'I If'J* I r 1. . . TL ance of the transformer bank paral-i norma l ov tage ton iltions In | nree- leled by capacitance between conductors

in series with line capacitance to ground,Pase ystem Produced ~high voltage to ground at the transformerPhaseSystems Produced by bank terminal or terminals of the open

phase or phases will result for certainSing e-Phase Switc ing ratios of the transformer exciting react-ances to the line capacitive reactances.

The capacitances of the transformer bank,,EDITH CLARKE H. A. PETERSON P. H. LIGHT bushings, and bus structure, with no inter-

MEMBER AIEE ASSOCIATE AIEE ASSOCIATE AIEE vening length of line, may be sufficient to,produce high voltages on small banks,such as potential transformers. The mag-

T has been recognized for some time high voltages may occur with one and nitude of the overvoltage is a function ofthat in three-phase circuits, high volt- two conductors open, with and without the ratio of the line capacitive reactance

ges may cause damage to equipment faults, and suggests rules for the applica- to the transformer magnetizing reactance;following the blowing of a fuse or the non- tion of fuses and single-pole switches. or, approximately, the length of line left:simultaneous opening or closing of switch Interpreting this reference to single-pole connected between the break and the.contacts.'-5 Motorsonsuchcircuitsmay switches very broadly, one may include transformer for a given transformerreverse their direction of rotation. Field any type of switching device, such as kilovolt-ampere and circuit voltage..observations of the reversal of direction single-pole disconnecting switches, oil The higher the voltage and the smallerof rotation of motors have been repro- circuit reclosers, horn gap switches, oil the transformer kilovolt-amperes, theduced in the laboratory, and the results circuit breakers, etc. The interval of more likely are high voltages to occur forreported.6 Failures of lightning arresters, time between the closing or opening of the a given line length.under conditions involving blowing of first and last phases may vary widely de- The solidly grounded and isolated-fuses or the opening of switches, appear to pending upon the type of switching device neutral systems were studied in detail.have occurred because of high sustained and also upon the circuit being energized Systems with neutrals grounded throughfundamental - frequency overvoltages or de-energized. For example, in cases of resistance or through reactance (includ-.above the arrester ratings. Cases of the fuse operation, one phase or two phases ing the ground fault neutralizer) were notbreakage of a line conductor with result- may be open for a relatively long time. included as part of this investigation.ing abnormal voltages have been noted. On the other hand, in the case of an oil Single-phase, and three-phase shell-type.Abnormal voltage conditions have been circuit breaker, the interval of time dur- and core-type transformers, with different:observed on potential transformers.7 The ing which one contact would be closed and primary and secondary connections, andV.D.E. Standard,8 in effect since October two opened, for example, would be very the effect of one and of two open con-1, 1925, points out the danger of using short. The time required for the transi- ductors, with and without faults tofuses and single-pole disconnecting tion to the condition of sustained voltages ground, are included in the study.switches under certain conditions, and depends upon the circuit constants andstates that the breakage of a conductor, the characteristics of the interrupting Basis of Studywhere the end on the system side falls to device. It may vary from a few cycles toground, may give sustained voltages to a relatively long period of several seconds. Since the highest voltages are apt to,ground on the portion of the conductor The phenomena to be discussed in this occur with the unloaded or lightly loadedsevered from the main system as high as paper may be associated with any inter- transformer bank, the study was made,3V\times normal line-to-neutral voltage. rupting device which has characteristics with the bank unloaded. Voltage condi-With the renewed interest in fuses at such that the interval of time between the tions were obtained with transformer

high voltages, and the possibility of using opening or closing of the first and last banks made up of three single-phase unitssingle-pole switches, the problem of de- phases is of sufficient length to permit the and with three-phase transformers of bothtermining the conditions under which ultimate steady state voltage conditions the shell-type and core-type, the wind-these devices may be safely used in cir- to be attained. ings on the line side and also the second-cuits supplying ungrounded transformer The phenomena encountered may be aries being connected alternately in deltabanks has attained new importance. This summarized as follows: When one con- and wye for each case.paper gives the results of an investigation ductor of a three-phase circuit supplying Figure la gives a one-line diagram ofto determine the conditions under which an ungrounded, unloaded transformer the system studied, which consists of a

bank is open, there is a path for currents three-phase power source supplying aPaper 40-105, recommended by the AIEE com- from the closed conductors through the transmission circuit and transformer bankmittee on power transmission and distribution, andpresented at the AIEE summer convention, exciting impedance of the bank and the through fuses or single-pole switches. ItSwampscott, MWass., June 24-28, 1940. Manu- capacitance between conductors to the will be assumed that thethe-asscript submitted October 24, 1939; made availat)lethe-asfor preprinting May 10, 1940. open conductor, and thence to ground system is large relative to the kilovolt-.EDITHs CLARKE, H. A. PETERSON, and P. H. LIGHT through the capacitance-to-ground of the ampere rating of the bank, and there-are in the engineering division of the central-station opnconductor.Teeiasmlrpth fecnbeersnedyanquvetdepartment of the General Electric Company, pn.Teei iia ah fr a erpeetdb neuvlnSchenectady, N. Y. with two conductors open. When the generator with balanced line-to-line volt-.The authors wish to express their appreciation to three-phase system is grounded, the path ages and negligible positive- and negative-.F. A. Hamilton, Jr., MS. B. Crary, and C. Concordia throufor their helpful suggestions and to 0. K. Carter iS completed thogh the system grounded sequence impedances. If the system iS.and MW. J. Kirby for their assistance in making the neutrals; if ungrounded, through the grounded, the equivalent generator will

tests.~~~~~~~capacitance-to-ground on the system side also havre balanced line-to-ground voltages1.aFor.alnmee eeece,selstedo of the opening. With the inductive react- and its zero-sequence impedance will be

1941, VOL. 60 Clarke, Peterson, Light-Abnormal Voltage Conditions 329

EQUIVALENT THREE- 00-PH ASE POWER SOURCE Figure 1 (left). 2.0 1TRANSMISSION CIRCUIT System studied 0 No1|| |||N 1L

FUSES OR SINGLE- i,}tn -lnPOLE SWITCHES TRANSFORMER 1.6BANK diagram>No.

