05055943.pdf

Calculation of Synchronous Machine Constants-Reactances and Time Constants Affecting Transient

CharacteristicsBY L. A. KILGORE*

Associate, A. I. E. E.

Synopsis.-Recent advances in the theory of synchronous lation is discussed in which the reactances are accurately resolvedmachines have introduced a large number of new constants. The into components. Comparisons of test and calculated values aremethod of symmetrical components requires sequence reactances, given. The formulas for salient-pole machines and turbineand an accurate theory of transients req-uires transient and sub- generators are given in Appendixes A and B.transient reactances and time constants. Most of the published The principle of superposition is applied to resolve accuratelydiscussion on the constants has been concerned with the application, the reactances into components which can be readily calculated.rather than the calculation of values. The induced currents in the -field and additional damping circuits

In this paper, all of the most significant constants are calculated, are accounted for simply by applying the constant interlinkageexcept the subtransient time constant. A general method of calcu- theorem.

I. INTRODUCTION component (reacting on the axis between poles).UNTIL recently, the theory of synchronous ma- Where the constants are quite different in the two

chines has required relatively few constants. A axes, this method requires quadrature-axis constants:single value of reactance, (usually called armature (Xq, x" and in some cases xq', Tq' and Tq") as well

leakage) was used to calculate the initial short-circuit as the corresponding direct-axis constants (Xd, Xd",currents, and standard decrement curves used to Xd', Td', and Td"), described above.determine the decay. Recent advances in theory have These new methods consist essentially in resolvingintroduced a large number of new constants. the actual currents and voltages into their essentialThe method of symmetrical components' applied to components, and defining a sufficient number of con-

stants to determine the initial and final values and therates of decay. These constants have all been definedsequence reactance (X2) and zero-sequence reactance (x0),in other papers;*"", however, since the general

as well as the positive-sequence reactances, which method of defining the reactances is vital to the methoddepend on the condition of application of current. of calculation, it is well to state it here. The reactanceAn accurate theory of transients6 recognizes a sub- of a synchronous machine is the ratio of fundamentaltransient (rapidly decaying component due to currents reactive armature voltage produced by the giveninduced in additional damping circuits), as well as a component of armature current applied to the valuetransient component of symmetrical armature current. of that component of current at rated frequency, andThis requires several (positive-sequence) reactances, with the rotor running at synchronous speed.the subtransient reactance (Xd"), the transient reactance These theories have been developed on the basis of(Xd') and a synchronous reactance (Xd). The rate of no saturation, or at least assuming fixed permeability,decay of the transient components of current is deter- but in practical calculations the effects of saturationmined by time constants. The time constant of the cannot be neglected. These effects are most readilyasymmetrical component is termed the armature time dealt with in most cases by modifying the constants.constant (Ta). The time constant of the rapidly decay- The effects of saturation on each of the constants ising component of symmetrical current is the sub- discussed in the companion paper'. It is shown thattransient time constant (Td'), and the time constant for for most practical purposes, only one value of eachthe main transient component of current is the transient constant is necessary for a machine. Except for thetime constant (Td'). synchronous reactances and open-circuit time con-As a reference, it is desirable to use the time constant stant, the final values calculated will be the "saturated

of the field, which is called the open-circuit transient values," corresponding to the currents obtained ontime constant (TMo') sudden short-circuit from full voltage.A most complete solutionM'7 may require resolution Most of the published discussion of these constants

of currents into a direct-axis component (reacting has been concerned with the application, rather thandirectly on the main field axis,), and quadrature-axis the calculation of the values. However, all designers

*Power Engineer, Westinghouse Elee. and lMfg. Co., East *Proposed definitions of these constants are included in aPittsburgh, Pa. report of the Transmission and Distrib-ution Committee of the

9. For references see Bibliography. A. I. E. E. This report, which was prepared by Prof. G. Dahi,Presented at the Summer Convention of the A. I. E. E., Asheville, will be presented at the Winter Convention, New York, N. Y.,

N. C., June 22-26, 1931. Jan. 25-29, 1932.

1201

31-105

1202 KILGORE Transactions A. I. E. E.

of large synchronous machines have had to meet this tion might be stated as follows: The armature leakageproblem of calculating the constants, and those who reactance is the reactance due to the difference betweenuse them are interested in how well they can be cal- the total flux produced by the armature current actingculated. alone, and the space fundamental of the "flux in theA recent paper8 treats armature leakage reactance air gap." To make this definition as useful as possible,

quite thoroughly, but does not cover the other con- the flux in the air gap will be understood to include thestants. Another4 describes an accurate method of fringing flux at the ends which enters the rotor, sincetreating synchronous motor constants by constructing this is very nearly the component -of the end leakagea complete equivalent circuit. In this paper, formulas field which is mutual with the rotor circuits. Forare developed for calculating the following constants: turbine generators with magnetic retaining rings, theXd, Xq, Xd', Xd", Xq ,x2,X2o Tdo' Td' and Ta. This end zone flux entering the retaining ring surface shouldincludes all of the more important constants, except the be included in the end leakage, and not in the air gapsubtransient time constant (Ta"wTd") which can be esti- flux.mated from the test values given in the companion Defined in this way, the armature leakage can bepaper.' The quadrature transient reactance (xq') is shown to be very accurately a component of all thenot included, since it is quite commonly assumed equal positive-sequence reactances. This may be seen fromto Xd', in short-circuit calculations. the fact that the fundamental flux in the air gap is

II. GENERAL METHOD OF CALCULATION determined almost entirely by the space fundamentalof the m. m. f., and that the remaining componentTo calculate any of the reactances as defined, it 15S osssol nso n n ekg,adhroi

only necessary to determine the flux produced by the flxs intgph areno ape affectedobycomponent of current applied in the given manner, and v . vto calculate the resulting armature voltage. The the condition of application of current. The effectsresiiltant field of flux may be complicated, but by of saturation on the armature leakage are generally

6.resultant negligible, except for the increased end leakage due toapplying the principle of superposition,6 theeresultant . . rb.fields~~~~ca bersle.nocmpnnswihcnb magnetic retaining rings in turbine generators.Form of the Results and Formulas. The most usefulmore readily calculated.Componentsai ealcuatane. It is convenient to divide. forms for expressing the reactance coefficients are per

cent and per unit, on a machine kva. base. In thisall the positive-sequence reactances into two com-ponents, one of which is the armature leakage (x1) paper, per cent notation is used, since this is the formdefined in a definite manner. The synchronous re- most generally adopted when specifying the values ofactance is then the sum of the armature leakage and reactances. The time constants are given in seconds.the reactance of annature reaction: Xd = X I + Xad

