1928 the Calculation of the Armature Reactance !!

The Calculation of the Armature Reactance ofSynchronous Machines

BY P. L. ALGER*Member, A. I. E. E.

Synopsis.-This paper presents new and simplifiedformulasfor given. The definitions of armature leakage and armature reactionthe armature leakage reactance of synchronous machines, and com- reactances used are those proposed by Doherty and Nickle inpares the results obtained with tests on 100 machines of varied types. 1926, and as a result the leakage reactance has a much smaller valueThe new form?nulas are characterized by novel expressions for the than has heretofore been assigned to it, thus requiring a ratherend leakage and for the "air-gap" or "differential leakage" reac- fundamental revision of the ordinary conceptions of the flux den-lances. Forimulas for zero phase-sequence reactances are also sities existing in the various parts of a loaded synchronous machine.

I. INTRODUCTION of armature m. m. f., is midway of the interpolar space.T HE growing interest of operating engineers in The armature leakage reactance is defined as the

machine reactances, (as system stability and short- difference between the total, or synchronous, armaturecircuit phenomena become continually more reactance and the above defined reactance of armature

important), together with the recent advances in the reaction. The distinction between the leakage and thetheory of synchronous machines, make renewed con- armature reaction components of the total reactance issideration of the subject of reactance calculation quite an artitrary one, since no winding can have aopportune at the present time. Papers on this subject definite value of leakage reactance except with respecthave appeared in the JOURNAL'S pages at intervals over to another winding. Some distinction of this charactera period of more than 20 years, each paper marking a is very desirable, however, for convenience in calculat-further refinement of methods and an advance in ing the transient reactance, and for other purposes.accuracy. At first, it was customary merely to estimate The armature leakage reactance as above defined, isthe numbers of leakage lines per ampere inch of em- as nearly equal as may be, to that part of the totalbedded and free conductor, and multiply them by the armature reactance which remains the same underrespective lengths to obtain the reactance. Later, the transient as under steady conditions of operation.reactance was segregated into several distinct elements Following Adams' treatment of the induction motor,that were separately calculated. These calculations the armature leakage reactance of a synchronousdue to their greater importance and in this case, the maehine will be divided into four parts; the slot, end,possibility of more accurate tests have reached greater zigzag, and belt leakages. Formulas for each of theseprecision in connection with induction machines will be derived in turn. As the slot leakage has beenthan with synchronous machines. Adams"11'3'14 first quite satisfactorily treated by Adams"1 already, thedeveloped formulas for induction motor reactances, and discussion of it here will be abbreviated as much aslater Fechheimer9 and Doherty and Shirley7 developed possible, consistent with the derivation of a little moreformulas for synchronous machines. In 1926 Doherty exact formula, and its extension to apply to zero phase-and Nickle3 presented some fundamental extensions sequence currents. None of the other three elements ofof the theory of synchronous machines, and proposed the leakage has been adequately treated by pre-new, more precise, definitions of armature leakage vious writers, so they will be given more detailedreactance and of armature reaction reactance. The consideration.present paper adopts these new definitions and develops The four divisions of the reactance found useful forsimple formulas for the accurate calculation of the two induction machines are followed for the two reasonsquantities, for salient pole machines. that they fit in very well with the new conceptions ofThe reactance of armature reaction is here defined as leakage reactance, and that their adoption will tend to

the reactance due to the fundamental sine wave of unify the theirtyption wind to

air-gap flux produced by the armature current acting particular the total "air-gap leakage, due to harmonicsalone. It is calculated by means of the curves derived of. the air-gap flux, is segregated into the two partsby Wieseman2 from flux plots. There are two distinct called zigzag and belt leakage, respectively, in order tovalues for it, the direct axis value occurring when thecaldzgganbetekg,rsptilyinoerovalues for it the direct axis value oceurring when the permit the effects of the independent variables, numberaxis of armature m. m. f. coincides with the pole axis; of slots, and numbDer of phase belts to be separatelyand the quadrature axis value, occurring when the axis taken into account.

*Assistant Engineer, A-C. Engineering Department, General All previously published formulas for zigzag leakageElectric CSompany, Schenectady, N. Y. hv novdtecnieaino vrapn rmr

For numbered references see bibliography,.aeivle h osdraino vrapn rmrPresented at the Winter Convention of the A. I. E. E., New York, and secondary teeth, and so have inadequately taken

N. Y., Feb1ruary 13-17, 1928. into account the effects of flux fringing in the air-gap,493

28-30

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494 ALGER: SYNCHRONOUS MACHINES Transactions A. I. E. E.

and have been inapplicable to synchronous machines. field, but the determination of this flux was erroneouslyAlso, older formulas for end leakage have been frankly based on the linkages of the armature by the air-gapempirical in nearly all cases. All new formula for flux due to the field.7 The remaining part of the totalthese elements will be derived by considering the leak- air-gap flux produced by the armature was then calledage flux to consist of revolving magnetic fields, divided tooth tip leakage. As the new definition of armatureinto a fundamental and a series of harmonics. In each reaction includes most of this latter flux, we havecase, the coefficient of the resulting formula will be abandoned the name "tooth tip" and called the smallderived from theoretical considerations alone, and no remaining part, the synchronous frequency voltage-empirical multipliers will be used to bring the results producing harmonic fluxes, "differential leakage," asinto agreement with practise, although simplifying just described.assumptions will be freely made as the occasion Bearing in mind these conceptions of the leakagedemands. reactances which we are going to calculate, we can now

Before proceeding to the derivation of the formulas, proceed to the derivation of the formulas. The fourit is desirable to get clearly in mind the distinctions components of the reactance will be taken up in thebetween the four elements of the leakage that have order of their difficulty, and subsequently the resultsbeen specified. The end leakage reactance is due to the of their application to actual machines will be comparedflux linking the end windings alone. The slot leakage with tests.comprises all of the flux crossing the slots due to the II. SLOT LEAKAGE REACTANCEarmature current, but does not include flux passing Only two-layer barrel-type armature windings infrom tooth to tooth in the air-gap space. Evidentlythere is a slight error here, because the flux lines near op slots il cnd inu this paper, isicethe mouth of the slot do not pass straight across, butbulge outwards into the gap. The zigzag leakagecomprises all of the space harmonics of the air-gapflux, due to the armature current in a one slot per poleper phase winding which induces fundamental frequencyvoltages in the armature. This includes the flux whichcrosses from tooth to tooth in the air-gap and interpolarspaces without actually reaching the field surface, and IXso takes up the leakage flux at the point at which the Itslot leakage left it. The belt leakage then comprises Lall the remaining fundamental frequency voltage pro-ducing space harmonics of the air-gap flux due toarmature current. This last reactance is the additionalreactance that an actual winding has above that whichit would have if there were as many phases as slots perpole. Thus, the zigzag lealtage reactance is due to the FIG. 1-SLOT LEAKAGE FLUX PATHSdeviation of the armature m. m. f. wave from a sinusoid,caused by the limited number of slots, and the belt of single layer windings have reduced them to a placeleakage reactance is due to thQ further deviation caused of only historical importance in polyphase a-c. machinesby the limited number of phases. of American manufacture.The belt and zigzag reactances together have been Consider the slot shown in Fig. 1. All the flux

appropriately called5 the "differential leakage," since crossing the slot returns below its closed end through a

they are due to the deviations of the armature m. m. f. path of negligible reluctance, and none of it links anyfrom the ideal sine wave. They may also be called of the other slots. The leakage reactance in ohms per"air-gap leakage," since they represent the difference phase, due to this flux is then equal to the product of thebetween the total and the armature reaction com- following factors:ponents of the fundamental frequency voltage producing 1. The permeance of the path across unit lengthair-gap flux. As the differential leakage reactance of slot.varies with the relative positions of the axes of the 2. The embedded length of slot, or Larmature m. m. f. and the poles, the armature leakage 3. The number of slots in series per phase, or S/qreactance varies likewise, and has a slightly higher 4. The square of the number of series connectedvalue in the quadrature than in the direct axis.3 / qZ \2

Formerly, the air-gap flux due to the armature current conductors in each slot, or ( Swas divided into two parts, that due to the armature /reaction, and that due to "tooth tip leakage."7 The 5. The rate of change of the flux, or 2 r ftarmature reaction part was intended to represent the 6. A constant to reduce linkages per second toair-gap flux produced by the armature that links the ohms, or 10-8


Feb. 1928 ALGER: SYNCHRONOUS MACHINES 495

The permeance of the path for flux A linking all the and that of II is proportional to:conductors in the slot of Fig. 1 is: 1 r d1- d2 d1-d21

40 ( wd lines per ampere per cm. length of slot 4 ld3+ 2

And if the winding is full pitch, so that the currentsI

- [d3 + 1 ±dllin coil sides I and II are identical in magnitude and 3phase, the linkages per ampere produced by the flux in while their mutual inductance is proportional to:paths B and D are equal to the expression: 1 [ (di - d2)

4 7r did-d2 4 Ld+ 410 ( d wd ' Hence, the total linkages due to the two currents

since with uniform current distribution over the height 00 out of phase acting together are proportional to:di, the flux density distribution is linear and the linkage 1 di - d2 1 2 di d2distribution is parabolic, while the average height of a - d3 + 6 + d3 + +parabola is one-third its maximum height. As all the 4 6 41 3 3flux through path C links one-half the total current, the (4 d3 + d- d2) cos 6linkages due to it are one-quarter as much as they would + 8be if this flux linked all the current. The linkages dueto C are, therefore: Or, the slot constant for this case is:

l10 ( 4w) 2(w1+cos 0) + 2w (5 + 3 cos 0)

and so the total slot reactance of a balanced, full pitch, d2polyphase winding is equal to: 24w (3cos 0-1).