(a)~~~~~(b)- Miniaturesystem equiva- 0

SINE WAVE GEN. lent circuit 4 1.0 3

5N\N..tI njXTTTo TRASOMER_jjC~C_rC7~ BANoKr

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34TIMES NORMAL EXCITING CURRENT ( R.M.S.VALUES)

negligible. Figure lb gives the miniature Figure 2. Transformer saturation curves used (b). Wye-connected on line side, delta-connectedsystem9 equivalent circuit for figure la, in making calculations and in the miniature on load side.with which the study was made. The system 1. Core type.impedance Sb thebsine-wave generator is . 2. Shell type.impedance~~~~~oftesn-aegnrtri 1-Miniature system single-phase transformers B w hssoe-ofutvery low relative tothe exciting reactance -135 rms volts normal-1,185 ohms normal B w hssoe ofutOf the unloaded transformer hank and the mt impeancline capacitive reactances used, so that adgneilzing mpdneSame asfor A.balanced line-to-line voltages are applied 2-Assumed curve having more saturation thanto the circuit at a'b'c' under all conditions. curve 1. Used in making calculationls to de- II. POWER SOURCE UNGROUNDED

The syste becomes groundedone withtermine the effect of this factor LOAD TRANSFORMERS UNGROUNDEDbalanced line-to-ground voltages when 3-Miniature system three-phase core-type A. One phase open-no fault.switch SN is closed. Resistance and transformer-158 rms volts line to line normalreactance are neglected in the transmis- wye-840 ohms per phase normal magnetizing 1. Single-phase load transformers.sion circuit since they are insignificant impedance (a). Delta-connected on line side, wye- or delta-relative to the capacitive reactances fcr 4-Miniature system three-phase shell-type conteonlasi.the lengths of line under discussion. Cl' transformer-144 rms volts line to line normal B. Oneconductoropen-fault-to-groundand C0' represent the positive and zero- wye-520 ohms per phase normal magnetizing on system side of open conductor.sequence capacitances, respectively, on impedancethe system side of the fuses or single-pole 5-Miniature system special transformers 110 l. Single-phase load tranlsformers.switches; C1 and C0, the positiveand zero- rms volts normal-13,200 ohms normal mag- (a). Delta-connected on line side, wye- or delta-sequence capacitances of the circuit sup- netizing impedance. Single-phase trans- connected on load side.plying the ungrounded, unloaded trans- formers (b). Wye-connected on line side, delta-connectedformer bank on the transformer side,onlasie

which in figurelbisshown as wye con-~~~~~~~(c). Wye-connected on line side, wye-connected onwhlch ln figurelb1Sshown as wye con-~~~~~~~~~~~loadside.

nected on the line side and delta con- reproductions to scale of typical powernected on the load side. transformers. All saturation curves give C. One conductor open-line-to-groundTo obtain various degrees of saturation the ratio of the applied sinusoidal rms fault on load side of open conductor.

for the miniature, system transformer voltage to the exciting rms current, with 1. Single-phase transformers.abanks, it was only necessary to vary the voltage and current expressed in terms ofnormal operating voltage level in the normal voltage and normal exciting cur- (a).nelta-onnectdsoiindid,we-oedlaminiature system. In obtaining the re- rent, respectively.lsults herein summarized, the miniaturesystem single-phase transformers were Cases Studied Resultsoperated at a normal voltage correspond-ing to the saturation curve shown by I. POWER SOURCE GROUNDED-LOAD Curves showing the voltages to groundcurve 1 of figure 2. Curve 2 of figure 2 TRANSFORMERS UNGROUNDED of the open phase or phases as a functiongives a transformer saturation curve with of the system constants were obtaineda slightly higher degree of saturation than A. One phase open -no fault. from the miniature system by varying1 of the miniature system. Curve 5, for l. Single-phase load transformers. the capacitances, Cl and C0, but keepingthe specially built miniature system trans- their ratio constant. Data were obtained-formers, gives an intermediate value of (a). Delta-connected on line side, wye- or delta- for C =C0 and Cl-=2C0 for the solidlysaturation and because of its higher im- (b.Wecnetdo iesd,dlacnetd grounded system. Calculated curvespedance, made it possible to cover a on load side. were obtained by the methods given ingreater range of system constants. These (c). Wye-connected on line side, wye-connected on the appendix.curves are thought to be fairly typical Of load side. In order to demonstrate the effect ofpower transformer saturation. The satu- 2. Three-phase load transformers, various factors in producing overvoltagesration curves of the three-phase core-type and phase reversal, case 1 will be dis-and shell-type transformers of the minia- (a). Delta-connected on line side, wye-connected cussed first in detail. Throughout theture system are given by curves 3 and 4, 1. Core type. following discussion, Xm is the magnetiz-respectively, of figure 2. These units are 2. Shell type.

-ing reactance of the transformer bank on

3:)0 ~~~~~Clarke, Peterson, Light-Abnormal Voltage Conditions AIEE TRANSACTIONS

3 4 _ 4_ ^.bS-F4 lll 11 XIlllll11111110 >0 3-J

ci ~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~c

zU z _

H~~~~~~~~~~~~~~~ Ho

Xcl/Xm Xc1/Xm

Figure 3a. Steady-state maximum voltages on I. SOURCE GROUNDEDA. ONE PHASE Figure 36. Steady-state maximum voltages onopen phase with one conductor open. No OPEN-NO FAULT open phase with one conductor open. Nofault on system. Power source solidly fault on system. Power source solidly

grounded 1. Sigle-Phase Transformers groundedCl = Co. Transformers delta-connected on line (a). Delta-Connected onl Line Side, C1=2C0. Transformers delta-connected onside and delta- or wye-connected on load side Wye- or Delta-Connected on Load Side. line side and delta- or wye-connected onCurve 1-Saturation curve 1, Figure22 The curves of figure 3a give the steady- load side

' ~~~~state maximum peak voltage of the open Cre1Strto uv ,FgrCuve2-atrtin ure2,Fiur conductor on the transformer side of theCurve 3-No saturation opening in terms of normal line-to-neutral ..Curve 2-Saturation curve 2, figure 20-Test points-miniature system with satura- peak voltage with the transformer bank ..Curve 3-No saturation

tion curve 1, figure 2 delta-connected on the line side, and 0-..OTest points-miniature system with satura-wye- or delta-connected on the load side, . .tion curve 1, figure 2versus Xci/Xm for C0= C1. Curves 1, 2,

the load side of the open conductor atnormal voltage. In more general terms, ..and 3 are calculated by methods given init is the equivalent magnetizing reactance ...the appendix. Curve 1 gives calculatedFigure 3c (lower left). Steady-state maximum .. rof the total transformer kilovolt-amperes voltages on open phase with one conductor ..voltages using saturation curve 1 Ofon the load side of the fuse or single-pole open. No fault on system. Power source fi..gure 2, which corresponds to the satura-switch location. Thus, the single-load solidly grounded . .tion of the miniature transformers.transformer of figure lb may be considered C1=C.Tasomrwy-oecdonle

as te euivaentof everl salle trns- side, wye- and delta-connected on load side ..Figure 3d. Calculated steady-state maximumformers all on the load side of the open Cuv1-auaincre1Fiue2 (a votgsnopnhsewhoecndtrconductor. Likewise, the positive- and opren N-durtonuvfagure2()..vltaeon system.asPwihoersonurceozero-sequence capacitive reactances, xcl Wy-we (b). woidygoudeltand XGO, respectively, should be inter- Curve 2-Saturation curve 2, figure 2. Wye- sldygone

pree in a moegnrl manrt o-wye ..C1=2C0. TransFormers wye-connected on linerespond to total positive and zero-se- Curve 3-No saturation ..side and delta-connected on load sidequence capacitances C1 and CO on the Curve 4-Neutral displacement saturation ..Curve 1-Saturation curve 1, Figure 2load side of the open conductor. Thus curve 1, figure 2. Connection wye-wye ..Curve 2-Saturation curve 2, figure 2the results assume much broader signifi- Curve 5-Neutral displacement no saturation ..Curve 3-No saturationcance and are much more widely appli- 0-Test points-miniature system with satura- ..Curve 4-Neutral displacement saturationcable. tion curve 1, figure 2. Connection wye-delta ..curve ii figure 2