The reactance formulas must, of course, involvefor the direct-axis, andxr =xu +Xae for the quadra- either directly or indirectly the armature turns, theture-axis. The reactance of armature reaction (Xad) frequency, and certain dimensions of the machine. Theis defined as the reactance due to the space d actual form of the equation makes considerable differ-fud-ence in the usefulness. Some writers have used themental of air gap flux produced by the armature current cncepin ofacting alone concepton of effective permeance per unit length of

TeuautdrnX, is cslot, and others the permeance per phase belt per unitThe unsaturated transient reactance (xdU') iS calculated oXteu oterarlkgaiof length. In extending these conceptions to theas the sum of the armature leakage and an effective calculation of all the reactances, it was found desirable

field leakage"(rF')p (XF) is not the true leakage of to change the conception to a permeance per pole perthe field with respect to the armature a ite inch of length, since the flux across the gap is mostcomponent of the total leakage of the armature with conveniently visualized as a flux per pole.respect to the field. The saturated valueofmthe theran- All the component reactances may be calculated assient reactance (Xd')is calculated by multiplying the a product of a reactance factor (X), and a specificunsaturatednvale from by an empi onstant permeance (X) for the given component, multiplied by

(Fas), determined from test in the companion paper1 flux distribution coefficients when necessary.as0.88. The specific permeance (X) is defined as the effectiveThe subtransient reactance is calculated as the sum flux per pole per inch of core length produced by unit

of the armature leakage and the "effective" damper ampereturnsperpole.winding leakage: Xd"' = xl + X'Dd, and Xq" = xl + XDq' The reactance factor (X) used here is a group of termsXDd' and XDq'X are components of the total leakage of common to all the reactance formulas, but it is chosenthe armature with respect to the damper winding deter- in such a way as to have a definite physical significance.mined by assuming no net change of interlinkage with It is the per cent reactance for unit specific permeance,any of the damping circuits. or the per cent of normal voltage induced by a funda-Armature Leakage Reactance Defined. Several recent mental flux per pole per inch numerically equal to the

writers8 have treated armature leakage as an arbitrary fundamental armature ampere turns at rated current.component of the synchronous reactance. The defini- The reactance factor can be written in several ways.

December 1931 CALCULATION OF SYNCHRONOUS MACHINE CONSTANTS 1203

In terms of the turns, frequency and rated phase voltage It is shown' that using the saturated value of Xd' inand current, this factor can be shown to be: this relation gives a good approximation to the effective

X=100%~~Iph 2~m ~ saturated value of Td'.X = 100%( E )f ImP q C kp kd 10- (1) The armature time constant (Ta) is the time constantpE~h LX

of the asymmetrical component of armature currentThe factor can be expressed in terms of the ratio of which is due to the flux trapped by the armature wind-

effective ampere conductors per inch (A kp) to the ing at the instant of short circuit. The effective re-fundamental air gap density (B1): actance to the average asymmetrical current is the same

(Ak kd kd \{Akp\ as for negative-sequence current (x2), and the resistanceX = 100% '\V2 B ) = V2 Ci Bg) (2) is the d-c. armature resistance (ra,) in per cent. Hence,

This is very useful if the effective ampere conductors(A kp) and air gap density (Bg) are thought of as Ta - 2fundamental quantities in design, the ampere conduc- 2 rf rators per inch being a measure of current loading, andthe air gap density a measure of the magnetic loadingof the machine. III. COMPARISON OF TEST AND CALCULATED VALUESThe chief advantages of this method are that it ex- The formulas of Appendixes A and B have been

presses the reactances as products of two factors, a checked by the numerous tests described in the com-specific permeance which involves only the proportions panion paper.' Tables showing the tests and cal-of the machine, and a reactance factor which is a simple culated values are given below thetestan calfunction of the fundamental quantities of design. values were obtained from special sudden short-circuitThis makes it possible to see at a glance the effect of lues were obtindifrom ia sde shor-ici. or locked tests, as indicated in the tables. The syn-changing any of the proportions or factors. Also, it chronous reactance (Xd) can be obtained from the com-simplifies the work of calculation.Camplculastiewon kofTmcalculaonst. Teonc ui mercial tests, hence a great many more checks on theCalculation of Time Constants. The open-circuittime constant of the field may be calculated as the self synchronous constants have been obtained than thoseinductance of the field (in henrys) divided by the field listed below. Also certain other special tests' were

resistanceinof thms:field (In henrys) dlv by the field made to determine the effects of saturation, but willresistance in ohms:nobeicudhr.not be included here.

Tdo' = Lf Table I gives the results of sudden short-circuit testsRf on salient-pole machines. Table II shows the results

The short-circuit transient ti"me constant is calculated of locked tests obtained on other salient-pole machineswith dampers, and Table III shows the results ofsimilar tests on machines without dampers. Table IV

X d= shows the results obtained from sudden short-circuitTd Xd T do'* tests on turbine generators.

TABLE I-CONSTANTS OF SALIENT-POLE MACHINES; COMPARISON OF TEST AND CALCULATED VALUESData from Three-Phase Short-Circuit Tests

All x's Are Per Cent Values All T's Are in Seconds

Kva, 5,000 100 750 30,000 20,000 15,000 331 240 % error

Kind......... Gen. Gen. No damper Cond. Cond. Cond Motor MotorH. W. G. Max. Avg

Frequency.... 60 60 25 60 60 60 60 60Volts . ...... . 7,600 2,300 6,600 5,000 11,000 11,500 440 2,200R. p.m....... 900 1,200 187.5 720 720 900 1,200 277

xd Test. 90 .... 125 ... 120 .. 150 .. 160 .. 165 .. 160 .. 105 6.8........ 3.6Calc.... 90 .... 130 ... 115 .. 160 .. 170 .. 170 .. 160 .. 110 ......

xd' Test.... 22 .... 24 1 37 45 34 23 .. 42 12.5........ 6.0Calc.... 21.5 .... 27 ... 39 .. 34 .. 40 .. 33 .. 23 .. 40

xd"' Test 13 .... .16.5 ... .. .19 .. 27 .. 21 .. 16 .. 32 . .. .11.0 . ....8.0Calc..14 .... . 15.0 ... 40 .. 21 .. 30 .. 19.5 .. 15 .. 30 ...

X2 Test13..... 1 .... 17.0 ... 62 .. 15 .. 26 .. 20 .. 16 .. 33 ...8.0 . ....5.0..Calc.. 13.5 ........16.0 .... 57 .. 19 .. 28 .. 19 .. 16 .. 33 ...

x0 Test.. 6.3 .... .3.30 ... .. 7.20 .. .. 11.0 .. .. . ....33.0 . 18.0Calc. . 5.5 .... 2.20 ... 15.0 .. 8.40 .. 13.5 .. 11.0 .. .9.3 .. 18.5 .....

