Induced Voltage of Electrical Machines

Induced Voltage of Electrical MachinesBY L. V. BEWLEY1

Associate, A. I. E. E.

Synopsis.-The object of this paper is to describe and discuss a ductors, or on the rates of pulsation and rotation of the flux; butgeneral equation for the induced voltage of electrical machines these may vary in any arbitrary manner which can be given ahaving parallel coil sides, and which includes as special cases single suitable analytic expression. The several methods for the reduc-and polyphase induction motors, synchronous generators, d-c. gen- tion of harmonic voltages are classified and their limitations dis-erators, synchronous converters, and static transformers. The cussed in such a way as to leave in mind a vivid picture of theapplication of the general equation to most of these cases is illus- process. Tables and curves have been prepared for comparing thetrated, and a number of interesting problems which may be solved effects of the skew, pitch, distribution, and phase connection har-by means of it is pointed out. A characteristic of this equation monic reduction factors. A new method for summing the finiteis that no restrictions are placed on the velocity of the moving con- series of the distribution summations is given in Appendix II.

I. INTRODUCTION currents, m. m. fs., permeances, fluxes, and voltages.T HE purpose of this paper is to derive and discuss Incidentally, a study of the general equation derived

a general equation for the induced voltage of in this paper permits making a critical review andelectrical machines having uniformly distributed classification of the available means for suppressing

parallel coil sides moving through a distribution of harmonics in the induced voltage by special arrange-flux which can be represented by a Fourier series. The ments of the windings. These special arrangementsequation is general in that the circuit considered may give rise to certain functions known as harmonic reduc-have any number of phases connected in series in any tion factors, which enter the equation as ordinaryarbitrary manner, and any particular phase may con- coefficients. They are called, corresponding to thesist of any number of uniformly distributed, fractional particular arrangement of the winding to which theypitch, skewed coil sides moving at variable speed apply, the skew, pitch, belt distribution, and phasethrough a distribution of flux which may pulsate and connection harmonic reduction factors respectively.rotate at different rates. This equation includes as These same factors have an almost exactly analogousspecial cases the procesess of induction found in the effect in suppressing the harmonics of armature reac-more familiar types of electric power apparatus, such tion, but their consideration in that respect is outsideas synchronous motors and generators, indu tion the scope of the present paper. The full limitations ofmotors, d-c. motors and generators, synchronous con- these factors are not generally appreciated. Forverters, and static transformers. Bescause of its instance, it is usually assumed that it is impossible togenerality, this equation may prove awkward and have a third harmonic in the line voltage of a three-unwieldy in dealing with a particular type of machine, phase synchronous generator if it is connected Y, orsince it was not derived with the idea in mind of has coils of 23 pitch, or 120 deg. phase belts, or con-adapting it to some special condition of symmetry ductors skewed 120 deg. with respect to the pole; be-affecting the choice of a coordinate system, or to the cause each of these conditions is supposed to completelynecessity of correlating with the related phenomena of eliminate the third harmonic and its multiples. Yetcurrents, m. m. fs., and fluxes. There is an essential such a harmonic will appear in the line voltage of adistinction between developing an equation for machine having all of these conditions, if there happensthe complete analysis and determination of character- to be an even time harmonic in the air-gap flux. On theistics of a particular type of machine on the one hand other hand, a third harmonic due to such a cause canand deriving an equation whose purpose is to study the be eliminated by a suitable arrangement of the winding.variation in process of some specific phenomena which Now this is not a new discovery, but only the specialistoccurs in several types of apparatus, on the other. would notice the fact in a treatise on the harmonicIn the latter case, greater flexibility is required, and the analysis of synchronous machines.expression should be purged of all possible detail II. GENERAL DISCUSSIONirrelevant to an understanding and classification of If it is possible to express the distribution of flux inthe ideas under consideration. For instance, certainfacts become m asked and obscured by temsi an electrical machine having parallel coil sides by thearray of symbols associated with the complete harmonic oresre canalysis of a synchronous machine, involving the ( xspecification and manipulation of the Fourier series of B =~̂ ,S sink~t T + z J (1)

1. General Transformer Engg. Dept. General Electric Co., hntems eea qainfrteidcdlnPittsfield, Mass.'thntemsgeea qaonfrhendc le

Presented at the Winter Convention of the A. I. E. E., New York, voltage, regarding x0, A*k, and 'yk as functions of time,N. Y., Jan. 27-31, 1930. is shown in Appendix I to be

456

30-11

April 1930 BEWLEY: INDUCED VOLTAGE OF ELECTRICAL MACHINES 457

T ncN' If the conductor is skewed or spiralled so that halfEli, =- 2 L T, 108 Cck Cdkl Cpk Csk of it is cutting through a positive loop, and the other

1 half through a negative loop of the harmonic of space

r1 dkXnk ±7r ,+

distribution, then the voltage induced in the two halvesLk d t sin k ( + 7k + lk' + k X of the conductor are always in direct opposition and

IcT therefore completely cancel within the conductor it-

7dk d xow xo Xr+td t + d t o + k+k+ k

(17)where the C coefficients are the harmonic reductionfactors depending on the arrangement of the windings,defined in Table I.