4OO-JT .... / 3

z Z

0~~~~~~~~~1

L-3H 2`o 3 l111J III1111111111111

0.1 0.5 1.0 2.0 5.0 10.0 20.0 100 01 0.5 1.0 2.0 5.0 10.0 20.0 100Xcl/Xm xcl/xm

1941, VOL. 60 Clarke,Peterson, Lig;htAbnormal Voltage Conditions 3. 331

Points obtained by test on the miniature crease and remains negative on both cal- (b). Wye-Connected on Line Side,system are indicated. The general shape culated and test curves; but before xc,I/xm Delta-Connected on Load Side, (c) Wye-of calculated and test curves is the same, = 1 is reached, two other steady-state Connected on Load Side. Figure 3c foralthough calculated voltages are higher values of voltage are possible as indicated C1 = C0 and figure 3d for C1 = 2Co, withexcept for large ratios of xCi/Xm where by the calculated curve. Both are posi- the transformers wye-connected on theboth calculated and test voltages are less tive. Only one of these other indicated line side and delta- or wye-connected onthan normal. values was obtained in test, the lower of the load side, are similar to figures 3aThe reference vector for calculated the two, until Xc,/Xm was decreased to and b. Test values from the miniature

voltages is the voltage of the open phase about 0.2 but for values of xc,/x,n=0.2 system for C1=CO and a wye-delta con-on the system side of the opening, so that or lower, three different values were ob- nection do not differ materially fromwith no capacitance in the circuit, or tained by test. Figure 4 shows oscillo- those for the transformer delta-connectedx,,==o, the voltage on the transformer grams of the voltages and currents for the on the line side except that only two of theside of the opening is -0.5 and the voltage three possible stable conditions, any one stable voltage conditions indicated byat the load is single-phase. As the ratio of which may occur, depending upon the calculation for values of Xc,/xm in thexci/xm is decreased, the voltage of the phases of the voltages at the instant the neighborhood of 1 and below, were ob-open phase increases in magnitude in the switch is opened. With the switch tained in test. Figure 5 shows oscillo-negative direction and there is phase opened at random a number of times, the grams of the voltages for the two stablereversal at the load. At xc,/x,n=2 for all high negative voltage or the low positive conditions obtained at x,/xm= 0.15, withcalculated curves and about 1.5 for the occurred more often than the medium C1=C0. Notestsweremadefor C1=2Co.test curve, the voltage-to-ground of the positive voltage.-open phase is -2 and the line-to-line Curve 2 of figure 3a was calculated, 2. Three-Phase Transformersvoltages at the load are balanced normal using transformer saturation curve 2 of (a). Delta-Connected on the Line Side,voltages of reverse phase rotation. (For figure 2. Curve 3 is for no saturation. Wye-Connected on the Load Side. Figurethe calculated curves, Va= -2, Vb= - 1/2 Figure 3b is similar to figure 3a except 6a gives test values of the voltage-to--jv'3/2, V,= - '/2+j/3/2). As XcJ/Xm thatC1=2Co. Thevoltagesabovenormal ground of the open phase for both core-is further decreased the voltage-to- of figure 3b are lower than those of figure type and shell-type transformers, delta-ground of the open phase continues to in- 3a for the same values of xc1/xI'. connected on the live side anid wve-con-

Figure 4 (left).Delta -connectedload transformer;co= C, 135 voltsline-to-line,lVxix= 0.212

Calibration: Cur-rent-Times nor-mal crest trans-former excitingcurrent. Volt-age-Times nor-mal line-to-neu-tral crest voltage

Figure 5. Wye-connected load'ttransformer, C. =Cj, 120 volts

X.=0.15Ca I i b r a t i o n:Times normal line-to-neutral crest Zi aw

voltage ~ iE J

332 Clarke, Peterson, Light-Abnormal Voltage Conditions AIEE TRANSACTIONS

.1MtSA-- ca| 4tWffl :2.ith.ki64 -C,64 40-12;] .5 .0 2O i.C ( 0 2t0 10 -42z4 zz +z -

.-J~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-o2 x-2 - o

3~~~~~~~~~~~~~~~LU-3-

0.5 1.0 2.0 5.0 100 200 100-4. 0.5 LO0 2.0 5.0 10.0 20.0 0Xci/Xm /Xc/

Figure 6a. Steady-state voltages on open phase with one conductor Figure 6b. Steady-state voltages on open phase with one conductoropen; no fault on system; power source grounded; C1 = Co, three- open; no fault on system; power source grounded; C1= Co; three-phase load transformer, connected line delta-load wye, ungrounded phase load transformer connected line wye-load delta, ungroundedNormal line-line voltage=91 volts for core- Normal line-line voltage=79 volts for core-

type, 83 volts for shell-type type, 72 volts for shell-typex-Shell-type; middle leg unopened 0-OCore-type; middle leg openedFigure 6c (below). Steady-state voltages on+-Shell-type; end leg unopened open phase with one conductor open; no fault r-Core-type; end leg openedO-Core-type; middle leg unopened on system; power source grounded;C=2Co; x-Shell-type; middle leg openedr1-Core-type; end leg unopened three-phase load transformer connected line +-Shell-type, end leg opened

wye-load delta, ungroundednected on the load side, obtained from Normal line-line w 4the miniature system with saturation voltage = 79 volts 3curves 3 and 4, respectively, of figure 2 for core-typeand with C1 =Co. Calculated values of O-Core-type; mid- 2voltage for saturation curves 3 and 4 of dle leg openedfigure 2 are not given, but they would not t 1differ materially from those for saturation rn-Coretpene endcurve 1 of figure 2, given by curve 1 of le pndwfigure 3a. In general, the curves of 6a J-1agree with those of 3a, except that themedium voltage of the three calculated 0voltage conditions indicated by calcula-tions for values of x,, /Xm in the neighbor-hood of 1 and below, does not appear. -401 05 L. 2.0 5.0 100 200 100Instead, for x,1/x, lower than 0.7, higher xcl/xmvoltages than indicated by calculationsappear, giving three stable voltage condi-. ...tin.a hshiving volestales htave po- as outlined in the summary of cases side of the opening in per unit of normalniounc.Thedsuharmonichvtan thaepror studied, detailed results will not be peak line-to-neutral voltage, with XcI/Xmcoulcednotubehpreictd"bydtheretor shown. Instead, a composite curve show- as abscissa and x8'I/x,l as parameter, foroused Howbevprer, ctheyd but littlemethig ing the range of values obtained for all Cl= CO, and the voltage of the open con-thed.Howevc thed volagesot ce 1,gh of the conditions studied using single- ductor on the system side as referencethan the calculated voltages of curve 1, phase transformers is shown in figure 7. vector. Test points from the miniaturefigure 3a. The narrow range of values obtained system with saturation curve 1 of figure 2

