0= peripheral co-ordinateurrent oneets Lquiva ent to End- 0=phase angle, center of coil0O= angle between reference axis 0=0 and

center of belt 5, of conductors which

fin ing Currents of Turbine-Generator t= carry outgoing (+z) current at z=0or~~ ~~~~~~~~~~~=angle of the pole face, radiansPo = radius of rotor iron at the end region, mP1= mean radius of stator end winding, mStator and Rotor P2=radius of outer cylindrical boundary of

end region, mPm=mean radius of rotor conductors, m

J. A. TEGOPOULOS pg = radius (on plane z=0) of the circle onASSOCIATE MEMBER AIEE which the stator and rotor returncurrents are placed (mean air-gap

radius), mo= angular frequency=27rf, radians/second

Summary: The steady-state end-winding bl=coefficient of the stator space sinecurrent distribution of turbine generators fundamental componentis replaced by appropriate current sheets br=axial length of straight portion of rotor Assumptionsof the sinusoidal type. By assuming that end winding, mthe end-winding coils of stator and rotor b, = axial length of straight portion of statorclose through the air gap, their respective end winding, m The end winding of a turbine generatorcurrent distributions are represented by C=number of parallel circuits per phase is assumed to have the shape shown insystems of closed currents. Then, the Cr = ar b Fig. 1. The following assumptions aretotal current distributions for stator and cs =a. - Fasbs drotor are each separated into four com- i =stator instantaneous current, amperes/r made:ponents; peripheral, axial, and radial cur- i stator fundamental current flowing in 1. The currents flowing in the conductorsrent sheet, and return current. positive z direction, amperes/m of the end winding of the stator and rotor

i- =stator fundamental current flowing in are replaced by equivalent current sheetsnegative z-direction, amperes/m of the sinusoidal type. These current

eHE PURPOSE of this paper is to ig = stator peripheral current sheet com- sheets are assumed to lie on cylindricalreplace the end-winding current dis- ponent, amperes/m surfaces of radius P1 and Pm for stator and

i-stator axial current sheet component, rotor respectively. p, is arbitrarily chosentribution in the steady state of turbine amperes/m equal to the radius to a point lying on thegenerators with equivalent current sheets i4=stator radial current sheet component, conic surface midway between the twoof the sinusoidal type. These current amperes/m layers and 1/3 the distance from thesheets can then be used conveniently in irg =rotor peripheral currenit sheet com- beginning of the slanted portion of the endponent, amperes/m winding.' The mean radius of the rotorthe solution of problems concerning the izm=rotor axial current sheet component, conductor cross section is taken as Pm.end zone of turbine generators, such as amperes/m The conical form of the stator end windingdetermination of flux impinging on the irmi=rotor radial current sheet component, is thus replaced by a cylindrical one ofend plate, losses on the iron surfaces of amperes/m proper radius; see Figs. 2 and 3.the eud region, calculation of the leakage Io = stator peripheral current sheet coeffi- This replacement is made on the basisth end region,calculation of the leakage cient, amperes/m that the conductor body in the stator endreactance end component, calculation of I =stator axial current sheet coefficient, winding is bulky and forms an angle ,forces exerted on end-winding conductors, amperes/m with the horizontal, usually not exceedingetc. Ir = stator radial current sheet coefficient, 20 degrees. From a field point of view,An analysis has been made in the past amperes/m the approximation is expected to give goodIAoi=rotor peripheral current sheet coeffi- results at points relatively far from thefor the determination of the peripheral cient, amperes/m conductors. For points in the neighbor-

and axial current sheets of the stator end Izm-rotor axial current sheet coefficient, hood of the actual conductors this arrange-winding in reference 1. This paper uses amperes/r ment should be modified (see also refer-the same analysis to determine also the Irm-rotor radial current sheet coefficient, ence 2).stator radial current sheet and return cur- bu= stator return current, amperes 2. The effect of the air gap is considered