Ta Test..... 0.095.. ... 0.180 .. 0.130 .. 0.210 .. 0.180 .. 0.033 .. 0.048... . ..38 ......16.0Calc..... 0.100.. ... 0.200 .. 0.180 .. 0.250 .. 0.160 .. 0.030 .. 0.040....

T.d' Test..... 0.90 .. 0.22 ... .. 2.80 .. .. 1.85 .. .. . . ....11.0O. 7.3...... 1.00 .... 0.24 ... 0.100 .. 2.70 .. 2.60 .. 1.75 .. .. 0.270....

1204 KILGORE Transactions A. I. E. E.

TABLE II-CONSTANTS OF SALIENT-POLE MACHINES; COMPARISON OF TEST AND CALCULATED VALUESData from Locked Tests

All x's Are Per Cent ValuesMachines With Dampers

Kva. 7,000 3,000 30,000 7,500 7,500 1,000 30,000 30,000 22,500 15,000 5,000 1,250 % errorKind W.W.G. W.W.'G. Cond. Cond.. Cond. Cond. Motor Cond. Cond. Cond. Cond. Cond.Frequency 60 50 60 60 60 60 50 60 60 60 60 60 Max. Avg.Volts 6,600 6,600 13,600 4,800- 4,000 2,300 11,500 5,000 12,500 11,950 12,470 480R. p. m. 450 750 720 720 900 900 600 720 300 900 900 1,200

xd test 97 160 170 175 145 160 78 170 120 155 170 190 6.3 2.0calc. 96 160 170 10 150 160 80 170 120 165 -175 200

Xd" test 26 16.5 23 24 21 18 13 20 29 20 27 23 7.5 4.6calc. 25 17.0 24 23 20 18.5 13 21 27 18.5 25 22

X2 test 25 16 20 23 19 16 12 18 28 19 27 22 10.0 3.7calc. 23 16 22 22 19 17 12 18.5 27 18.5 25 22

x0 test 14.5 3.60 11.0 12.0 3.0 3.0 7.2 15.5 3.1 4.4 31.7 12.5calc. 13.5 3.45 11.5 13.0 2.45 6.55 3.95 8.2 18.0 2.8 3.9 8.0

TABLE III--CONSTANTS OF SALIENT-POLE MACHINES; COMPARISON OF TEST AND CALCULATED VALUESData from Locked Tests

Machine Without Dampers

Kva.........13,750 ....7,000 .....4,250.....2,330 .... .30,000 .....187......730 ..... % errorKind........W.W.G ....W.W.G.....W.W.G.....W.W.G.....Gein......Gen......Motor.....Frequency. .60 ....50.....60 ..... 50 ....60.....60......60...... Max. Avg.Volts..~~~6,600 ....6,600 .....2,300 .....6,600 .....11,500 .....240 ......2,300 .....

R.p.m....... 90 ....94 .....600.....125 ....600 .....200......1,200.....

xd test......110 ....115.....135.....115 ....115.....94......130.... 4.2.1.calc......110 ....120 .....135 .....115 ....120 .....94......135.....

xdu" test......43 ....44.....37.... 45 ....33.....41 ......30......6.0......2.5calc......44 ....43.....38 .....44 ....35.....42......30 .....

X2 test......58 ....61.....62 .....65 ....58.....55......55......10.3......4.5calc......57 ....60.....60 .....60....52 .....57......53 .....

x0 test...... 25 ....24 .....24 9.3..... .....10.4......5.7calc......25.5 ....23.5..... ....21.5 .... 8.5 .....

TABLE IV-CONSTANTS OF TURBINE GENERATORS; COMPARISON OF TEST AND CALCULATED VALUESData from Three-Phase Short-Circuit Tests

All x's Are Per Cent Values All T's Are in Seconds

Kva.......7,500 ....8,575 ... .9,375 ... .12,500 .. .18,750 ... .43,750 ... .68,750 ... .75,000 %... 0errorFrequency....60 ... 60 ... 60 .. 60 .. 60 ... 60 ... 60 ... 60Volts.......6,600 ... 4,150 ... 6,600 .. .11,000 ... 13,800 ....13,800 ....13,200 ... 11,600 ..... Lax. Avg.R.p. m......3,600 ... .3,600 ... 3,600 ... 3,600 ..3,600 ..1,800 ... 1,800 ... 1.800

xd test 12.....12 96 ... 110 ... 100 .. 115 ... 130 ... 97 ... 110 2.0.....0.6calc.....120 ... 97 ... 109 .. 98 ..116 ..130 ... 97.5 ... 110

xd' test.....16.5 ..12 ... 14 .. 13.5 .. 14 ... 25 ... 21 ... 24.....10.8.....4.1calc..... 16.0 ... 12 ... 13 .. 13.0 ... 15.5 ... 26 ... 21 ... 23

xd" test.... 9.2 . 8.4 .. 7.5 ... 8.8 .. 8.5 ... 17 ... 12 ... 14 ....14.4.....7.4calc.....9.6 ... 7.2 ... 8.1 ... 7.9 .. 8.1 ... 16 .. 13 ... 14.5....

x0 test..... 1.80 ... 1.60 .. 1.60 ... ... 3.1 ... 5.1 ... 5.9 ... 10.0 ....18.7 .....7.1calc..... 1.95 ... 1.50 ... 1.30 .... 2.7 ... 5.0 ... 5.9 ... 9.8....

Ta test.....0.043 ... 0.11 ... 0.085... 0.11 .. 0.13 ... 0.28... 0.25 ... 0.26....53.3.....17.4calc.....0.066.. 0.11 ... 0.086... 0.105... 0.12 .. 0.23 ... 0.31 ... 0.34....

Td' test.....0.56 ... 0.44 ... 0.49 ... 0.54 .. 0.59 ... 1.00.. 1.60 ... 1.60....16.4.....8.0calc.....0.48 ... 0.415 ... 0.41 ... 0.55 ... 0.57 ... 1.10.. 1.70 ... 1.70....

