There seems to be no limit to the applica-tion of cathode-ray oscillography nor anylimit to the imagination that one mayapply to conditions of traveling waves ontransmission systems. The graphics andthe analogue have saved the cost of endless

full-scale experimentation. It should notbe inferred, however, that this particu-lar approach is a complete substitute forall experimentation since it can be readilyunderstood that either is an adjunct to theother.

Unfortunately some of the referencesin the paper do not seem to be generallyavailable to the readers. They do, how-ever, make valuable source material, andshow the wide interest and application oftraveling wave phenomena.

Voltage Unbalance in Delta Secondaries

Serving Single-Phase and 3-Phase Loads

A. S. ANDERSON R. C. RUETEFELLOW AIEE ASSOCIATE MEMBER AIEE

made in innumerable such instances wherethe cost of other corrective measureswould have resulted in lower costs andmore satisfactory installations. Accept-ance of some degree of voltage unbalanceas a fair allocated apportionment of theservice factor, resulting in a reasonableapproach to a minimum over-all costfor all concerned, should be the aim of arepresentative national group.

IT IS a fairly widespread practice in theutility industry to serve both single-

phase and 3-phase loads simultaneouslyfrom 120/240-volt 4-wire delta secondarycircuits. The unprecedented growth of3-phase residential and small commercialair-conditioning load in our southernstates is extending this practice veryrapidly. The practice creates unbalancedloads on the 3-phase portions of the powersystem, hence customer service voltageson these systems are generally unbalancedto some degree. A reasonable amount ofvoltage unbalance should be tolerated toachieve optimum economy, taking intoconsideration all interests of the user, themanufacturer, and the supply utility.The purposes of this paper are three-

fold:

1. To present equations for determiningthe unbalances in voltage in 4-wire single-and 3-phase delta secondaries.2. To stimulate interest which may leadto acceptance by manufacturers and utilitiesof some degree of voltage unbalance inthe supply.3. To promote a definition of voltageunbalance in a 3-phase system which shouldeventually receive industry acceptance.

Voltage and Current Unbalance

Unequal voltage drops in conductorsand transformers caused by simultaneoussingle-phase and 3-phase load currentsproduce unbalanced load voltages. Sincein general, the negative-sequence imped-ance of rotating 3-phase equipment issmall compared with the positive-se-quence impedance, very little voltage un-balance is required at the load to producerather large current unbalance. Forexample, if the negative-sequence im-pedance is 15 per cent of positive- sequenceimpedance, and negative-sequence volt-age is 2.5 per cent of positive-sequencevoltage, the negative-sequence current is16.7 per cent of positive-sequence current.

Negative-sequence current reducestorque of a motor slightly and increaseslosses in a more significant degree. Whenit is present the three line currents are un-equal. The largest of these may tripout thermal overload protection whilethe over-all heating of the motor may beonly slightly in excess of normal.

Unbalance Factor

Voltage unbalance has been defined invarious ways, none of which seem to befully accepted as a standard in thiscountry. In this paper, voltage un-balance is defined as the ratio of negative-sequence voltage to positive-sequencevoltage and is called the voltage un-balance factor. This should be a suitabledefinition since it is based on the theoryof symmetrical components, which con-cept has greatly simplified calculationsinvolving unbalance. It is suggested thatthis definition be considered for adoptionas an industry standard. (In the case ofcurrent unbalance, it is suggested thatthe term "current unbalance factor"be used and that it be defined as the ratioof negative-sequence current to positive-sequence current.)

In applying motors engineers areaware, and frequently make use, of theservice factor which is a part of the ratingof a motor and which allows for devia-tions in operation from rated values oftemperature, voltage magnitude, fre-quency, and mechanical load. It is cus-tomary, however, for manufacturers andmany utility engineers to assume that thesupply voltage is balanced in magnitudeand phase angle. In many cases of un-satisfactory motor operation, the utilityhas been blamed for unbalanced voltagewhere other deviations from rated condi-tions, frequently not under control of theutility, also existed. Large utility invest-ments in transformer capacity have been

Equations to Determine VoltageUnbalance Factor

Equations for voltage unbalance factor(as defined in the foregoing) in terms ofthe secondary supply system and loadimpedances have been developed for thethree most usual transformer connec-tions applicable to a 4-wire delta second-ary.