S&OW C .0 I

TABLE ISeat of the

Reduction factor reduction _---aLo

sin k X/2Skew factor...kCi = /2In the coil side

Pitch factor Cpk= sin - Between coilCO/

__ 1 2 ] sides of the coil J CO

Distribution fac- sin k c d/2tor of belt.... Cdk * / Between coils ofob]t:Cdk=C. sinlk 6/2 .|the phase belt

°h, *s,

Connectionr

acC= ( sin k Or)2 + (z cos k Or)2 Between phases Con n O

n

If the n phases of the connection factor are uniformly FIG. 1-SEAT OF HARMONIC REDUCTIONdisplaced by the angle ¢ then

sin kn¢/2n sin kc/

which is then in the same form as Cdk, and the same setof curves will serve for both. Moreover,

sink 6c/2 sinkn 2Cdk k ac/2 if ->OandC0, = kn-/2 if +0. 4aA

Thus under these limiting conditions, the three reduc-tion factors C,k, Cdk, and C,k have the same form. 03u

Associated with Cdk, and Cck are the distortion anglesk,,' and l,k defined in Appendix I, and these cannot, in

general, be made to disappear by a convenient choiceof reference axes. In other words, a lack of symmetryand uniformity in the windings, or in the way that they lare connected in series, will cause a distortion of thevoltage wave by an angular shift in the relative posi-tions of the harmonics, as well, as by the reduction inamplitudes characteristic of symmetrical arrangementsof the windings. FIG. 2-THE SKEW COEFFICIENTWhile these reducing factors have different names,

they are in fact due to the same essential cause-the self. The next simplest method of cancellation is toarrangement of the circuit so that the harmonic volt- use a coil of fractional pitch so that both coil sides areages produced in different parts are in partial or com- cutting through positive ioops of flux, but the voltagesplete opposition and thus tend to cancel out over the generated thereby cancel in the coil. In this case thecomplete circuit. Fig. 1 illustrates how this is accom- coil may be either of short or long pitch. The distri-plished in the case of a third harmonic. bution of the winding in more than one slot per phase

458 BEWLEY: INDUCED VOLTAGE OF ELECTRICAL MACHINES Transactions A. I. E. E.

gives rise to a reduction of the harmonic voltages be- As a rule it is desirable to suppress as many harmonicstween the coils which make up the phase group. as possible at the least cost to the fundamental.Finally, it is possible to eliminate those harmonics For all harmonics are more or less objectionable,which are multiples of the number of machine phases,by connecting phases in series.

Representative curves of the skew, pitch, and distri- '11bution factors are shown in Figs. 2, 3, and 4 respectively, 1I9

and the phase connection factor for a few simple cases OA 9 t

FIG. 3-THE PITCH COEFFICIENT

iS given in Table II. Only a sufficient number of har-monics has been plotted to indicate the general nature FIG. 4THE DISTRIBUTION COE:FFICIENTof the functions. It will be noticed that the maximumvalue which any of these coefficients may have is unity, but when they are suppressed in such a way as toor more appropriately, their values lie within the range reduce materially the amplitude of the funda-(- 1 <C ' ± 1), because they have been defined as mental it means more flux in the air gap, more excita-geometrical averages. tion, and possibly higher losses, or even a larger sized

TABLE IITHE PHIASE CONNECTION COEFFICIENT AND ANGLE

Connection| k^= 1 3 5 7 9 11 13 115 117 19

3 fDelta Cek 1 1 1 1 1 1 1 1 1 1

M~ {k 0 0 10 10 0 0 0 0 0 0

,9 IkM +300 0 +30° 1- 30° 0 +30°0 -30° 0 +30° 30°

2 / Cck t 1 1 1 1 1 1 1 l 1 1 1

.-JT Ck I+45° 45°0 +450 450 +45° | 450 +450 | 45°0 +450 | 45°

6 fStar Cck 1 1 | 1 | 1 | 1 1 1 | 1 | 1 | 1

X ~2 2 2 1 1 2 2 1 2 2

3 fZigzag |Cck |3 0 0 0

6'E 4 4 4 4 4 11 4 4A k 60° 0 +600 -60' 0 ±60° -60° 0 ±60°' -60'


machine. It must be understood that other considera- constant velocity; then only those time harmonics cantions than the relative efficiency have a decided bearing appear in the induced voltage which are present in theon the choice of a method of harmonic reduction. space distribution of the flux, and which are not wipedThus if either skewing or fractional pitch is an available out by the harmonic reduction factors. In this case,method in a given case, the fractional pitch would if any reduction factor for the kth harmonic be madeundoubtedly be the choice; because it is less expensive equal to zero, then the kth time harmonic will beto make straight slots and coils, there is considerable canceled out of the induced voltage. But everysaving in copper and copper losses on account of shorter harmonic, including the fundamental, will sufferend windings, and for the same reason a possibly shorter a reduction in magnitude if the slots are skewedmachine with less windage loss. or if the coils are made either short or long pitch, or

