IEEE Transactions on Power Apparatus and Systems, Vol. PAS-100, No. 4, April 1981

A CALCULATION METHOD FOR VOLTAGE DISTRIBUTION IN A LARGE AIR CORE POWER REACTOR

M.M.A. SalamaElectrical Fngineering Dept.

Amn bhams UniversityCairo, Egypt

AbstractThis paper presents a calculation method

for the voltage distribution of a multiple-package air core power reactor. The nece-ssary equivalent circuit elements requiredfor the calculation of the impulse and thesteady state voltage response were develo-ped. The results of the experimental inves-tigation are in a good agreement with thecalculated impulse voltage distribution. Amore precise analysis of the steady statevoltage distribution is presented,which hasproduced satisfactory results, judging fromthe correspondence between the calculatedand the measured steady state voltage dist-ribution.

Introduct ion

In the late 1950's the main applicationof an air core inductance was as a current-limiting reactor. The standard method ofconstructions used by all major electricalequipment manufactures, comprised a cablewinding supported in Cast concrete form. Inthe early 1960's, a break-through in coilmanufacturing was achieved by innovating theuse of new materials, such as continuousfilament glass fibers iman epoxy resin mat-rix to provide the physical strength of theair core reactor. The cable winding wasreplaced by aluminum conductor insulated bypolyester fiim. MWith the increasing demandof high rating power reactor, the windingwas divided to a large number of parallelconcentric layers of conductors. The conceptof fractional turns in each layer was intro-duced to balance the current flow. Two tofive layers are completely encapsulated inthe epoxy-fiberglass composite material, toform a winding package. Each package isseparated from adjacent ones by axial coo-ling ducts. This resulted in further impro-vments in strength, and protection fromcontamination (Fig. 1).

(1) Current Limiting Reactors, are used toreduce the short-circuit current to avalue within the rating of the equip-ment after the reactor.

80 SM 625-4 A paper recommended aiid approved by theTEEE Transformers Committee of the IEEE Power En-gineering Society for presentation at the IEEE PESSummer Me.ting, Minneapolis, Minnesota, July 13-18,1980. Manuscript submitted January 21, 1980; madeavailable for printing May 19, 1980.

Fig.

(2)

1. Typical Air Core Power Reactor.

Filter Reactors, are used, in serieswith a shunt capacitor bank,to providea tuned series resonant LC circuit forspecific harmonic currents.

(3) Neutral Grounding Reactors, limit theline-to-ground fault current to thesame value as the line-to-line faultcurrent.

(4) Capacitor Reactors, are designed to beconnected in series with a shunt conne-cted capacitor bank either to limitinrush current on energization, or tocontrol the resonant frequency of thesystem due to addition of the shuntcapacitance.

(5) Smoothing Reactors, are used to reducethe magnitude of the ripple current ina direct current systea. They areoften required in HVDC schemes,and maybe required in series with rectifier-fed large DC motors used in the steel,mining and process industries.

(6) Shunt Reactors, are used to compensatefor capacitive VARs generated by longlightly loaded transmission lines .Other applications are as VAR compensa-tors, where the reactive VARs are con-tinuously controlled by thwristors.

(7) Duplex Reactors, provide a low reacta-nce under normal conditions and high

©C 1981 IEEE

1752

reactance under fault conditions.Thesereactors can be applied to any systemthat has two separate feeder circuitsthat will remain isolated under allcircumstances*.

(8) Motor Startiag Reactors, limits thestarting current of AC motors.

(9) load Balancing Ieactors,serve to cont-rol the curtent flowing in two or moreparallel circuits.

The distribtition of the impulse voltagein a multiple-package air core power reactorhas an important effect on the design ofhigh voltage reactors. Information on cal-culations on multiple-package reactor surgeperformance is sparse. Most of the work(2,3),that has been reported in literature onsurge performance of power reactors arebased On the experimental studies. Air coremultiple-paokage power reactors of a 4.4 ldVArating, have been successfully tested (4)for AEP at 550 KV BIL (Basic Impulse Level),reactors of oneor two packages have alreadybeen successfully tested at a voltage levelOf 1050 KV BII.