Fent.Burthermore, the samne four corn- 'mg =srotor return current, amperes by assuming a return of the currents in therent. ~~~~~~~~~~~~~~~~~~~~endwinding of the stator and rotor in theponents, i.e., peripheral current sheet, If= excitation current in rotor conductor, center of the gap or at radius pp; see Figs.axial current sheet, radial current sheet, amperes 2 and 3. According to this assumption,anld return current, are also determined Imax =stator maximum crest current per the end coils of stator and rotor are con-and reurn curent,are alo deteminedcoil, amperesseparately for the rotor end winding. Ia=stator maximum amplitude of current sidered to be closed circuits. In Fig. 2, a

in belt number 5, amperes/m stator end-winding coil is shown schemati-Nomenclature Ia = stator rms current ampere per turn cally as well as its fictitious return in the

I phase/C center of the air gap and on the end-core

a= rotor's total axial length of end wind- Kd=distribution factoring, meters(m) ~~p-fractional coil pitching, meters (in) '

as = stator's total axial length of end P p7/2windi,ng,mrn Qs= no. of armature slots r

a1 =coefficient of the stator space cosine Q no. of rotor slots STATOR CORE -,. ~~~~~~~qi=no. of conductors in a regular rotor slot STATOR END WINDINGfundamental component ~~~~ q = no. of conductors in a short rotor slot END CORE PLANEm ROTOR END WINDING

Paper 62-187, recommnended by the AIBE Rotating t = timne in secondsRORSHFMachinery Committee and approved by the AIEE T=number of turns per coil AIR GAP-i- ROO SHAFTTechnical Operations Department for presentation z2 axial co-ordinate /gat the AIEE Winter General Meeting, New York, ,8 = angle between end winding and the IP m-N. Y., January 28-February 2, 1962. Manuscript . 0submitted October 9, 1961;* made available for horizontal Zprinting December 8, 1961. z=-angle between coil sides and @ axis, ROTORJ. A. TaCoPosULOS is with the Westinghouse Electric radiansCorporatio~n, East Pittsburgh, Pa. 5=-phase group number Fig. 1. SImplified end region configuration

FEBRUARY 1963 Tegopoulos-Current Sheets Equivalent to End-Winding Currents 695

plane. As can be seen in Fig. 2, in the end uniformly distributed over the surface ofcoil portions AB and ED there is only h n i r s v Isaxial current flowing; in portion BC and, .

27rCD axial and peripheral. Return current peres per unit width of the band. A num- ^]of peripheral nature is flowing in part ber of bands equal to the number of con- Cos (4 +09)JFG along paths FHG and FKG. The ductors per pole per phase form a beltcurrent divides itself between these two of bands and the collection of the belts of For balanced currents, Ii=12=.. I.paths. The currenlt in each path is in- aUl three phases forms a current sheet The first term in parentheses adds up to

path. Finally, the portions EF and GA lying on the cylindrical surface of radius zero for a 1 to 6, as long as the angles 0Ecarry radial current. Collection of all such P1. to 0 are symmetrically distributed in 2w,coils constituting the end winding of a tur- The current outgoing at a=0 and, per- so that bi becomesbine generator, complemented as described, taining to the general phase belt 6, is 31will give current sheets corresponding to . . . b W+ (2the different currents mentioned previously, gven by equation 1 i amperes per meter. 1 --cos + (2)Thus, peripheral currents flowing in portions i=Ilo sin (4 +Oo-G5) amperes/in () Tecefcetmftesaecsnlike BCD will form a peripheral current The coeffiient of the space cosinesheet. Axial current flowing in portions where h maxium amplitude of the fundamental component in a Fourierlike ABCDE will form an axial current analysis is found similarly by summing thesheet. Currents flowing in radii like EF current, amperes/m; 0a= angle between contribution to this of al six belts:and GA will form a radial current sheet. 0 = 0 and the center of belt 6 at z=0;IDFinally return currents will be assumed 0O,= general phase angle with respect to +tto flow in the circle KFHG which is con- some abitrary reference. An example a,= v