While3k, yk, and xo in Equation (17) may be any if the total turns are distributed, or if phases are con-arbitrary functions of time, yet the electromagnetic and nected in series-in brief, any departure whatsoevermechanical characteristics of electrical apparatus are from a straight-sided, full-pitch, concentrated windingsuch that each of these items may, for practical purposes, will cause a reduction of all harmonics. In particularbe taken as composed of a series of terms of the type: if the fundamental is canceled by any reduction factor,

Ok =O130k + l3k fa + 132k sin w t +±33k Ebt sin co t (22) so will every harmonic be canceled.d 'Yk (b) The presence of exponentials will cause eitherdt = V°k + Vlk 6"' + V2k Sin w' t + V3k fb ' sin ' t (23) damped or cumulative oscillations, depending on the

d t sign of the exponents, to appear in the line voltage.The abrupt change of the excitation on a machine,

d xo either by rheostat adjustment or by an automaticdl= Vo + V,Ifat + V2 sin w" t + V3 eb't sin co" t voltage regulator, causes the flux to build-up (or decay)

d t exponentially. This change in 0 and its derivative is(24) reflected in the equation for the induced voltage, as

Thus immediately after an unsymmetrical short cir- exponential-trigonometric products, that is, as dampedcuit on a synchronous motor the flux in the air-gap or cumulative oscillations. An exponential variation

of 3k may also be caused by a symmetric short circuit,or abrupt change of load occurring on a generator. Ifthe torque of the prime mover driving a generator, or

____\___,__ the mechanical load on a motor changes, an adjustment7' _-\ r / in speed takes place. This speed transient will contain

an exponential term due to the mechanical inertia ofthe rotating parts. In the case of power suddenly

rft5411T {TH2shut off from an open-circuited generator, the speedt- t7- t-Xt yzr- decrement is practically a pure exponential (not rigor-

ously so, because the friction is not constant throughout/|a :-/-2..< ' lthe speed range). These exponentials in speed cause

SIDr*-tJtO the amplitudes of the harmonics in the induced voltageto change proportionally, and the wavelengths of theseharmonics either contract or expand according to an

consists of terms such as given in (22). If the short exponential law.circuit is severe enough to cause hunting, the constant (c) Perhaps the most important class of time varia-velocity of the armature will have superimposed there- tions in flux and speed are those of a purely oscillatoryon an oscillation which may, if the conditions are right, nature. Periodic fluctuations in flux are caused bycontinue indefinitely, die out exponentially, or even sustained unbalanced short circuits and unbalancedincrease in amplitude until the machine falls out of step. loads on polyphase machines, the time harmonics ofThe oscillatory passage of the teeth and slsacross the polyphase armature reaction; variation in the air-gappole faces will cause the distort . - flux. Thus reluctance due to the passage of teeth and slots, and theEquations (22), (23), and (24), represent >tetual possi- elliptial rotating fields of Single-phase motors.bilities. The direct substitution of (22), (23), and (24) Any possible product of sine and cosine terms may bein (17) yields a representative equation for the induced expressed as the sum or difference of sines or of cosines,voltage in electrical machinery, but for the purpose of for examplethis discussion it is not necessary to carry out the actual 2 sin w t sin k w0 t = cos (Ic wo-w) t-cos (kc wo + w) tsubstitution. It is sufficient to notice the following It is of importance to observe from relationships of thisfacts: type that it is quite possible to produce a harmonic of

(a) When the flux distribution contains only rigid any order K (integer or fractional) in theinduced volt-space harmonics, and when the coils are moving at a age, even though the reduction factors have been

460 BEWLEY: INDTJCED VOLTAGE OF ELECTRICAL MACHINES Transactions A. I. E. E.

adjusted for its complete cancellation. These possibili- and taking the armature as the rotating element, thereties are given by is:

woK = (k wo0 w)For instance, suppose that the reduction factors will Xo = 2 fo r t = coo t wheref0 = frequency.

wipe out the 5th time harmonic due to a 5th spaceharmonic of flux distribution. But as a possibility d /kld t = 0 since the flux distribution is steady.