December 1931 CALCULATION OF SYNCHRONOUS MACHINE CONSTANTS 1205

sient characteristics. In Appendixes A and B formulas Appendix A-Formulas for Salient-Polehave been given for all the more important constants, Machinesexcept the subtransient time constant. Comparison I. ARMATURE LEAKAGE REACTANCE (X1)of test and calculated values shows that they give suf- S

* n ' ~~~Slot Leakage. Complete formulas have been given'ficiently good results over a wide range of types andsizes. for calculating slot leakage, derived by assuming the

Thze formulas are based on a physical conception offlux to go straight across the slot. With good approxi-The formulas are based on a physical conception of mation, Mr. Alger showed that the "slot constant"

the actual currents and flux fields involved. An attempt (effective permeance per cm. of slot) could be calculatedhas been made to obtain simple formulas for design ascalculations, and still retain all the most significantfactors, so that the formulas will cover a wide range of kx h2 +types and sizes of machines. The formulas are definitely \ bs 3b,!'limited to the usual two-layer type of winding. Only and kx could be determined aslaminated-pole salient-pole machines, and solid rotorturbine generators are considered. The methods used _3Y_can readily be extended to cover other special cases. 4 mq JThe general method of resolving the reactances into

components has advantages both in calculating and in for three-phase, and as ( 8 I for two-phase.visualizing the flux fields involved. The method of \ q /expressing the component reactances as a product of a I.-reactance factor, and specific permeances and flux o-0- I - i/ - - - ,distribution coefficients for the several components, has 0-8been found very useful. 0O.7 -

ACKNOWLEDGMENTS a I JThe author wishes to thank Messrs. C. J. Fechheimer, X 1 xPHA /

C. R. Soderberg, S. L. Henderson, M. W. Smith, C. M. 03 - -_ - -Laffoon and J. F. Calvert for many valuable suggestions. 0.2 \ /

Bibliography 02- ZI0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1. Determination of Synchronous Machine Constants by Test, COIL THROW AS A FRACTIONOF FULL PITCHS. H. Wright. A. I. E. E. TRANS., Dee. 1931. p. 1331.

2. End Connection Reactance of Synchronous Machines, FIG. 1REDUCTION Or SLOT REACTANCE DUE TO CHORDINGAlfred Still, A. I. E. E. JOURNAL, JUlY, 1930. k= factor for positive- and negative-sequence reactance

3. Transformer Ratio and Differential Leakage, R. E. kxo= factor for zero-sequence reactanceHellmund and C. G. Veinott, A. I. E. E. TRANS., Vol. 49, T r e1930, p. 1043. These results expressed as a specific permeance give:

4. Starting Performance of Salient-Pole Synchronous Motors, 20 h2 hi_T. M. Linville, A. I. E. E. TRANS., 1930, p. 531. = Cm L b + 3 bs

5. "Graphical Flux Mapping-I, II, III, IV, V and VI,"J. F. Calvert and A. M. Harrison, Electric Journal, Vol. 26, 1928. where

6. The Reactances of Synchronous Machines, R. H. Park and kxB. L. Robertson, A. I. E. E. TRANS., Vol. 47, 1928, p. 514. CZ = 2 k

7. Synchronous Machines IV, R. E. Doherty and C. A. p dNickle, A. I. E. E. TRANS., Vol. 47, p. 457. See curves of Figs. 1 and 2.

8. Calculation of Armatutre Reactance of Synchronous Tooth Tip and Zig-Zag Leakage. In the paperMachines, P. L. Alger, A. I. E. E. TRANS.,Vol. 47, p. 493. Tooth To and formuas Leagen the and

9. "Symmetrical Components," C. F. Wagner and R. D. referred to above, formulas were given for belt andEvans, Electric Journal, March, 1928, p. 151; April, 1928, p. 194; zig-zag leakage, due to flux produced across the gap byJune, 1928, p. 308; July, 1928, p. 359. the harmonics of armature m. m. f., but these did not

10. Graphical Determination of Magnetic Fields, R. W. include any increase in leakage due to flux going fromWieseman, A. I. E. E. TRANS., Vol. 46, p. 141. tooth to tooth. In the following development, the

11. Reactalnce of Synchronous Machines and Its Applications, principle of superposition is used to establish anaccurateR. E. Doherty a,nd 0. E. Shirley, A. I. E. E. TRANS., Vol. 37, basis for calculating all of these components.

p. 1209.Thtoareutnfiloffu,(ihntece12. Self Starting Synchronous Motors, C. J. Fechheimer, Thtoareuanfiloflx,(tinhece

A. I. E. E. TRANS., Vol. 31, p. 529. length) produced by the armature current may beaccurately resolved into components produced by two

1206 KILGORE Transactions A. I. E. E.

imaginary sets of currents which, when superposed, harmonics8 can be shown to be 0.002 Xad for three-phase,give only the real current. The first component is due and 0.015 Xad for two-phase. (This was derived assum-to a set of currents flowing on the armature surface, ing a uniform gap and no damping.) For a smallereach uniformly distributed over a slot pitch and equal number of slots, these formulas are not very accurate,to the current in the slot; the second component is but they are sufficiently good since the value is small.that due to a set of currents flowing in the conductors The case of fractional slots per pole is difficult if a com-and returning uniformly distributed over the slot pitch. plete solution is desired. However, the unbalance inThe m. m. f. due to the first set of imaginary currents m. m. f. resulting from fractional slots is kept to amay be analyzed as a fundamental and harmonics. minimum, and the additional reactance may be neg-The fundamental of this m. m. f. produces the funda- lected in normal machines.mental flux; and the harmonics produce the belt leak- Expressing these results as a specific permeance,age, which is due to the concentration in a finite number and assuming that the two-phase leakage varies withof phases. The second set of currents produces the

slot, tooth tip and zig-zag leakage fields. The T-I-s

portion of the current of one slot, which returns over a TOOTH SLOT TOOTHhalf tooth width on each side, produces a field as shownin Fig. 3.

2.07-

1.4-7_j

1.3_k X _ - ~ ~FIG. 3A-TOOTH TIP LEAKAGE FIELD DUE TO DIFFERENTIAL

I. -E? -X i A X CURRBENTLARLGE AIR GAP

2PHASE

0.4 0.5 0.6 0.7 0.8 0.9 1.0COIL THROW AS A FRACTION OF FULL PITCH

TOOTH SLOTFIG. 2-SLOT LEAKAGE FACTOR (CX)

A number of such plots shows that the effective per- |0I~ -meance per cm. of slot (not included in the slot leakage) /|may be approximated as ,[

[8Tg +035-b]j /

Unaareae=06irus FIG. 3B-TOOTH TIP LEAKAGE FIELD DUE TO DIFFERENTIAL

CURRENT-SMALL AIR GAP

(0.2 + 0.07 b. chording in the same way as that part which is due tog ~~~~~thethirdharmonic:

This includes the tooth tip and the so-called zig- XB = 0 (negligible) forthree-phase;zag leakage. 3 y 7r 2

It is convenient to calculate a specific permeance d rsinX, for the slot tooth tip and zig-zag leakage combined: XB = 0.095 Pg9'L km 2|frtopae ()

Si= m, q [ b+ b+ 0. 0 (la) End Winding Leakage. The end leakage formulagApresented here iS essentially the same as that developed

Belt Leakape. As defined, the belt leakage is due to by Mr. B. G. Lamme. Professor Alfred Still has re-the harmonics of armature m. m. f., determined by uni- cently published a paper2 in which he develops a formulaform current distribution. For a full pitch winding, onthesame basis.where the slots per pole per phase (q) are integral and The method referred to consists in treating the endlarge, the reactance calculated by summing up the winding leakage as a revolving field in air. The pres-

