase-to-phase voltages of 235, 230, and 222volts, the equation shows that the ratio of the magnitude of V2 toV1 is 3.30 percent. In comparison, the percent voltage unbalancefor these voltages using the NEMA definition is 3.06 percent.
A third definition for percent voltage unbalance is 100 times theper unit negative-sequence voltage, where the per unit value isthe actual value in volts divided by the system nominal voltage.The voltage unbalance calculated in this fashion does not differsignificantly from 100 times the ratio of V2 to V1, as V1 is close to1.0 per unit under loading conditions. In the curves in section V-D,voltage unbalance is quantified in terms of negative-sequencevoltage.
73
�����C. Causes of Voltage Unbalance
Voltage unbalance in secondary distribution systems is causedby dissymmetries in either the primary system, distribution trans-former 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 con-ductors in each phase with the conductors arranged such that themutual impedances between its sequence networks are zero. Anunsymmetrical circuit is one where the same size conductor is notin each phase, or the same size conductor is used but the spac-ings are such that the sequence mutual impedances are not zero.With a symmetrical primary system, the open circuit voltages atthe transformer are perfectly balanced.
Whenever a symmetrical transformer bank supplies a perfectlybalanced three-phase load (one where there is no couplingbetween the sequence networks) through a symmetrical second-ary circuit and the voltages at the primary terminals of the bankare balanced, the voltages at the secondary terminals also will bebalanced. That is, a negative- or zero-sequence component willnot be present in the secondary phase-to-neutral voltages. How-ever, when a symmetrical transformer bank supplies a perfectlybalanced three-phase load and a single-phase load, the second-ary voltages will be unbalanced regardless of the symmetry of thesecondary circuit. Furthermore, when an unsymmetrical transformerbank supplies only a balanced three-phase load, or both a balancedthree-phase load and a single-phase load, the secondary voltageswill be unbalanced irrespective of the secondary circuit symmetry.
The main parameters which can affect voltage unbalance in sec-ondary systems are transformer bank connection, transformerimpedance, primary system impedance, secondary circuit char-acteristics, three-phase and single-phase load magnitudes, loadpower factors, and primary system voltage unbalance. Any one ofthese parameters can have a significant effect on voltage unbal-ance in secondary systems, with the only exception being primarysystem impedance which usually has a minor effect. This isbecause the impedance of the primary system typically is muchsmaller than that of the distribution transformers.
A complete discussion on the effect of all of these parameters onvoltage unbalance is beyond the scope of this guide. Illustrated inthe following is the effect of distribution transformer impedanceon voltage unbalance for the situation where the unsymmetricaltransformer bank supplies only balanced three-phase load througha symmetrical secondary circuit, is supplied from an infinite busprimary 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 imped-ance on the maximum negative-sequence voltage in percent whichcould appear in the secondary system with balanced three-phasenominal loading on the bank. They apply respectively to the delta-delta, floating wye-delta, and open delta transformer banks. For agiven bank, each transformer has the same kVA rating. Nominalloading occurs when the positive-sequence current in the sec-ondary windings equals winding rated current. Since, in general,
74
a negative-sequence component of current is present under nomi-nal loading conditions, the actual winding currents may besomewhat greater than or less than winding rated current. Thecurves apply to banks supplying balanced three-phase loads. Forthese loads, the positive- and negative-sequence impedances areeither the same or different, but there is no coupling between thesequence networks representing the load. Loads with these char-acteristics are three-phase induction motors, or impedances ofequal magnitude and angle connected in either wye or delta. Forthe three-phase induction motor, the negative-sequence imped-ance is less than the motor’s positive-sequence impedance,whereas these impedances are equal for loads made from eitherwye or delta connected impedances. Furthermore, the curves areplotted assuming the primary system voltages are of equal mag-nitude and 120 electrical degrees displaced from each other. Also,the impedance angles of the transformers in a bank were assumedequal, with the impedance magnitudes being the same or differ-ent. It is emphasized that the curves give the maximum, or greatestupper bound on negative-sequence voltage at the load in the sec-ondary with nominal loading (three-phase) on the bank. Actualunbalance can be considerably less, depending upon the rela-tionship between the transformer bank impedances, secondarycircuit impedances, and the three-phase loads negative sequenceimpedance.
1. Delta-Delta and Floating Wye Delta Banks
The curves of Figure 5.2 and 5.3 respectively show on the ordi-nate the maximum negative-sequence voltage in percent in thesecondary system with balanced three-phase load supplied froma delta-delta bank and floating wye-delta bank. The curves applyto banks with transformers of the same kVA and voltage ratings.Two transformers in the bank have the same leakage impedance,designated as ZP, and the third unit’s impedance is ZL. Given onthe abscissa is the ratio of ZL to ZP.