With the very large power reactor nowbeing built, an accurate assessment of thevoltage stress likely to occur under surgeconditions is essential in order that insu-lation dimensions may be kept to a minimum.The expensive test procedure can also beminimized if a more appropriate method ofcalculating the voltage distribution isused in the power reactor design.

A typical winding constructions used inmultiple-package air core reactor is shownin Fig. 1. The equivalent ladder networkfor the reactor winding is shown in Eig. 2.

I

P

1

T

L I I

-HZZHF

_R ~~- - -

-'~-I PKGn

HLII-

I

rz566L PKG1

_-HF-II

1753This network is composed of lumped constantssuch as self and niutual inductances. Seriescapacitances and capacitance to earth (r5).

The actual response of the winding to theapplied voltage is a transient phenomena,but the electrostatic treatment of the pro-blem is a valid approximation (5-). There-fore, the actual winding can be replaced bya system of capacitances which has the samedielectric properties under the conditionof the initial voltage distribution (Fig.3).

Fig. 3.-Reactor Package Equivalent Circuit,for the Transient Condition.

Figure 4a, shows a cross sectional areaof a typical package of a large power reac-tor. This package consists of many layers,to increase the current rating of the reac-tor. To insure the inductance equilibriumof the reactor, each layer has differentnumber of turns (l)* In the following ana-lysis it is assumed that, for all layers ofa reactor package, turns of equal voltagelevels, are physically placed at the sameheight from the ground, Fig. 4. This assu-mption is practically acceptable, becausethe conductor diameter of the multiple-pac-kage air core reactor winding is small.Figure 4b shows the approximated representa-tion of a package winding. The set of turnswhich has the same voltage level and placedat the same height fromthe ground is calleda package element.

V3 ___

Fig. 4. A Package Winding Representation.

11140/,,-f 7z Z/ /-Z// Z7Fig. 2. Equivalent Circuit for n-PEG Air

Core Reactor.

{}e-A

I

-.III I

T

1754Experimental studies (2), have revealed

that the outer package surge perIformance isthe most predominant design factor. Thispakage suffersfrom a relatively large groundcapacitance coupling. In the following ana-1ysis, only the outer package surge perfor-mance was considered. At the impulse condi-tion, the outer package can be replaced by asystem of capacitances. This system consi-sts of the lumped- ground capacitance fromeach outer package element to ground,and ofthe lumped series capacitance between theends of the adjacent outer package elements,Fig. 3. Let a surge voltage B1 be appliedat the reactor end P and the other end F isgrounded. Tbhe voltage E to ground at anydistance x from the tip of the reactor isgiven by (6)

E - a(l - (1)

whereL the outer package length.N the number of the package elementsC5 the series capacitanceCg the ground capacitance

a gN(C /C

Calculation of the Series Capacitance CsThe series capacitance of the outer pack-

age elements is mainly determined by thecapacitance between the winding conductorsof the two adjacent outer package elements.C'S can be calculated easily from the modelof the parallel plate electrodes.

wheree the dielectric constant of the insu-

lating -material.A the effective cross sectional area

of the package element.t the thickness of the insulating med-

ium.'

The effective cross sectional area A isgiven by

2_

2A = rr(r - rI) (3)where

ro the outer radius of the outer pack-age

r the inner radius of the outer pack-age.

The thickness of the insulating medium t,physically represents double the thicknessof the polyester film that wrapped the win-ding conductor.

Calculation of the Ground Capacitance Cg

The ground capacitance of each packageelement is assumed to be equivalent to thecapacitance of a horizontal toroid havingthe same height from ground as this packageelement. The capacitance of the equivalenttoroid is given by (7)

g p

whereC the total capacitance to groundC66 the capacitance of the toroid well

above groundCp the additional capacitance due to

the proximity of groundCp = nI 6 D (D + in (1 + t)p h (4)

= 2 rr C (2 3± S) (5)Therefore, the ground capacitance of a pa-chage element is given byCg = nC0 iD( + in (1+ E)) + 2 ( ,3where (6)

S the diameter of the winding conduc-tor,

D the diameter of the reactor outerpackage,

h the height of the package elementabove ground.