December 1931 CALCULATION OF SYNCHRONOUS MACHINE CONSTANTS 1207

ence of the iron is neglected in the derivation and, if ampere turns (Cm) is the ratio offundamental air gap fluxnecessary, an empirical correction is applied to the produced by the fundamental of armature m. m. f. to thatresult. Both Messrs. Lamme and Still assumed that produced by the field for the same maximum m. m. f.the flux went from pole to pole without spreading out Cdl is the ratio of the fundamental air gap flux producedaxially. Further, in order to integrate the flux, each by the direct-axis armature current to that which would beassumed that the flux paths were arcs of circles. produced with a uniform gap equal to the effective gap over

This method of analysis can be carried one step the pole center. CQI is the corresponding coefficient forfurther by using the exact solution8 for a sinusoidal the quadrature-axis.m. m. f. distribution in air, assuming a two-dimensional All of these coefficients may be determined by ac-field. The exact solution gives the very simple result curate flux maps, but it is often more convenient to usethat the field is sinusoidally distributed, and the flux formulas, except for C, and Cl. These two constantsper pole per inch is 3.19 times the fundamental ampere are determined directly or indirectly in almost anyturns. design procedure, by laying out the field form eitherThe pitch of the winding for the straight coil exten- from a flux map or other accurate method.

sion (l12) is the same as for the gap, but it drops off to A formula for Cm may be derived assuming the fluxzero at the end of the coil. An effective length 1/2 (lei) to go straight across a uniform gap over the pole face.will be used as in Lamme's formula. The formula is much more accurate than might appearThe end winding field is not truly a two-dimensional

field as assumed, but spreads out axially. This effect .Owas investigated further by making approximate plots T - -of the three-dimensional field in the several views. -l - /The increased leakage due to local flux closing about the 9VGIindividual conductors was calculated by resolving the cfield into components in a manner similar to that de- 0.8 _ 7scribed for the tooth tip leakage. It was found that - -these factors, although small, are not negligible, and 0.7 . -| -may be approximately accounted for by a constantfactor 1.25. 0.6

Expressed as a specific permeance, the end windingleakage is (1.25) . (3.19) times the ratio of effective end 0.5 - - -winding extension on both ends, to the core length ' A _

Xe = 4/1 (2 le2 + lel) (3a) 04

The armature leakage reactance is calculated as theproduct of the reactance factor (X) determined by 0.3 _equation (1) or (2), and the specific permeances, _equations (la), (2a), and (3a). Q2_. __ ._

0.5 0.6 0.7 0.8 0.9 1.0Xi = X(Xj + Xe + XB) (4a) POLE EMBRACE (cc):

RATIO OF POLE ARC TO POLE PITCH

II. REACTANCES OF ARMATURE REACTION (Xad, Xaq) FIG. 4-FLuX DISTRIBIUTION COEFFICIENTS FOR SALIENT-POLEAND SYNCHRONOUS REACTANCES (Xd, xq) MACHINES Cm AND Cql

The specific permeance of the air gap (Xa) iS mostconveniently determined on the assumption that the from the assumptions stated, since the fringing andgap is uniform, and the actual flux distribution may be increase in gap at the pole tip affect both quantities incalculated by the use of flux distribution coefficients the ratio about equally. This formula was checked bygiven below. The effective gap (g') is determined by accurate flux plots on a wide range of pole shapes frommultiplying the actual gap (g) by Carter's coefficients 4 to 88 poles.for stator slots and vents, and by a reduction factorfor end fringing flux. Cmn = (6a)

_dX \4 sin o J\a=3.19( )g (5a) 2

Where at is the pole embrace. See curve, Fig. 4.The flux distribution coe;fficients used here are defined From the definition of Cm, it may be seen thatasfollws C1 is the ratio of thefundamental to the actual Cm = Cdl/Cl. Having determined Cman C1 smaximum value of the field form. (The field form iS the descrbed aboewave of flux density due to the field only.) The "pole Cdlr= Cm Cove,constant" (Cr) is the ratio of average to maximum of the d=mI(afield form. The ratio of equivalent field to armature An approximate formula for CQi may be derived on the

1208 KILGORE Transactions A. I. E. E.

assumption that the flux goes straight across a uniform The side leakage expressed as a specific permeancegap over the pole face. based on the above assumptions is:

F sinia w 4 3 ±hg+9- .O55Tr,)Cqi a - XFs (3. 19) - L ( bh)

making some allowance for end fringing and an averageincrease in gap'at the pole tips, a modified formula was hf, ±3hfO 1T I

10obtained which checked sufficiently well with the series + +.Tr p ]of flux plots mentioned. W(dr 2hh-0.4hfI)- (15a)

4 a + 1 sin a7r ] P

This may appear rather long, but it will be noted thatThe flux distribution coefficients which have been it is expressed directly in the main dimensions of the

designated here as C1, Cdl and Cqi can be calculated for pole, and does not require a pole layout. This formulaa particular pole shape from the curves given in a paper10 was found to check well with the results obtained byby R. W. Wieseman. flux maps on a wide range of pole shapes for machines

Reactances of armature reaction may be calculated as: from 4 to 88 poles.

XCl - X Cql Xa (9a) and (10a) To determine the end leakage field, a number of two-Xad = X Udl a Xaq - 2a ( a) an ( a dimensional flux plots was made in two views, whichAs an alternate method: Xad may be calculated from the show the flux which fringes from pole to pole and thegap ampere turns Ma and the demagnetizing ampere flux which closes about the field coil ends. The resul-turns (M. = Cm MA). tant field is obviously not two dimensional, but an

Mm C(l approximate formula based on these plots should beXad = 1%0 a-M= (Ila) and (12a) sufficiently accurate.

Synchronous reactances are calculated simply as the XFe = 3.19 4(h - 1) + 2hfI + 0.5 bp1 (16a)sum of armature leakage and reactances of armaturereaction: The specific permeance for the effective pole leakage is:Xd = XI + Xad xq = XI + Xaq (13a) and (14a) XF = XFs + XFe (17a)

III. FIELD LEAKAGE AND TRANSIENT REACTANCE The effective field leakage reactance (XF'), whichThe transient reactance is actually the total leakage of added to armature leakage (xi) gives the transient re-

the armature with respect to the field (assuming no actance, may be calculated as follows: With unit funda-additional damping circuits). As explained in Sec- mental armature ampere turns suddenly applied ontion II of the paper, it will be calculated as the sum of the direct-axis, an initial field current (If) will betwo arbitrary components, the armature leakage (xi) induced of such value that the net interlinkage with theand an effective field leakage (XF'). field produced by both the armature and field currents

The specific permeance for field leakage (XF) will be the is zero. The interlinkage (per inch) produced by theside and end leakage flux per pole per inch for unit field armature current isampere turns. The side leakage flux per inch may be 7rdetermined accurately by a flux map for the field only, Cl Xa Npysimilar to those shown in Fig. 1 and Fig. 50 in a series ofarticles5 by J. F. Calvert. Each element of flux is and by the field currentmultiplied by the fractional part of the turns linked.