8wr2 / q Z \ 12 d3 + 4 d -d When the winding pitch is 100 per cent, all the slots=10, (Sq 12' ] q- 1Xslot = 109(fL) (S/q) t S / L 12w ] have 0 = 0 deg.; when the pitch is , all the

q

0.79 f L q Z2 [ 12 d3+ 4 di - d2 1 180 deg.10'S L 12w j slots have 0 = , and when the pitch is

qohms per phase (1)

For dimensions measured in inches, the constant - all have 0 g For intermediate0.79 becomes 2.006. The expression outside thebracket is a quantity containing factors common to pitches, the slot constant varies linearly betweenmany of the elements of reactance, and so it is con- these values. Thus, by plotting the function KSvenient to give it a special symbol, M, in future equa- against per cent pitch, where KS is a broken linetions. The bracketed expression is the "slot constant." function havihg values equal to

For a fractional pitch winding, the currents in the m 7r qq-mtwo coil sides in some slots are not in phase, and so, a KS= (1+cos at pitch, (2)more detailed analysis must be made. If the current in 2 qcoil side II lags that in I by 00, then in the slot carrying m being any integer, (as shown in Fig. 2), and substitut-the return conductor of I, in the bottom of another ing it in the expression for the slot constant, we haveslot, there will be (under balanced conditions) a top as the final equation for slot reactance:conductor carrying current 0Qahead of the current inl. d3 __

diAs the mutual inductance of I, with respect to II, is Xsi0t = M KS ±+ 8 + (1-Ks)equal to that of II with respect to I, the out-of-phase W 3w J 12w (component of voltage induced in each coil side by thleother will be canceled by the corresponding out-of -___ {Ks- ~-()Jphase voltage in the return coil side, leaving only the 4w 3JJin-phase components of voltage to be considered. Th thr ter in thi exrsio.snglgbe

The otalsel-indctane o I i proortonalto: except for zero phase-sequence currents, unless there iS1r d1-d2 - a large space between the top and bottom coil sides,4d+ J as in the case of a multi-speed machine with two

. ~~~~primary windings. The second term is very small,



unless the pitch is below two-thirds, so that only the TABLE Ifirst term is ordinarily of importance. VALUES OF Ks FOR USE IN EQUATIO-N (3)

The slot reactance given by (3), with the coefficients Winding pitchKs of Fig. 2, applies when balanced polyphase currents Connection Type of winding 0 1/3 2/3 1flow in the windings. It is sometimes desirable to Ki valuesknow the slot reactance for other conditions, however, Balanced three-as for single-phase, or for zero phase-sequence currents. phase......... three-phase, 60-deg. belts 0 0.250 0.750 1.00These reactances can all be obtained from equation (3), Line to neutral,s one-phase... three-phase, 60-deg. belts 0 0.500 0.500 1.00

if proper values of Ks are used. The appropriate val- Line to line, one-phase......... three-phase, 60-deg. belts 0 0.250 0.750 1 .00

Zero phase ....... three-phase, 60-deg. belts 0 1.00 0 1.00Balanced, three-phase.......three-phase, 120-deg. belts 0 .375 0.750 0.750

Line to neutral,one-phase.one-three-phase, 120-deg. belts 0 0.250 0.500 0.500

Line to line, one-phase......three-phase, 120-deg. belts 0 .375 0.750 0.750

Zero phase.....three-phase, 120-deg. belts 0 0 0 0Balanced, two-

phase ..;.. two-phase, 9C-deg. belts 0 0.333 0.667 1 .00Line to line, one-phase.... two-phase, 90-deg. belts 0 0.333 0.667 1.00

Zero phase.....two-phase, 90-deg. belts 0 0.333 0.667 1.00

For intermeldiate va1i s of pitch.Kh varies linearly between the tabu-lated values.

reactance, plus one-third of the zero phase sequenceI vI ~~~~reactance.

iII. END LEAKAGE REACTANCEThe leakage reactance of the end windings will here

be considered as due to two kinds of field. The firstit it j iz1 + < 91| kind is a revolving magnetic field of the same nature

as those that are produced in the core length. Thisfield flows in planes perpendicular to the shaft, and is

FIG. 2 2X/)(~

ues can readily be derived by the methods alreadyemployed, and therefore to save space the details ofthe work will be omitted here. The following table of zwv 4/,special values of Ks covers most of the cases that are met :_:- = -c C0,9X/,V4 eZ,9rlv/-with in practise. opc.7 t.,8r

It will be noted that the single-phase line to neutral FIG. 3 AXIAL LEAKAGE FLUX OF END VVINDINGSslot reactance of a three-phase winding is not half thereactance between lines, unless the winding pitch is due to the axially directed and winding currents only.exactly 0, 50 per cent, or 100 per cent. This is so Hereafter it will be designated as the "peripheralbecause the mutual slot reactance between any two leakage field." The second kind is due to the periph-phases, A and B, is only zero when there are as many erally directed end winding currents, shown in Fig. 4,slots containing an A coil side in the bottom and a which produce a leakage field flowing in radial planes,B in the top as there are slots containing a B coil side as indicated in Fig. 3. This will hereafter be called thein the bottom and an A in the top; a condition that "axial leakage field." The peripheral leakage pre-always exists with two-phase, but that only occurs dominates for synchronous machines with barrel typ\ewith pitches of 0, 50, and 100 per cent with three- windings. With single layer chain type windings, thephase windings. As shown by Park,1 the line to axial leakage flux is greatly increased, and so becomesneutral reactance of a three-phase winding is always the greater of the two. For induction machines, theequal to two-thirds of the positive phase sequence peripheral leakage is greatly reduced by the proximity



of the primary and secondary windings, and so the The first of these factors is given by the familiaraxial leakage usually predominates. expression:The procedure will be, then, first to find the per- V 7

meance of a cylindrical air core to a sinusoidally dis- ¢ f Kp Kd Z 10-8 (8)tributed encircling m. m. f., second to find the effective 2m. m. f. of this character produced by the end windings, The second factor is:third to determine thence the reactance due to the 8 7 D Lperipheral end leakage flux; and, finally, to determine A = gP (9)the additional reactance due to the axial leakage. This gcompletes the story for a salient-pole synchronous and the third* is:machine, where there is no mutual inductance between A 2 q Z Kp Kdthe armature and field end windings. For induction (10)machines, however, the further step of subtracting the 7r P / 2mutual reactance of the two windings from the sum The product of these three factors gives the magnet-of their total reactance is necessary.As shown by earlier writers3'6 l0, the total flux per

pole produced in a circular air core of one cm. axial - --length by a sinusoidal m. m. f. distributed around theperiphery is equal to just twice the maximum m. m. f. -in gilberts. Or: A ,

44wA 7r=2 (10) (4)

This equation assumes the flux to lie entirely in air | lon one side of the periphery of the core, and entirely %--in iron of high permeability on the other side. It makes _ Crno difference whether the air is on the inside or the out-side of the core, and neither does it make any differencewhat the radius of the periphery of the core is. If there _7 -<

is no iron present, so the flux paths lie entirely in air, -both inside and outside of the m. m. f., the reluctance -- .9is doubled, and so:

(.1)2= (5) ACE/DA91. £4'O WZ/M ws10 ,9 S/NU4 A ;' o ?flhVy10,3'r f,aurA-oe:PLO0W //A. U19Lh94, YS'P?9c_o ,9,V?'E1 4,qrA ,7.

8649'S 9A,9q,Vqg ,,v 73*7o 1-WIRAS Po,4r /i v0 7Wo Co,'vPa rsWr -

The flux per pole produced by the same m. m. f.acting on a uniform air-gap of length g is equal to: = 'Acz';:/

.8 7r A D r4'P" LS<44y

3= g P (6) FIG. 4 END WINDING CUTRRENT RELATIONSHIP

And, therefore, the ratio of the flux produced by izing reactance of a polyphase armature winding ina given sinusoidally distributed m. m. f. acting on the ohms per phase:periphery of an air cylinder of diameter D, with air 8 7r q f D LZ2 KP Kd2outside, as well as inside the cylinder, to the flux X0= P2 g 109produced by the same m. m. f. acting on a uniform air-gap of length g, is equal to: 0.319 Kp2 Kd2 D SM

-1)2 Pg(7) (11)q53 2 D 'The ratio of the axial length of the end windings on

where P is the number of poles of the m. m. f. both ends of the machine to the core length is:The magnetizing reactance due to the air-gap flux iv p D tan ae

of an induction motor with uniform air-gap g is equal to P L'(12)the product of the following factors:

1. The volts per phase due to unit flux per pole; where p is the fractional pitch, and at is the angle2. The flux per pole due to unit armature reaction; indicated in Fig. 4.3. The maximum of the sine wave of armature *R. R. Lawrence, "Frinciples of Alternating-Current

reaction per ampere in one phase. Machines," page 108.



Multiplying together (7), (11), and (12), the leakage harmonics may be approximately taken care of byreactance of the end windings due to the fundamental simply omitting the factor Kd2 in the first member ofof the peripheral field only is: equation (13).

[0.319Kd2DS1D Pg9 The component of end leakage reactance due toXEP ML p2g Jt 2D ) axial flux may be evaluated by considering the periph-

eral components of the end winding currents at any7r p D tan al [pD S Kd2 tan al fractional distance x from the outer end of the winding,L PL J F (p) = M L 2 p2 L jF (P) as indicated in Fig. 4. The peripheral currents in the

upper and lower layers of the winding are out of phase(13) in time by an angle equal to p 7r x, which differs by 7r

where F (p) is that function of the winding pitch that from the phase difference between the axially directedcorresponds for the coil end projection length to the components of the same currents. Thus, the resultantfunction Kp2 over the core length. The distribution peripheral current at any point is proportional tofactor Kd is the same for the end windings as for thecore portion, due to the unvarying widths of the phase cos P The enter of gravity of this current areabelts. 2