5 = (k + wlwo) d-Y kld t = 0 since the flux distribution is fixed.from which it is (kidentthat the

± w/ tim) harmonic Substituting these values in Equation (17) there isfrom whlch lt 1S evident that the 5th time harmonic ,x,may nevertheless appear in the line voltage if the air n N' ,gap flux contains a space fundamental, k = 1, pul- Eline = - Lfo- 108 CckCdk Cpk Csk Osating at wlwo = 4 times normal frequency, or a 3rd 1space harmonic pulsating at w/wo = 2 times normal cos k (co t + 7Y±+ ;k' + Ak)frequency; or in general a kth harmonic pulsating at Now in commercial machines the field pole flux isw/wo = (k - 5) times normal frequency. More gener- rigid and symmetrical. The space harmonics are there-ally, since the periodicity of the flux pulsation may be fore of odd order and fixed in a symmetrical position.arbitrary, it is possible to produce frequencies in the in- Consequently for a machine on open circuit (or supply-duced voltage which are not multiples of each other. ing zero power factor current), k is odd, and eitherThe only common sinusoidal fluctuations in speed 7k = 71 or 7k = (71 + w). When the machine is

that occur in practise are those due to sustained supplying an energy component of load the air-gaphunting of synchronous machines, and prime movers flux is distorted and unsymmetrical, so that the zeros ofor drives with cyclic torques (reciprocating engines, the flux harmonics are not necessarily coincident;compressors, etc.). and only consideration of the effects of armature

It is seen from Equation (17) that such cyclic varia- reaction and air-gap permeance will permit /k and ktions in speed affect both the amplitudes and wave- to be completely specified.lengths of the induced voltage. If all phase belts of an alternator are alike (that is an

integer number of slots per phase per pole) and if Q is(d) Finally there iS the class of time variations in the number of phases per pair of poles, then fromflux and speed represented by damped (or cumulative) Equation (9) of Appendix Ioscillations. Each of these appears in the voltage as apair of damped (or cumulative) oscillations of different sin (2 k - 1) r/Qfrequencies. They are caused by hunting, pulsating cdk - C sin (2 k - 1) v/c Qspeed regulation by the governors of prime movers,

a uy ecsriuoi. = 0 if (2 k - 1) = multiple of Q but not of (c Q)and unsymmetrical short circuits of polyphase mnachines.1i 2k-1 utpeo cQIII. APPLICATIONS ~~= 1 if (2 k -1) = multiple of (c Q)

III1. APPLICATIONS It is rather interesting that the distribution factor ofUnder this section of the paper, the application of a Q phase machine will eliminate the multiple of Q

Equation (17) to some of the more familiar types of harmonics if there is more than one coil per phase belt,electrical machines will be illustrated. In order to but will not eliminate those harmonics which are also aconserve space, certain simplifying assumptions will be multiple of the product of the number of phases byadopted in making these applications. But it must be the coils per phase belt, (c Q). For instance, if a trueborne in mind that the generality of the equation is not three-phase machine (120 deg. phase belts) has threelimited by any such restrictions as may be invoked in coils per phase belt, then the third harmonic will bethe interests of simplicity and brevity. An extension completely wiped out, but the ninth harmonic and itsof the analysis to include more detailed considerations multiples will not be reduced by the distribution.will be obvious from the simple specifications given However, in a double layer winding the ninth would behere. wiped out by the pitch factor in this case. A single

Synchronous Generator. If all of the phenomena layer wi-nding would have to be used to take advan-which occur in a synchronous machine are to be included tage of tiv-in its analysis, then every term of Equation (17) will be Then"the ior Z1line given above is based on therequired for the description of the induced voltage. assumptionl that the armature is the revolving member.Thus an unsymmetrical short circuit will cause a If the field revolves, then x0 = 0, and 7k = Yk+ 00 tdamped pulsating armature reaction with a correspond- (since the flux distribution moves as a rigid system),ing variation of flux; the passage of teeth and slots so that d 7k/d t-=c0. Making these substitutions, thecause the teeth harmonics of flux distribution to sweep same equation is obtained as that given above for theacross the pole face; and the hunting of a synchronous case of a revolving armature, except that 7k replacesmachine superimposes on the constant speed of rotation 7k as the constant angle fixing the relative position ofan oscillation which may be damped, sustained, or even the harmonics in the rigid distribution of flux.cumulative. But neglecting such abnormal conditions, Polyphase Induction Motor. Consider for the sake


of simplicity only the space harmonic of fundamental possibility of taking either sign is necessary in order tofrequency f in the revolving field of the air-gap flux, neutralize a change in sign caused by the harmonicand let fo be the mechanical frequency of rotor rotation. reduction factors.If the direction of rotation of the revolving field be Since the "phase belt" of a d-c. machine spans a poletaken as positive, then the angle -yi in the Fourier series pitch, the distribution factor isfor the flux distribution, Equation (1), is negative, so sin k 7r/2 2 k7rthat Cdk = . sin if c + OCt yh =-27rft, .. dyl/dt -27rf Cdncknk7r/2c - k r 2

Xo 2 ir fo t; .-. d xo/d t = 2 rfo Thus even harmonics in the flux distribution will notSubstituting these values in Equation (17) there is contribute to the induced voltage. For a full pitch,

infinitely distributed winding, without skew,nN'Erotor = 4 L - 108 CCl Cdl Cp1 Csl l . kwC,k -sin2 and Csk=1.