These transformer connections are: un-grounded wye-delta, herein called wye-delta; open-wye to open-delta, hereincalled open-wye; and open-delta toopen-delta, herein called open-delta.With the open-wye and open-delta con-nections, both the leading and laggingphase relations of the two transformersare considered. The connection is con-

sidered leading when the transformercarrying the single-phase load (lightingtransformer) is connected to a voltagewhich leads by 120 degrees the voltage to

which the other transformer (power trans-former) is connected. The connection isconsidered lagging when the lightingtransformer is connected to a voltagewhich lags by 120 degrees the voltage towhich the power transformer is connected.

Assumptions used in the derivation ofthe equations are:

1. Primary voltages are balanced (orsource impedance is zero).

Paper 54-195, recommended by the AIEE Trans-mission and Distribution Committee and approvedby the AIEE Committee on Technical Operationsfor presentation at the AIEE North EasternDistrict Meeting, Schenectady, N. Y., May 5-7,1954. Manuscript submitted October 29, 1953;made available for printing February 18, 1954.

A. S. ANDERSON is with Ebasco Services, Inc.,New York, N. Y., and R. C. R UETE iS with theGeneral Electric Company, Pittsfield, Mass.

The authors wish to acknowledge the assistanceof Chase Hutchinson, Henry Rudolph, C. R. Joy,and others of Ebasco Services, Inc., in the prepara-tion of this paper. Special credit is due to Prof.J. G. Tarboux of the University of Michigan forhis assistance in the initial stages of the study.

Anderson, Ruete- Voltage Unbalance in Delta Secondaries AUGUST 1954928

2. The transformers have identical turn-ratios.

3. Three-phase motors are represented byequivalent positive- and negative-sequenceimpedances, the zero-sequence impedancebeing infinite since the neutral is not con-nected.4. The single-phase and 3-phase loads areconnected at the end of the secondary lines.5. The 120-volt single-phase loads arebalanced and no current flows in the single-phase neutral.

The equations for three transformerconnections are given as equations 1, 2and 3, in which the loads are representedby

ZL = single-phase load impedanceZp = 3-phase load positive-sequence im-

pedanceZn =3-phase load negative-sequence im-

pedance

The impedances of the lines and trans-formers for the open-wye connectionsare represented by

Za = impedance of secondary line andlighting transformer

Zb=impedance of secondary line commonto both transformers

Zc=impedance of secondary line and powertransformer

The impedances of the lines and trans-formers for the wye-delta connectionsare represented by

Z= lighting transformer impedanceZ2, Z3=power transformer impedanceZa, Zb, Z,=secondary line impedances

Voltage Unbalance Factor for Open-Wye or Open-Delta LeadingConnection

EabnEabp

1(aZa+a2Zb+Zc)±+ (aZa-Zb)

a(Za+Zb+Zc+3Zn)- (aZa-Zb)Zn ZL

(1)

The equivalent circuit is shown inFig. 1. Equation 1 applies to the open-wye leading connection and to the open-delta leading connection since the onlycircuit change would be in the connec-tion of the high-voltage windings of thetransformers

Voltage Unbalance Factor for Open-Wye or Open-Delta LaggingConnection

EabnEabp

1 V/gfj301(a2Za+aZb+Zc)++ (aZ -Za)

ZP ZL

(Za+Zb+Zc+3Zn>)- (aZb-Za)Zn ZL

(2)For the equivalent circuit refer to

Fig. 1. For lagging connection the phasesequence is the reverse of that shown,i.e., the sequence of voltage vectors ischanged from Eab, Ebc, and Eca as shownto Eab, Eca and Ebc, This may be doneby changing the primary connection fromline C to line A.