ImAulse Voltage Distribution Results

A test reactor was built for the purposeof comparing the experimental finding withte calculated impulse voltage distribution.The proposed method was used in this calcu-lation. Figure 5 shows the impulse voltagedistribution alongthe test reactor surface,obtained experimentally. The test reactorhas an outer package diameter of 2.31 m(91 in), reactor length of 1.14 m (45 in)and it was raised above ground a distance of2.18 m (86 in). The diameter of the windingconductor was 2.54 x 10-3 m(0.1 in) and thethickness of the wrapped mylar was 5.08 x105 m (0.002 in). The applied impulsevol-tage was the standard full wave 1.5/40. Thepeak applied voltage was 550 kV.The voltagewas then calculated using the proposed meth-od of analysis, the result i shown in Fig.6. The calculated -voltage distribution ag-rees reasonably well with the measured val-ues, Fig. 6.

3.5 u.s

% age of the winding lengthFig. 5. The Measured Impulse Voltage Dist-

ribution Along the Winding Length.Vact = Actual Measured Voltage.

-Vapp = Applied Voltage.

1755

0.

Cuco

C-)-Cu

o

. MeasuredaCalculated

0 10 20 30% age of the winding length

Fig.6. Impulse Voltage Distribution Alongthe Winding Length at T = 0.5 .a sec.

The proposed method is valied, only, inthe first few u sec twhere the electrostaticeffect is predominant. In practice the firstfew u sec are the most important time dura-tion. Figure 7 shows the measured electri-cal stress distributions along the reactorsurface, at different time intervals. Theseverest electrical stress occurs arroundthe first vu sec. (2, 5).

0.6 T=O.5A-C

\0.4v) 0.2 ¢ Tw2T-1.5us

0 2 4 6Axial winding length (-in)

Fig. 7. The Measured Electric Stress Dist-ribution Along the Reactor AxialLength.

The other parameter that may affect theaccuracy of the calculation is the reactorheight above ground. If the reactor is placedvery close to the ground plane, Cg will bedifferent for each package element (due toproximity effect), and expression (1) is nolonger hold for this particular case. Fbr-tunately, this situation is rarely happendin practice. To allow a reasonable electricand magnetic clearances, the power reactors

are usually placed at a suitable heightabove ground.

Steady State Vo ltagDiributionThe steady state voltage distribution of

the multiple-package air core power reactor,determines the design limitation. Experi-mental studies, (4) have shown that the safesteady state electrical stress is in orderof 0.3 kV/cm (750 Volt/inch) along the win-ding. Also, in the multiple-package reac-tor, the voltage across each package has tobe accurately the same. Any slight unbala-nce in the package voltages will produceheavy circulating currents in the winding.The effect of these currents is to increaseboth the intensity and the number of the hotspots on the reactor surface. This may dama-ge the insulation of the reactor winding.The failure of the winding insulation maylead to turn-to-turn reactor failure.

In the followinS section,a more accuratecalculation method for the steady state vol-tage distribution is presented. The deve-lopment of this method was conducted throughthe following steps:

(1) To determine a precise multiple-package equivalent network composed of suchsubdivided elements as one turn coil(Fig.8).

14 L,M,M

Fig. 8. Equivalent Circuit for theMultiplePackage Air Core Reactor,at SteadyState Condition.

(2) To determine the self and mutualinductances very accurately. Milutual induct-ance of element i is due to the elements ofall packages except element i. The methodproposed by Fawzi et al (8) for the calcu-lation of self and mutual inductance of coa-xial circular coils, has been used in thisanalysis. This method is exteremely effi-cient and produces very accurate results.