The curves show that if all three transformers in the bank haveequal impedance, regardless of impedance magnitude, the bankwill not produce voltage unbalance with only balanced three-phase 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 ZL to ZP isbetween 0.5 to 1.5, and ZP is 3.0 percent or less on nameplaterating. Thus, considering voltage unbalance when serving bal-anced three-phase load, it is not necessary that all units havethe same impedance or low impedance in closed delta banks. Ifone unit fails in a bank made from three “old” units of “low”impedance, it could, in most cases, be replaced by a “new” unitwith a higher impedance without creating objectionable unbal-ance. It is unduly restrictive from a voltage unbalance standpointto require the same impedance for all units in the delta-deltaand 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, considerthe following situation. A utility had some 1500 kVA delta-deltabanks made from 500 kVA units with nameplate impedances of2.2 percent. The utility decided to order several spare units andrequested units with 2.2 percent impedance. When the supplier
75
quoted units with an impedance of 1.9 percent, the utility objectedbecause it was thought that voltage unbalance problems wouldbe created by the “low impedance” units. If two units in the bankhave an impedance of 2.2 percent (ZP) and one has an imped-ance of 1.9 percent (ZL), the curves of Figure 5.2 show that themaximum negative-sequence voltage possible at nominal loadwould not exceed 0.13 percent. If two units have an impedance of1.9 percent (ZP) and one unit has an impedance of 2.2 percent(XL), the maximum negative-sequence voltage possible at nomi-nal load would not exceed 0.1 percent. For this situation, theimpedance differences will not cause significant voltage unbalance.
However, when one unit in the delta-delta bank has a differentimpedance, it may be necessary to derate the bank for thermalreasons. Section IV-B of this guide presents derating factors forthe delta-delta bank made from 3 units of equal capacity supply-ing perfectly balanced three-phase load where one unit has adifferent impedance. A perfectly balanced three-phase load, forpurpose of derating is defined as one which draws only positive-sequence current. Figure 5.4 is a plot of derating factor as afunction of the ratio of ZL to ZP for the delta-delta bank.
2. Open-Delta Banks
The curves of Figure 5.5 show the maximum negative-sequencevoltage (voltage unbalance) in percent which could appear in thesecondary with balanced three-phase load supplied from eitherthe open wye-open delta or open delta-open delta bank. Thecurves apply to banks made from two single-phase units with thesame kVA and voltage ratings. The impedance of one unit is ZP
and that of the other is ZL. For the open delta bank, nominal loadis that which makes the positive-sequence current in each sec-ondary winding equal to rated current of the winding. Practically,nominal load is when the positive-sequence kVA of the balancedthree-phase load equals 1.732 times the kVA rating of one trans-former. When both transformers in the open delta bank have thesame impedance, the upper bound on negative-sequence volt-age in percent at nominal load is the impedance in percent dividedby the square root of 3.
A comparison of Figure 5.5 with either Figure 5.2 or 5.3 showsthat the maximum negative-sequence voltage at nominal load ismuch greater for the open delta bank than for the closed deltabanks. Even when both transformers have the same impedance(ZL = ZP), the maximum negative-sequence voltage at nominalload can be appreciably above 1.0 percent. Also, for any ratio ofZL to ZP, the impedance of the transformers in the open deltabank must be low if the negative-sequence voltage (voltageunbalance) is to be limited to less than 1.0 percent.
For example, consider an open delta bank made from units with3 percent impedance. This impedance is typical in units purchasedtoday by some users. From Figure 5.5, the voltage unbalance atnominal load due to the open delta bank is 1.73 percent. Thiswould be the total unbalance (upper bound) at the load if theprimary system voltages were perfectly balanced. Recognizingthat the voltage unbalance of the primary system could be in the1 to 2 percent range, a worst case upper bound on the voltageunbalance at the load is in the range of 2.7 to 3.7 percent.
76
As another example, one utility was serving a 460-Volt load, pre-dominately motors, from an open delta bank with two 1000 kVA,5.8 percent impedance transformers, with a load of 1360 kVA orabout 79 percent of bank rating. Measured voltage unbalance atthe service entrance was 2.3 percent. In comparison, the upperbound on voltage unbalance with just three-phase load, assum-ing a balanced primary, is calculated as 5.8 x 0.79/ CFF3 or 2.65percent. When the open delta bank was replaced with three 500kVA units with 4.8 percent impedance, the measured voltageunbalance at about the same loading was less than 0.5 percent,due mainly to primary system unbalance.