(f - fo) cos [2 r (fo - f) t + V/i' + V/]This equation shows that at standstill, fo 0, the N' (RPM)voltage induced in the rotor is of stator frequency, f. Eirusies --2 i L 108 60 (Poles)If the rotor is running at synchronism the voltage iszero and direct current. For speeds above synchronism 2 co (2 k - 1)7r 2 k-the voltage increases in magnitude and frequency with r siw- 2 2k-1the speed.The equation for the stator counter e. m. f. is, of cos (2 k-1) ('2k-1 ± 4)

course, of the same form as that derived in the previous N' (RPM) N'-2 (Poles) 4= -4 (fcase of a synchronous generator. 108 60 Jos

D-C. Generator. The exact calculation of the voltageacross the brushes of a d-c. generator would take into whereconsideration such things as the variation of the air- 'rL rgap flux due to the moving teeth and the pulsation in > = 4 > 1(2k-31) sin (2 k-1) ('Y2k-1 + ) d 4armature reaction caused by the oscillation about itsmean position of the belt of conductors between 2 ____brushes; the sector of brush overlap; the slight cyclic 2 2k-1 Cos (2 k-1) ( y2 I +variation of field excitation when under automatic Wr 2k-iregulator control, etc. Assuming, however, that thenumber of armature conductors per pole is reasonably

Fo(fu btweenb du toltheoddeharmncs)

large, so that the effects of the belt oscillation may be From the standpoint of telephone interference, aneglected, and disregarding the teeth pulsations, brush rather interesting problem in the design of d-c. commu-overlap, etc., the following conditions obtain: tating machines is the calculation of the voltage pulsa-

tion due to the oscillation of the armature conductorsd _Yk d t = 0 since there iS no movement of flux... about their mean position with respect to the brushes.d ,Okld t = O since there iS -no pulsation of fluxdo 71/d t = 0=sinceth breishnopulsat o These ripples may be computed by allowing 4 to varyx0 7r/T = 4 = angle of brush shift-... over the oscillation range, in the above equations.

d x/dt = (no.iyofpoles)t(rev, prmn/0=2' While the segregation and ultimate specification of- velocity of conductorsVlk = Vlk = 0 since there is only one belt between these pulsations as Fourier series is usually desirable,

space will not be taken up here for such a study.brushes.Substituting these values in Equation (17), the induced Synchronous Converter. The a-c. and d-c. induced volt-

voltage across an adjacent pair of brushes is found to be ages of a synchronous converter are found from theequations for the Synchronous and D-C. Generators

Ebrukes- 2~ LN' (RPM) (oe)respectively.108 60 Substituting in the equation for the Synchronous

Generator the values

1 n=1, 4k =O, V/k'=O, N'==cN, Cdk'=Cdk= C ink 3/200

=T L i 8" CdkCp kCOkfcCos k ( yk + 4) (36) where 6 = 2w/c Q = slot angle; the phase voltage isevdn Ephase = - 4LTf0N10 8

It is eintthat the voltage is a maximum when ~ ,sink wr/Qzb= -(4±+mw), for then cos k QYkb+4) = ± 1 2> sink5/2 CPkCSk1k COSk (X0t±+ yk)

and each harmonic contributes a maximum. The


The effective value of the fundamental (the applied of a standard converter supplied with a-c. field excita-e. m. f.) is tion, or for that matter, an excitation varying in any

4 sin 7r/Q arbitrary way.E1 = - L fN 10-8 sin 6/2 C1PI Csl (31 The Transformer. There is no practical advantage

V/2 in showing that Equation (17) includes the special caseSubstituting in the equation for the D-C. Generator of the static transformer. The demonstration is in-

sin k 7r/2 cluded here merely for the sake of completeness and tothe value Cdk= si k6/2 gives for the d-c. voltage show the general validity of the equation. Consider an

ideal transformer having a rectangular core of sectionEd = --4 L foNl-8 (N L); and neglect the leakage flux. Then at any in-

o . stant the flux density is uniform over the core, and thesin k 7r/2 Cpk C,k /k cos k (yk + t) equation of its distribution in the direction of r is thatsin kI/2 of a rectangular wave. But in a rectangular wave the

Therefore the ratio of the d-c. induced voltage to harmonies are all of odd order, and their amplitudes arethe effective value of the fundamental component of inversely as their order, so that Equation (1) becomesthe a-c. induced voltage-the ratio of conversion-is oT

Ed- c CosQ(y1-I B= 2k- 1si(2 1E1 sin 7/Q 10

4_ sin(2k-1)7rx/rsin k ir/2 . sin 6/2 . Cpk C0,kk k

k

(2 k - 1)5msi rQ . sin k /2 . CplC5,u1 cos k ('y1 ± t) J1

2 Since the turns are full pitch, and the conductors are= /2 A/sin (7r/Q) not skewed or distributed with respect to the distri-For a detailed discussion of the voltage ratio of a con- bution of flux, it follows that C,k = 1, Cpk = sin k r/2,

verter, based on this equation, reference may be made Cdk = 1, 41, = 0. Both the conductors and spaceto The Synchronous Converter by T. T. Hambleton and distribution of flux are stationary, so that d xo/d t =0L. V. Bewley, A. I. E. E. TRANS., Vol. XLVI, 1927, and d yk/d t = 0. The coordinate to the midpointp. 60. of the coil is xo = T/2. The induced voltage therefore