al ~~~~sin (wt+Oo-08) cos edOsidered to be a filament. , t s

It will be noticed that each one of the of the space currents outgoing at z=O r1current sheets does not form a closed loop for values of 08, as shown in Fig. 5, equalof current but that all current sheets to Al, 02, is,Sgiven i Fig. 5. The v otogether constitute a closed system. This wave shown in Fig. 5 is analyzed subse - 2- [sin (wt+Go)+assumption is applied in a like manner to quently by Fourier series. Only the a-ithe rotor end winding, fundamental terms are being considered. sin (cat+Go-28&)13. Only fundamentals of the current sheet The space fundamental is a sinusoidalspace and time waves are considered. . . The second term in parentheses addswave with 0 as variable measred from an4. The generator has a 3-phase winding origin 0=0. Its components will be up to zero for 6=1 to 6, so that a, be-with 60-degree coil groups and double .

s e t . comeslayers (number of coils equals number of determmed subsequently at z 0. Thearmnature slots). coefficient of the space sine fundamental 31I

componen 'in a ouriesinalysi+sis)f(3n5. Only 2-pole turbine generators are component in a Fourier analysis is found aconsidered in this paper. by summing the contribution of all six

belts: The resultant fundamental wave repre-6 senting the outgoing currents at z=O,

Stator End-Winding Current Sheets I*a'6 can be derived now by combining the2 sin (cot+Oo -0) sin odO two fundamental space wave components

STATOR PERrPHERAL AND AXIAL CURRENT a4 =a, cos J+bi sinGSHEETS*

The stator end winding is arranged =3I [sin (w+Go) cos e-cos (W +Go) sin e1basically as shown in Fig. 4, where z= 0 I

is the end-core plane, b, the straight part 3Iof the protruding conductors, while c5 is l -sin (wt+Go-8) amperes/r (4)the length along the z axis of the skew K NTCRE PLANEpart. The sum c,+b,==a,. The con- The space fundamental wave at s=O,ductors which carry concentrated current equation 4 is sketched in Fig. 5 as a sineare assumed to be replaced by thin wave with maximum value equal to 31/w.bands carrying the same total current, The returning current i. at z=O has

* The development in tbis section is from referencethe same form as the outgoing 4, but

1, with minor chanxges. l l l llJit is displaced from it by fw radians,where p is the fractional coil pitch.~~~~~~~~11

END CORE PLANE 0r s-. =-sin (Qwt+0o-+w0p) amperes/m (5>^SX,5TAT ENDCORE LANEZCr -

G~~~~~~~~~~ARGK~~~~~~~~~~~~~~~~~~~~~~

Fig. 3. End-zone region showing cylinders 8 1-Y| Y4

C'i on which peripheral and axial current sheets -P >Px\ ~~~~~flow(P1, Pm) for stator and rotor respectively. rp |

Return current circle (radius par) is also shown.Fig. 2. Stator's end-winding coil with Radial current sheets flow on the end-core

fictitious return plane Fig. 4. Stator's end-winding conA;guration

696 Tegopoulos-Current Sheets Equivalent to End-Winding Currents FEBRUARY 1963

On the basis of equations 4 and 5, If for convenience (Oo+pr/2) is set equal b[ < -Ico P 1 z-b)j,an analysis will be made to determine the to ir/2, then equations 8 and 9 become C'speripheral and axial current sheet com- cos(wt-0) (17)ponents from the resultant current sheet. i6 cos zcos - cosThe peripheral component is along the L2\ C8 i =IIsin[P(1b-s)0-axis, the axial component along the z- (oI-t) (10) ca

axis. By the help of Fig. 4, equations 4 sin(wt-6) (18)and 5 will be modified so that they can i [ (1- .jf sin The varation of the maximum of thebe valid for the general point P in terms L2 C8 peripheral and axial current sheet com-of z and x. (,wt 0) (11) ponents, represented by equations 15-18,

In Fig. 4, a general point P is shown at is shown in Figs. 7 and 8 respectively.the intersection of an outgoing and a re- The value of I can be derived on the isw ngturning current. According to Fig. 4 basis of Fig. 6, and is given by equation STAToR RETN CUENTequations 4 and 5 become respectively 2.