It is often more convenient to use an approximate / rformula. The following formula was derived assuming Np If t 2 Cp Xa + XF NP;that the flux goes straight across the interpolar space(at right angles to the center line between poles), henceexcept near the armature surface where the flux isassumed to go straight across half way from the pole r CliXa ]head to the center line between poles, and then to spread N Ifout so that the upper boundary intersects the armature L rcp Xa + XF 'surface and the center line between poles at 45 deg., as it 2should theoretically. A further correction was made toallow for the shortening of the path of part of the flux The net fundamental air gap flux (per inch) producedwhich terminates on the under side of the pole tips. by both armature and field is then

December 1931 CALCULATION OF SYNCHRONOUS MACHINE CONSTANTS 1209

these conditions, the distribution is such that the total__ C12_ a

2 flux is a good approximation to the fundamental. Com-Cdl' 4 j bining these permeances and multiplying by the frac-

d J tional part of the ampere turns acting, the resultant2 formula for the specific permeance is:

The effective field leakage is: XDd= [COS(b-l)Tb JL][( + Npt)± N]

NC2Xa2 (23a)XF= X Cdl -a X Dd = X XDd (24a)

Cp Na + XF J The direct-axis subtransient reactance (Xdi') of a2 machine with a damper winding is calculated as the sum

r C, of the components (4a) and (24a):_4~F1 _ ~c~ 1 Xd" = Xl + XDd' (25a)

Xad C+ NF j (18a) For a machine without dampers:

2 Xa Xd = Xd (26a)Thusauatreactance is: The quadrature-axis damper leakage is determined on

The uardathe same base. In this case, the flux goes tangentiallyXdu' = Xl + XF' (19a) down the gap and across the damper bar slots. The

The saturated transient reactance (Xd') is determined by effective permeance per cm. for one slot ismultiplying by an empirical factor (F,s). The test hb2 rresults described in the companion paper' indicate that 0.5 + b2 hbl +this factor may be taken as a constant Fst = 0.88, with bb2 3bbl Tbsufficiently good results on normal machines. The part of the fundamental ampere turns acting across

Xd' = Fst . xdxu (20a) oneslotis

IV. DAMPER WINDING LEAKAGE (XDd', XDq/) AND .T bSUBTRANSIENT REACTANCES (Xd"', Xq") T J

The initial flux set up by suddenly applied direct-axis and the flux is very nearly sinusoidally distributed;armature current is limited by the damper winding, hence,The flux maps of this field were constructed on the

2

assumption that there is no net change of interlinkage 20Tb F hbl hb2 -

with any of the damper winding circuits. The field is X Tr I + 3 bbl bb2 + Tbquite complicated, but it was found that the funda- In this case, the dimensions of the slot and bar in'themental flux could be calculated with fair approximation pole center should be used. The effective damperfrom the permeances of the main parts of the flux paths. winding leakage is:

Approximately xDqI X NDq (28a)

(nb- 1) Tb Wi] The quadrature-axis subtransient reactance (x,,') forITr 2-i machines with dampers is calculated as the sum of the

part of the armature m. m. f. acts across the field, leak- components (4a) and (28a):age permeance (NF) in series with the permeance of the xq = x + xDq' (29a)end damper bar (Nb) combined in parallel with the For machines without dampers (12a) or (14a):permeance of the gap over the pole tip (NPt). xq"t = xq (30a)

Nb = 6.38 (0.5 + hb2 hb) (21a) V. NEGATIVE SEQUENCE (X2)bb2 +3 bbl

As explained in Section II, the negative sequenceWhere 0.5 is an approximation to the tooth tip perme- reactance may be calculated as the average of the sub-ance per cm. of damper slot. For round bars, use 0.62 transient reactances.

hbl 1

instead of 3 bb . x2 = 2jxd" + xq"t) (31a)

Using an average increase in gap at the pole tip:b (nb1~~~~~ VI. ZERO-SEQUENCE REACTANCE (XO)

NP = 6.38 [ bh b (- / ] (22a) Since the zero-sequence currents are all in phase, theg fundamental m. m. fs. of the three phases cancel out,

Although the flux is not sinusoidally distributed under and there is no appreciable fundamental flux across the

1210 KILGORE Transactions A. I. E. E.

gap. Hence, it consists in a slot and belt leakage, with The unsaturated value of the (short-circuit) transienta small belt leakage in the end windings. Mr. Alger' time constant can be shown to be exactlyhas derived a factor for the effect of chording on zero- /Xd,sequence slot reactance. This factor will be designated X-) Tdo'.here as kxo. See curves, Fig. 1. The specific permeance d

for slot and tooth tip leakage can be shown to be: Tests' show that a good approximation to the saturatedk( o) 20 \ hi +2 h, value may be had by assuming the transient time con-

0 = Ni(kz1 + mqk 2k, 1 stant to be reduced by the same factor as the transientkx m q k, k,0 12 b, reactance; hence:(32a) Xd'

For machines with a damper winding, the belt leak- Td = - Tdo' (37a)age flux over the pole face is restricted in the same man-ner as the quadrature subtransient reactance flux. The armature time constant may be calculated asNeglecting the reluctance of the flux path across the explained in Section II:gap, the specific permeance for a full pitch winding / x2 \wouldbeNDq, equation(27a). Sincethisleakagevaries Ta 2 7r f ra (38a)with the throw in very nearly the same manner as theslot leakage, it may be written: Where ra is the d-c. armature resistance in per cent at

k X 75 deg. cent.NB0 = ( j )N0k (33a) Appendix B-Formulas for Turbine

For machines with no dampers, the effective per- GeneratorsDue to the differences in construction, many of the

meance per cm. of phase belt would be , , if the formulas for turbine generators are different from those36 g for salient-pole machines, buit most of the conceptions

winding was full pitch and the gap was uniform. Ex- involved are the same. Hence, only the differences inpressed as a specific permeance, this would be the formulas will be discussed.