(3) To determine the optimal number ofmutual inductances to be calculated. InFig.9 the multiple-package air core reactoris represented by a matrix form. Each rea-ctor package is subdivided to N elements.The total inductance of the element is wri-tten as follow

DIV I

DIV 2

DIV N

PKG.1PKG.2

the multiple-package air core reactor. Atest reactor was built and tested at TrenchElectric Ltd power laboratory, ¶LTronto,Canada. The test data along with the calc-ulated results are shown in lig. 10. It isevident from Fig. 10, that the proposedmethod , calculates very accurately thesteady state voltage distribution of the aircore power reactor.

PKG. KN

Fig. 9. Mutual and Self Inductances for KN-PKG Air Core Reactor.

N

-k,k' Em=lmi

KN

Xi,m I

n=1MAK

N

m=l(7)

. MeasuredACalculated

%age of the wind ing I engthIK K the self inductance of the elementMi the mutual inductance between the

im element under consideration andother element at the same package.

Mm the mutual inductance between the' element under consideration at

package K and the elements of allother package except the package K

KN the total number of the packagesin the power reactor.

(4) To determine the package voltage VK,N

VK =jw

m=l

EN

n=ln K

IK (,E +

N

i=li,xm

m,i)

N

in ( Im,I3=1

(8)

and the element voltage en of the package K

Nen = jw 'K (1'k, + M

i=l

i,m

KN N

+ IX 'n "m3 (9)n_1 j=lnoK

(5) To varify the accuracy of the methodby comparying the calculated and the measu-red steady state voltage distributions of

Fig. 10. Steady State Voltage Distributionfor a Mult ip le Package Air CoreReactor.

ConclusionResearch work on the air core power rea-

ctor is very limited in the literature. Amethod for the calculation of the voltagedistribution of the multiple-package aircore reactor is developed.

When seeking the transient voltage of theair core reactor, the equivalent network ofthe multiple-package reactor winding wasreplaced by a system of capacitances. Amethod for calculating these capacitanceswas proposed. The reactor surge voltage wasthen calculated. The calculated and themeasured results are in good agreement.

The steady state voltage distribution wascalculated, based on the assumption thatonly the self and the mutual inductances ofthe winding were considered. The validityof this assumption was confirmed by theexcellent agreement between the calculatedand the measured steady state voltage dist-ribut ion.

References(1) Dry-Type

100-05,Toronto,

Air Core Reactors, Bulletin1978, Trench Electric Ltd,Canada.

(2) M.M. A. Salama and R. F. Dudly, "Experi-mental Data on Impulse Voltage Testfor Trench Power Coil"l, Research andDevelopment Report 56 19?7, TrenchElectric Ltd, Toronto, 6anada.

1756

I,3

where

(3) R1.F. ]Xdly, "Current Distribution,Imp-ulse Voltage Studies and New Ak?plica-tionas, National Research Council,Pro-ject-Induotance 158-1973, Toronto, Can-ada.

(4) Dry-Type Air Core Reactors, BulletinT100-35-02, 1977, Trench Electric Ltd,Toronto, Canada.

(5) K. Okuyama, "A Numerical Analysis ofImpulse Voltage Distribution in Trans-former Windings', Elect. Eng. Japan,Vol. 87, No. 1, 1967, pp. 80-88.

(6) H.H. Skilling, "Electrical EngineeringCircuits", Book, John Wiley and Sons,1967.

(7) P.M. Maruvada and N. Hylten-Cavallius,"Capacitance Calculations for SomeBasic High Voltage Electrode Configura-tionsl", I Trans., Vol.PAS-94, No.5,1975, pp. 1708-1718.

(8) T.H.Fawzi and P.E.Burke, "The AccurateComputation of Self and Mutual Inducta-nces of Circular Coils" IEEE Trans.9Vol. PAS-97, No.2, 1978, pp. 464-468.

DiscussionR. F. Dudley and G. Gela (Trench Electric Limited, Scarborough,Ontario, Canada): Compliments should be extended to the author on aconcise but positive contribution to the study of multi-package air-corepower reactors. The author was at Trench Electric on a NRC post-doctoral fellowship from 1977-1978. Since the author's departure fromTrench Electric Ltd., discussors have advanced the understanding ofthe topics addressed in the paper, and they would like to share their ex-perience with him.The paper deals with two broad topics, namely, transient response of

a reactor winding to fast (lightning) impulses, and the steady-state(power frequency) behaviour of the same winding. The following com-ments will also, for the main part, address these topics seperately.