Synchronous Converter with Polyphase Field. Con- is, by Equation (17) and the above conditions:sider a synchronous converter armature with a fre- T N' 4 dB sin2(2k-i) w/2quency f2 applied to its slip rings, revolving in the poly- E =-2 L t (2 k-1)2phase stator field of an induction motor supplied with acurrents of frequency fl. If fo denotes the frequency of N' d B N' d fmechanical rotation of the armature, then from the - Tr - -equation for the rotor voltage of a Polyphase Induction 108 d t 108 d tMotor, there is (considering only the fundamental) 8 1

N' since (2k 1)2 = l and (r L B) =¢Eroor=4L rn 108 Cdl CpiCs,ii(fi-fo)cos2 7(fo-fi)t 1

Although rather irrelevant to the subject matter ofThe condition that this voltage must have the same this paper, a discussion of the theory of induction in afrequency f2 as that applied to the slip rings, requires transformer by "cutting action" may not be too muchthat out of place.

4 f2 = (fO -fl) or fo = (fi 4 f2) In the above equations B is the instantaneous valueThen xo = 2 r fo t, d xold t = 2 r fo of the flux density pulsating at frequency f, so that

y -22wf,t, d oyi/dt = -22wf B = Bosin wct and thereforeThe voltage at the brushes, as in the analysis for the 4 sin w t . sin (2 k-1) x 7rlr

D-C. Generator is B = Bo Tw (2 k - 1)

N'cEF,e=-z LAr wr 10 CdlCPlCl-3(fO-fl)eos (t-2f17rF t) -B0 'A {COS[ Cot- (2k- 1) - ]

Thus the voltage at the brushes is of the same fre- 1quency as that of the field, but the amplitude is pro- -r TX 1 1portional to the frequency applied at the slip rings. - cos L X t + (2k-i) T J = E + z (3If fi = 0, the field is d-c. and the equations reduce tothose for the ordinary Synchronous Converter. The term [co t ± (2 k -1) iv x/1-] = constant defines

Equation (17) is also readily applicable to the case the position of any given point on the cosine wave at


any time t, and this point moves across the core section exaggerated cyclic torque. Segregate out the time

dx +: -, harmonics for different amplitudes and periods ofat a velocity d = ( thus inversely oscillation.d t (2 k- 1) 7r '

(d) Each phase of a two-phase induction motorproportional to the order of the space harmonic. The stator is excited at a different voltage and frequency.pulsating flux wave has thus been decomposed into two Find the slip ring voltage induced by the stator fluxsystems E ,B' and E 3" of traveling waves moving at in a three-phase Y-connected rotor driven externallyequal speeds in opposite directions. The flux density at constant speed.at the conductors, located at x = 0 and x = r, is found (e) A d-c. generator field is excited with alternatingby substituting these limits in the expression for B. current. Calculate the voltage at the brushes.The induced voltage, calculated by the (B L V) (f) The field of a separately excited d-c. generatormethod, then is is closed as the machine starts to accelerate from

standstill. For a reasonable field time constant and- N rate of acceleration plot the voltage at the brushes.

e 108 -(BLV) (g) Study the effects of slots and teeth in producingNL voltage ripples, by analyzing flux plots and specifying

-_N_X=0 + Ox_=-_x=o_x=_ as Fourier series. In particular, decide whether these108 7r 2k-I flux harmonics are standing or traveling waves.

(h) Calculate the voltage ripples of a d-c. generator-Nw 2 4 cos w t when the oscillation about its mean position of the belt= 7 L Bo2 ( -)2 of conductors between brushes is taken into account.

i Specify as Fourier series.Nw (r LBo) cos Xo t The author wishes to thank Mr. S. T. Maunder who-N co N d o verified the equations with independent derivations.

- 108 f Cos w t 108 d t LIST OF SYMBOLSB = instantaneous flux density

If the core section is not rectangular it may be re- Bo = maximum flux densityplaced by any rectangular section having the same effec- c = coil sides per phase belttive area, r L = A, and operating at the same density. C,, = skew factor for the kth harmonicAlthough the correct result has been obtained by Cr, = pitch factor for the kth harmonic

considering the standing waves to be composed of two Cdh= belt distribution factor for the kth harmonicseries of traveling waves, and then computing the Cdk' = equivalent distribution factor for dissimilarinduced voltage as due to "cutting action," yet at the phase belts in seriesconductors themselves the flux density is zero, for C,, = phase connection factor for the kth harmonic

O =0 + Ox== 0 em = voltage induced in the mth coil of a phase beltx=±+ 0x eg = voltage induced in a single-phase group