(b). Wye- Connected on Line Side indicates that the transformer connection for various values of x,'I/x,l are indicated.Delta-Connected on the Load Side. Figures and its construction (core, shell, or single The voltages are independent of x,l'.6b for C,= CO and 6c for Cl=2Co are-bfrC CadcfrC=2 ar phase) have little effect on the over-

similar to figure 6a except that the trans- voltages obtained. Figure 8 shows someformers are wye-connected on the line typical oscillograms for one case with B. ONE CONDUCTOR OPEN-FAULT-TO-side and delta-connected on the load side.side~~~~~~~Aan.et-once ntela ie single-phase transformers. GROUND ON THE SYSTEM SIDE OFOnly two stable voltage condiltions wereobtained for values of Xci/Xm in the vicin- II. POWER SOURCE UNGROUNDED THOPNC DU ORSGL-AEity of 1 and below. The voltages of LOAD TRANSFORMERS UNGROUNDED TRANSFORMERSfigure 6b are substantially the same as A. ONE PHASE OPEN-NO FAULT Figure 10 shows the range of voltagesthose of figure 3c for the same values of .obtained for the vrarious transformer con-Xc/xm 1. Single-Phase Transformers-(a) Delta- nections. Here again the transformerC1/ m

~~~~~~~~Connected on the Line Side, Wye- or connections make little difference exceptB. TWO PHASES OPEN-NO FAULT Delta-Connected on the Load Side for very low ratios of Xci/Xm. The wye-While numerous factors were investi- Figure 9 gives the calculated voltages- delta connection tends to reduce the

gated in connection with this condition to-ground of the open phase on the load overvoltages somewhat in this region.

1941, VOL. 60 Clarke, Peterson, Light-Abnormal Voltage Conditions 333

4..1

I..

0Ian4Z

02in

-4t -0.1 0.2 OA 0.6 O's 1.0 2.0-- 4.0 6.0 aL0 10

XC1/Xm

Figure 7. Steady-state voltages on openphase with two conductors open. No fault on

system. Power source solidly groundedCurve 1-Calculated using curve 1, Figure 2 Figure 8 (right).for C1=Co and transformers delta-connected Wye-connectedloadon line side and wye-connected on load side transformer; Co=

Curve 2-Same but Ci=2Co Cl, X.,/X. = 0.89.Test points as follows-Miniature system using Phases b and c opensaturation curve 1, figure 2. Single-phase Calibrations: Cur-

transformers rent-Times normal*-Cs = Co, transformers delta line wye load crest transformer ex-a-C1=2C, transformers delta line wye load citing current.x-Cl= Co, transformers wye line wye or Voltlge-Times nor-

delta load cres vletagneuta-C,=2Co, trdnsformers wye line wye or

delta load

C. ONF CONDUCTOR OPEN-LINE-TO- nation of system and load-side zero-se- losses, the high-voltage condition did notGROUND FAULT ON THE LOAD SIDE OF quence capacitances, and the magnetizing occur for values of x(,+C,')/x,x below 0.2,THE OPEN CONDUCTOR-SINGLE-PHASE reactance of the transformer respec- approximately. This elimination refersTRANSFORMERS DELTA-CONNECTED ON tively shown in figure lb. only to the steady-state condition, how-LINE SIDE-WYE- OR DELTA-CON- If the total capacitive reactance of the ever. In many cases, long transient con-

system (capacitance on both sides of the ditions were observed, which could beNECTED ON LOAD SIDE opening) is large, no high voltages are quite hazardous to insulation or con-Figure 11 shows the voltages obtained produced. As this reactance is decreased, nected apparatus. It is important to bear

for this assumed condition. It should be voltages are increased. Only one voltage this in mind in conlection with the pro-noted that the severity of this system value is obtained until the ratio of total tective criterion offered in the latter partfrom the standpoint of overvoltages can zero sequence system capacitive react- of this paper.be defined for all practical purposes by ance to magnetizing reactance,x('0+c')IIxm, Figure 11 indicates that phenomenathe ratio of total zero-sequence capacitive reaches a value of about 5.0. Below this such as this can take place in a relativelyreactance to transformer magnetizing value, three stable voltage conditions large percentage of ungrounded systems.reactance. The ratio of xe,,/x, has rela- were obtained in some cases. As the Perhaps the occurrence of faults followedtively little effect. capacitive reactance was decreased, in- by the blowing of fuses may account forThe basic circuit which produces the creasingly higher maximum voltages were much of the heretofore obscure transient

high voltages shown for this condition obtained. With a relatively low-loss phenomena occurring in isolated-neutralis somewhat more complicated than those transformer (curve 1 of figure 2), the systems.circuits investigated in the other cases. high-voltage condition could be obtainedThe principal elements of the circuit are for values of X(co+c.,)/Xm down to 0.03, Discussiona capacitance in series with a parallel cir- although the low-voltage condition oc-cuit composed of another capacitance and curred more often. In other words, the The manner of connecting the windingsa saturable reactance. These three ele- higher the voltage, the less would be the of an ungrounded transformer bank,ments can be identified with the zero- probability of reaching that voltage. whether delta-delta, delta-wye, wye-wye,sequence capacitance on the system side With the smaller transformers (curve 5 or wye-delta, does not materially affectof the open conductor, a parallel combi- of figure 2) which had slightly higher the magnitudes of the voltages-to-ground

334 Clarke, Peterson, Light-Abnormal Voltage Conditions AIEE TRANSACTIONS

obtained on the open phases except in the on the open conductor, or conductors, positive, but usually greater than one-halfvery high voltage region. The voltages- on the system side of the opening, or open- the positive.to-ground of the open phases obtained ings, does not affect the sustained voltages From figure 7 it is seen that calculatedwith three-phase transformers of both at the transformer bank terminals. A voltages in the region of 1.73 timescore-type and shell-type are approxi- fault-to-ground on the transformer side normal line-to-neutral voltage are greatermately the same as those obtained with of the opening reduces the voltage there than test voltages. Allowing for this,banks of. single-phase units, with the to zero. The highest voltages are ob- and with C> Co0> C1/2, voltages abovesame saturation curve, for corresponding tained when two conductors are open 1.73 times normal will probably not occurvalues of XcI/Xm. and there is no fault. Such a condition with transformers of the usual degree ofThe method of calculating sustained can exist when a line-to-line fault is saturation if the ratio Xcl/Xm is 6 or