Considering voltage unbalance, successful operation of the opendelta transformer bank supplying balanced three-phase loadfrequently is enhanced by the use of low impedance distributiontransformers. As transformer impedance decreases, the negative-sequence voltage due to transformer bank dissymmetriesdecreases. When primary voltages are balanced the negative-sequence voltage at the load due to the dissymmetries of the opendelta bank will be reduced by either closing the bank (adding athird transformer), by using transformers with the same kVA ratingbut with lower impedance, or by using transformers of a higherkVA rating. Using transformers with a higher kVA rating (samepercent impedance) corresponds to loading the bank to less than“nominal load.”
When primary system voltages are unbalanced, the total voltageunbalance (negative-sequence voltage) at the balanced three-phase load fed from the open delta bank is the vector sum of thatdue to the transformer bank dissymmetries and the voltageunbalance of the primary system. Decreasing the component ofvoltage unbalance due to the transformer bank by reducing trans-former impedance may either increase or decrease the totalnegative-sequence voltage at the load. The effect of reducingimpedance depends upon the relative magnitude of the primarysystem voltage unbalance and the voltage unbalance due to trans-former bank dissymmetries, plus the angle between these sourcesof voltage unbalance. Under conditions where the two sources ofvoltage unbalance are in-phase, reducing transformer impedancereduces voltage unbalance at the secondary load.
Figure 5.1: Fractional and integral-horsepower inductionmotor derating factor.
77
Figure 5.2: Maximum negative-sequence voltage at nomi-nal load in the secondary system with balancedthree-phase load supplied from a delta-deltabank made from three transformers with thesame kVA and voltage ratings.
Figure 5.3: Maximum negative-sequence voltage at nominalload in the secondary system with balancedthree-phase load supplied from a floating wye-delta bank made from three transformers withthe same kVA and voltage ratings.
78
Figure 5.4: Derating factor for a delta-delta bank made fromthree units of equal capacity (kVA rating) withthe impedance of one unit being different fromthat of the other two. Derating factor is approxi-mate as it assumes perfectly balanced three-phase load drawing only positive-sequencecurrent.
Figure 5.5: Maximum negative-sequence voltage at nominalload in the secondary system with balancedthree-phase load supplied from an open deltatransformer bank made from two transformerswith the same kVA and voltage ratings.
79
Vl. Reference Data Page
Solid and Concentric Stranded Aluminum and Copper Conductors . 80Temperature Correction Factors for Resistance of AluminumConductors .................................................................................... 81Logarithm Tables ........................................................................... 83Nominal Direct-Current Resistance, Ohms per 1000 Feet,at 20°C and 25°C of Solid and Concentric Stranded Conductors .... 85Natural Functions of Angles .......................................................... 86TypicaI Isokeraunic Map ............................................................... 87Selected Sl Equivalents ................................................................ 88
80
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61
41.6
2/0
133100
364.8
122.5
402.8
19
83.7
0.4
18
125
.4
1137
60
.061
46
.7
3/0
167800
409.6
154.4
507.8
19
94.0
0.4
70
157
.5
18
37
67
.361
52
.4
4/0
211600
466.0
194.7
640.5
19
105.5
0.5
28
199
.6
53
37
75
.661
58
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25
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23
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26
16
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52
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30
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……
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79
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28
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56
17
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57
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35
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32
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61
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91
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40
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61
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2469
91
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127
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1000
……
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61
128.0
1.1
52
939
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91
104.8
127
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1250
……
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91
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27
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1500
……
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91
128.4
1.4
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1408
.4632
127
108.7
169
94.2
17
50
……
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……
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….
12
711
7.4
1.5
26
16
43
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169
101.8
217
89.8
2000
……
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….
127
125.5
1.6
32
1877
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169
108.8
217
96.0
•IP
CE
A S
tandard
Public
ation S
-66-5
24, N
EM
A W
C 7
-1971.
81
Temperature Correction Factors for Resistance of AluminumConductors*
Temperature, Multiplying Factors for Reduction to Degrees °C 20°C 25°C
0 1.088 1.1105 1.064 1.085
10 1.042 1.06315 1.020 1.041
20 1.000 1.02025 0.980 1.00030 0.961 0.98135 0.943 0.962
40 0.925 0.94445 0.908 0.92750 0.892 0.91055 0.876 0.894
60 0.861 0.87865 0.846 0.86370 0.832 0.84975 0.818 0.835
80 0.805 0.82185 0.792 0.80890 0.780 0.796
The correction factors given in this table are satisfactory for most appli-cations. They are based upon aluminum having 61 percent conductivityand are derived from the formulae:
R1 = R2
248
228 + T2
R3 = R2
253
228 + T2
where R1 – Resistance at 20°CR2 – Measured resistance at test temperature, T2
R3 – Resistance at 25°C
*IPCEA Publication S-66-524, NEMA WC 7-1971
82
Temperature Correction Factors for Resistance of CopperConductors*
Temperature, Multiplying Factors for Reduction to Degrees °C 20°C 25°C
0 1.085 1.1075 1.063 1.084
10 1.041 1.06115 1.020 1.040
20 1.000 1.02025 0.981 1.00030 0.962 0.98135 0.944 0.963
40 0.927 0.94545 0.911 0.92850 0.895 0.91255 0.879 0.896
60 0.864 0.88165 0.850 0.86670 0.836 0.85275 0.822 0.838
80 0.809 0.82585 0.797 0.81290 0.784 0.800
The correction factors given in this table are satisfactory for most appli-cations. They are based upon copper having 100 percent conductivityand are derived from the formulae:
R1 = R2
254.5
234.5 + T2
R3 = R2
259.5
234.5 + T2
where R1 – Resistance at 20°CR2 – Measured resistance at test temperature, T2
R3 – Resistance at 25°C
For more accurate determination of resistance, allowing for differentconductivities, see
“Copper Wire Tables,”National Bureau of Standards,Handbook No. 100.