20 / i.\ I. ARMATURE LEAKAGE (xl)m kd2 \ 36 g',)O.1 The reactance of armature reaction of turbine

generators is calculated by assuming the flux to gobut It should be reduced by about 0.7 for an average straight across the gap. Therefore, it is necessary topole embrace. To take into account chording, it may include the fundamental flux going tangentially in the

be multiplied by ( ) Hence, for machines with gap as a part of the armature leakage. This componentbe mltlpled y k 2 §Hne o ahnswtp P / a

is most conveniently accounted for by adding 2-'- tono damper winding:

kx0_ the permeance per cm. of slot. The specific permeanceNB0 = k 2 (. 07 Na) (34a) for the embedded portion is then:

p

In the end windings, the fundamental m. m. fs. xi _ C 20 [ h2 hi 0_2_-9_ 39acancel, but the local flux about the phase belt is equiva- mm q L bs + 3bs +. 2± (3)lent to about 0.2 of the positive sequence end windingleakage. The zero-sequence reactance will be cal- get fed cilting rings iscvery siml toculated as the sum of the slot, belt, and end leakage tatifient-polemachries bu an addto

components: ~~~~~~~that for salient-pole machines, but an additionalcomponents: empirical factor of 1.2 is used for the increased axialXo =NX(TANT+ NBdo0 °+02Ne) (35a) divergence. The formula may be further simplified

VII. TIME CONSTANTS (Tdo'),^ (Td't) (Ta) by using average proportions:The self inductance of the field in henrys is

13 dLf=NP2plI1O8 ( 2CN±N )F) N\e' = p1 (40a)

As eplanedin ecton I, te oen-ircit ime For machines with magnetic retaining rings, theconstant may be calculated as a ratio of Lf to the lekg une staysaecniin.sicesdbresistance of the field (RF); the resistance (Rf) should tepsncofherg. Une nomlcndtosF l ~~~~~~~thegreatest effect of saturation iS in the tapered sectionbe the value in ohms at 75 deg. cent. next to the rotor body, and a rough approximation to

T,__Lf ~~~36a the resultant field may be had by assuming that thedo 1?R( effect is equivalent to cutting out this tapered section.

December 1931 CALCULATION OF SYNCHRONOUS MACHINE CONSTANTS 1211

Numerous two dimensional flux plots constructed on Krs-I sin y_ 7r(47athis basis indicate that the increased leakage may be C,q Krs[ + I (47a)

/13 d\accounted for by adding p ) to the specific per- The factor Cf may be calculated as:

meance. Hence, for machines with magnetic retaining CF= [(1 - y) + 3K ] (48a)rings, it is necessary to use a different specific permeance rs

for the end winding leakage for steady state conditions: The reactances of armature reaction are calculated13 d by the same formulas as for salient-pole machines:

Xe = XeI + (41a) (9a) and (lOa) or (Ila and 12a); using the coefficientscalculated by (44a), (45a), (46a), and (47a) given above.

Under initial sudden short-circuit conditions, this The synchronous reactances are given by equationsadditional leakage path is completely saturated, and (13a) and (14a)the increased leakage will be neglected. III. FIELD LEAKAGE AND TRANSIENT REACTANCEThe total leakage reactance for transient and sub- (Xd') FIELD LEAKAGE SPECIFIC PERMEANCE (XF)

transient conditions is: The rotor slot leakage permeance per cm. is similarXl' = X [Xi + XB + Xe'l (42a) to the stator, but expressed as a specific permeance

For machines with non-magnetic retaining rings, the for the field it is:leakage reactance for steady state conditions is the 4 (3.19) P [hr2 hri gsame as (xi') given above, but for a machine with mag- XF, = Q L ± 3 b7 +0.2± 2 T ] (49a)netic retaining rings:

Xl = X [Xi + XB + Xe] (43a) The end leakage was calculated by determining the

II. REACTANCES OF ARMATURE REACTION (Xad AND Xaq) LOB

AND SYNCHRONOUS REACTANCES (Xd AND Xq) 1.04 0 cJ I

The flux distribution coefficients may be defined in the E 1.02same way as for salient-pole machines, but a factor u

(Cf) is required in place of the average of the field form z

(Cp). Cf is the ratio of the interlinkage with the field < 0

to that which would be produced with a uniform gap ' O.1and a concentrated field winding. 094

The constants will be calculated assuming the flux 092

togo straight across the air gap. On this basis: 0.65 0.66 0.67 0.68 0.69 0.70 0o71 0.2. 0.73 0.74 0.75 0.76 0.77 0.78X =RATIO OF SLOTTED PORTION

TO TOTAL CIRCUMFERENCE OF ROTOR

8 sin Y 2 FIG. 5-FLUX DISTRIBUTION COEFFICIENTS FOR TURBINE

Cl72Krs \ /GENERATORS

4 Krs-c

7r flux per pole per inch closing about the coil ends.+ ( Krs cos Y 2 (44a) The retaining ring saturation was accounted for by

7 Krs 2 assuming the effect equivalent to cutting out theSee curves, Fig. 5. tapered section of the ring, as discussed for (Xe). AnThe exact equation for Cm is long, but if the increase approximation based on these assumptions and using

in gap due to slotting is neglected, a very good approxi- average proportions was found to be:mation is obtained. 35 d

2 ( X Fe = (50a)

8=-( ) The specific permeance for field leakage is:2 XF = (XF + XFe) (51a)

See curve, Fig. 5. The coefficient Cdl is exactly The unsaturated transient reactance (Xdu') can be de-K1 r~~~~~~~rived in the same manner as for salient-pole machines,

Cdl=[-I _Kr-1 sin -y 7r 1 except that Cf should be used instead of Cp. However,L Krs IT -J for any normal distribution of field turns, a simpler

but itmaylsobe calulated as:solution may be obtained by neglecting the harmonicsof the turns distribution of the field. Such harmonics

Cdl = Cm C1 (46a) produce entirely negligible mutual interlinkage, andThe quadrature-axis coefficient might be treated as additional field leakage, but for

1212 KILGORE Transactions A. I. E. E.

normal m-achines this is also negligible. All the flux The quadrature-axis effective damper leakage de-across the gap may then be assumed mutual flux, and pends mainly on the flux penetration in the solid poleif the mutual reactance were very high, the fundamental center. The magnitude and frequency of the pulsationampere turns of the field would equal that of the of ampere turns per inch acting over the pole centerarmature. The actual field ampere turns would be Cm vary with conditions of short circuit. For a line-to-times the fundamental of the armature, and the line fault, a double frequency pulsation of ampere turnsfundamental flux across the gap required to produce is produced by the symmetrical current and a rated

4 frequency pulsation due to asymmetrical current.zero net interlinkage is Cm times the actual field The resulting maximum ampere conductors per inch

are a little more than half the value used above, but atleakage. Hence, the true field leakage with respect to double frequency the penetration is essentially thethe armature, expressed as an armature reactance, is: same (6q = 1.2) for 60 cycles (and 6q = 1.8 for 25