31 1M..Q8 According to assumption 2, the return4 =3 sin Iwt+Oo-0+I-x (6) I= . amperes//m (12) current is assumned to flow in a fictitious

L \_ \ 2/J 2-plcsin y circular filament of radius p, equal to the

3I [,( Pr i 7 where, I ,=maximum crest current per mean radius of the air gap. The circular- sin LXt+OoOmk +X)(j coil, Q,-number of stator slots, and filament is lying on the end-core plane;T 2(2rpi/Q) sin= yu width of a thin band, see Figs. 1, 2, and 3. Also, in Fig. 2, it is

Tlhe value of x can be determined by the over which the coil currenta in ban shown that the return current divides

use of Fig. 4, considenrng that at any sumed to flow, itself between two paths, FHG and FKG,value of z, such that b, z. a,a, there is a It is added here that the magnitude around the circular filament. The cur-correspondence of an x such that of 'y can be determined in terms of rent in each path is inversely pro-

p,,l s-b,\ pi and cs. If P=pir/2, it is portional to the length of the path.x= 2 t1- ) This situation, so far as current flowing

2 CAP =pn.- Cs in the circular filament is concerned, isAccording to this equation, P. Pi equivalent to the situation where the

p,, Now, let total coil return current flows only in theat z=b, x- one path of the circular filament, i.e.,

2 6I FHG. Consequently, the return currentatzs=a, x=O I=- cos y amperes/m flowing in the circular filament is a wave

61 similar to the peripheral current sheet butBy substituting z= b, in equations 6 and I, sin y amperes/m with opposite direction and its magnitude7, equations 4 and 5, respectively, are T in amperes is the integral of the peripheralobtained. . . h . . current sheet io flowing on the cylindrical

Equations 6 and 7 are now used in Considering the chaacteristris of the surface of radius Pi, from b. to a.. Thedeterm'ining the peripheral and axial stator return current B2pliscomponents of the current sheet. If tor and equation 12, Ie and 1h becomey is the angle of the skew portion of the 2Q,KdVRIaT cot l Irg= - r iodz amperesconductor with the 0-axis, the peripheral amperes/m (13)component is

2QsKd v'2IaT where io is given by equation 17.i.=(+- s 1,= amperes/mn (14)t0= (i+ +- ) cos zY 27rpl 1 a= I /os-b,\=-{sin +00 +- coS where Kd = distribution factor, Ia= rms J Cs

Irz-b current per stator turn in amperes, T= cos (cot-9)dx

PT-1-z-b

cos y (8) number of turns per coil, and Q,= number_2 CScs /_ Jof stator slots. This integration yields:

The axial component is To sulmmarize, the equations of the 1peripheral and axial current sheet com- 1-1* sin P cos (ct-0) amperes

i= (4 -i ) sin 'y ponents, for the two ranges of variation 0 (19)611 1 P\s. to b, and b, to a., are as follows:

- cos 2 J +0+)The stator's peripheral current sheet

_P 0.O<z<b, i8=O (15) and return current are represented in2 (1- |=h sin P sin(9-0) (16) space schematically in Fig. 9.

9 L = T. qin P qin (ad - 0) (161~~~~~~~~~SOT TOTH PTC

.t *.s6+3~6+ 846+33 ~6+43 ~6ttS S=O0(1+) Fig. 6. Current band flowing through a pitchFEBRUARY 1963 Tegopoulos-Current Sheets Equivalent to End-Winding Currents 697

ie space are shown schematically in Fig. 9. AXAL CURRENT SHEET

PERIPHERALCURRENT

Iecos(27) Rotor End-Winding Current Sheets / H SHEET

Ie ROTOR PERIPHERAL CURRENT SHEET CURRE

The configuration of the rotor end wind-pi

ing is shown in Fig. 10. The peripheralcurrent sheet is formed by the rotor end

bs4 Cs z winding currents flowing in the partS CIS

ABCD. For both poles these currentsRADIA

are shown in Fig. 11. If If= field cur- CURRENT SHEET

Fig. 7. Maximum of stator's peripheral rent, q =number of conductors per reg-

current sheet component ular slot, q2=number of conductors per Fig. 9. Schematic representation of the dis.short slot, and Q,=total number of slots tribution of the sttor's four current sheetof the rotor, then the rotor's peripheral components