Voltage Unbalance Factor forUngrounded Wye-DeltaConnection

EabnEabp1/ Z1 aZ2 a2Z3\(aZa +a2Zb +z,-_Z _ +23

ZKa +bZ 3 -3 3/

,-.r3d3o(aa -Z1(I 2Z1+c2Z2+aZ3\

ZL (aZa-Zb)+Z -3

Z1 z2 z3Za+Zb+Zc+3Zn+ ± ±+--)zn ~ 3 3 3/'r&j0(a,,, 1

Z( 2Z,+a2Z2+aZ3\

ZL (aZa-Zb)-z (- 3 + +-ZL ~~ZL 3 31(3)

For the equivalent circuit refer to Fig.2. The transformer impedances cannotbe converted to an equivalent wye be-cause the division of current in the deltais independent of transformer imped-ances. The primary wye connection isisolated from ground; therefore balanced3-phase currents divide evenly and thesingle-phase current divides two-thirdsin the lighting transformer and one-thirdin each of the other two transformers.Equation 3 applies only for phase rota-tion Eab, Ebc, and Eca.

Effect of Transformer Connectionon Unbalance Factor

The voltage unbalance factor for open-wye or open-delta leading connection,

the open-wye or open-delta lagging con-nection, or the wye-delta connecti on maybe determined from equations 1, 2, and 3respectively. These equations are longand cumbersome so that general solutionshave not been attempted. However,calculations were made for certain as-sumed conditions, and the data obtainedare summarized now. The assumedconditions were as follows:

1. Single-phase and 3-phase load eachranged from 0 to 100 kva.2. Maximum length of secondary of 300feet. Length of 300 feet assumed orlength reduced to keep unbalance factorbelow 0.025 for leading connection.3. Maximum size of copper conductor of1,000,000 circular mils.4. Maximum transformer loading of 150per cent.

5. Maximum voltage drop across lightingphase of 16 volts.6. Maximum conductor loading 2/3 ampereper thousand circular mils.7. The following combinations of powerfactor:

a. Single-phase load power factor of0.95 and 3-phase load power factorof 0.80.

b. Single-phase and 3-phase power fac-tors of 0.80.

c. Single-phase power factor of 0.70and 3-phase power factor of 0.90.

A summary of the conclusions drawnfrom partial calculations for voltage un-balance factors for installations coveredby the limitations and range of assump-tions given in the foregoing are as follows:

1. The 2-transformer (open-wye or open-delta) connection causes lower unbalancefactors than the 3-transformer (wye-delta)connection for power factor conditions a,except where the ratio of 3-phase load tosingle-phase load is three or more; however,the unbalance factor is within reasonablysmall limits even where single-phase load iszero.

2. The unbalance factor is lower for theleading connection than for the laggingconnection with load power factors between0.70 and 0.95 with one exception. Thisexception is where the ratio of 3-phase tosingle-phase load is greater than approxi-mately one and the power factor of 3-phaseload is greater than that of single-phaseload. Where the unbalance factor is largerfor the leading connection the differenceis usually not great enough to be signifi-cant.

C

NEUTRAL

A

I Zol a

Z3 AAz Zp1 2 Za Z I-

Z3 P. b :1 Zn

3 vy

Fig. 1. Equivalent circuit for open-wye connection Fig. 2. Equivalent circuit for wye-delta connection

Anderson, Ruete- Voltage Unbalance in Delta Secondaries

A

AUGUST 1954 929

Conclusions

1. There is need for a definition of un-balanced voltage which will receive industryacceptance.2. The ratio of negative-sequence topositive-sequence voltage appears to be themost logical definition of unbalancedvoltage.3. The use of equations for unbalancedvoltage as given in this paper should aid indetermining unbalanced voltage in 4-wiredelta secondaries supplying single-phaseand 3-phase loads, including inductionmotors.4. The open-wye leading connectiongenerally has a lower unbalanced voltagethan the open-wye lagging connection.

from the voltage drops in the line andtransformer impedances.The line currents can be found by adding

negative- and positive-sequence currents ofthe 3-phase load and the current of thesingle-phase load

Ila =IaOp+IaOn+IL

I2b IbOp+IbOn-IL

I3C = IcOp+IcOn

where IL = single-phase currentfrom a to b.From symmetrical components

Eaop = Eabp _j30

Eaon = ~Eab 3

(12)

(13)

(14)

flowing

unbalance for the different transformerconnections will be found.

Referring to equivalent circuits andequations

ZL= 1.38 /18.2 ohms. /18.2 denotespositive angle of 18.2 degrees.