Thus in an application of the (B L V) method, the ep = voltage induced in a complete phasecomponent flux densities and their individual velocities Eiine = induced line voltagerelative to the conductor must be considered separately, f = frequencyand only those flux densities superimposed which have k = order of the harmonicequal velocities and in the same direction at the L = effective length of stackingpoint of superposition. This interesting situation was m = position of coil in a phase grouppreviously pointed out by Dr. H. B. Dwight in the April n = number of phases in series between lines1928 issue of the Electric Journal. N = number of turns per coil

Additional Applications. For those wishing to be- N' = number of turns per phasecome familiar with the use of the general equation, p = per cent pitchthe following problems may prove interesting and P = number of polesinstructive. q = number of phase groups per phase

(a) Calculate and plot representative ranges of the Q = number of phases per pair of polesvoltage wave of a synchronous generator on open cir- Rev. per min. = revolutions per minutecuit brought up to speed by a prime mover delivering t = timeconstant torque, assuming friction and windage inde- v = velocity of moving fluxpendent of speed. V = velocity of moving conductors

(b) Plot the open-circuit voltage of a synchronous x = coordinate of any point in the direction ofgenerator supplied with field excitation from a recti- motionfying device having a second time harmonic. x0 = coordinate of the midpoint of the first coil group

(c) Plot the open-circuit voltage of a synchronous y = coordinate of any point perpendicular to thegenerator driven by a reciprocating engine having an motion


a = mechanical angle of skew c = number of coils in the coil group/k = instantaneous amplitude of the kth harmonic of u- = distance between adjacent coil sides = slot pitch

flux density a = angle of skew'yk = instantaneous displacement angle of the kth y = coordinate at right angles to the direction of

harmonic of flux density motion, reckoned from midpoint of the coil5 = electrical angle between coil sides side.E = base of Naperian logarithms The total flux included by the mth coil is, by Equa-

= angle between adjacent phases tions (1), (2), and (3),6' = displacement angle of a phase group L/2 Xm'6 = displacement angle of a phase r=X = electrical angle of skew O B d y dx

= angle of brush shift -L/2 xm

Of = slot pitch distance 7 3k x 7rr -pole pitch distance 2L 2 k Cpk C,k sin k Tb ==flux 1'Pk = shift of the kth harmonic due to phases in series

shift of the kth time harmonic due to dissimilar C + -2 k (4)belts

= 2 wr f = angular velocity wheresign of summation, effective as implied. L = effective length of the stacking

Appendix I sin k (L r tan a/2 r) sin k X/2GENERAL EQUATION FOR THE INDUCED VOLTAGE OF sk k (L 7r tan a/2 r) k X/2

ELECTRICAL MACHINESThe following analysis applies to electrical machines duction factor for the kth harmonic (5)

having uniformly distributed parallel coil sides moving p 7rthrough a distribution of flux which is uniform in a C2J = sin k = pitch reduction factor for the kthdirection in the plane of the coil perpendicular to itsmotion. harmonic (6)

Let the space distribution of flux density due to the Now in general, X0, Ok, and 'Yk are all functions ofcombined effects of field excitations, armature reactions, time t; so that the induced voltage of the mth coil ofand leakage reactances be specified by the Fourier N turns isseries: -N d m -N O/m d xo

co e =D - _

BX ±U)108 d t 108 xo d tB = Ok sin k t r + ah J 0)1 , a m dYjk , akm dIk l

where + b dt dtk = order of the space harmonic 1

Ok = amplitudeofthetkthrharmonic Nx = coordinate of any point =V 108 A

= wavelength of the fundamental = pole pitch 1#yk = displacement angle of the kth harmonic. 1 dk. . 7r c + 1-2 m o-

If the coil sides situated in this field are skewed with k dtsintk Ix2. -rrespect to the poles, the equations of the trailing andleading coil sides of the mth coil, counting from the (___ + 07d ) COS k(trailing end of the group of coils, are respectively (see d t d tFig. 5)

c+1-2m r-wIr c- I + ' 7

xm =xo-p 2- 2 a-+(m-1) -+ytan a (2) 2 T

2The total voltage for a phase group consisting of cr c-i such coils uniformly distributed by the pitch cf is, by

Xm' =x0+P 2- 2 a-+(m-1) L+Y tan a=Xm+p r the method of summation given in Appendix II,

where ~~~~~~~~~~~~~*Thisresult is obtainable by direct substitution in Equa-where ~~~~~~~~~~~~~~tion(19) of tlie autbor's paper Flux Linkages and ElectromagneticO -coordinate of the midpoint of the coil group Indulction in Closed Circusits, A. I. E. E. Quarterly TRANS.,

p = pitch of the coils expressed as a fraction Vol. 48, April 1929, p. 327.


c~~~~~e= em = -2 L Cdk Cpk Csk Eiine = 2 L - 108n C,N Cdk Cpk C8krr108 0 c Ck p s

1 1 1

[1ldf3k k Xo1r ± .Yk 1 df3k x 'r_k dt sint k (r + k ) [ k dt sin k ( _ + tk + 4k + 41k