STATOR RADIAL CURRENT SHEET current sheet coefficient is

According to assumption 2 and Fig. 2, im qi±(. q) If amperes/rn rm=1 4 l.sI [r )(zbr)+c 1l

the radial current sheet component is CT_24 LYC_made of those fictitious parts of end- (22) cos(wt-6) amperes/m (25)winding coils, which connect the pro-truding part of the coil with the return- in amperes per meter of the width C7 of the Its maximum is shown in Fig. 12.ing part of the coil. Consequently, the current band covering the rotor end pe- ROTOR RETURN CURRENTradial current sheet lies on the end-core ripherally. The developed rotor end withplane (z = 0), and is made of the axial the peripheral currents is shown in Fig. The rotor's return current is also as-current flowing in the straight part of 11. sumed to flow in the circular filament of

the protruding conductors by changing In the two poles the peripheral currents radius p,, equal to the mean radius of thedirection from axial to radial. have opposite directions so that they can air gap. It is again a wave similar to theThe stator's axial current sheet at z= o be represented by a cosine wave. This is rotor's peripheral current sheet but with

is given by equation 16: being done subsequently by Fourier anal- opposite direction. Its magnitude inysis. Only the fundamental component amperes is the integral of the peripheral

jg=Ig sina P sin (oe rfaB) will be considered. The space funda- current sheet i9m flowing on the cylindricalamperes/rn of periphery of radius P1 mental of the cosine expansion of the surface of radius Pm from br to a.- The

The radial current sheet at z= 0 is formed wave, for any value of x (radians), such rotor's return current I'mg, isby the same axial current by changing that < x.< (Fig.11) has the coefficient ra?direction. Hence, 2 2 = - J iemdz amperes

2~I9m2Jbir = Ir sin P sin (cot- ) al = le.*cos OdO=- IemXsirJw 2 where iom is given by equation 25.amperes/m of periphery of radius pi Z/2 4 x

(20) sin =- lam sin - ar 4IOM97 - Io,, sin

is the radial current at any angle a and 2. . .Jsnrtime 1. Also, 1=.For any other where 1 is the pole-face arc in radians. r

Hence, the fundamental component of -X -)(z-br)'+Cr cos("t- )dzradius p, such that ~qpo Pi,l the radial '~ L.72cZ-i %c-(J-Bdradius .such tha p, '.pplthera the cosine wave corresponding to thecurrent sheet is given by equation 21. plane perpendicular to the z-axis and This integration yields:irf(at p) =Ir sin P( ." sin (wti-B) passing through x is 4 I#m

\p 4 xg" -- cos- cos(Wt-B)4 x 1 2ampei es/m of periphery of radius p iGm - '9m sin - cos (wti-) amperes/tn (23) - (T-0

(21)This wave is sketched qualitatively in the amperes (26)

The axial and radial current sheets in elevation part (dotted line) of Fig. 11.In order to find the equation of the

total wave, x is expressed in terms of z. F BFor this, considering r and x in radians,there is vl

_' E

rl~ ~~~~~~\z-Zb7 F A11 -lz.SIN(P2W) | I7r-t CT |LGI Dt Z

lt \ ~~~~~whence,L \ ~~~~~~~~~~~~~1lob.L T Z~~ x=-[(7r-1)(z-b7)+crfl (24) 1 t,.|y