To evaluate F (p), consider an elementary section of is accordingly located at a distance from the end of thethe end windings at a distance x from the outer end, as core equal toindicated in Fig. 4. If x is expressed as a fraction ofthe total axial winding projection on one end, the pitch - 1of the winding at any point is equal top x. Therefore, x Cos p 7 x dxthe square of the ratio of the m. m. f. at any point to its r pD tan a 106value in the core portion is, 2P Cos P 7 x dx

sin2 P J 2

7r-p D tan ae /2tanp 7r/4and accordingly the value of F (p) is: 2P tapW/4)

p p p (14) sinpDpp2F (p) sin2 2 dx= 2pw (14) _ pD tan a 1 + P5

0 /

To evaluate the additional end reactance due to theirregular peripheral space distribution of the end wind- approx. (15)ing m. m. f., we may proceed as follows: The ratio of We will now make the assumption that the axial endthe nth harmonic flux per pole in an air core to the leakage reactance is the same as if the peripheral currentfundamental flux per pole is n times greater than the were concentrated in a conductor of circular cross-same ratio for a uniform air-gap, (from equations (5) section at each end of the machine at the distanceand (6) ). For a full pitch winding with a large number from the core ends given by (15). The proper diameterof slots per pole, the reactance due to the nth harmonic of cross section to assume for these conductors is evi-is therefore approximately 1/n2 times that due to the dently greater than the depth of primary slot, and lessfundamental for an air core, and 1/n4 times the funda- than the axial length of the end projection on one end.mental for a uniform air-gap, the latter ratio being For reasons of simplicity, we will assume this diameterderived in the section of this paper on belt leakage. The to be one-half the axial projection at each end, or, fromsum of the values of 1/n2 for all the harmonics present (12):in a full-pitch polyphase winding with an infinite 7r p D tan anumber of slots is equal to 1/Kd2.t The actual effect 4 p (16)of the harmonics is less than this for windings withpitches a little less than 100 per cent, and greater for As the peripheral current extends over a radial depthvery short pitch windings, but these errors are not equal to the depth of primary slot, the equivalentimportant. The tooth harmonics due to the concen- diameter cannot be less than this. Hence (16) istration of the m. m. f. in a small number of conductors really based on the assumption that the primary slotare smaller in the end windings than in the core, since depth is about equal to one-third the axial end pro-the finite slot depth makes the actual length of flux jection, and if wide variations from this relationshippath much greater for the higher harmonics than in the occur, then (16) will be correspondingly in error. Theideal case, for which the formulas are derived, where the effect of this error is to make the calculated reactancem. m. f. is concentrated in a cylinder of zero thickness. too high for very deep primary slots and too low for veryThus, the increase in end leakage reactance due to shallow slots. However, since only the logarithm of

ISilberstein, "Synopsis of Applicable Mathematics," p. 87. this effective diameter enters the final equation, a



considerable error in the diameter makes only a small very closely. The substitution of 0.3 (3 p - 1) for theerror in the result. complicated function of pitch given in (20) is justifiedThe current in this conductor varies cyclically around by the comparison of the two values shown in Fig. 5.

the periphery, its r. m. s. value for a full-pitch winding Over the ordinary range of pitches from 0.5 to 1.2, the

being1 agreement is very satisfactory. We will, therefore,being , 2times the total axially directed current in adopt (21) as our final expression for the end leakage

2 v/ 2 reactance of a synchronous machine with a barrel-typeone pole pitch, or: primary winding.

(2 1) ( Kp Kd q I Z) ({2 a/2)\ Equation (21) gives the end leakage reactance, under

2 -\/ 2 J P Jt 7

qIZK,Kd

We will neglect the leakage flux that enters the endsof the armature core in an axial direction, as it islargely canceled by opposing eddy currents in thepunchings, that are entered broadside on. Then, the _peripheral currents at the two ends may be consideredas flowing in a pair of parallel wires separated by adistance twice that given by (15). _ YAssuming the air-gap diameter, D, large compared LeS 1

with this distance, the problem reduces to that of cal- |7/|| I

culating the reactive kv-a. of a single-phase trans-mission line consisting of round wires of a diametergiven by (16), separated by a distance twice thatgivenl by (15), of a length w D, and carrying a currentper conductor given by (17). The inductance of sucha line is equal to:

20D [12+ 2 1 n 4 (1 + p2/5)] henrys

which is practically equal to:

4wrD (8.2 +p2) (8(5) 109

The total reactive volt-amperes due to the axial endleakage flux are, therefore, equal to: FIG. 5-END LEAKAGE REACTANCE COEFFICIENTS

13.1 f q D Kp2 Kd2 Z2 (1 + 0.12 p2) (19) the condition of balaneed polyphase currents flowingXEA = 10° P2 in the armature winding. As the derivation of thisAdding (19) to (13), and dropping the Kd2 factor of equation is based on the assumption that the end

(13) as previously explained, we obtain for the total leakage fields are due to revolving m. m. fs., withend leakage reactance: nearly negligible space harmonies, the reactances for

p D S tan a F (p) other winding connections may be taken as proportionalXEnd = M p 2 p2 L to the corresponding values of armature reaction.

On this basis, the coefficient 0.3 of equation (21)Kp2 Kd2 D S (1+ 0.12 p2) - may be changed in accordance with the following table

+ 6 p2 L J (20) in the special connection cases:

We will take a = 60 deg., a value higher than the TABLE__________II__usual angle of the ends, but one that checks the mean ||Value of coefficientlength of conductor on most windings, and so one that Connection Winding for equation (21)allows something for the straight portion beyond the Balanced polyphase................... .........|two orthree-phase0.3core. Equation (20) then reduces to: Single-phase, line tO lineutral] two or three-phase 0.23

0.3 1M D 5 (3 p- 1) Zero phase .................three-phase, 60 or 120

End = p2 L Zero phase ....|two-phase . ....|0.3



IV. ZIGZAG LEAKAGE REACTANCE The distinction between belt- leakage and zigzagInasmuch as the stator windings of polyphase leakage is that the former is due to the concentration of

machines consist of coils arranged in a finite number of the m. m. f. in a definite number of phase belts, whileslots and connected in a finite number of phases, usually the latter is due to its concentration in a definite num-two or three, the shape of the wave of magnetomotive ber of slots. No accurate dividing line can be drawnforce along the air-gap surface is not sinusoidal, even between the two kinds of leakage, as they are mutuallythough the winding carries balanced sinusoidal poly- dependent on each other, and as in the case of one slotphase currents. At any particular instant the shape per pole per phase, for example, they coalesce into theof the m. m. f. wave is stepped, the agreement between same thing. The reason for making a distinction isthe stepped figure and a true sinusoid becoming closer that, for ordinary design proportions, a change in theand closer as the numbers of slots and phases are made number of slots affects only the zigzag leakage, while alarger and larger, until perfect agreement results when change in the winding pitch affects only the beltboth the number of slots and the number of phases leakage, so that the independent effects of number ofare infinite. slots and of pitch can be more easily dealt with whenAs shown in reference 3, the air-gap flux produced the two parts are considered separately.

by this stepped m. m. f. wave in a salient-pole syn- Independent formulas will be developed for the beltchronous machine consists of three elements: and zigzag reactances, therefore, but it must be recog-

1. A fundamental sine wave stationary on the rotor, nized that, whenever the number of slots per pole isinducing fundamental frequency voltages; small, the two are not separable, and the formulas

2. A series of harmonics moving with respect to become inaccurate. The only accurate way to obtainthe rotor and also inducing fundamental frequency the total differential reactance is to compute it forvoltages; and each case separately, and tabulate the results. The

3. Other harmonics inducing other than funda- computation of the total for any regular winding can bemental frequency voltages in the armature winding. simply carried out by the means suggested by Chap-Of these, 1 is the effect of the armature reaction, 2 man.' This has been done in Appendix A, wherein tablesconstitutes an element of the leakage reactance, and 3 for some of the more usual regular windings are given.may be disregarded. Thus, if we now derive a formula The definitions adopted will be as follows:for 2, we will complete the work of determining the The per cent differential leakage reactance is thearmature leakage reactance. total per cent reactance due to harmonic fluxes crossing

In the present discussion, to begin with, an ideal type the air-gap that are not of the fundamental number ofof machine will be considered. The air-gap length will poles. Or, it is the total excess per cent reactance, duebe assumed uniform, and the flux will be assumed to to air-gap fluxes, above that of a similar machine, withcross the gap radially. The slots will be assumed to an infinite number of slots and of phases.have no real physical existence which might entail a The per cent belt leakage reactance is the totalvariation in air-gap permeance, but will be assumed to excess per cent differential leakage reactance of a poly-be merely the points along the periphery of the machine phase armature winding above that of a squirrel-cageat which the current is concentrated and at which, winding with the same number of slots.therefore, the m. m. f. changes. Under these conditions, The per cent zigzag leakage reactance is the totalthe air-gap flux will vary along the periphery of the per cent differential leakage reactance of a squirrel-machine at any instant in a stepped wave shape exactly cage winding, or of a winding with as many phasesproportional to the m. m. f. at each point. as it has slots per pole.

If the flux wave produced by the stator currents In this section we will consider the zigzag leakage,alone be considered, it has been shown that the stepped or the harmonic fluxes produced by a squirrel-cagewave of flux may be analyzed into a fundamental winding. It is shown in Appendix A by Chapman'ssinusoidal distribution of flux having the same number method that the total inductance of a full pitch windingof poles as that for which the stator winding is con- with one slot per pole per phase, as a ratio to thenected, plus an infinite number of harmonic flux inductance of one full-pitch coil is:waves having different numbers of poles.3 There 1being no rotor currents by assumption, the magni- Q =- CSC2 (22)tudes of the fundamental, and of all of the har- s 2 smonic fields, are proportional to the current in the where s is the number of slots per pole.windings, the back e. m. f. produced by these fluxes in And, it is also shown in Appendix A that the totalthe stator winding is proportional to the current, and useful inductance of such a winding, due to the funda-the fluxes are truly self-inductive. mental flux it produces, as a ratio to the inductance ofThe reactance due to these harmonic fields consti- one full pitch coil is:

tutes the "air-gap leakage," or "differential leakage" 4 sreactance of the machine, which is usually divided into Qu= (23)two parts, the "belt leakage" and the "zigzag leakage."



Hence, the zigzag leakage, Xt, is equal to: KPnKdn (KPnKdn 2 KP =iKdn :=2Xe- (7 CcS1r2s-) 92d (f7 -d Po' + 1/2( Pi d p2N)Xt=Xa( ~- CSC2 ir/2 s -4 s2(24)pN+n±

where the plus signs are taken if the nth harmonicwhere X,, is given by (11). rotates backwards with respect to the fundamental,As 7r/2 s is a small angle, (24) may be developed into and vice versa. For the quadrature axis, the signs of

a series of which only the first two terms need be the P2 terms in both the fundamental and the harmonicconsidered, so that: magnetizing reactance expressions change sign.