In the analytical study of impulse voltage distribution, the authorconsiders only the last (outer) package of the multi-package reactor.Such an approach is justified in the paper by the assumption that "Thispackage suffers from a relatively large ground capacitance coupling".Since the coil is usually mounted vertically, ground capacitance willvary along coil length, as pointed out by the author. At the same time,however, the author maintains that such variation of groundcapacitance is usually inconsequential, since the coil is placed at a largeheight above ground. Hence, the discussors would like to inquire regar-ding the importance and effects of ground capacitance and of its varia-tion upon voltage profile along coil length. The question may be ex-plored further by identifying several types of capacitances presentwithin the reactor. Considering a single turn in a layer of winding withina package, one may, in simplified terms, identify the followingcapacitances:

a) capacitance of the turn in question with another turn of the samelayer, Ctt.

b) capacitance of the turn in question with a turn of another layer inthe same package CI.

c) capacitance of the turn in question with a turn in anotherpackage, Ce,.

d) capacitance of the turn in question with ground, C,.There is no doubt that C, is of greater significance in case of the outer

package than the inner ones, since the latter are "shielded" by theformer.

However, the discussors must point out that the only difference bet-ween the first and last packages are turn length and (usually) wirediameter. As a result, two questions come to mind. Firstly, based on hisstudies, can the author suggest some physical reasons for the apparentexperimental evidence referred to by him that the investigation of theimpulse performances of the outer package only is sufficient for design

1757

of the reactor? Secondly, can the author's computational method beemployed to shed light on relative roles played by the capacitances iden-tified earlier in shaping the impulse voltage profile of the coil? In par-ticular, recent investigations conducted by the discussors and aimed atthe development of a high BIL (1.6MV and up) and of series-woundcoils indicate that understanding of the various capacitances listed is in-valuable in proper design of each reactor. Furthermore, as individuallayers of a package may be constructed with different numbers of turns,there exists a possibility of large layer-to-layer stresses (determinedmainly by C,,) which would not be apparent from author's results. Asimilar situation may arise among packages. In addition, would adistributed-parameter model, or at least one incorporating inductancesand resistances as well as the capacitances calculated by the author, berequired to reproduce winding behaviour on the impulse wave tail?Author's comments on these points would be welcomed.The second topic addressed in the paper is that of steady-state

behaviour of the reactor at power frequencies. The first and most ob-vious comment regarding this work is that resistances have beenneglected in Fig. 8 of the paper. While such simplified approach may besufficient for determination of steady-state voltage profiles, theroeticalconsiderations and abundant experimental evidence clearly indicate thatresistances must be included in computations of current subdivisionamong layers and packages of the reactor. Conversely, would omissionof resistances account fully for the well-behaved, very small deviations,between calculated and measured results presented in Fig. 10? Also, hasthe author experienced numerical difficulties in performing the calcula-tions?

Lastly, the discussors wish to point out that, after all, it is one and thesame coil that has been represented by the author through twodissimilar models, each suited for a different application. In an attemptto comply with increasing demand for thorough understanding of high-frequency behaviour of multi-package reactors, the discussors have em-barked upon a program to consolidate the two models representing op-posing extremes of the frequency spectrum. Work on a travelling-waveapproach is also in progress. The author's thoughts on this topic will beappreciated for further fruitful discussions.

Manuscript received August 12, 1980.