voltages-to-ground when one or two con- cleared by the blowing of two fuses; and greater.ductors of the three-phase circuit supply- also on a circuit controlled by single-pole With a criterion of xci/Xm=6,ing an unloaded, ungrounded transformer switches when one phase is closed before 106 kvabank are open, gives values which approxi- the other two. xlxm= ImXc = Im 2 X =6mately check tests made on the miniature If the peak value of the sustained volt- 27rfCl kv2103system. Except where the calculated age-to-ground of the open conductor on I == kva 10/6(2fC )(voltages are less than normal, the method the load side of the opening in a solidly kv2gives values, which, with a few exceptions grounded system with ungrounded trans- wherein the very high voltage region, are too former bank is to be kept below 1.73 times 1= allowable length of line betweenhigh. The method can therefore be con- normal line-to-neutral peak voltage, the fuses or single-pole switches and loadsidered to give conservative values. approximate length of line, 1, between the transformer bank

While test points agree quite well with opening and the transformer bank should kva = rated kilovolt-amperes of thetransformer bankcalculated results, it may be well to point not exceed a value which can be approxi-out that for some of the conditions which mately determined from the normal volt- Figure 10. Maximum steady-state voltages togave rise to very high voltages, it was age of the line and the rated kilovolt- ground with one conductor open. Line-possible to obtain slight variations in the amperes and exciting current of the trans- ground fault on source side of open conduc-wave shape depending upon the initiating formers. tors. System neutral isolated. Ci = Co.conditions. For instance, the high volt- The positive-sequence capacity sus- These results are independent of Ci' and Co'.age might, for one opening of the switch, ceptance of overhead transmission cir- No load on systembe essentially of fundamental frequency. cuits, 2rfC1, in micromhos per mile at 60 Single-phase transformers, delta-connected onFor a second opening of the switch, it cycles varies from about 5 to 7, depend- line side-delta- or wye- on load side:might be essentially of the same mag- ing upon the diameter of the conductors Curve 1-Calculated voltages based on satu-nitude, but contain definite subhar- and the spacing between them. Since ration curve 1, figure 2monics.10,"' Some of these subharmonics these, in turn, depend upon voltage, Curve 2-Calculated voltages based on satu-observed were of very low frequency. 27rfC1 at a given voltage and frequency ration curve 2 of figure 2

For the ground system or the isolated can be estimated with fair accuracy. The Curve 3-Calculated voltages based on nosystem effectively grounded through its following values will be taken as typical saturationcapacitance to ground, a fault to ground at 60 cycles: 0 Test points-Miniature system saturationFigure 9. Steady-state maximum voltages on curve 1, figure 2open phase on load side with one conductor Single-phase transformers, wye-connected onopen. No fault on system. Power source KY 27rfCi Micromhos line side:

ungrounded Per Mile Curve 4-Calculated voltages based on satu-C1=C0. Transformers delta-connected on line 230-115 ..... 5.2 ration curve 1 of figure 2 with wye-connected

, , . .......69-34.5 .s..s. load sideside and wye-connectedl on load side 6934.51 .........8. odsdsiean1yecnece.o.oa.ie.45-386 Curve 5-Same as 4 but delta-connected load

xco Ixc= 0 = + Test points-Minia- Below 13.8 .6.5= 0.1 = turesystem with satu- side= 0.5 = oEl ration curve 1, figure (D-Test points-miniature system-saturation= 1 .0 = 0 2 The zero-sequence capacitance of over- curve 1, figure 2-delta-connected load side= 10 = x J head transmission circuits is less than the x-Same but wye-connected load side

No-3

191 O.6 lre eesn LgtA norma _S,ag CnitinI335

0ccLii~~~~ttl,-~

kv = normal line-to-line voltage of the tance of power transformers are given in 25.0 approximately, depending upon thecircuit in kilovolts a convenient form by L. V. Bewley."2 In ratio xn,,,/x,. In the light of our first

= average per unit rms exciting cur- figure 8 of his paper he gives the capaci- restriction on xcl/xm, this can be con-27rfCl= positive-sequence capacitive sus- tance between windings in magneto- servatively interpreted to impose the

ceptance of the circuit in micro- motive force per phase which may be con- restriction that xcox/Xm must be less thanmhos per mile sidered as the capacitance to ground, as 0.04 or greater than 40.0 approximately

The transformer exciting current 'm lies the capacitance of the low-voltage wind- Figure 12 can be used to interpret thebetween 1 per cent and 7 per cent, with ing to ground is usually larger, while the first criterion in terms of total transformer3.5 per cent not an unusual value. Figure capacitance of the high-voltage winding kilovolt-amperes, system voltage, and12, drawn with Im==3.5 per cent, gives to ground is small. The largest values of maximum allowable lengths of line on thethe approximate maximum allowable capacitance given are on the order of transformer side of the open conductorlength of line at various voltages supply- 0.01 microfarad which would be equiva- location. This figure is based on a ratioing a transformer bank of given kilovolt- lent to one mile of line, or less under most of xc/Xmx= 6.0 and since for the isolatedampere rating and 3.5 per cent average conditions. For the higher voltages, the system this must be increased to approxi-rms magnetizing current which can be internal capacitance would be equivalent mately 25.0, it is necessary to divide theoperated by fuses or single-pole switches to considerably less than a mile of over- miles of line shown by approximately 4.Jwithout the risk of voltages-to-ground head line. Bushing capacitance, being in order to conform to the requiredexceeding line-to-line voltages when two very small relatively, can be neglected. criterion.conductors are open. This allowable In case the transmission circuit con- In figure 14 the second criterion, im-length of line varies directly as the per tains some cable, its equivalent in terms posing the restrictions on the lengths ofunit rms exciting current of the bank and of miles of overhead line may be used. line on the system side of the open con-inversely as the ratio xeI/xm selected. For For that purpose, figure 13 has been pre- ductor, is shown. This curve shows thatan rms exciting current of 7 per cent, the pared which makes it possible to evaluate overvoltages may be expected as a resultallowable lengths of line would be doubled; this equivalent readily. This figure has of faults and subsequent fuse blowing forfor an exciting current of 1 per cent, they been reproduced from reference 13. wide ranges of system constants.would be only about one-third those of For the ungrounded system, it is impor- In applying these criteria, it should befigure 12. tant to note that the criterion for the pointed out that these are approximate