Zp = 5.76 /36.8 ohms

Zn = 0.749 /90 ohms

Open-Wye or Open-Delta Connection

Using transformers rated 37.5 kva and5 kva

(15) Za= 0.0718 /62.5 ohms

Zb=0.0293 /65.4 ohms

(16) Zc = 0.397 /32.7 ohms

Appendix 1. Derivation ofEquation for Open-Wye Leading

Connection

The derivation of the equation for un-balance factor in the open-wye leadingconnection is as follows:

Refer to Fig. 1 for the equivalent circuitand symbols. Double subscript notationis used throughout. Additional subscriptsn and p refer to negative- and positive-sequence quantities respectively.

ZL = single-phase load impedanceZp =3-phase load positive-sequence im-

pedanceZn =3-phase load negative-sequence im-

pedanceejO=vector rotation of 0 degreesa= j120

From the definition of negative-sequencevoltage

3Eabn = Eab+a2Ebc+aIEca (4)

From the equivalent circuit

Eab =Eai +E12+E2b (5)

Ebc =Eb2+E23+E3C (6)

Eca= Ec3+E31+Eia (7)

Substituting equations 5, 6, and 7 inequation 4

3Eabn =Eal+El2+E2b+a2(Eb2+E23+E3c)+a(EC3+E31+Eia) (8)

Since primary voltages are balanced andturns-ratios are equal

E12+a2E23+aE31 = 0 (9)

Substituting equation 9 in equation 8and rearranging

3Eabn = (a-1 )Eln+( 1-a2)E2b+(a2-a)E3c(10)

Substituting the identities involving theoperator a

3Eabn = /3Einae j50+ UE2 bE30 ±X'+vE3E3CC270(11)

Equations 9 and 11 indicate that if theprimary voltages are balanced the negative-sequence voltage at the load can be found

IaOp =Eaop= Eabp >-j30 (17)Zp Zp

'non-Eann Eabn j30la 4= (18)ZP -\-Zn

I Eabp+Eabn ( 19)

The zero-sequence voltage is zero sincethe sum of three line voltages is zero

(20)

(21)ITbop = a2laopIcOp = aTaop'bCn = aIaon

IcOn = a2Iaon

The voltage drops in line and transformerare

Eia IiaZa (24)

E2b =I2bZb (25)

E3c 13cZc (26)

Substituting equations 24, 25, and 26in equation 11

3Eahn = -\/-3liaZaZEj+ 3I2bZb Ej3° +-/3cZcei270 (27)

Substituting equations 12 to 23 in equa-tion 27 and rearranging, the equation forEabn/Eabp is obtained as given here andshown as equation 1.

Equations 2 and 3 may be obtained in asimilar manner.

Appendix II. Example

The following example is worked out toillustrate the use of the equations and thecomparisons between the different trans-former connections.

Consider a secondary circuit 300 feetlong made up of two 300,000-circular-milcopper conductors carrying single- and3-phase load and one American WireGauge no. 4 conductor carrying 3-phaseload only. This secondary circuit serves40 kva of single-phase load and 10 kva of3-phase motor load. Power factors areassumed as 0.95 and 0.80 for single- and3-phase load respectively. The voltage

LEADING CONNECTION

EabnEabp

(aZa+a2Zb+Zc)+ (aZa-Zb)4t ZL

1 -\/32E330(Za+Zb+Zc+3Zn)- (aZa-Zb)zn ZL

Eabn 0.0585 /-3.0+0.1118 /211.3Eabp 3.45 /-8.5+0.1118 /31.3

0.0718.=50.0203

(22) LAGGING CONNECTION

(23) EabnThabp

(a2Za+aZb +Zc)+3

(aZb-Za)ZP ZL

-(Za+Zb+Zc+3Zn)-> (aZb-Za)Zn ZL

Eabn 0.0654 /-13.1+0. 1140 /238.6Eabp 3.45 /-8.5+0.1140 /58.6

0.1122- =0.03213.50

Wye-Delta Ungrounded Connection

Using 25-kva, 15-kva, and 10-kva trans-formers for the lighting phase, the phaselagging the lighting phase, and the phaseleading the lighting phase respectively