*IPCEA Publications S-66-524, NEMA WE 7-1971
83
Logarithm Tables
Four Place Mantissas for Common Logarithms
Proportional Parts
N 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
10 0000 0043 0086 0128 0170 0212 0253 0294 0334 0374 *4* 8 12 17 21 25 29 33 37
11 0414 0453 0492 0531 0569 0607 0645 0682 0719 0755 4 8 11 15 19 23 26 30 34
12 0792 0828 0864 0899 0934 0969 1004 1038 1072 1106 3 7 10 14 17 21 24 28 31
13 1139 1173 1206 1239 1271 1303 1335 1367 1399 1430 3 6 10 13 16 19 23 26 29
14 1461 1492 1523 1553 1584 1614 1644 1673 1703 1732 3 6 9 12 15 18 21 24 27
15 1761 1790 1818 1847 1875 1903 1931 1959 1987 2014 *3* 6 8 11 14 17 20 22 25
16 2041 2068 2095 2122 2148 2175 2201 2227 2253 2279 3 5 8 11 13 16 18 21 24
17 2304 2330 2355 2380 2405 2430 2455 2480 2504 2529 2 5 7 10 12 15 17 20 22
18 2553 2577 2601 2625 2648 2672 2695 2718 2742 2765 2 5 7 9 12 14 16 19 21
19 2788 2810 2833 2856 2878 2900 2923 2945 2967 2989 2 4 7 9 11 13 16 18 20
20 3010 3032 3054 3075 3096 3118 3139 3160 3181 3201 2 4 6 8 11 13 15 17 19
21 3222 3243 3263 3284 3304 3324 3345 3365 3385 3404 2 4 6 8 10 12 14 16 18
22 3424 3444 3464 3483 3502 3522 3541 3560 3579 3598 2 4 6 8 10 12 14 15 17
23 3617 3636 3655 3674 3692 3711 3729 3747 3766 3784 2 4 6 7 9 11 13 15 17
24 3802 3820 3838 3856 3874 3892 3909 3927 3945 3962 2 4 5 7 9 11 12 14 16
25 3979 3997 4014 4031 4048 4065 4082 4099 4116 4133 2 3 5 7 9 10 12 14 15
26 4150 4166 4183 4200 4216 4232 4249 4265 4281 4298 2 3 5 7 8 10 11 13 15
27 4314 4330 4346 4362 4378 4393 4409 4425 4440 4456 2 3 5 6 8 9 11 13 14
28 4472 4487 4502 4518 4533 4548 4564 4579 4594 4609 2 3 5 6 8 9 11 12 14
29 4624 4639 4654 4669 4683 4698 4713 4728 4742 4757 1 3 4 6 7 9 10 12 13
30 4771 4786 4800 4814 4829 4843 4857 4871 4886 4900 1 3 4 6 7 9 10 11 13
31 4914 4928 4942 4955 4969 4983 4997 5011 5024 5038 1 3 4 6 7 8 10 11 12
32 5051 5065 5079 5092 5101 5119 5132 5145 5159 5172 1 3 4 5 7 8 9 11 12
33 5185 5198 5211 5224 5237 5250 5263 5276 5289 5302 1 3 4 5 6 8 9 10 12
34 5315 5328 5340 5353 5366 5378 5391 5403 5416 5428 1 3 4 5 6 8 9 10 11
35 5441 5453 5465 5478 5490 5502 5514 5527 5539 5551 1 2 4 5 6 7 9 10 11
36 5563 5575 5587 5599 5611 5623 5635 5647 5658 5670 1 2 4 5 6 7 8 10 11
37 5682 5694 5705 5717 5729 5740 5752 5763 5775 5786 1 2 3 5 6 7 8 9 10
38 5798 5809 5821 5832 5843 5855 5866 5877 5888 5899 1 2 3 5 6 7 8 9 10
39 5911 5922 5933 5944 5955 5966 5977 5988 5999 6010 1 2 3 4 5 7 8 9 10
40 6021 6031 6042 6053 6064 6075 6085 6096 6107 6117 1 2 3 4 5 6 8 9 10
41 6128 6138 6149 6160 6170 6180 6191 6201 6212 6222 1 2 3 4 5 6 7 8 9
42 6232 6243 6253 6263 6274 6284 6294 6304 6314 6325 1 2 3 4 5 6 7 8 9
43 6335 6345 6355 6365 6375 6385 6395 6405 6415 6425 1 2 3 4 5 6 7 8 9
44 6435 6444 6454 6464 6474 6484 6493 6503 6513 6522 1 2 3 4 5 6 7 8 9
45 6532 6542 6551 6561 6571 6580 6590 6599 6609 6618 1 2 3 4 5 6 7 8 9
46 6628 6637 6646 6656 6665 6675 6684 6693 6702 6712 1 2 3 4 5 6 7 7 8
47 6721 6730 6739 6749 6758 6767 6776 6785 6794 6803 1 2 3 4 5 5 6 7 8
48 6812 6821 6830 6839 6848 6857 6866 6875 6884 6893 1 2 3 4 4 5 6 7 8
49 6902 6911 6920 6928 6937 6946 6955 6964 6972 6981 1 2 3 4 4 5 6 7 8
50 6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 1 2 3 3 4 5 6 7 8
51 7076 7084 7093 7101 7110 7118 7126 7135 7143 7152 1 2 3 3 4 5 6 7 8
52 7160 7168 7177 7185 7193 7202 7210 7218 7226 7235 1 2 2 3 4 5 6 7 7
53 7243 7251 7259 7267 7275 7284 7292 7300 7308 7316 1 2 2 3 4 5 6 6 7
54 7324 7332 7340 7348 7356 7364 7372 7380 7388 7396 1 2 2 3 4 5 6 6 7
N 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
* Interpolation in this.section of the table is inaccurate.
84
Logarithm Tables
Four Place Mantissas for Common Logarithms (Continued)
Proportional Parts
N 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
55 7404 7412 7419 7427 7435 7443 7451 7459 7466 7474 1 2 2 3 4 5 5 6 7