2 Xa 77.2 V4 It may easily be shown that the product KP KPn2xt = 102 l1±+ 20 2± + 4S (25) is always either negative or very small, and that

KdnKdn,2is always negative for the low order (beltor, approximately: leakage) harmonies, but is positive for the tooth

5 Xa 5 ' P X (zigzag leakage) harmonics. Hence, the numericalxt = 6S2 =6 S2 J (26) values of the two terms in the foregoing expression are

additive for the belt leakage, but are subtractive forEquation (26), then, gives the desired result, the the zigzag leakage, and the contrary is true for the

total differential leakage reactance of a regular winding quadrature axis.with one slot per pole per phase, or, by definition, the It is quite feasible to carry through the indicatedzigzag reactance of any winding with the same number numerical work for any particular case, using theseof slots per pole. Clearly the error in the formula is general methods, but to do so for the general case issmall for large values of s, but it makes the calculated quite impractical. We will, therefore, make somevalue a little low when s becomes less than 3. The bold assumptions at this point, with the object ofvalue of Xt thus found applies to a single winding, and obtaining a simple formula accurate enough for generalto the case of a uniform air-gap length. use. First, we will assume average values for the

For the case of a salient pole synchronous machine, permeance coefficients, corresponding to a two-thirdsthe zigzag leakage can be found from the same formulas ratio of pole arc to pole pitch, a minimum air-gap twoif an equivalent uniform value of air-gap length can per cent of the pole pitch, and a maximum air gap 1.5be ascertained. It has been shown by Doherty and timesthe minimum, as follows:Nickle3 that the reactance due to the nth harmonic of pil 2/3pmnIu 12 nd N =13 Nprimary m. m. f. depenids on the average permeance, On3thisbs, h nt2hroni anetiin reactaNcand on the second harmonic of permeance variation On this basis, the nth harmonic magnetizing reactanceonly, and that these two permeances have different varies about 15 per cent, above and below its average,values for each harmonic. The effect of the variation of between the direct and quadrature axes. We shallpermeance due to the salient poles is to make the leave this relatively small variation out of account,fundamental armature flux and the belt leakage flux therefore, and calculate only the average reactance,greater in the main axis than in the quadrature axis, assuming it to be the same in both axes. Then, thebut to make the zigzag harmonic leakage fluxes greater ratio of the magnetizing reactance for the nth harmonicin the quadrature axis than in the main axis. This is to the direct axis magnetizing reactance of the funda-true because the predominating portions of the high mental is approximately equal to:order harmonic fluxes are produced at the points of Xan .9 Kpn2 Kdn2maximum armature current, and these come over the 2 2K 2pole faces (low reluctance) when the poles are in the Xad n K K I

quadrature axis, while they come over the interpolar Finally, we must take into account the fact that ourspaces (high reluctance) when the poles are in the main ideal assumption of true rectangular m. m. f. waves foraxis. each coil is not exact. Actually, the m. m. f. of eachThe average permeance increases with the order of slot is distributed across the slot opening, instead of

the harmonic, since the shorter the harmonic pole pitch, concentrated at a point in the center of the slot, and sothe greater the relative permeance for the peripheral the steps in the m. m. f. waves are not vertical, butflux that reenters the primary iron without ever reach- slightly sloping. This reduces all the harmonies ap-ing the pole face. For the same reason, the second preciably, especially the higher order ones. We shallharmonic of the permeance variation is less the greater assume that the average effect of this error is to reducethevalueofn. the magnetizing reactance to 5/6 of its ideal value.The same authors have shown that the fundamental Multiplying the last expression by this factor, therefore,

magnetizing reactance in the direct axis is proportional we find the proper average value of Xa to substitute into: equation (26) is:

Kp,2 Kd,2( Po' ± 1/2 p2'), 3and that on the same basis the direct axis magnetizing Xa = Xad (27)reactance for the nth harmonic is proportional to:



V. BELT LEAKAGE REACTANCE and rotor belt leakage must be added to obtain the

The belt leakage reactance is due to the increase total.in differential leakage caused by reducing the number of As a first step, it will be necessary to calculate thephase belts per pole, q, to a small number. When the Xainumber of slots per pole is very large, this effect can ratio Xa where X is the magnetizing reactancebe readily calculated, but as the slots are made fewer,

the eltleakge ncresesto wo o thee tmesitsof the nth harmonic. A measure of the effectiveness ofthe belt leakage increases to two or three times itsthsaorwnignpodcgfuamtlfuxslimiting value. When the slots finally become equal the stator winding in produeing fundamental flux isto one per pole per phase, the belt leakage becomes zero, the number of turns in series per phase times the pitchby definition. As the effect of the number of slots is and distribution factors for the fundamental number offurther complicated by variations in the winding pitch, poles orit is not considered practicable to present any adequate N1 =K1 Kd1 Nformula for the belt leakage in the general case. How- in the same way the effectiveness of the stator windingever, as the belt leakage is small by comparison with the in producing nth harmonic flux is measured byzigzag leakage, unless s is large, and as in this case the Nn = KPr, Kd N,belt leakage formula is fairly accurate, these errors are where KP,,. Kdn are the pitch and distribution factors,not important. For accurate results, it is necessary to respectively, of the stator winding for the nth harmoniccompute the total differential leakage reactance, and flux.tabulate the results for all useful windings, as outlined

zen 1X ~~~~~~~~~Ifa voltage iS applied to the primary winding and iSin Appendix A. first assumed to be opposed entirely by fundamentalThis is especially necessary in the case of fractional flux and then entirely by nth harmonic flux, the ratio

slots per pole, as here the belt leakage increases greatly, of the magnetizing currents in the two cases will be theand as it varies considerably, depending on the partic- ratio of Xa to Xa.ular sequence of large and small numbers of slots per If nis the total flux per pole, thenphase belt employed.The harmonic fields corresponding to the belt leakage = 4 N1

are of lower orders than those of the zigzag leakage, n

as the former are primarily due to the phase belts which and if B is the air-gap flux density, thenextend over several of the slots to which the latter aredue. Further, since all the harmonic fields must N1create fundamental frequency voltages in the producing B = B1Nwinding, and since the nth harmonic has n times thefundamental number of poles, it follows that each since the area of each pole for the nth harmonic is 1/nharmonic must revolve with respect to the producing timesthatforthefundamental.winding at one nth of fundamental speed. Therefore, Since for the nth harmonic there are n times as manythe harmonic fields due to the primary winding induce poles to magnetize as there are for the fundamental,high frequency voltages in the secondary, and, if the the magnetizing current in the second case is:latter is short-circuited, produce damping currents in it.A regular phase winding, or a salient-pole field winding, Imn n2has so little admittance for these voltages that this N2ndamping effect can be neglected. But, in the case of asquirrel-cage, the induced currents are sufficient to And, as the magnetizing reactance is equal to thegreatly reduce the belt leakage harmonic fluxes. The ratio of the applied voltage to the magnetizing current,zigzag leakage fluxes are only slightly reduced in usual tcases, as the squirrel-cage admittance for these high 1 / KPn2Kd,z2 (2order harmonies is small. %X 2aKn 22 2

Therefore, the primary belt leakage can be practicallyneglected in all machines with integral slots per pole The total belt leakage reactance is equal to the sumper phase, having squirrel-cage secondary windings, and of the magnetizing reactances of all the harmonics ex-so we will assume for this case: isting with a definite number of phases, a definite

(28) winding pitch, and an infinite number of slots. It isXB =O()well known that a winding with q phase belts per pole

For regular windings with an infinite number of slots produces only those harmonics whose orders are equalper pole, the belt leakage reactance is the sum of the to 2 k q ± 1. And, for all these harmonics, the dis-magnetizing reactances of all the harmonic m. m. fs. tribution factor with an infinite number of slots isFor synchronous machines the stator only need be simply equal to one nth of the fundamental distributionconsidered, while for induction machines the stator factor. Hence, we may write:



ne Go 2The actual value of XB for any pitch for eachXB = Xa > ' (30) type of winding can not be expressed by any simpleP1 formula, but must be found by actually carrying

where n takes only the values 2 k q ± 1; k = 1, 2, 3, ... out the summation indicated in (30). This has beenFor a full pitch two-phase winding with 90 deg. phase done, and the results are shown in Fig. 6.

belts, this is: The value of XB for a salient pole synchronous ma-chine is also approximately given by the foregoing

XB90o = Xa ( 3 + 54 + 74 + * )equation, if 3/4 the direct axis magnetizing reactance isused for Xa, as shown in the previous section. Sum-ming up, therefore, the belt leakage reactance is equal

7F4 \to:96 - 1 Xa = 0.0147 Xa (31) XB = 0 (28)

for any machine with a squirrel-cage winding and inte-For a full-pitch three-phase winding with 60 deg. gral slots per pole per phase;

phase belts, it is: 31 1 1 XB - Xad (KB1) (33)

XB60= Xa + + 1 13 (33/74 14 14for salient-pole synchronous machine without squirrel-

1B(1 1 ) 7r14)X cage; and with integral slots per pole per phase; and=(I + XBg0) t1- 34 - Xal 486 X p2

XB = XT= 5 Xd (34)= 0.00214 Xa (32) 8 S

for a salient-pole synchronous machine with fractionalI 1B 114It ~~~~slots per pole per phase. This last equation is arbitrary,

ezat_a tJW X _ t ias XB for fractional slot windings varies very widely,but in a general way XB in such cases is larger the largerXT S, and it is on t e average of the same order ofmagnitude. From some unpublished work of ProfessorA. A. Bennett, the differential leakage reactance forsix different fractional slot windings averaged 1.82times XT. Hence, our assumption of 2 for this factorin the general case is not unreasonable, and should besatisfactory until such time as an adequate table ofdifferential leakage reactance values for irregular

[tlI ___Q__C _ _ _ r - - - - - -1-- Nwindings is available.Since some of the harmonics of belt leakage flux for

fractional slot windings are of fractional, and many areof rather low, orders, a squirrel-cage winding will have

To<'- 1"1 t _ l / _ _ _ _ L t _ C I __ l _ _ _demagnetizing currents induced in it, which may reduceI4L4- 1t1t 1 111 \1ll F 4 ZFXXF the belt leakage reactance considerably. The magni-

_00161, lav, ~~~~~~~tude of this effect is extremely variable, however, as itXI I depends on many factors. It is the writer's opinion

that the demagnetizing action of a squirrel-cage on thebelt leakage reactance due to fractional slot windings is

Ai+t

V[ / small for usual design proportions, and so it will be

IrbrE l ll 3l nreglected in the present paper.In these equations, we have derived expressions for

lo 3* Jo 0SXO .5- /Po /40 40/AoF all the elements of the reactance, and we may now turnto their application to actual machines.