M. M. A. Salama: The author appreciates the discussion and the com-ments of the discussors.For the analysis of the transient phenomena in the multiple package

reactor, the suitability of the method for calculating the initial impulsevoltage distribution is judged by its capability for representing the win-ding constants to give sufficient results [1,2]. Different winding con-stants may be proposed for each calculation method. Such method isconsidered appropriate for calculating the impulse voltage distributionas long as it gives good results. In that sense the discussors inquiryabout the effect of varying C, upon the calculated voltage profile alongthe coil length (using the author model), shows a misunderstanding ofthe philosophy of the proposed technique. In the paper, the equivalentcircuit of the reactor winding has been solved by using expression (1).The author indicates very clearly on page 4, that expression (1) nolonger hold for the case of varying C,. On the other hand, the discussorsproposal for introducing several type of capacitances is a prematuresuggestion, for the following reasons:

First, if they are going to consider these capacitances along with thecapacitances in the author's model, they should have explained,

(1) How these capacitances are going to be connected to theequivalent circuit of the reactor winding.

(2) What complication of the analysis is required.(3) What improvement in the initial voltage distribution is expected

after adding all these sophistications to the model.Second, if they are proposing a new method of calculation, one

would like to know.(I) The reactor winding equivalent circuit with all these capacitances

in it.(2) The methods for calculating the values of their capacitances (C,,,

Ci., C, and C,). This should be the core of their work, for anyvaluable research in this field.

(3) The computer storage and execution time, since they are propos-ing turn-to-turn calculation (number of turns in a single packageis 200). For multiple package reactor of five packages, the size ofthe problem is 1000 package elements, i.e., 8000 different capaci-tive elements).

1758

(4) The expected improvement in both the calculated initial voltagedistribution and the calculated stresses between layers.

The author does not agree with the discussors that the only differencebetween the first and last packages are turn length and wire diameter.The fundamental difference is the package diameter. Bearing this inmind, the experimental evidence (that shows the impulse performanceof the outer package only is sufficient for the design of the reactor) isself explanatory. Consider the case of a multiple package reactor thathas the outer package diameter of 2.31 m and the inner packagediameter of 1.0 m, the other design parameters are the same as in thepaper. Therefore the ground capacitance of the outer package,Ca = 96.4 PF and that of the inner package, C, = 39.3 PF. Let a surgevoltage E,, be applied at the reactor tip, and the other end is grounded.The voltage E to ground at distance 0.1 L from the tip of the reactor isas follows (using expression (1)),E/E, = 0.139 for the outer packageE/E, = 0.283 for the inner package

This means that, the upper 10% of the outer package is subjected tomuch more voltage (about 14.4% E,) than the upper 10% of the innerpackage. In general, expression (1) shows that the outer package surgeperformance is highly non-linear. The high ground capacitance coupl-ing (as shown in the paper) is the physical reason for the experimentalevidence.

Concerning the second inquiry of the discussors about the steadystate behaviour of the reactor, the author wishes to indicate that thetype of reactors (T. E.) discussed in the paper are very low power lossreactors (typical ratio between the inductive current component to theresistive current component is about 200). Therefore the error in thecalculation, due to the omission of resistance is very small and in many

cases much less than 1%. The author can see no reason for thediscussors to worry about any numerical difficulties in performing thecalculation. It is well-conditioned problem and similar inductive circuitbehaviour has been studied by many researchers [3,4] in the areas ofpower transformer and electric power systems, without experienced anynumerical difficulties.The author wishes to encourage the discussors to continue the

research work, they have started after his departure from T.E., for thebenefit of more understanding of the air core reactor transientresponse.

REFERENCES

(1) P. I. Fergestad and T. Henriksen, "Transient oscillation in multi-winding transformer winding", IEEE Trans., Vol. PAS-93, pp.500-509, 1974.

(2) A. Miki, T. Hosoya and K. Okuyama, "A calculation method forimpulse voltage distribution and transferred voltage in transformerwindings", IEEE Trans., Vol. PAS-97, pp. 930-939, 1978

(3) P. I. Fergestad and T. Henriksen, "Inductance for the calculationof transient oscillations in transformers", IEEE Trans., Vol.PAS-93, pp. 510-517, 1974.

(4) S. R. Sagardia and P. D. Smith, "A polarization current approachto the calculation of inductances and the solution of the field pro-blems involving magnetic materials", IEEE Trans., Vol. PAS-97,pp. 1402-1410, 1978.

Manuscript received November 18, 1980.