56 7482 7490 7497 7505 7513 7520 7528 7536 7543 7551 1 2 2 3 4 5 5 6 7
57 7559 7566 7574 7582 7589 7597 7604 7612 7619 7627 1 2 2 3 4 5 5 6 7
53 7634 7642 7649 7657 7664 7672 7679 7686 7694 7701 1 1 2 3 4 4 5 6 7
59 7709 7716 7723 7731 7738 7745 7752 7760 7767 7774 1 1 2 3 4 4 5 6 7
60 7782 7789 7796 7803 7810 7818 7825 7832 7839 7846 1 1 2 3 4 4 5 6 6
61 7853 7860 7868 7875 7882 7889 7896 7903 7910 7917 1 1 2 3 4 4 5 6 6
62 7924 7931 7938 7945 7952 7959 7966 7973 7980 7987 1 1 2 3 3 4 5 6 6
63 7993 8000 8007 8014 8021 8028 8035 8041 8048 8055 1 1 2 3 3 4 5 5 6
64 8062 8069 8075 8082 8089 8096 8102 8109 8116 8122 1 1 2 3 3 4 5 5 6
65 8129 8136 8142 8149 8156 8162 8169 8176 8182 8189 1 1 2 3 3 4 5 5 6
66 8195 8202 8209 8215 8222 8228 8235 8241 8248 8254 1 1 2 3 3 4 5 5 6
67 8261 8267 8274 8280 8287 8293 8299 8306 8312 8319 1 1 2 3 3 4 5 5 6
68 8325 8331 8338 8344 8351 8357 8363 8370 8376 8382 1 1 2 3 3 4 4 5 6
69 8388 8395 8401 8407 8414 8420 8426 8432 8439 8445 1 1 2 2 3 4 4 5 6
70 8451 8457 8463 8470 8476 8482 8488 8494 8500 8506 1 1 2 2 3 4 4 5 6
71 8513 8519 8525 8531 8537 8543 8549 8555 8561 8567 1 1 2 2 3 4 4 5 5
72 8573 8579 8585 8591 8597 8603 8609 8615 8621 8627 1 1 2 2 3 4 4 5 5
73 8633 8639 8645 8651 8657 8663 8669 8675 8681 8686 1 1 2 2 3 4 4 5 5
74 8692 8698 8704 8710 8716 8722 8727 8733 8739 8745 1 1 2 2 3 4 4 5 5
75 8751 8756 8762 8768 8774 8779 8785 8791 8797 8802 1 1 2 2 3 3 4 5 5
76 8808 8814 8820 8825 8831 8837 8842 8848 8854 8859 1 1 2 2 3 3 4 5 5
77 8865 8871 8876 8882 8887 8893 8899 8904 8910 8915 1 1 2 2 3 3 4 4 5
78 8921 8927 8932 8938 8943 8949 8954 8960 8965 8971 1 1 2 2 3 3 4 4 5
79 8976 8982 8987 8993 8998 9004 9009 9015 9020 9025 1 1 2 2 3 3 4 4 5
80 9031 9036 9042 9047 9053 9058 9063 9069 9074 9079 1 1 2 2 3 3 4 4 5
81 9085 9090 9096 9101 9106 9112 9117 9122 9128 9133 1 1 2 2 3 3 4 4 5
82 9138 9143 9149 9154 9159 9165 9170 9175 9180 9186 1 1 2 2 3 3 4 4 5
83 9191 9196 9201 9206 9212 9217 9222 9227 9232 9238 1 1 2 2 3 3 4 4 5
84 9243 9248 9253 9258 9263 9269 9274 927Si 9284 9289 1 1 2 2 3 3 4 4 5
85 9294 9299 9304 9309 9315 9320 9325 9330 9335 9340 1 1 2 2 3 3 4 4 5
86 9345 9350 9355 9360 9365 9370 9375 9380 9385 9390 1 1 2 2 3 3 4 4 5
87 9395 9400 9405 9410 9415 9420 9425 9430 9435 9440 0 1 1 2 2 3 3 4 4
83 9445 9450 9455 9460 9465 9469 9474 9470 9484 9489 0 1 1 2 2 3 3 4 4
89 9494 9499 9504 9509 9513 9518 9523 9528 9533 9538 0 1 1 2 2 3 3 4 4
90 9542 9547 9552 9557 9562 9566 9571 9576 9581 9586 0 1 1 2 2 3 3 4 4
91 9590 9595 9600 9605 9609 9614 9619 9624 9628 9633 0 1 1 2 2 3 3 4 4
92 9638 9643 9647 9652 9657 9661 9666 9671 9675 9680 0 1 1 2 2 3 3 4 4
93 9685 9689 9894 9699 9703 9708 9713 9717 9722 9727 0 1 1 2 2 3 3 4 4
94 9731 9736 9741 9745 9750 9754 9759 9763 9768 9773 0 1 1 2 2 3 3 4 4
95 9777 9782 9786 9791 9795 9800 9805 9809 9814 9818 0 1 1 2 2 3 3 4 4
96 9823 9827 9832 9836 9841 9845 9850 9854 9859 9863 0 1 1 2 2 3 3 4 4
97 9868 9872 9877 9881 9886 9890 9894 9899 9903 9908 0 1 1 2 2 3 3 4 4
98 9912 9917 9921 9926 9930 9934 9939 9943 9948 9952 0 1 1 2 2 3 3 4 4
99 9956 9961 9965 9969 9974 9978 9983 9987 9991 9996 0 1 1 2 2 3 3 3 4
N 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
85
No
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al D
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Cu
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t R
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1000 F
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r______________________________________________
_____________________________________________
_______________________________________
Aw
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20°C
25° C
20°C
25° C
So
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Str
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Cla