FIG. 6 CHART OF VALUIES OF KB THE BELT LEAKAGE VI. APPLICATION TO SALIENT POLE SYNCHRONOUSCONSTANT MACHINES

The leakage reactance in ohms of a salient poleFor a full-pitch three-phase winding with 120 deg. synchronous machine with integral slots per pole, and

phase belts, it is also 0.00214 Xa, but in this case it with a squirrel-cage winding, has been determined byincreases rapidly as the pitch changes from unity, due (3), (21), (26), (27), and (28), to be:to the resulting dissymmetry of the winding, while in r S-_ ___d \ 0.3 (3 p-i) D) Sthe other cases XB decreases as the pitch changes from 271= M LK5 I + 3wJ+ p

unitJr.~ ~w wjP2



5 pP Finally, therefore, our formula for theper cent leakage+ 8 ( )2 Xad (35) reactance, expressed as a decimal, of a salient-pole

8 / synchronous machine with a barrel-type armatureThe belt leakage given by equation (33) or (34) winding, and fractional slots per pole, in inch units, is:

should be added in case the machine has fractional 20 A P L 3p + 1\ d3 dislots or has no squirrel-cage winding. The second XI KP2Kd2 LS 4 A w + 3w)order terms of the slot reactance (equation (3)) have P d

been omitted as being non-essential for normal syn- 0.3 (3 p- 1) D 1.1 Achronous machines. + p ] + F ((P/S)2 + 0.6 KB) (37)

Expressed as a fractional voltage drop due to full-load current, (34) becomes by (8), (10), and (11): If integral slots per pole are used, the (P/S)2 term

7.90 A r PLKS / d3 di should be multiplied by 0.5 If in addition the machineX KI 2K2 + 31 j has a squirrel-cage winding, the KB term is to be

omitted. The reactance is assumed to be the same in

0.3 (3 p- 1) D 5 p 2 A K\ both axes, but actually it is greater in the quadrature+ i]+ 8 ( S a( F--) (36) axis than in the direct axis by about 1/4 of the A/Fp8Sf F / term.

for lengths measured in cm., the constant 7.90 becoming As the value of X1 found from test is totally different,20.0 if inch units are used. The value of F to be used depending on how the armature reaction is con-in the last term of (36) is the number of normal voltage verted into equivalent field ampere-turns, it is neces-

sary to consider the matter before proceeding to"ef=/o I ZLUX A2NDF,TetqL ,q () the comparison of test reactances with those given by

r0e.,s^6ecF&i,F0 227A/E .W3Wieseman2 has derived accurate coefficients by flux0oe9,* sea~ 70 ,qae-: >sE 0J7 ^fJ 0AI

_Z&

_ /_7- _Z7 11}- plotting for the calculation of the fundamental sine1.04 _ /I _ _ ___>4$ ~wave of flux produced in a salient pole machine by the

74.2-I~~~~~<t- VAO ,- X t=tconcentrated field turns, and by a sinusoidally dis-tributed armature reaction. By taking the ratio ofthese two coefficients, the necessary factor, Ka,, toconvert the armature reaction ampere-turns into

B,06lr, t e-equivalent field ampere-turns, can be derived. This has0ov ! > | been done by Mr. Wieseman, and the result is shown in

/.021 _ _ T s_ - 0& SelFig. 7. By substituting the quadrature axis coefficients,0. _060 0_ a 0.l75 also given in reference No. 2, for those of the direct axis,

the armature reaction and the total, or synchronous,reactance in either axis can be accurately calculated.The usual way to determine the armature leakage

07e o.07 voa ao3 004 005 reactance by test is first to take open and short-circuitcharacteristic curves, which give armature voltage on

FIG. 7 open-circuit and armature current on sustained shortcircuit, respectively, as functions of field current.

field ampere-turns, and Ka is the constant required to Then, the field current corresponding to full loadconvert armature reaction ampere-turns into equiva- armature current on short circuit, divided by the fieldlent field ampere-turns. Ka depends on the particular current corresponding to normal voltage on the air-gapshape of field pole and length of air-gap used, but it has line, is equal to the synchronous reactance, expressed asan average value a little less than 0.9, and is given by a fraction. By subtracting the calculated armatureFig. 7. reaction, also expressed as a fraction of the no load air-A further simplification of (36) can be made by gap field ampere-turns, from the test value of synchro-

3 p + 1 nous reactance, the test value of armature leakageputting Ks = _, which is exactly correct for reactance is found. The calculated armature reaction

ampere-turns are given by (10), multiplied by the valuethree-phase windings with pitches between 2/3 and 1, of Ka from Fig. 7.and which is slightly higher than correct for all other As the errors of the test value of synchronous reac-windings, (Fig. 2). As two-phase windings have higher tance probably average one per cent, and as Fig. 7 alsobelt leakages, and as the neglected second order terms involves slight errors, it is evident that this method ofin the slot reactance become appreciable when the pitch obtaining the leakage reactance is inherently veryis below 2/3, the small errors so introduced are on the inaccurate. On high-speed machines, the leakageright side. reactance is very small by comparison with the total



reactance, so that the results are particularly inaccurate TABLE IVMEDIUM HIGH SPEED MIACHINES

in thiscase.TetXIt is possible to measure the total leakage reactance ______Calculated Tes TestXd

of a synchronous machine at standstill, just as for an Poles Kv-a. Freq. Xid Xad Xd Xd Calc. Xdinduction machine, but this requires the inclusion of the 8 560 60 0.067 0.815 0.882 0.884 1.002reactances of the field winding and squirrel-cage, if 8 800 60 0.077 1.338 1.415 1.416 1.001

pre-sentwhich do not enter the calculation of steady 8 1200 60 0.086 1.073 1.159 1.150 0.992present, d o n te 8 2500 60 0.143 2.100 2.243 2.230 0.994

state performance. It is also possible to measure the 8 5000 60 0.161 1.750 1.911 1.928 1.009armature reactance with the rotor removed and to 10 1100 '60 0.097 0.983 1.080 1.098 1.019determine the leakage reactance as the difference be- 10 1880 60 0.061 1.139 1.200 1.214 1.012tween the total and the calculated reactance due to the 10 7500 60 0.080 0.853 0.933 0.922 0.988

fundamental of the flux produced in the air core. This 12 1000 25 0.081 0.810 0.891 0.905 1.01612 1900 25 0.089 0.758 0.847 0.842 0.994

last method involves large errors in the amounts of 12 2400 60 0.159 1.103 1.262 1.245 0.986zigzag and end leakage, however, and is not as accurate 12 2500 60 0.093 1.452 1.545 1.564 1.012

12 5000 60 0.132 1.560 1.692 1.667 0.985as the usual short circuit test method of measurement. 12 5000 60 0.101 0.750 0.851 0.852 1.001Finally, it is possible to insert exploring coils on the 12 5300 60 0.135 1.280 1.415 1.414 0.999

12 7000 50 0.115 0.900 1.015 0.987 0.972armature surface and by their means measure the net 12 12500 50 0.110 1.048 1.158 1.172 1.012flux existing during a short-circuit test, which gives a 12 15000 60 0.086 0.667 0.753 0.766 1.017measure of the leakage reactance. The dissymmetry 12 15000 60 0.137 1.065 1.202 1.208 1.005

and wave form errors introduced, unless the exploring 16 3500 50 0.114 0.969 1.083 1.071 0.989Average ...........1.000

coils exactly follow the armature coil grouping, together Average error from mean......3-.010with the fact that the slot leakage part of this flux links Averageerrorfrom 1..010only a portion of the winding, make this method of no TABLE Vpractical use. For these reasons, the usual method of MIEDIUM LOW-SPEED MACHINESfinding X1 from open and short-circuit tests has been Test Ndfollowed in ehecking the results obtained with the __i _Calculated Test -formulas derived in this paper. PolesKv_a. Freq. Xld Xad Xd Nd Cale. Xd

18 1.650 40 0.108 0.844 0. 952 0.993 1.043

TABLE III 18 1,650 60 0.123 0.961 1.084 1.100 1.015

HIGH SPEED M\ACHIN'ESHIGH_SPEED_____ACHI_ES 20 2,250 40 0.148 0.975 1.123 1.136 1.011Test Xd 20 14,444 50 0.123 0.896 1.019 1.023 1.004

Calculated Test __ _

Poles Ky-a. Freq. Xld Xad Xd Xd Calc. Xd 22 2,000 60 0.123 1.078 1.201 1.188 0.989-____ 1.2____________ _______-_____.........22 7,900 60 0.162 1 017 1.179 1.198 1.0164 90 60 0.105 0.955 1.060 1.043 0.9844 105 50 0.054 1.220 1.274 1.250 0.981 24 1,100 60 0.132 0.992 1.124 1.152 1.0254 1S0 60 0.061 1.002 1.063 1.032 0.971 24 4,000 50 0.136 1.027 1.163 1.178 1.0134 600 60 0.060 2.163 2.223 2.195 0.987 24 6.880 50 0.131 0.688 0.819 0.826 1.0094 940 I 25 0.128 1.770 1.898 1.895 0.998 24 tO,700 50 0.120 0.618 0.768 0.761 0.9914 2000C 25 0.198 1.562 1.762 1.793 1.018 24 30,00 60 0.128 0.617 0.775 0.803 1.036

4 3000 25 0.117 2.300 2.417 2.355 0.9744 5700 25 0.109 1.857 1.966 1.937 0.980 28 10,125 60 0.147 0.919 I.Q66 1.089 1.0226 150 25 0.102 1.040 1.142 1.092 0.9546 187 69 0.063 0.998 1.061 1.068 1.007 30 3.125 60 0.106 0.973 1.079 1.080 1.001

6 2C0 60 0.055 0.804 0.859 0.872 1.0156 4Cl00 60 0.069 1.250 1.319 1.275 0.967 32 750 60 0.138 0.820 0.958 0.951 0.9936 435;) 60 0.077 1.127 1.204 1.172 0.973 32 2,250 40 0.180 0.918 1.098 1.113 1.014

6 500 60 0.151 2.065 2.216 2.212 0.998 32 9,375 60 0.142 1.064 1.206 1.202 0.9976 500 60 0.105 1.610 1.715 1.678 0.9786 1000 60 0.127 1.932 2.C59 2.040 0.991 36 500 60 0.155 0.853 1.008 1.003 0.9956 1500 60 0.C92 2.025 2.117 2.145 1.013 36 1,50() 60 0.117 0.881 0.998 0.988 0.9906 :315 25 0.098 1.019 1.117 1.133 1.0146 5000 25 0.097 1.060 1.157 1.161 1.003 40 750 60 0.162 0.756 0.918 0.906 0.9876 5000 25 0.088 1.592 1.680 1.745 1.039 40 1,375 60 0.088 0.619 0.707 0.716 1.013

Average ..................... 0.999 Average ............. 1.008Average error from mean ..... 0).017 Average error from mean. 0. 013Average error from I ......... 09.012 Average error from 1.0 .014

In Tables III to VII inclusive, the calculated and inherent in Fig. 7 are considered. If the value of thetest values of synchronous reactance are compared for armature reaction reactance, Xad, and the test value offive groups of machines of different types. The total reactance, Xd, are assumed to be exact, themachines were selected at random, and the tests were test value of leakage reactance can be found by sub-all made in the regular commercial routine. The tracting the one from the other. This has been doneaverage absolute error forthe entire listofl100machines for each group of machines, and the average errorsis 0.4 per cent, and the average numerical error is 1.7 between the test and calculated values of X1 for eachper cent, so that the results are as accurate as could case are given in Table VIII, together with the cor-reasonably be expected when the errors of test and those responding errors in Xd.