Za= 0.0293 /65.4 Z1 =0.0618 /55.5

Zb=0.0293 /65.4 Z2=0.1041 /51.3

Zc = 0.0920 /22. 1

Eabn _

Eabp

Z3=0.1632 /42.5

1 /Z, aZ2 a2Z3\+ aZ,+a2Zb+Zc-3--> -3+

W/e(Z3 Z)±1 _-2Z3 a2Z+ aZ3ZL ZL\ 3 3 3

It Zl z2 zAZa+Zb+Zc+3Z+n+ 3+-3-Zn\ 3 3 3/

N/30 1 (2Z1 a2Z2 aZa)ZL ZL 3 3 3

Anderson, Ruete-Voltage Unbalance in Delta Secondaries

habp

AUGUST 19549.30

0.0160 /-12.0+0.0638 /227.2+Eabn 0.0581 /200.3

Eabp 3.23 /-4.3+0.0638 /47.2+0.0581/20.3

0.1082= = 0.0326

3.33

The unbalance factor for the open-wyeleading connection was 0.0203; for theopen-wye lagging connection, 0.0321; andfor the wye-delta connection, 0.0326.

DiscussionJ. E. Gerngross and H. M. Bankus (GeneralElectric Company, Schenectady, N. Y.):We would like to congratulate the authorson their contribution to the literature onthe subject of voltage unbalance whichoccurs with unbalanced transformer con-nections serving combined single-phase and3-phase loads.We agree with conclusions 1 and 2 that

there is need for a definition of unbalancedvoltage which will receive industry ac-ceptance, and that the ratio of negative-sequence to positive-sequence voltage ap-pears to be the most logical definition.This is the definition which we used incalculating voltage unbalance in our recentAIEE paper.'

Equations 1, 2, and 3 should be veryuseful to the distribution engineer forchecking the performance of an occasionaltransformer bank. However, we agree withthe authors that these equations are longand cumbersome and therefore are notsuitable for calculating a large number ofpoints such as in a general solution.

It is interesting to note that equations1, 2, and 3 agree except for sign equation53 of reference 1, when the proper changesin notation are made. Reference 1 derivesequations for voltage unbalance whichare suitable for use when a large number ofcalculations are to be made. The equationsderived can easily be solved by means ofmodern automatic digital computers whichare available at various locations.Power systems engineers are becoming

more and more conscious of the advantagesof using this type of equipment, both fromthe standpoints of saving time and money.2,3The economics of time and money dependvery much upon the individual problem,but in general the more complicated theproblem, and the more personnel timewhich would be involved in computing tosolve the problem longhand, the moreeconomic advantages can be gained bygoing to some type of digital computer.An approximate picture of the savingsinvolved in using digital computers isprovided in reference 3.

Judging from the data given in AppendixII, and also from the result of the studypresented in reference 1, the general orderof magnitude of voltage unbalance whichmay be attributed to the transformer andsecondary conductors is of the order of 1, 2,or 3 per cent for many common combina-tions of transformers and loads. Whileany unbalance is definitely undesirable, itwould be rather difficult to economicallyjustify any substantial expenditure to

further reduce very small unbalances.However, it must be remembered that thevoltage unbalance given by equations 1,2, and 3 does not include that alreadyexisting in the primary. The primaryvoltage unbalance must be added to thesecondary unbalances according to thephase relationships of primary and second-ary negative-sequence voltages. Underthe worst conditions these would adddirectly.With increasing voltage unbalance in a

circuit, currents become unbalanced at aneven more rapid rate. Therefore a rela-tively small primary unbalance added tothe voltage unbalance occurring in thesecondary circuit will cause reduced motortorque and excessive motor heating to suchan extent that it may become economicallyfeasible to consider remedial measuresother than derating the induction motors.An indication of the derating which mustbe applied to motors operating with un-balanced voltages was provided in a recentAIEE paper by J. E. Williams.4To reduce the problem of voltage un-

balance to a minimum, the primary systemshould be designed in so far as possible sothat good voltage balance is obtained atpoints where 3-phase load is tapped off.In actual practice it will be impossible toachieve perfect balance at all times. How-ever, if the unbalance is consistently in agiven phase (i.e., one phase voltage isconsistently low) there are a number ofdifferent possibilities which may be con-sidered to improve the situation. Two ofthe most direct approaches are as follows:

1. Rotate the phase connections on theprimary side of the 3-phase transformerbank until a minimum total voltage un-balance on the secondary at the loadappears.