By introducing the value of x in equation c1i.Fig. 8. Maximum of stator's axial current 23, the equation of the total peripheral

sheet component wave becomes Fig. 10. Rotor's end-winding conAguration

698 Tegopoulos-Current Sheets Equivalent to Enzd-Winding Currents FEBRUARY 1963

Iz iAXIAL CURRENT RADIAL CURRENT SHEET

I _ _ _ I RETURN CURRENT

OI,l,:>>_>r I15"Ea ;A

_1 r~~4I

Fig. 13. Schematic representation of the

br I~~~~~~~~~~~~~~~~~~~~~~~~~b

Fig 1 1 Plan view and elevation of the rotor distibution of the rotor's four current sheet Fi 14. Plan view deeaino h ooperipheral current components wandelrrevation ofth.rotFr

the sine wave corresponding to the plane 4 gThe rotor's peripheral current sheet and perpendicular to the z-axis and passing irm =- 'Tm C05 2 sinl (46-)

return current are represented in space through x issceaial in Fig 13. amperes/rn of periphery of radius Pm

izm= Izm cos - sin (4t-0) amperes/mn (31)ROTOR AXIAL CURRENT SHEET 2 '28'i~~~~~~~~~~~~~~~~~,0s the rotor's radial current sheet in spaceAccording to Fig. 10, the rotor's axial ''} and time. Also Irm=I2zm For any other

current sheet is formned by the rotor end This wave is sktched qualitatively in radius p, such that Pm P Pg, the radialwinding currents flowing in the part the elevation part (dotted line) of Fg. current sheet is given byequation 32.ABFE and DCHG for the one pole. If 14. Equation 28 is valid for the range 4the rotor end is developed, the axial cur- bT.<z.<a,. By substituting the value Of irm 'T r-cos 2 sin (4t-0)rent sheet will have the shape shown in x from equation 24, equation 28 gives 7 ~

Fig. 14. ~~~~~~the fundamental component of the total ames/noprihyofadupThe rotor's axial current sheet coeff- rotor's axial culrrent sheet wavre. (32)

dent hasthemagnitude4 v1 _ The rotor's axial and radial current sheetslem __ /'ez+Qr _ 8XI z z o <2lt¢(§b)6¢ in space are shown schematically in Fig.'zm 2 i)ri4

tzm I* c si t-)aprs(9 13 together with the rotor's peripheralamperes/rn (27) current sheet and return current.

When 0< z bT, equation 29 takes theThe actual axial current sheet corre- form Conclusions

sponding to the general position x, asshown in the elevation of Fig. 14, will be izm= 'zm, cos 2 sin (4t-0) amperes/rn The derived equations of sinusoidalsubsequently replaced by a sine wave by 7T 2(3) tp reesncuetshtsaro-use of Fourier analysis. Only the funda- mael eqiaettAheata urnmental of the sine wave expansion will be The maximum of the rotor's axial cur- sheso ubn eeao n einconsidered. The coefficient of the funda- rent sheet is shown in Fig. 1a~. as described in thebodyof this paper.mental iS In fact, the actual culrrent sheet of

2 C0 2 f+w/2RTRRDACU ENSHT each componenlt IS represented only by itSb=- 3 I:,, sin 0ds- - J kzm As was the case with the stator, the fundamental. While it is possible to add

w2 IT ° rotor's radial current sheet is also made any number of harmonics, it is felt that

4 x~~~~~~~~~~~~~~~~~~~

sinOde= - Izm cos - of those fictitiouspartsofrotorend-wind- the idealized representation with only2 ing coils, which connect the protruding the fundamental sinusoidal wave, is an

Hence, the fundamental component of part of the coil with the returning part of approximation compatible with the as-the coil. Consequently, the radial cur- sumptions necessary for the solution ofrent sheet lies on the end-core plane problems concerning the end region.

Therotor's peripheral currentsheetand p(ze=0) and is made of the axial currents The sinusoidal form of the current sheetsreturn current auflowing in the straight part of the pro-

^ I ~~~~~truding conductors, by changing direction/ t ~~~~~fromaxial to radial. The rotor's axial 1zm

w~~~~urrent sheet at z =0O is given by equationr

I flrki": | | _, ~izm-4Izmcos sin(4-0) lTIZVRC4 11schematicayinFig.13ee/ oaperes/m of periphery of radius p, P

The rotor's radial cu2rrent sheet at z =0 isFig. 12. Maximum of rotor's peripheral formed by this axial current by changng Fi. 15. Maximum of rotor's axil current

current sheet component direction. Hence, sheet component

FEBRUARY 1963 Tegopoulos-Current Sheets Equivalent to End-Winding Currents 699

can be used conveniently for the solu- the generator is operating. It is thus T. Smith. AIEE Transactions, Pt. III (PowrApparatus and Systems), vol. 77, Aug. 1958, pp.