The average absolute error of the leakage reactance each one uses. The formulas as given were derivedfor the 100 machines on this basis is 0.6 per cent, and the straightforwardly from theoretical considerations, andaverage numerical error is 14.6 per cent. The worst no attempt has been made to make empirical cor-errors occur for the high speed machines, and inspection rections, although simplifying assumptions have beenof Table VIII indicates that the calculated value of Xd freely used.is appreciably too high for high speed machines and a The causes for the varying magnitudes of error shownlittle too low for low speed machines. By empirical in Table VIII are brought out more clearly by a study

TABLE VI TABLE VIIILOW SPEED MACHINES Percent Numerical

Calculated Test Xd errorl__l_________________d__ TesTest Test Per cent -_ _ _

Poles Kv-a. Freq. Xld Xad Xd Xd Calc. Xd - Absolute From_ Type of machine Calc. error From 1 mean

56 3,500 60 0.144 0.798 0.942 0.960 1.019 -56 4,500 60 0.147 0.860 1.007 1.013 1.006 Errors in values of XI56 5,720 60 0.159 0.648 0.807 0.775 0.960 High-speed (II) ......... ........ 0.86 -13.6 31 3060 750 60 0.177 0.699 0. 876 0.902 1.030 Medium high-speed (III) ......... 1.01 1.0 11 1160 1,500 60 0.155 0.871 1.026 1.041 1.015 Medium low-speed lV) .......... 1.06 6.3 11 1060 1,500 60 0.231 0. 587 0.818 0. 882 1.078 Low-speed (V) ......... ......... 1.06 5.7 12 1060 2,500 60 0.130 0.724 0. 854 0. 849 0.994 Small low-speed (VI) .............. 1.04 3.6 8 7

Errors in values of Xd64 625 60 0.168 0.782 0.950 0.907 0.955 High-speed (II) ......... ........ 0.992 - 0.8 1.9 1.764 6,250 60 0.128 0.870 0.998 1.013 1.015 Medium high-speed (III) ......... 1.000 0.0 1.0 1.064 8,000 60 0.254 0.776 1.030 1.043 1.013 Medium low-speed (IV) .......... 1.008 0.8 1.4 1.368 1,500 60 0.192 0. 766 0.958 0.961 1.003 Low-speed (V) ......... ......... 1.011 1.1 2.2 1.8

Small low-speed (VI) ........ ..... 1 .019 1.0 1.91.772 750 60 0.239 0.818 1.057 1.065 1.00872 875 60 0.223 0.818 1.041 1.068 1.02672 1,000 60 0.228 0.740 0.968 0.962 0.994 of the relative proportions of the various elements of the72 2,250 60 0.147 0.813 0.960 0.983 1.02472 12,250 60 0.213 0.782 0.995 1.015 1.020 total reactance for the different types of machines.72 32,500 60 0.166 0.926 1.092 1.130 1.035 Table IX gives the average ratios of slot, end, and80 3,500 60 0.187 0.810 0.997 1.020 1.023 differential leakage to the total leakage reactance; and82 2,250 60 0.162 0.745 0.907 0.898 0.990 also the average ratio of the total leakage reactance to

100 1,250 60 0.127 0.510 0.637 0.652 1.024Average ..................... 1.011 the armature reaction reactance for each case. Since inAverage error from mean........ 0.018 the high-speed machines, Xi averages only 6 per cent ofAverage error from 1 .......... -0 .022

Xd, while in the small low-speed machines it averages

adjustments of the coefficients of the end and differential 25 per cent, it is clear that a given error in the mea-

leakage components of the reactance, it is evident that surement of Xi makes more than four times as great an

these errors could be reduced. Such adjustments of the apparent error in Xi in the former as in the latter case.

constants should be made by each manufacturer forhimself, however, as they will depend upon the par- _TABLE_IXticular end winding constructions, and so forth, that Average ratios

Xslot XEnd XDit- Xl XI newTABLE VII _

SMALL LOW SPEED -MACHINES Type of machine Xl XI Xl Xd Xi old

Calculated Test Xd High-speed ............. 0.37 0.60 0.03 0.059 0.56PoesK r Xd XMedium high-speed.. ........0.570.35 0.08 0.087 0.58

poles Kv-a. Freq. Xld Xad Xd Xd Cale. Xd Medium low-speed...... 0.60 0.32 0.08 0.132 0.661 228 1 07 ~Low-speed .... -..... -- 0.67 0.16 0.17 0.189 0.70

Oe 200 60 0.249 0.970 1.219 1.228 1.007Smalllow-speed . ........

6.58 0.16 0.261 0.2521 0.72

36 240 60 0.244 0.833 1.077 1.076 1.001 Saio-pe . .8 01 .6 022 0740 80 60 0.281 0.660 0.941 0.962 1.02244 62 60 0.284 0.576 0.860 0.888 1.03336 100 60 0.302 0.704 1.006 1. OCS 1.002 This partially accounts for the relatively large numerical80 120 60 0.245 0.793 1.038 1.017 0.980 erosiXifrhg-p dma in40 360 60 0.208 1.013 1.221 1.256 1.029 errorsinX1forhigh-speedmachines.42 160 60 0.293 0.961 1.254 1.307 1.042 The last column of Table IX gives the average ratio44 50 60 0.430 0.972 1.402 1.460 1.041 of the new value of leakage reactance, to the value44 62 60 0.251 0.619 0.870 0.906 1.04144 62.5 60 0.143 0.417 0.560 0.552 0.986 given by the older formulas derived by Doherty and5406 1040 660 0. 277l 0.4698 0.69475 0.69658 1.010° Shirley.7 This ratio varies from a little over one-half52 140 60 0.207 0.611 0.818 0.838 1.024 for the high speed, to nearly three-quarters for the low-60 60 6.0 0.1008 0.3012 0.409 0.o404 0.988 speed machines. The percentage error between test60 250 60 0 .2909 0 .9012 1 .192 1 .183 0.992 and calculated values of Xl iS thus magnlified on the new62 280 60 0.195 0.6579 0.8521 0.826 0.969 b.asis, so that a direct comparison of the accuracy given74 375 50 0.224 0.781 1.015 1.005 1.010 by the two formulas cannot be made. The older

Average . ........ 1.010 formula had an average apaeterror of about 16 perAverage error from mean..........~.0.017 aprn

_______________ Average error from 1 ....... = 0.019 cent, so that taking into account the test errors, which


Feb. 1928 ALGER: SYNCRHONOUS MACHINES 507

cause a considerable part of the dispersion, the new machines, and the values compared with tests, with theformula is indicated to have about half the true error following results:of the old.

Zero-phase

VII. ZERO PHASE-SEQUENCE REACTANCE Assumed Sequence ReactanceCALCULATIONS PoWinding value Calcu- Per cent

CALCULATIONS ~~~~Poles Kv-a. Freq. pitch of Ko lated Test error

The zero phase reactance is important in the calcula- ---6 435 60 2/3 0.026 0.021 -24

tion of single phase short circuits, circulating currents in 36 20,000 60 0.80 0.25 0.070 0.0675 4delta windings, and in other cases. The slot portionof this reactance is given by the complete equation (3), VIII. CONCLUSIONSusing the appropriate values of KS from Fig. 2. For It is believed that equation (37) is at once the mostusual three-phase windings, KS for the zero phase accurate and the simplest comprehensive formula forsequence reactance is equal to 3 p - 2. The end the armature leakage reactance of a synchronous ma-leakage portion is very small, and will be neglected. chine that has been published. It requires no curves,The differential leakage portion is given by the ratio of no logarithms, and no tables for its use, but only a fewequation (12a) or (13a) to (14a) in Appendix A. Com- slide rule operations, and it gives a value of leakagebining these expressions, we derive the following formula reactance which, added to the armature reactionfor the zero phase reactance of a three-phase, 60 deg. reactance derived from flux plots, quite accuratelybelt winding, with a pitch between 2/3 and unity, checks the test values of synchronous reactance for theexpressed as a decimal: entire range of salient pole synchronous machines in

20 A PL commercial use.Xzero phase K2 K The leakage reactance so determined averages about

npad (p W oS,/two-thirds of that given by the widely used formulasderived in reference No. 7, since the latter included as

(3 p -2) (d3) + (9 p 5) 12 -(9 p-8) 12 leakage reactance a part of the fundamental sine wave1 of air-gap flux due to the armature, which links the

field winding in the direct axis, and thus constitutes a4 A Ko (p-2/3) part of the true armature reaction. The new value of

F Kp2 Kd2 leakage reactance, added to the squirrel-cage reactance,2 checks observed standstill reactances of synchronous

(P + + 7/18 ( -2/3)-( -2/3)2 (38) motors; and added to the field reactance checks thes 27 values of transient reactance found from oscillographic

tests on synchronous generators. With the old re-The corresponding formula for a winng pitch actance formulas, rather arbitrary reductions of the

between 1/3 and 2/3 is: calculated field reactance and tooth tip reactance

20 A / L were necessary before checks with transient and stand-Xzero phase = ) still reactances could be secured. Finally, with the old

formulas, the calculated increase in internal voltage of a

di d synchronous generator under load gave much higher[(2 - 3 p) d3 + (7-9 P) 12 -(4 - 9 p) 12 flux densities in low power factor machines than

L actually do occur, so that arbitrarily reduced values of

4 A Ko (2/3 p)field leakage under load were used in calculating satura-

+ A ( / - tion curves, thus establishing a series of compensatingF K 2 K,2 errors.