2. Connect secondary shunt capacitorsto the low-voltage phase at the load.

Both of these solutions can be used con-currently if the situation warrants it. Thecapacitance which would be required willdepend upon the individual conditions.It can be calculated from symmetricalcomponents theory or can be determinedby trial and error.

REFERENCES

1. UNBALANCED LOADING AND VOLTAGE UN-BALANCE ON 3-PHASE DISTRIBUTION TRANS-FORMER BANKS, H. M. Bankus, J. E. Gerngross.AIEE Transactions, vol. 73, pt. III, April 1954,pp. 367-76.

2. DIGITAL COMPUTERS AID ENGINEERS, T. L.Lee, A. P. Fugill. Electrical World, New York,N. Y., March 8, 1954, pp. 89-91.

3. PROGRESS IN THE APPLICATION OF DIGITALCOMPUTERS TO POWER SYSTEM PROBLEMS, F. J.Maginniss. Proceedings, American Power Con-ference, Chicago, Ill., 1954.

4. OPERATION OF 3-PHASE INDUCTION MOTORSON UNBALANCED VOLTAGES, J. E. Williams. AIEETransactions, vol. 73, pt. III, April 1954, pp. 125-33.

Bela B. Mohr (Oklahoma Gas and ElectricCompany, Oklahoma City, Okla. ): Theauthors have presented valuable equationsfor calculating the voltage unbalancecaused by different transformer secondarydesigns. The practice of serving 3-phaseand single-phase loads simultaneously froma 120/240-volt 4-wire delta secondary isusually adopted for economic reasons. The

advantage of this system as compared to aseparate light and power system and a 4-wire wye system is the savings in distribu-tion transformers, secondary services, andmeters. These savings come at a sacrificeof voltage unbalance. The savings can besizable where the open-wye open-delta orthe open-delta open-delta transformer con-nections are used. This problem of voltageunbalance is similar to flicker voltage dropand steady voltage drop and will affectmotor manufacturers, motor applicationcompanies, dealers of electric equipment,electrical utilities, and the users of electricequipment or the customer.

In commercial areas our company usesthe 4-wire delta and the 4-wire wye systems.The 4-wire delta system is used on over-head distribution where the load is largely3-phase motors and a relatively smalllighting load. The 4-wire wye is used onthe overhead system where the load is pre-dominantly lighting or single-phase wherethe 3-phase load is small. The 4-wiredelta provides a higher voltage to thecustomer but the transformer bank is alsoa source of some voltage unbalance. The4-wire wye transformer bank does notcreate any additional voltage unbalance,but the voltage is usually 120/208. Smalldry-type autotransformers have been usedin specific cases to improve motor opera-tion. Neither the delta nor the 120/208wye system seems to be the solution forall cases.

In residential areas we use 120/240-volt3-wire secondary. An economical methodof providing 3-phase service for light satura-tions of 3-phase motors has recently beenadopted. This method provides service bythe addition of one transformer on the polewhere the 3-phase service is required. Thistransformer, with the existing single-phasesecondary, provides a 120/240-volt 4-wire delta supply. This design has theusual voltage unbalance that is found inany 4-wire delta supply. The amount ofvoltage unbalance will depend on the sizeof the lighting transformer, the distancefrom the 3-phase service to the lightingtransformer, the leading or lagging con-nection of the lighting transformer, and therelative per cent of 3-phase load to thesingle-phase load. In most cases voltageunbalance will be no worse and in manycases better with this split type of installa-tion. The best voltage balance will beobtained if the power transformer is assmall as the 3-phase load will permit.Some voltage unbalance troubles have

been experienced in the last few years.One source of trouble has been the packagedair-conditioning equipment designed sothat any deviation from optimum serviceconditions resulted in unsatisfactory opera-tion. Another source of trouble is the 2-phase thermal protection of a 3-phasemotor. If the largest phase current hap-pens to be in the unprotected phase, themotor will burn up before there is anyindication of trouble. This is not a seriousproblem where the motor is not overloaded.We have had two specific cases in whichthis condition occured. The motor wasoverloaded in both cases.Our company has been using the "maxi-

mum deviation from average" method ofdefining voltage unbalance since May1952. We find it satisfactory because theper cent of voltage unbalance can be