ss B
____________________
____________________
____________________
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___________________
kc
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Alu
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Alu
min
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Co
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Alu
min
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Co
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min
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Co
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20° C
25°C
20°C
25° C
20
16
.61
0.1
17
.01
0.3
……
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0.4
……
……
10
.61
0.6
10
.811
.011
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18
10
.56
.39
10
.76
.51
……
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.53
……
……
6.6
66
.66
6.7
96
.92
7.0
5
16
6.5
84
.02
6.7
24
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……
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……
……
4.1
84
.18
4.2
64
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4.4
4
14
4.1
42
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4.2
22
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……
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……
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2.6
22
.62
2.6
82
.68
2.7
3
12
2.6
01
.59
2.6
61
.62
2.6
61
.62
2.7
11
.65
1.6
51
.68
1.6
81
.72
10
1.6
40
.99
88
1.6
71
.01
81
.67
1.0
21
.70
1.0
41
.04
1.0
61
.06
1.0
8
81
.03
0.6
281
1.0
50.6
404
1.0
50.6
41
1.0
70.6
54
0.6
46
0.6
59
0.6
66
0.6
79
60.6
48
0.3
952
0.6
61
0.4
029
0.6
61
0.4
03
0.6
74
0.4
10
0.4
07
0.4
15
0.4
19
0.4
27
40.4
07
0.2
485
0.4
15
0.2
534
0.4
15
0.2
53
0.4
24
0.2
59
0.2
56
0.2
61
0.2
64
0.2
69
30.3
23
0.1
971
0.3
30
0.2
010
0.3
29
0.2
01
0.3
36
0.2
05
0.2
03
0.2
07
0.2
09
0.2
13
20.2
56
0.1
563
0.2
61
0.1
594
0.2
61
0.1
59
0.2
66
0.1
62
0.1
61
0.1
64
0.1
66
0.1
69
10.2
03
0.1
239
0.2
07
0.1
264
0.2
07
0.1
26
0.2
11
0.1
29
0.1
28
0.1
30
0.1
31
0.1
34
1/0
0.1
61
0.0
9825
0.1
64
0.1
002
0.1
64
0.1
00
0.1
68
0.1
02
0.1
01
0.1
03
0.1
04
0.1
06
2/0
0.1
28
0.0
7793
0.1
30
0.0
7946
0.1
30
0.0
795
0.1
33
0.0
811
0.0
798
0.0
814
0.0
827
0.0
843
3/0
0.1
01
0.0
6182
0.1
03
0.0
6303
0.1
03
0.0
630
0.1
05
0.0
642
0.0
633
0.0
645
0.0
656
0.0
668
4/0
0.0
803
0.0
4901
0.0
820
0.0
4998
0.0
820
0.0
500
0.0
836
0.0
509
0.0
502
0.0
512
0.0
515
0.0
525
25
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……
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……
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……
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……
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0.0
69
40
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0.0
70
80
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0.0
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00
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49
30
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……
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……
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0.0
57
80
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53
0.0
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36
70
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35
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0.0
49
50
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0.0
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0.0
31
40
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20
40
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0.0
43
40
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64
0.0
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20
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27
20
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78