2 There is no space to demonstrate the validity of\

( ) -(2/3-p)2l (39) these statements here, and they are simply made toS 2 show that the acceptance of the new values of leakage

reactance involves a fundamental revision of theThe coefficient Ko is introduced in (38) and (39) to generally adopted design constants of synchronous

allowforthereductionoftheharmonicfluxes,especially machines. In writing the paper, an effort has beenthe third, by induced currents in the rotor circuits. made to make the new formulas and the new definitionsAs all the terms except the first in the last brackets of such that the further refinements which the future willthese equations represent third harmonic fluxes, thesebrneabededwtotginleigouenterms are generally reduced to a fraction of their appar- brpiong can bhe addeds wtoutmagainud althern ouaracon-stient values, and K0 should ordinarily be taken as less ceptonsts or teoders o antdeo h haatrsithan 0.5. The coefficient 4 of the last bracket in (38) cntnso einincludes the factor of 3/4 for the average permeance of NOMENCLATUREa salient pole machine, as previously derived. A = Maximum ampere-turns per pole ofThe zero phase reactance has been calculated for two armature reaction



di, d2, d3, w = Slot dimensions shown in Fig. 1 ductance of a regular, distributed, polyphase windingD = Gap diameter with s slots and q phase belts per pole (s/q a wholef = Frequency number), and a uniform air-gap length, followingF = No-load air-gap field ampere-turns per Chapman's method. In doing this, the self-inductive

pole voltage produced in the set of coils forming the innerg = Air-gap length (radial) halves of two adjacent phase belts in the same phase byKB = Belt Leakage constant, given by Fig. 6 their air-gap flux is considered as the sum of the sKP, Kd = Pitch and distribution factors of arma- voltages produced by the fluxes in the s teeth under one

ture winding (less than unity) pole. The voltage due to any tooth is proportional toKs = A function of winding pitch given by the square of the number of turns linking it, as the

Fig. 2 reluctance of each tooth pitch is the same, while bothL = Gross core length the flux per ampere and the volts per unit of flux are

2f L q Z2 proportional to the number of turns. It is most con-M - 1O7 S for inch units venient to make all the calculations in relative terms,

and, therefore, to take as the unit of inductance thep = Winding pitch expressed as a decimal self-inductance which the two adjacent half-phaseP = Number of poles belts of one phase would have if all their turns wereq Number of phases concentrated in one pair of slots exactly one pole pitchs = Slots per pole = SIP apart.S = Total number of slots Assuming the winding pitch to be b slots short ofV Volts per phaseXa = Armature reaction reactance -llI-,XB =Belt leakage reactanceofpiAXd Total, or synchronous, reactance in P_j

direct axis 2345 17 76''4 3

XI Total armature leakage reactanceFlxwt PasB

XI = Zigzag leakage reactanceZ - Series connected conductors per phasea =Angle of end windings as shown in Fig. 40 = Angular phase difference between cur-

rents in upper and lower coil sides in 4 7t 321 Aone slot

:~_PaeB=Flux per pole in c. g. s. lines

Appendix AEXACT CALCULATION OF DIFFERENTIAL LEAKAGE FIG. 8 IDFALMM\F. DIAGRAMS FOR POLYPHASE WINDINGS

REACTANCEChapman* has shown that the total inductance of a full pitch, (b being not greater than s/q), and referring

winding due to the air-gap flux can be very conve-niently calculated by adding the inductances due to the to Fig. 8 there will be (q + 1b central teethrectangles of flux produced by successive pairs of sym- qmetrically located slots. With this method the smallest that are linkedj2by all the turns in the two half belts ofunit of flux is that produced by a uniform m. m. f. phase A, and hence the total inductance due to theseacting over one slot pitch, so that the permeance varia- teeth will be:tions due to the slot openings are quite satisfactorily 1 s (q 1)taken into account by merely using the average per- 1 (q-bmeance, as in calculating the magnetizing reactance. s\ q )Thus, the point of view leads to the immediate con- ,clusion that all the harmonic fluxes; (hence the zigzagand belt leakage reactances) are exactly proportional onswl.elne by s-q ofteoalursanto the fundamental magnetizing reactance. And so sthe elaborate analyses of the permeance for the zigzag wl aeaprene2so h oa emac,sleakage flux as distinct from that for the fundamental ta hywl otiuet h nutneflux, that have frequently been employed heretofore,are shown to be unnecessary. 2 (s_- )We shall now proceed to calculate the total in- S*Reference No. 13. Similarly, consecutive pairs of teeth will each have



qls less turns than before, until the bth pair is reached. Of the remaining teeth there are b - 1 that carryThe next pair of teeth after that (1 and 1') will be full flux, but are linked by successively fewer coils of

s- 2q- bq the B phase. These contribute an inductance equal to:linked by of the total turns, so that they 1

coCOS [(S-q) + (S-2 q)will contribute an inductance equal to: cs [ + -

2 + . . . + (s -(b-1) q)]S3(s - 2 q-b q)2 There are also b - 1 pairs of teeth, (3' and 3") one

of each pair of which carries positive flux and the otherThe following pairs of teeth will have consecutively an equal negative flux, the former of which is linked by

2 q/s less turns than the last, until the outermost pair all, and the latter by successively less of the coils ofis reached. Hence, the total self-inductance of Phase A phase B. Each pair is linked by successively less ofis: the coils of phase A. Hence, these teeth contribute

1 [ S3 (q1) +an inductance to phase B of:L = L + S2 (1- b) +2 (s-q)2 + 2 (s-2q)21

S3 cos [q (s- (b- 1) q) + 2 q (s- (b- 2) q)s3 q+...-+ 2(s-bq)2+2 (s-2q-bq)2+2 (s-4q-bq)2+...I

. ~~~~~~+* . . + (b-1) q (s -q)]As the sum of the squares of the first n integers is The remaining teeth consists of pairs (6, 5-7, 4"-

n (n + 1) (2 n + 1) 1', 1"-2',2") carrying equal and opposite fluxes and6 , this may be reduced to: linking the same numbers of coils of phase B, so that

they contribute nothing to the inductance of phase B.r 6 2 z S1 bi Hence, the total increase of inductance of phase B

[63L 2 S s- q + 1- b due to phase A is:

-su+22 (S 1) (1) (281) cCos r/q[S2 (S- q b + 1

+ s(s-q+s-2 q + . . . + s- (b-1) q)+2q2 (S _ b-b) (s b)b)s -b-)] +wq(s-o(b-)q)+2q(s-(b-2)q)

an thi reue to [ q-2 bb q b2b212 q LAq owq s

++ 3 + (b)- 1) q (s-q)

andthisredueesto: ~~~~~~~~LAB= cos r/q [ -6 (b2-1)]

LAA 21 + qS (2s- 3b2S- bq+ Vq), ob q(a

< , zero if it is negative, since the inductance is always

q-1 positive.all for a winding pitch not less than q The corresponding expressions for values of b between

s q and 2 slq, corresponding to winding pitches betweenThe linkages produced by phase A with phase B one and two phase belts short of full pitch, are, if q is

are similarly found, except that, as the currents in the greater than 2:two phases are out of time phase by 7r/q degrees, and 1 1 2 sas conditions of symmetry make the out-of-phase com- LAA 1-3 q - S2 (b s-q/3), s/q < b 7r q (3a)ponents of the linkages cancel with the correspondingcomponents due to the other phases, the effective andcurrent in phase A must be reduced by a factor cos ir/q. r 1 q 3 bThe number of teeth that are linked by all of phase B LAB = cos 7r1/q L 2 q + 2 82 + 28

as wellasbyallofphaseAis (s q - b + 11, and b2qbsq bq 1J 3S (4a)

these will create linkages with B equal to:

(+)ceos (s- q -b-+Fl) s/'q <b <



And, the value of the self-inductance of one phase 2 / 2 7r 3 7rfor winding pitches greater than zero by not more than = 1-Cos 7r/s + 2 cos s + 3 cosone-phase belt width is: s

LAA= q (s ) [q+(s-b)(3s-qs+b q)] + .+(s-1)eo (s) w)

s <bs (5a) = cs c2 7r/2 s (a)

There is no need of considering values of q greater It is interesting to compare the total reactances of thethan 3, except for full pitch windings, as three-phase several types of windings by means of these formulas.belts per pole are all that are used in practise. A For a regular three-phase winding, equations (9a) andsquirrel-cage winding always has one slot per pole per (lOa) show that:phase, so that its differential leakage ratio is the same 10 2for all values of pitch. For this case s,/q = 1, and L60 = 9 + S" for full pitch(2a) reduces to:

s- 2 149 7LAB = S COSwr, S (6a) 144 + 4 s2 for 5, 6 pitch

and the corresponding expression for the nth phase is: 5 3

s-2 (n-1) (n-1)w - 6 + 282 for 2,3 pitchLAN = COS (7a)

s 85 1

The total inductances per phase are obtained by 9 + 2 for 1 2 pitchadding the self and effective mutual inductances, asgiven by the foregoing expressions. For q = 2, (2a) is 5 1zero, so that the total inductance of a two-phase 18 + 2 2 for 1/3 pitchwinding is from (la):

1 For a regular two-phase winding (8a) shows that:Lgoo =- 2/3 - 3 (4s-6b2s-4 b + 4 b3) (8a) 2 4

Lgoo = 3 + 3 2 for full pitchFor q = 2, (la) and (5a) are identical, so that the s

single equation (8a) is valid over the entire range of 9 1pitches from 0 to 1, for two-phase windings. 16 + 2 for 3//A pitch

For a three-phase winding, q = 3, and the total -inductance of one phase is given by the sum of (la) and 1 2twice (2a) for pitches between 2/3 and 1, and by the = + 3 for 1,2 pitchsum of (3a) and twice (4a) for pitches between 1/3 s

and 2,/3: 5 1

L60 = 10/9 + 2s (4s-6b2s-3 b+ 3 b3), 48 + 32 for4pitch2sl For a single-phase winding made by using two legs

0 < b < s/3 (9a) of a regular three-phase winding, the total inductance isand found by substituting 1 for cos 7r,/2 in (2a) and adding

1 (la) to it, which gives identically the same equationL60= 19/18 + 2 (5 s + b S2-6 b -9 b2 s + 6 b3), (9a) as for the original three-phase winding. Thus,

2sl the per cent line-to-line reactance of a three-phasewinding is always identical with the per cent three-

2 s phase reactance. For a single-phasewinding consistings/3 < b < 3 (lOa) of one leg of a three-phase winding only, the total

inductance is given by (la), (3a), and (5a) ,whence:(9a) and (lOa) become identical if b = s/3* Line to neutralFor a squirrel-cage winding, the total inductance of 7 2

one phase is, from (la) and (7a), placing b = 0 and L600 = 9 + s2 for full pitchq = s in the former:

s-2 s-4 2wr 17 3LK =1 +. <;COSw/S + S COSs + - 24 + 2 s2 for 5/6 pitch



5 1 reactance will give the desired values of total differential- +± 2 for 2/3 pirch leakage reactance.