Anderson, Ruete-Voltage Unbalance in Delta SecondariesAUGUST 1954 931

determined from any set of measurementsby simple mathematics. The simplicityof application and understanding makes thismethod more desirable for nonprofessionaloperating and service personnel. The"symmetrical components" method, asrecommended by Mr. Anderson and Mr.Ruete, requires the reference to a chart todetermine the per cent of voltage un-balance. If a chart is required, it willhave to be made available to all serviceand operating personnel when and where itis needed. The chart would also add thepossibility of an error in its use. If dueconsideration is given to the large numberof nonprofessional people who are goingto be operating and servicing polyphasemotors, the simple method would certainlybe adopted. It seems that the mostpractical solution would be the adoptionof the "maximum deviation from average"as the standard with the "symmetricalcomponents" method adopted for use indesign problems. The average differencebetween the two methods is 4.7 per cent.The solution to the voltage unbalance

problem requires an economic study in-cluding initial cost and operating costcovering distribution system, services, me-ters, customers wiring, and polyphasemotors. The system that would providethe necessary horsepower for the customerat the lowest annual cost should be adoptedby all. To properly consider all of thefactors involved would require a great dealof time and work. However, a look at theother sources of voltage unbalance at thistime would seem desirable.The other sources of voltage unbalance,

in addition to the transformer bank andsecondary system pointed out by theauthors, are the unbalanced loading on thedistribution primary and unbalanced load-ing of the customer's wiring. The authorsassume balanced primary voltages but ourcompany experience is that distributionprimary system almost always containssome voltage unbalance. The primaryunbalance can be traced to several things.The basic cause is the type of load. Theload is mostly single-phase on a largemajority of feeders and it is impossible tomaintain a perfect balance between theprimary phases. If the loads are balancedin the afternoon the night load will bedifferent, so the result is a compromiseto give the best continuous operatingbalance.Modern feeder regulators are designed to

operate in a range of + 1 volt. It hasonly recently been reduced from i 1.5volts. Voltage balance is not improvedby the regulator. At any point on the 3-phase distribution primary where anunbalance in load occurs, there will be anunbalance in voltage due to this unbalanceddemand. Since the regulators operate in a2- or 3-volt band, it is possible to have over1.5-per-cent voltage unbalance at the mostperfectly balanced point on the primaryfeeder with single-phase regulators and noovercompensation. It is possible to havea considerable amount of voltage unbalanceas the load changes throughout the day.To assume that the primary distribution

system is balanced is to assume that all ofthe electric equipment has a balanced 3-phase demand. Since it appears that single-phase equipment will be in use indefinitely,

Table 1. Motor CharacteristicsFixed Kilovolt-Ampere MotorGeneral-Purpose 4O-Degree-Centi!

Volt Motors

Applied voltage.90 ...Service factor' ...............1.04.Maximum heat dissipated

(service factor) squared ... 1 .09 .

Approximate voltage un-balance equal to servicefactor, per cent .......... .2 ...

we should expect to find some N

balance in the distribution primAnother source of voltage unb<

the customers' wiring and is prthe customers' changing requirelthe design of the customers' systeout engineer balances the simultaas well as he can estimate themand. New equipment and meresult in altogether different usdividual circuits by the custcvoltage unbalance from this sourin excess of 1 per cent.The design of the transformer

produce from 2- to 3-per-cent Nbalance as shown in the paper.from all these sources could beit were all added together and thpen in some cases, but one unbalaican be used to reduce the otlthe total voltage unbalance can bTo maintain the voltage balatolerable limits requires a methitinuous checking and supervisioloading by the utility. With thmind, it seems desirable that a dizones (acceptable and undesiralbe adopted. Corrective measurmade when the voltage unbalanciundesirable zone.The characteristics of the