4 cycles). For a short circuit from load, where quadra-XF = X Cm2 XF (52a) ture-axis flux is involved, the resultant ampere con-

W ductors per inch are about the same as given above forThe effective field leakage is the true field leakage (XF) the direct-axis, and the effective penetration is again

added in parallel with the mutual reactance, which on 6, Expressing these relations as a specific permeance:the above assumptions is Xad. 3.19

XDq [g + 2 4q] (55a)XF XF Xa )

(53a)d

'\ZXad + XF The effective damper leakage reactances (XDd') and (XDq')The transient reactance is determined as for salient- are calculated as for salient-pole machines, equations

pole machines, equations (19a) and (20a). (24a) and (28a).IV. EFFECTIVE DAMPER LEAKAGE (XDd') AND (XDq') V. SUBTRANSIENT AND NEGATIVE-SEQUENCEThe solid iron and metallic wedges of a turbine REACTANCES

generator rotor act as an effective damper winding. The subtransient reactances xd' and xq" may beThe actual penetration of flux into the solid iron de- calculated as for salient-pole machines with damperspends on the resultant ampere turns acting and the (25a) and (29a). The negative-sequence is deter-frequency of pulsation. The value of the reactance mined as the average of Xd" and xqV, equation (31a).which is effective in the first instant may be approxi- It will be seen that there is very little difference betweenmated by assuming no decrement and by using the Xd" and xq', and the tests described in the companionmaximum ampere turns per inch to determine an paper' show that they do actually become very nearlyeffective depth of penetration. For most turbine equal at high currents. For most practical purposes,generators, the maximum ampere conductors per inch, it should be necessary to calculate and test only Xd".corresponding to initial symmetrical short-circuit cur- VI. ZERO-SEQUENCE REACTANCErent, is around 16,000.The direct-axis damper leakage is determined by the For turbine generators, the zero-sequence belt leak-

age may be approximated by using the direct-axisdepth of flux penetration in the rotor teeth and wedges ame lakage speciic prence d)recedabyunder sudden short-circuit conditions. The ampereconductors per inch over the center of this section pul- the rt kotheatio'rofor chording.sate between zero and twice the value, due to the sym- k,metrical current. The effective permeability at thesurface, for these maximum ampere turns, is about 2. o=koAssuming this constant throughout, the formula given XBo = k XDd (56a)by Steinmetz for flux penetration in deep solid partscan be applied. For 60 cycles, the flux penetrates to The mainer of the fm )about 0.6 inch, which is equivalent to a path in air ofdepth d =1.2 inches at 60 cycles; (for 25 cycles, rated Xo = X (Xio + XBo + 0.2 Xe') (57a)frequency bId= 1.8). The direct-axis leakage may be VII. TIME CONSTANTS Td0', Ti, AND Tdstill further restricted by the slot. The flux across the Tesl nutneo h il nhny sslot will be assumed to penetrate to the bottomof the wedge. Half of the leakage flux going tangen- L= pp - [ (-__,__ A](5atially in the gap was included in thearmature leakage. \ gfExpressing these relations as a specific permeance:

The open-circuit time constant will be calculated, neg-

XD dX1 Pfg + 6d + hr2] (54) sameasequation (36a)

December 1931 CALCULATION OF SYNCHRONOUS MACHINE CONSTANTS 1213

The short-circuit time constant Td' and armature hbl = depth of damper bartime constant (Ta) are calculated as for salient-pole hb2 = depth of slot above the damper barmachines (37a) and (38a). hfl = depth of field coil

SYMBOLS hf2 = distance between field coil and pole headh = depth of conductors in the rotor slot

NOTE: The symbols for the characteristic reactances hr2 = depth of rotors ao the onductand time constants are given in the introduction, h2 = depth of cotorsl ina tor sothi = depth of conductors in stator slotSection I. The symbols for the components of reac- h2 = depth of stator slot above the conductorstance are given in Section II. The symbols for the h = dspecific permeances and the flux distribution coefficient h = ratep etwinding currentare given as they are defined and calculated in Appen- kd = driti frsao wind

dixes A and B. ~~~~~~~kd= distribution factor for stator windingds A ad B. .kp = chord factor of the stator winding

For symbols representing machine dimensions, see kx = reduction of slot reactance due to chordingalso Fig. 6. kzo = reduction of zero sequence slot reactance due to

chordingEND WINDING krs = Carter's coefficient for rotor slots

bCb5tjtI>1 = core length (including vent ducts)i;iE! . 7!le2% lb = length of damper bar

Ile = extension of bent section of end windingh3+| h -1,2 = length of straight section of end winding

lh = length of pole head (including magnetic coil.=. LFe, supports)

STATOR SLOT e. Lf = total self inductance of the fieldm = number of phasesMA = maximum fundamental armature ampere turns

JSTATOR M = air gap ampere turns at no-load and ratedL 9 lbh H voltage

M ho i 1 t hh Mm = "demagnetizing ampere turns;" field ampere

7 FJT1] w1C71 hf2 turns required to balance armature reactionV- P r-V VL p atrated current (zeropowerfactor)

tAll PL P AlA f, NAt nb = number of damper bars per polePOLE I . i iiiLiLLAZJILv..nL == number of conductors per slot

Np = number of field turns per poleFIG. 6-DETAILS p = number of poles

Ps3 = perneance per cm. of slotA = ampere conductors per inch (amperes per slot q = slots per phase per pole

* slot pitch) Qr= total rotor slotsbbl = width of damper bar ra = d-c. resistance of armature in per centbb2 = width of slot above damper bar Rf = field resistance in ohms at 75 deg. cent.bh = width of pole head X = reactance factor, per cent reactance for unitbp = width of pole specific permeance. See Section IIbrs = width of rotor slot y = coil throw (number of slots spanned)bs = width of stator slot a = "pole embrace" ratio of pole arc to pole pitchbt = width of stator tooth y = ratio of slotted portion to total circumference ofBg = air gap density over pole center (at rated a turbine generator rotor

voltage) ad = depth of an equivalent path in air for the pul-B, = maximum fundamental density at rated voltage sating flux in the rotor teethC = number of parallels in the stator windingd = inside diameter of armature (armature bore) aq = depth of an equivalent path in air for the pul-dr = rotor diameter sating flux in a solid pole center

p = ratedphasevoltage (phasewindingvoltage) X = "specific permneance:" effective flux per polef = rated frequency per inch of length for unit ampere turns.Ft= ratio of saturated to unsaturated transient re- See Section II

actance (an empirical constant used here as Xr = polepitchon statordiameter0.88) Tb = pitch of damper bars

g = actual (single) air gap in pole center tr, = pole pitch on rotor diameterg' = effective air gap, including increase in gap due Tr8 = rotorslotpitch

to stator slots and vents and end fringing rs = stator slot pitch