The ratio of the useful to the total flux generated by a7 1 single full-pitch coil is 8/7r2, as this is the ratio of the18 + s2 for 1/2 pitch area of a sine wave to the area of a rectangle whose

fundamental it is. This factor, multiplied by the2 1 squares of the pitch and distribution constants of the9 + s2 for 1/3 pitch winding, and by the ratio, q/2, of the total revolving

field produced by all the phases acting conjointly to5 1 the field made by a single phase, gives an expression72 + 282 for1/6pitch for the reactance due to the useful flux. Or:

Thus, the line-to-neutral air-gap flux reactance of a 4 q3 sin 27r/2 q sin2 7r/2 (1 - b/s)three-phase winding operated single-phase is 7/10 of its Lm 2 S2 sin2 ,/2 a)value in three-phase operation for full and half-pitchwindings, but is only 2/3 of its three-phase value for Putting q = 2 in (14a), dividing it into (8a), and2/3 pitch windings. For a 1/3 pitch winding, it is subtracting unity from the result, we have for the total4/5 of its three-phase value. These variations are due differential leakage reactance of a two-phase winding,to the triple harmonic fields that are produced by each expressed as a ratio to the fundamental magnetizingphase winding separately, but that are canceled out reactance:by a 2/3 pitch, or by a three-phase connection.

Since the zero phase sequence reactance is equal to XD90° 1r2 (s-b) (S2+ b s-2 b2 + 2) sin2 w/2 sthree times the line-to-neutral, less twice the balanced Xm' 24 s sin2 7r/2 (1- b/s) -1three-phase reactance, of a three-phase winding, theforegoing expressions lead to the following expressions (lSa)for the zero phase sequence inductance: If s is large, and b is zero, corresponding to the caseZero phase of a full-pitch two-phase winding, (15a) gives:

L60,. - + 1 (2 s- 3 b2s- 6 b + 6 b3), XD,, w2 (82 + sin2ir/2s-r=99 s3 ~~~~~~~Xm24 n 2s-1 960 b s/q (12a)

5 1 74 . 687- -5 3 (2s +4bs2-6b-9b2S+6b3) +4882 (.412- .S2+61 (16a)

9 s348s

-= b8

(13If s is very large, (16a) reduces exactly to (31), thus

q < < q confirming the latter result. For small values of s,

Hence: equation (15a) is the only convenient means ofobtaining exact values for XD60. Numerical values' of it

Zero phase L60~ = -+ ~2 for full pitch for some usual windings are given in Table X.Zero phase L60 9 + s2 for full pitch

TABLE X

-18 + for 5/6 pitch DIFFERENTIAL LEAKAGE REACTANCE OF TWO-PHASE18 s2 WINDING EQUATION (l5a)

0 for 2/3 pitch Pitch s/2 = slots per pole per phasedeficiency

1 1 15 1 2 3 4 5 6

- 4 for 1/2pic18 + s2 / pitch 0 0 2337 0.0840 0.0468 0.03:30 0.0264 0 02291 0.2337 0.0583 0.0334 0.0251 0.0213 0.01932 0.0840 0.0285 0.0177 0.0147 0.0138

1 2 3 0.0840 0.0468 0.0189 0.0116 0.0096+ -2 for 1/3 pitch 4 0.1019 0.0330 0.0148 0.0089.

9 s-2 -----50.2337 0.0649 0.0265 0.01306 0.1222 0.0486 0.0229

All of this zero phase inductance is leakage flux, since 7 l 0.2337 0.0839 0.0398the fundamental m. m. fs. of the three-phases cancel 8___ - ___ 0.1382 0.0650with this connection.The useful, or fundamental, flux produced by the Putting q = 3 in (15a), dividing it into (lOa) and

same windings can be readily calculated from the usual (lla), and subtractx,ng unity from the result, the totalpitch and distribution factors, and the difference re- differential leakage reactance for a three-phase, 60 deg.maining after the useful is subtracted from the total phase belt winding is:



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5. 1922, Siegel, E., "Leakage Calculations for Three-Phasexm Induction Motors with Phase-Wound Rotors," (in German),

2(19 S3 2St3\ ~Elekt. u. Masch., Vol. 40, pp. 217-22, May 7, 1922. Sci. Abstr.,i7r2(19 S3+45t +9 b S2-54 b-81 bs+54 s)1n1 2 s 1 See. B, Vol. 25, pp. 502-03, Oct 1922, Abstract translation.486 S Sjn2 7r/2 (1 - b/s) 6. 1920, Alger, P. L. and Eksergian, R., Induction Motor

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:For a full-pitch three-phase winding with a large 8. 1916, Chapman, F. T., "Air-gap Field of the Polyphasenumber of slots per pole, (17a) may be reduced to: Induction Motor," Electrician, Lond., Vol. 77, pp. 663-67, 705-08,

730-34, 904-06, 936-38; Vol. 78, pp. 9-12, 49-50. Aug 18-XD6O_ 2wr2 (5 + 9)in2i-/2s~ 1 Sept. 1, Sept. 22-29, Oct. 6-13, 1916.Xm 243 9. 1912, Fechheimer, C. J., Self-Starting Synchronous

Motors, A. I. E. E. TRANS., 1912, Vol. 31, pp. 529-85; Discussion,5 w4 w44 0.672 pp. 586-604.

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Alternating-Current Machines," (in German), Elekt. u. Masch.,If s is very large, (19a) reduces exactly to (32). Vol. 27, pp. 201-08, Feb. 28, 1909. Sci. Abstr., Sec. B, Vol. 12,Numerical values of (17a) and (18a) for some usual pp. 160-61, Apr. 1909, Abstract-translation.

three-phase windings are given in Table XI. 11. 1907, Adams, C. A. and others, Fractional Pitch WindingsFor a squirrel-cage winding, q = s and b = 0, 50 for Induction Motors, A. I. E. E. TRANS., 1907, Vol. 26, pp.For a quirrl-cag windng, q= s ad b =0, so

14805-1503; Discussion, pp. 1525-26.TABLE XI 12. 1907, Hellmund, R. E., Zigzag Leakage of Induction

DIFFERENTIAL LEAKAGE REACTANCE OF THREE-PHASE Motors, A. I. E. E. TRANS., 1907, Vol. 26, pp. 1505-24.WINDINGS'' '

EQUATIONS (17a) AND (18a) 13. 1905, Adams, C. A., The Design of Induction Motors-Pitch s/3 = slots per pole per phase With Special Reference to Magnetic Leakage, A. I. E. E. TRANS.,

deficiency 1905, Vol. 24, pp. 649-84; Discussion, pp. 685-87.b 1 2 3 4 5 6 14. 1904, Adams, C. A., "The Leakage Reactance of Induc-0 0.0966 0.0284 0.0140 0.0089 0.0064 0.0052 tion Motors," Int. Elee. Congress, St. Louis, Trans., Vol. pp.1 0.0966 0.0235 0.0115 0.0074 0.0055 0.0045 706-24; Discussion, pp. 725-28, 1904.2 0.0966 0.0284 0.0111 0.0063 0.0044 0.0035 15. 1904, Hobart, H. M. and Punga, F., A Contribution to the3 0.0284 0.0140 0.0089 0.0041 0.0030 Theory of the Regulation of Alternators, A. I. E. E. TRANS., 1904,5 0.0137 0.0092 0.0064 0.0040 Vol.. 23, pp. 291-322; Discussion, pp. 323-336, 340-43.6 0.0140 0.0089 0.0068 0.00527 0.0083 0.0067 0.00568 0.0089 0.0062 0.0055 Discussion9 . ____ 0.0057 0.0052

______0_____0057_________0____0052_______ - B. A. Behrend: In connection with Mr. Alger's very com-

that by substituting these values in (15a) and dividing plete investigation of the factors which determine the armaturereactance of a synchronous machine the question may be raisedwhether it is more satisfactory for the designer to derive his

is found to be: characteristic data from comparison with known similar designs

X 2 or by a fresh calculation for each particular case. As a ruleXm = 4 CS C2 7r/2 s - 1 (20a) the designs vary comparatively little, thus making it possiblexm 4 S2 to obtain design data by comparison. The analysis of the

this aybedevelped nto aserie of onstituent parts of the tota.l leakage field as carried out byIf~ ~ ~~ ~ ~ ~ ~ ~ ~~~~~~~~~iiSlret1aedvlpdnoasre f M.Agr1n keeping with the desire to leave no part of thewhich only the first term need be considered: theory unexplored and to link together the leakage of syn-

XDK T2 / T2 cli\eronous and induction machines. In this very painstakingK 1=2- 2 | (21a) a.nd intricate procedure I doubt that a method which is evolvedXm 12S\ 20 S 504 S 4 by one designer and which suits his method of working is likely

to appeal equally to another designer. I am inclined to viewFor large values of s, (21a) agrees exactly with (25), such methods as to a great extent individual though notwith-thus confirming the previously derived formula for sta,nding this they are of the greatest importance in clarifyingzigzag leakage reactance. obscure problems and showing us the empirical basis of our


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1928 the Calculation of the Armature Reactance !!