Since for some cases, the maximum solidly grounded system is adequately values. In some cases it may be necessaryallowable length of line is very short, it conservative for conditions not involving to make a more detailed calculation.may be necessary under such conditions faults. See figure 9. However, should a Exact line configuration, exact magnetiz-to take into account the internal capaci- permanent fault occur on the system side, ing reactance, and more detailed knowl-tance of the transformer bank itself. the criterion must be changed if voltages edge concerning the particular trans-Generally this will not have to be done. are to be limited to maximum values of former saturation characteristics may beFor cases requiring this refinement, 3 times normal. Figure 10 shows that necessary. However, the curves serve torepresentative values of internal capaci- a minimum ratio of XCl/Xm= 25.0 approxi- indicate in general the range of system

mately is required. This criterion alone constants which can give rise to highFigure 11. Isolated-neutral-system steady- is not sufficient, however. If a perma-state voltages on open phase with one conduc- nent fault on the load side is assumed, Figure 12. Solidly grounded system-ap-tor open. Fault to ground on load side of then the total zero sequence capacitance proximate maximum allowable length of lineopen conductor curves calculated using satura- of the system must be very large or else between fuses or single-pole switches and load

tion curve I of figure 2 very small in order to prevent overvolt- transformer to limit line-to-ground voltage on0-)Test points for xo'I/x,o=0 and saturation ages. See figure 11. Therefore, in addi- open phase to v'_ times normal, with two con-

curve 1, Figure 2 tion to imposing the criterion of xc,Ixm_ ductors open25.0, restrictions must also be imposed on Xci/xm = 6

x xco'/xcou=0 ' the ratio of Xco'/Xm. = 3.5 per cent rms average exciting+ XCO,/XCO= 0o 1 From figure 11, the ratio of X(co+co')/xm currentA-xc'/x,O= 1 0 should be less than approximately 0.04 or NOTE: Allowable length of line varies di--xC0'/xC0= 10.0 greater than a value varying from 4.7 to rectly with rms average exciting current

4-

NOT N INT110~~~~~~~~~~~~~~~~~~~.

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BUT RATINININT TRNFRERBN-

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goo 7-- _ / 11) XLA I11Z _ _*_-U. 12w.00

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seun equpdivalusentofcabeso .mco

For~~~~~~~~~~~~~~rqiepapeinsulated cableso assme dielectricag o queletliele so sseFor varnished cambrictvediecio bu i ori lssthy irubberf us o insulationtc lcaio

_J 0 ulipyby135201~ 0 IVO 060 0,Ovrha IneeuvnIae npstv-TTLTASOMRKAO LOAD SIDEVOOPNCNDCO

tuImo prml egtv-euec oqe Iseqalt ta FIgr14 Isoae eta ytmapoiw t- TRIII eurd ominanroain ntepoi at ageo0quvln ln entsonsse

overvoltages n solated-neutral systems. I I I it s olUnder lightly loaded conditions, it would slow down and stop. They will then re- which may cause voltages greater than V/3

h 1ti 1 hi h ' t f verse their direction of rotation if the timegsInornmal witrh one colnducto openlandaidisolated-neutral systems are subject to negative sequence torque is great enough I

ovevolageasa rsul offus blwin or to overcome the positive sequence torque Xco'/Xm must be greater than 40.0 or less thansingle-pole switching under permanent and provide the torque required to start 0 04fault conditions. the motors from rest. With a motor run- Im=3.5 per cent rms average exciting current

Obviously, there are othercombinations ning, the positive sequence torque is NOTE: Allowable length of line varies di-of faults and open conductors possible in greater than the negative for equal pOSl- rectly with rms average exciting currentisolated systems which might tend to shift tie .n ngatv squec votge.A NOTE: This criterion alone is not sufficient

somehatthe ang of onsantscauing stadstill these torques are equal for for elimination of overvoltages. The criterionhigh voltages. In distribution circuits equal voltages. of figure 12 must be met also. Maximum per-involving single-phase feeders, a single The conditions under which loaded missible miles of line in figure 12 should beblown fuse can sometimes cause over- motorswillreversetheirdirection of rota- divided by 4.0 for the isolated-neutral systemvoltages if the system constants are of tion, will depend upon the characteristicscertain values. Generally such cases can of the motors, their loads and the voltagesbeiterretd i ters o thee-has at their terminals, the latter being in- three-phase core- or shell-type) make little

constants such that some one of the fluenced by the presence of the motors difference in the magnitudes of voltage or ini1ddi his aper and other loads on the transformers. thentregrions in which hgh votge ars n

can be used directly in determining the Since reversal of the direction of rotationovervoltages possible. has occurred in practice following the 3 High overvoltages can be eliminated in

o 136KV-,, o'lgroude sstm by goning= thIeurl

The effect of load on the transformers is to opening of a conductor, it iS reasonable Of the load transformers.reduce the voltages on the open phases tassum that the higher the negative 4. Theregions withinwhichabnormal volt-obtained with the transformers unloaded, voltage produced on the open phase with ages are obtained, are considerably extendedWith the transformers loaded and phase the transformer unloaded, the more apt by the effects of transformer saturation.vi open, Va, the voltage of the open phase is reversal of thle direction of rotation of 5 The regions wherein voltages in solidlyon the transformer side of the opening motors to occur. grounded systems in excess of rated line-to-line voltages occur, and therefore of impor-referred to the voltage on the system tac in th 'plctono. ihnngarses

, . . tance the appllcalon of ll.h0ngarrZsters

side, will have a quadrature component Conclusions are for the usual cases confined to ratios ofin addition to the positive or negative Xci/Xm less than three for one conductorin-phase component. If the in-phase 1. In athree-phase powersystem, high sus- open, and less than six for two conductorscmoetis negative and greater than tamned voltages may in some cases result open.component ~~~from the opening of one or two conductors 6. Toaodtepssblt of obann un--0-5, Va2 will be greater than Vai and which separates ungrounded transformers duly high atbnnormSasl voltages tin soglidly