45
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38
50
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24
20
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……
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0.0
34
70
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22
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……
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28
90
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39
1000
……
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……
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……
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……
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0.0
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30
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……
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……
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0.0
139
0.0
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0.0
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……
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….
0.0
0871
0.0
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1500
……
….
……
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……
….
……
…..
0.0
116
0.0
0705
0.0
118
0.0
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……
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0.0
0726
0.0
0740
1750
……
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……
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……
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……
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0.0
0991
0.0
0605
0.0
101
0.0
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……
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……
….
0.0
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0.0
0634
2000
……
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……
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……
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……
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0.0
0867
0.0
0529
0.0
0885
0.0
0539
……
….
……
….
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Natural Functions of Angles
Deg. Sine Cosine Tangent Cotangent Deg.
0 0.0000 1.0000 0.0000 oo 901 0.0175 0.9998 0.0175 57.2900 892 0.0349 0.9994 0.0349 28.6363 88
3 0.0523 0.9986 0.0524 19.0811 874 0.0698 0.9976 0.0699 14.3007 865 0.0872 0.9962 0.0875 11.4301 85
6 0.1045 0.9945 0.1051 9.5144 847 0.1219 0.9925 0.1228 8.1443 838 0.1392 0.9903 0.1405 7.1154 82
9 0.1564 0.9877 0.1584 6.3138 8110 0.1736 0.9848 0.1763 5.6713 8011 0.1908 0.9816 0.1944 5.1446 79
12 0.2079 0.9781 0.2126 4.7046 7813 0.2250 0.9744 0.2309 4.3315 7714 0.2419 0.9703 0.2493 4.0108 76
15 0.2588 0.9659 0.2679 3.7321 7516 0.2756 0.9613 0.2867 3.4874 7417 0.2924 0.9563 0.3057 3.2709 73
18 0.3090 0.9511 0.3249 3.0777 7219 0.3256 0.9455 0.3443 2.9042 7120 0.3420 0.9397 0.3640 2.7475 70
21 0.3584 0.9336 0.3839 2.6051 6922 0.3746 0.9272 0.4040 2.4751 6823 0.3907 0.9205 0.4245 2.3559 67
24 0.4067 0.9135 0.4452 2.2460 6625 0.4226 0.9063 0.4663 2.1445 6526 0.4384 0.8988 0.4877 2.0503 64
27 0.4540 0.8910 0.5095 1.9626 6328 0.4695 0.8829 0.5317 1.8807 6229 0.4848 0.8746 0.5543 1.8040 61
30 0.5000 0.8660 0.5774 1.7321 6031 0.5150 0.8572 0.6009 1.6643 5932 0.5299 0.8480 0.6249 1.6003 58
33 0.5446 0.8387 0.6494 1.5399 5734 0.5592 0.8290 0.6745 1.4826 5635 0.5736 0.8192 0.7002 1.4281 55
36 0.5878 0.8090 0.7265 1.3764 5437 0.6018 0.7986 0.7536 1.3270 5338 0.6157 0.7880 0.7813 1.2799 52
39 0.6293 0.7771 0.8098 1.2349 5140 0.6428 0.7660 0.8391 1.1918 5041 0.6561 0.7547 0.8693 1.1504 49
42 0.6691 0.7331 0.9004 1.1106 4843 0.6820 0.7314 0.9325 1.0724 4744 0.6947 0.7193 0.9657 1.0355 46
45 0.7071 0.7071 1.0000 1.0000 45
Deg. Cosine Sine Cotangent Tangent Deg.