motor as it relates to the voltageproblem are shown in Table I.analysis of this subject is given i2.An examination of Table I

motors manufactured to meeElectrical Manufacturers Associaards do not require perfectlv balages for successful operation.frequency in the present-day pois relatively stable, a large porservice factor could be allocalbalanced voltages. If 50 perservice factor is allocated tobalanced voltages. If 50 perservice factor is allocated tobalance at rated conditions antional service factor provided bat 10-per-cent overvoltage is alhto voltage unbalance, a motor sevolts could have 4 per cent for volance and still leave 2 per cent ftfrom rated conditions. A genemotor served at 240 volts shoisatisfactorily with 6-per-cent abalance for a period of time, ifplate ratings are not exceeded.The amount of the service f,

allocated to voltage unbalancefurther considered by all concehope the preceding comments wimore data to the discussion anifurther consideration of the "maviation from average" method

Based on

Demand.grade 220-

nition with the "symmetrical components"method recommended for design purposes.

REFERENCES1. STANDARDS FOR MOTORS AND GENERATORS.Pub. No. MGI-1949, National Electrical Manu-

100 ... 110 facturers Association, New York, N. Y., section1.15. . 1.26 4.15, 1949.

1.30 ... 1.60 2. See reference 4 of Mr. Gerngross and Mr.Bankus' discussion.

4 ...6

A. S. Anderson and R. C. Ruete: In com-paring the paper by Bankus and Gerngross

vroltage un- with ours it can be seen that an entirelyLary. different method of approach was used inalance is in each case. Their derivation is based on the-oduced by interconnection of positive- and negative-ments. In sequence networks. In our derivation them, thelay- positive- and negative-sequence currentsineous load were applied to the actual circuit directly.circuit de- We have not tried to compare the equationsthods may developed by the two methods; Mr.sage of in- Bankus and Mr. Gerngross have found themmer. The to be in agreement.ce could be We agree with the discussers that the use

of digital computers would be economicalbank could in fully exploring equations with as manyvoltage un- parameters as must be considered in thisThe total problem. However, we have developed

excessive if working tables showing transformer andLis will hap- conductor sizes which appear to be goodnce at times practical guides. Such tables can be de-her so that veloped in a reasonable time using system-ie tolerated. atic methods of calculations and assuming a.nee within minimum set of conditions to representod for con- practical cases.)n of phase We agree with all the discussers that theLese facts in problem of determining allowable unbalanceefinition by of voltage in circuit design and operation isble) should not simple. Utility engineers need to deter-es would be mine practical limits of primary voltagee was in the unbalance as caused by unbalanced loading,

flat spacing of primary conductors andpolyphase operation of single-phase regulators. Itunbalance is also necessary that manufacturers estab-A detailed lish allowable limits of voltage unbalance

in reference based on design of motors and protectivedevices. With a knowledge of practical

shows that limits of voltage unbalance that should bet National allowed on motors and that can be main-ition stand- tained in primary voltage, the utility en-lanced volt- gineer will then be able to design the second-Since the ary system using equations that have been

ower system presented in our paper.tion of the Mr. Mohr makes a plea for the acceptanceted to un- of the standard definition of voltage un-cent of the balance to be the "maximum deviation fromvoltage un- average." He does this on the basis thatcent of the such a definition is needed because "simplic-voltage un- ity of application and understanding makesd the addi- this method more desirable for nonpro-y operation fessional operating and service personnel."so allocated He further states that the definition advo-rved at 240 cated in the paper could be used for designItage unbal- purposes. We believe that just the reverseor deviation of what Mr. Mohr advocates should bezral-purpose adopted. A definition of unbalance shoulduld operate be adopted by the AIEE based on the bestvoltage un- mathematical procedure available to studythe name- unbalanced problems. The theory of sym-

metrical components fulfills this require-actor to be ment. Any short-cut method of determin-should be ing unbalance could then be adopted for

erned. We field use. There are several methods avail-ill add some able including the arithmetical definition of.d stimulate 'maximum deviation from average."lximum de- XVe are very grateful for the interestingas the defi- and informative discussions.

Anderson, Ruete-Voltage Unbalance in Delta Secondaries932 AUGUST 1954