there will be phase reversal. Running from the rest of the system. ruddsses igepl wthsomotors, however, will not slow down If 2. The transformer connections (wye or fuses should not be used where the length ofthe difference between the positive- and delta) and construction (single-phase, or line, or equivalent capacitance, between the

1941, VOL. 60 Clarke, Peterson, Light-Abnormal Voltage Conditions 337

break and ungrounded transformers supply- Xab, xac, xc = per unit effective transformer The capacitive reactance of the three-ing the load is greater than that given by exciting reactances of windings ab, ac, bc phase shunt, Cl-C0, which is in parallelequation 1. Figure 12 can be used directly of delta-connected bank corresponding with xt is (3/2)x,0x,i/(x,0-xc1). The twowhen the rms exciting current is 3.5 per cent. to voltages across them parallel paths are in series with the capaci-7. The highest voltages appear in an un- Xa, Xb, x, =per unit effective transformer tance Co between phase a and ground. Ia isgrounded system with the simultaneous exciting reactances of windings a, b, c independent of Cl' and Co'.occurrence of an open conductor and line-to- of wye-connected bank corresponding toground fault, such as might correspond to voltages across them Ia -0.5/[-jxco+j3xxcjxco/0 3xclxco-the breaking and falling to ground of a line xcl, xco', xcl, xc0=per unit positive- and zero- 2xt(xco-xc) l (6>conductor. sequence system and circuit line capaci-tive reactances, respectively, correspond- 3xcoxc1-2x1 (xc0- Ci)8. From the standpoint of open conductors ing to C1', C0', Ci and C0 of figure lb Va = -XcoIa 2xt (x. +2xco) - (without faults, the criterion given in con- tXC1-

puincmclusion 6 is always conservative for un- All reactances are in per unit on a common With saturation neglected, xt= 3xm/2 ingrounded systems. However, if the criteria kilovolt-ampere base. Effective reactances (3), (4), or (5) and (7) becomesare to include the effects of permanent of delta-connected transformer windings,faults, then it is necessary to impose two as defined above, are expressed in per with Xco= Xci, Va= xcl/xm/(3 - 2xcl/Xm) (8)approximate requirements, namely, xcl/Xm unit on normal line-to-line voltage. Theymust be greater than 25.0, and Xco/xm must are to be multiplied by 3 to be expressed on with x,c = 2xcl, Va= (2xcl/xm-1)be either less than 0.04 or greater than 40.0. normal line-to-neutral voltage. (5-4Xc/Xm) (9)Figures 12 and 14 can be used when the Vb and V, can be resolved into two com-rms exciting current is 3.5 per cent. ponents of voltage in quadrature with each V0, calculated from (8), is the ordinate of

.. ~~~~other, - i/2and -ja'3. The former sends curves 3 of figures 3a and 3c; calculated9. The voltages arising from open con- current to ground through phases b and c in from (9), the ordinate of curves 3 of figuresductors can be calculated with reasonable parallel in series with phase . The latter 3b and 3d.accuracy by consideration only of the funda- produces equal and opposite currents in Effect of Transformer Saturation. (a).mental-frequency components of voltage phases b and c Since the circuit is sym- Delta-connected primary windings, delta- orand current as outlined in the appendix. metrical about phase a and resistance is wye-connected secondary windings. Re-

neglected, the voltages between neutral and placing xtin (7) by3Xdo/2, and dividingterminals b and c of a wye-connected trans- both numerator and denominator by 3x0o Xco,

Appendix former bank will be equal in magnitude. Va [(xci/xab) (Xco/Xci) - (XcolXci - 1) ITherefore the impedances Xb and xc are [- xclxcl) -x)equal. In a delta-connected bank, the [(1 +2xlo/x1) -(2xl/xab) (Xco/xcl) ]

Analytic Determination of the Sus- voltages between a and b and a and c will be (10)taned Voltage to Ground on Open equal in magnitude, and therefore xab and

-

V-=_ 1 (11 j.Conductors in a Circuit Supplying Xac are equal. The component - j\/3 will ab2 -/ 2an Unloaded Transformer Bank produce no voltage at terminal a. With

resistance neglected, Va can be positive or _ 1 V 3pUsing figurcs lb, the following additional negative but will have no quadrature com- - \ 2+ 0/4 per unit

assumptions will be made: ponent. In a wye-connected bank, the normal line-to-line voltage (11)neutral will likewise have no quadrature

1. The sustailned voltages to be determined component of voltage. The following procedure can be used toare substantially sinusoidal voltages of Le obanagahot0vessXiX,we hfudmna frqeny Let obtain a graph of Va versus X,,i/x., when the

fundamental frequency. Ja= total current flowing from phase a on ratio of x,o/x,l is known.2. The effective exciting reactance of the transformer side of the opening to 1. Calculate Va for an assumed ratio oftransformer windings to be used in calculat- ground xCl/xab from (10).ing fundamental-frequency voltages can be It= current flowing from transformer bankdetermined from the saturation curve of the terminal a 2. Calculate Vabl corresponding to Vatransformer, determined by applying sinu- xt = effective exciting reactance of the trans- from (11).soidal voltages. The exciting reactance former bank to It 3. From the given transformer saturationvaries with the magnitude of the impressed curve, calculate xab/xm, the ratio of the re-voltage, but at any voltage, it is the ratio With delta-connected primary windings actance at voltage Vabl to the reactance atof the rms voltage to the rms current. and secondary windings either wye- or delta- normal voltage.3. Resistance in the transformers as well as connected, 4. Xci/xrn corresponding to Va is the prod-in the transmission circuit can be neglected. xt = 3xa0/2 (3) uct of Xcl/xab and Xab/Xm.I. POWER SOURCE SOLIDLY GROUNDED With wye-connected primary and second- Illustration: Let x,0=2x,0. PerformingA-Phase a Open, No Fault ary windings, the four steps of the procedure given above,

' ~~~~~~~~~~~~~~~~~~indicatingeach by number:Referring to figure lb, switches SN, Sb, SC X= Xa+Xb/2 (4) indiAtinge chlbynumber:

are closed and Sa and SL open. No voltage With wye-connected primary windings (1) Assuming XXb= 1.42, from (10),iS applied to phase a of the vhrltpase V0r. -2.7three-phase cir- and delta-connected secondary windings,cuit supplying the transformer bank. The neglecting leakage reactance, the sum of the (2) From (11), Vab0 =voltages to ground, Vb and V&, applied to three voltages to neutral of the wye must be 1phases b and c in per unit of normal line-to- zeITo -=V/(-2.21)2+0.75=1.37 per unit of nor-neutral voltage, with the applied voltage of z/3phase a as reference vector, are .. V0v = 2 (real part of VNb) mal line-to-line voltageTr=-C/-jV/2; VC=-/2j V3/2 (2) If IA is the circulating current in the (3) From saturation curve 2 of figure 2,