87
Ty
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Iso
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Ma
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88
b
Selected Sl Equivalents
LENGTH TEMPERATURE1 in = 25.40 mm 1°F (interval) = 5/9°C1 ft = 0.3048 m temp (°F) = (9/5) temp (°C) + 321 yd = 0.9144 m Ice . point at 1 atm = 32°F1 mile = 1.609 km = 0°C = 273.15 K
Triple-point of waterAREA = 0.01°C = 273.16 K
1 cmil = 506.7 um2
1 in2 = 6.452 cm2 POWER1 ft2 = 0.0929 m2 1 watt = 1 joule/sec1 acre = 4047 m2 1 Btu/hr = 0.2931 W1 mile2 = 2.590 km2 1 hp = 746.0 W
VOLUME POWER FLUX1 litre = 1 dm3 1 Btu/hr. ft2 = 3.152 W/m2
1 fl oz (US) = 29.57 ml1 gal (US) = 3.785 litres POWER DENSITY1 in3 = 16.39 cm3 1 Btu/hr.ft3 = 10.34 W/m3
1 ft3 = 28.32 dm3
1 yd3 = 0.7646 m3 SPECIFIC POWER1 hp/lb = 1.645 kW/kg
VELOCITY1 in/min = 25.4 mm/min STEFAN-BOLTZMANN CONST.1 ft/min = 0.3048 m/min o = 56.7 nW/m2.K4
1 mile/hr = 1.609 km/hrTHERMALCONDUCTIVITY
MASS 1 Btu/hr.ft F = 1.731 W/m.K1 oz (avdp) = 28.35 gram1 lb = 0.4536 kg THERMAL CONDUCTANCE1 short ton = 0.9072 Mg 1 Btu/hr.ft2 F = 5.678 W/m2.K
DENSITY VISCOSITY, ABSOLUTE1 Ib/ft3 = 16.02 kg/m3 1 poise = 0.1 Pa. 21 Ib/in3 = 27.68 Mg/m3 1 Ib/ft.sec = 1.488 Pa.s
1 lbf.sec/ft2 = 47.88 Pa.sFLOW RATE
1 gal/min = 63.09 cm3/s VISCOSITY, KINEMATIC1 ft3/min = 0.4719 dm3 1 stoke = 10 -4m2/s
1 ft 2/sec = 0.0929 m2/sMASS VELOCITY
1 Ib/hr . ft2 = 0.08137 kg/s.m2 ELECTRICITY1 coulomb = 1 ampere second
FORCE 1 volt = 1 joule/coulomb1 newton = 1 kg.m/s2 1 ohm = 1 volt/ampere1 Ibf = 4.448 N 1 farad = 1 coulomb/volt1 kgf = 9.8066 N 1 henry = 1 volt sec/ampere
PRESSURE MAGNETIC FLUX1 pascal = 1 N/m2 1 weber = 1 volt second1 mm Hg = 133.3 Pa 1 maxwell = 10-8 Wb1 in H
2O (60°F) = 248.8 Pa 1 kiloline = 10-5 Wb
1 in Hg = 3.377 kPa1 psi = 6.895 kPa MAGNETIC INDUCTION, B1 kgf/cm2 = 98.07 kPa 1 tesla = 1 Wb/m2
1 bar = 100 kPa 1 gamma = 10-9 T1 atm = 101.3 kPa 1 gauss = 10-4 T
ENERGY MAGNETOMOTIVE FORCE1 joule = 1 N.m 1 gilbert = 0.7958 amp-turn1 ft.Ibf = 1.356 J1 cal = 4.187 J MAGNETIC FIELD STRENGTH, H1 Btu = 1055 J 1 oersted = 79.58 amp-turn/m1 kW.h = 3.600 MJ1 MWD = 86.40 GJ DIELECTRIC COEFFICIENT
1 farad/m = 1 coul2/N.m2
SPECIFIC ENERGY1 ft.Ibf/lb = 2.989 J/kg PERMITTIVITY CONST.1 Btu/lb = 2.326 kJ/kg
O= 8.8542 pF/m
1 Btu/ft3 = 37.26 kJ/m3
1 Btu/gal = 278.7 kJ/m3 PERMEABILITY CONST.uo = 4n x 10-7 henry/m
SPECIFIC HEAT: ENTROPY c2uoeo = 11 Btu/lb.F = 4.187 kJ/kg.K
Gas constant:R = 8.314 kJ/kg-mol.K
3A49299H01
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