± (djk ± jOk) Okc s X Xk +o + j X 'Y)co k Xo 7(+ Yk+ ;+ 1

; -+ -dt JTo Yd tIr d t d t J

(8) (17)where where

sin k (c o-7r/2 r) sink c8/2 - Cci = c[(1E Cos k °r) + ( sin k Or) ] -con-Cdk = * = distribution nc sn (o ir/2 r) c sin k 8/2c sin kS1fl 7r/2 r) c sin k 6/2 nection reduction factor for the kth harmonic

reduction factor for the kth harmonic. (9) (18)If the complete phase is made up of q phase groups - ds

connected in series, but unsymmetrically arranged and 'Pk = tan- sin k °r)/( E cos k Or displace-with a different number of coils per group (as in thecase of fractional slots/pole/phase) then the phase mentangleforthekthharmonic (19)voltage is, by the method of summation given in If the n phases are displaced by the same amount ¢Appendix II. from adjacent phases, Equations (18) and (19) reduce,

as shown in Appendix II, toq c

N -2 L ~~r N' Nsinkn~/2ep= e,, 2 108 Cdk' Cpk C8k C,k = nsink //2 (20)

[ k d t sin k ( - +k+± k ) 1lk= k (2>) (21)Equation (17) is the most general equation for the

+ (d tk+ dx°)t , cos k ( + k + {k')] induced voltage of electrical machines, if xo, f3k, and 'Ykd t r d t rare perfectly arbitrary functions of time. Its applica-

(11) tion to the special cases of the more ordinary types ofwhere electrical machines is discussed in the main body

Cdk' = V\Ak' + Bk'/N' = equivalent distribution fac- of t paptor of the phase, for the kth harmonic Appendix II

46,b' = tan-' (Bk/Ak) = displacement angle of the kth THE DISTRIBUTION SUMMATIONSharmonic in the phase voltage (13)qf In the harmonic analysis of electrical machines it is

Ak = [c N Cdk],cos k 6,' (14) repeatedly necessary to find the sum of certain finite1 trigonometric series. Those required in this paper are

given below.Bk = I [cN Cdkl, sin k 0, (15) n-1

1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~n-q

Since 1+ Zn zN' = I [c N], = total number of turns in the phase r=0

1 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(1)(16) it follows, upon replacing e ja by z, that

0,' = position of the midpoint of phase group g in -1 n-Ielectrical radians reckoned from the reference sin (x ± r a) = i(x+ra) - i-A(x+ra)point on the armature. If all groups are 9 (alike, and displaced by multiples of 2wu, then r 2=nCdk' reduces to Cdk. EJX I I E--Jna e- x i 1-E-o \

Two or more phases may be connected in series, 2]j 11-ia J 21 t 1 - C-ia /either additive or subtractive, to form the line voltage. , i-a±&(+n-)-C(x+)-C-xIf the displacement angles of the n phases as connected = 1 r CaX+1Xa) -+ -j(x+naa)+C-ej(x+n ,-a)j Iin series are (6,, 62 ... °n) the total line voltage is, by =21 +e +- + +e<+the method of summation in Appendix II. L1-Cm"i- Ca< + 1


sin x-sin(x- a) +sin (x±n a- a)-sin (x+n a) By the symbolic operators of complex notation:2 (1 - cos a) 2 Er E Er eJOr = E Er COS Or+j z Er sin Or= Eo eiOo

(6)2 sin a/2 cos (2 x - a)/2 where-2 sin a/2. cos (2 x + 2 n a- a)/2 E2 = V( EEcos Or)' + ( Er5sin Or)2 (7)

4S Er. EsOr 7cos (2 x- a)/2- cos (2 x- a + 2 na)/2 00 = tan-' Ers O, (8)

2 sin a/2ErCSOOr transforming to trigonometric notation

sin n a/2 sin x + n 2 1 a (2) 2 Er sin (X 4 0r) = Eo sin (x+O0) (9)22 E, cos (x ± Or) = Eo cos (x + 0,) (10)

Replacing (a) by (- a) gives If E, = E2 = .E.= Er = E and 0, = r a thenn-1 (7) and (8) reduce to (2) and (9) as follows:

.sin (x - ra) sin n a/2 sin x - anI (3) n-l n- 2r=O ~~~~sin a/2 2 2 -12

Replacing (x) by (x + 90 deg.) gives Eo =E ( >Zcosr)+r(a sin r a)o o

X-n

sin n a/2 ( n-iI E sinna/2cos (x + r a) = sin a/2 cos x + 2 a) sin a/2

r =O

n-1 (\(cos n-i a)'+(sin2n- a)2N ~~~~~sinn ae/2 n- 1cos (x - r a) =Osinn/2 (x n

r=0 =sina/2 c E 2sinna/2(5) sin a/2

The more general sums, in which the angles do notdiffer by a regular amount, and in which different Ho = tan' (tan ni2 a ( 2 ) aamplitudes are involved, are found as follows:

Induced Voltage of Electrical Machines

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