Let ~~~~~~~~~delta,assumed opposite to It in phase a, Xab/Xm= rms voltage - 1.37/8.0=Va=voltage to ground of phase a on trans- X0(It-IA) =2X0(It/2+IA) and (4) xci/xm"'1.42XO.172=0.244former side of opening in per unit of nor- .~.IA=It(xa-Xl/(Xa+2Xb) The point, V0 =2.71, Xci/Xm=0.244, ismal line-to-neutral voltage plte incre2o iue3.Frn au

V =voltage to ground of neutral of wye- The voltage drop through the bank, from b raotedion, cub/rve=1 andfiue3bX 14.Frostheconnected transformer bank in per ulnit to a is ptoint V0/x=-.1, adXc /Xm = 1.42.iplTtediof normal line-to-neutral voltage pitV=27,XlX=14 spotdi

Xm=per unit effective transformer exciting j(It/2+IA)xbj(1t-1A)xa =j10[(9/2)Xaxb . curve 3of figure 3b.reactance of any winding at normal volt- (xa+2xb)] (b). Wye-connected primary and second-age, either wye or delta-connected .t.Xt= (9/2)XaXi/(Xa+2Xh) (5) ary windings. The current, It, through

338 Clarke, Peterson, Light-Abnormal Voltage Conditions ATEE TRANSACTIONS

the wye-connected bank produces voltage from the transformer saturation curve. Va not the case, the zero-sequence capacitancedrops between b and N, and N and a in the is calculated from (15). With V7a deter- of the power source, CO', in figure lb must beratio Xb/2 to xa. Replacing xt in (7) by mined, the corresponding value of xclXm can considered. Under the assumption of bal-xa+xb/2 and dividing numerator and de- be calculated from (7) if xt is replaced by its anced line-to-line voltages at a'b'c', thenominator by Xcl/xm, value from (5). positive-sequence capacitance, Cl', has no

effect upon the resulting voltages.Va B-Phases b and c Open, No fault The procedure used in case II is analogous(3Xci/xm) (x,o/Xl1)+ (2xa/xm+ * to that given for case I. The voltage which

/\'1 / 8 Refernng to figure lb. switches SN and SaXb/Xm) (1-XCO/cl) sends current to ground is 1.5 times normalare closed and Sb, Sa, and SL open. Case B line-to-neutral voltage applied in a loop cir-(2Xa/Xm+Xb/X,) (1 +2xco/xcl) - differs from case A, in that the only voltage cuit in which the ground can be considered a6(Xci/xr)(x,,/Ix,) applied to the circuit supplying the trans- snle point.(12) former bank is Va = 1; and 'a, the current g Pfrom phase a, flows to ground through Co of

The real part of phases b and c in parallel. Making these ReferencesVN IVIVNb 12-3/4 (13) two changes, with Va as reference vector,

2Vb = VC =Ia( -jxco/2) 1. THEORY oF ABNORMAL LINE-TO-NEUTRAL2 ______ VeI( i /)=TRANSFORMER VOLTAGES, C. W. LaPierre. AIEEVaN/(xa/xm) = ` / A V 12|3/4 3x,lxo - 2xt(x0cXC) (17) TRANSACTIONS, March 1931, page 328.

(14) 2xt(2xc1+x,O) -3xC1xC0 2. PHYSICAL NATURE OF NEUTRAL INSTABILITY,A. Boyajian and 0. P. McCarty. AIEE TRANS-

VaN = -real- VaN (a). Delta-Connected Primary Windsngs, ACTIONS, March 1931, page 317.Va = Vb- VNb- VaN= 2VV real- Va Delta- or Wye- Connected Secondary Wind- 3. INVERSION OF SYSTEM RESULTING FROM2 ings. ReplacingSxTin (N7) by 3xab/2, SWITCHINGeOPERATIONS, A. Boyajian and W. J.(15) ins Relcnx1i(1)b3a/2 Rudge. General Electric Review, July 1931, page436.A graph of Va versus Xcl/Xm for a given Vt, = VG 4. MAIHEMATICAI, ANALYSIS OF NONLINEAR CIR-

ratio of xc0 to xc1 can be obtained by the (Xci/XaD) (xCO/xC1)-(xCO/xC1-1) CUITS-PART 1, A. Boyajian. General Electric Re-ratio ofx* 0 to x_jcan_be obtained by the (X,iIXb) (X coIX ci)- (XcoIXe 1 1) (18) view, September 1931, page 531. Part II, Generalfollowing procedure: (2+xco/xxl) - (Xco/xcl (Xci/Xab) Electric Rcview, December 1931, page 752.1. Assume VNbi, the magnitude of the 5. ELEETRISCEE SCHALTVORGANGE (a book),voltage between transformer terminal b and Vba =- Va- Vb | - 1- Vb times R. Riidenberg. Third edition.neutral in per unit of normal. From the 6. Talk by F. A. Hamilton, Jr., at a meeting of thegvntransformer saturation curve, obtain (9 AIEE Toronto Section reported in Electrical News

givxn co pod t VNbl 'normal line-to-line voltage ( a9)nad Engineering, November 1, 1937.Usig (8)nd 19)intea of(10 an (1), 7. EXPERIENCES WITH GROUNDED-NiEUTRAL, Y-2. Calculate the real part of VNVb from (13). Uig (18) (1) (1) a (11), CONNECTBED POTENTIAL TRANSFORMERS ON UN-

respectively, the procedure for determining GROUNDED SYSTEMS, C. T. Weller. AIEE TRANS-3. Obtain the ratio VaN!(Xa/Xm) from a graph of Vbor V, versus xCl/xm for a given ACTIONS, March 1931, page 299.(14), and from the given tratisformer satu-(14),and from the given transformer satu- ratio xC0/x,j is analogous to that for one 8. LEITSATZE FUR DEN SCHUTZ ELEKTRISCHERration curve find values of VaNl and (xa . open conductor. ANLAGEN GEGEN UBERSPANNUNGEN, VDE, 0145/xm) which are in this ratio. The sign of VaN Wye-Connected Primary Windings. Sec- 1933, page 242.is the same as that of the real part of V_vb- ondary Windings (b) Wye-Connected, (c) 9. AN ELECTRIC CIRCUIT TRANSIENT ANALYZER,4. Calculate Va from (15). Delta-Connected. The procedure for obtain- H. A. Peterson. General Electric Review, Septem-

ing a graph of V0 or V, versus Xcil/Xm for a5. Knowing Va, calculate xci/xm from the known ratio, xco/x, is analogous to that 10. SUBDARMONICS IN CIRCUITS CONTAININGimpicieqatin (2),after replacing xco/xci sdfroeoe odutr xetteei IRON-CORED REACTORS, Irven Travis and C. N.by its given value. used for one open conductor, except there (S Weygandt. AIEE TRANSACTIONS, 1938 (August

no quadrature component of voltage in section).(c). Wye-connected primary and delta- VNb, and equation (17) is used instead of 11. ANALYSIS OF SERIES CAPACITOR APPLiCATIONconnected secondary windings. This case (7). PROBLEMS, J. W. Butler and C. Concordia. ELEC-differs from the preceding in that xt = (9/2) X TRICAL ENGINEERING (AIEE TRANSACTIONS), Au-XaXb/(Xa+2Xb) and II. POWER SOURCE UNGROUNDED gust 1937.

12. EQUIVALENT CIRCUITS OF TRANSFORMERS ANDVaN = 2 VNb real (16) If the capacitance to ground of the power REACTORS TO SWITCHIsNG SURGES, L. V. Bewley.

source with ungrounded neutral is large rela- AIEE TRANSACTIONS, volume 58, 1939, pages 797-The procedure to determine a graph of Va tive to that of the transmission circuit and 802.versus xil/xm is similar to that of case (b) bank, the power source can be considered to 13. PROTECTOR TUBES FOR POWER SYSTEMS, H.except that (16) is used to determine VaN, have its neutral effectively grounded. R. LUdWig. AIEJE TRAgSACTIONS, volume 59, 1940and xa/xm corresponding to VaeN is obtained Case II then becomes case I. When this is (May section).

1941, VOL. (;0 Glarke, Peterson, Light-Abnormal Voltage C'onditions 339