tion of such problems. suggested that the effects of the two 838-47.The two groups of sinusoidal waves groups of current sheets be determined 2. EDDY CURRENTS INN TE END PORTION 0O

pertaining to stator and rotor end wind- separately and then combined appro- TURBINE-GENEBRATOR STATOR WiNDINOS, G. W.ings have been dealt with separately and priately8.SELeCTIC MCIR (book),A.K.4tgead

independently of ech other. Actua*y8. BLIRCTIUC MACHINBRY (book), A. B. FItztgerad,indePenden1tlY Of eaCh Other. Actually C.Xlgsley, Jr. McGraw-Hil Book Company, Inc.both are rotating synchronously and References New York, N. Y., 1952.there is a phase difference betwen them 4. TuB NATURB 0F POLYPEAS2 INDUCTION

1. I3NV COMPONENT OF ARMATuitz LBAIKAGB MACHINBS (book), P. L. Algr. John Wiley &depending on the power factor at whhich REACTANCE OF ROUND-ROTOR GNBERATORS. R. Sons, Inc., New York, N. Y., 1951.

ri S n f * g=auxiliary constant for computation of

Fiux Impinging on the End Plate or elliptic integrals (every time has avalue as indicated)H=magnetic field intensity, AT (ampere

Turbine Generators H=axial component of magnetic fieldintensity, AT/m

h distance between end-core plane andJ. A. TEGOPOULOS end bellASSOCIATE MEMBER AIEE ie=stator peripheral current sheet com-

ponent, amperes/mX= stator axial current sheet component,

Summary: The flux density on the end- ar= rotor's total axial length of end winding, amperes/mcore plane of turbine generators is de- meters (m) a=-stator radial current sheet component,termined by the Biot-Savart law applied a, = stator's total axial length of end amperes/mto the current sheets representing the end- winding, m o = rotor peripheral current sheet com-winding current distribution of stator and A =constant of elliptic iMtegrals ponent, amperes/mrotor, as well as on their mirror images b=auxiliary constant for computation of ,=,rotor axial current sheet component,with respect to the end-core plane. It is elliptic integrals (has in every case amperes/massumed that there are no other iron a value as indicated) irm =rotor radial current sheet component,boundaries present. The effects of stator b =axial length of straight portion of rotor amperes/rmand rotor are determined separately and end winding, m Ie=stator peripheral current coefficient,must be combined vectorially for determi- b, = axial length of straight portion of stator amperes/mnation of the total effect. end winding, m I,=stator axial current sheet coefficient,

Bzs PER = maximum axial flux density at amperes/ma point at a distance r on the end- Ih=stator radial current sheet coefficient,

' HE PURPOSE of this paper is the core plane due to the stator's pe- amperes/mlyi determination of the flux ripheral current sheet, lines per square Ien = rotor peripheral current sheet coeffi-analy teal determnatio of th fluxinch cient, amperes/m

densities on the end plate of turbine gen- Bzs RET =maximum axial flux density at a I'm-=rotor axial current sheet coefficient,erators under any load conditions. point at a distance r on the end-core amperes/mThe method employed for the deter- plane due to the stator's return I,, = rotor radial current sheet coefficient.

mination of flux densities is the Biot- current, lines per square inch amperes/mBZS RAD= maximum axial flux density at a Iog = stator return current, amperes

Savart law applied to each one of the end- point at a distance r on the end-core I'mg =rotor return current, ampereswinding current sheets as determined in plane due to the stator's radial 'a = stator rms current per turn = I phase/C,reference 3, and also to their mirror images current sheet, lines per square inch ampereswith respect to the end-core plane. This Bzsr=algebraic sum of the three stator If= excitation current in rotor conductor,method is used assuming that there are no components; total stator effect, lines amperes

per square inch k = modulus of elliptic integralsother iron boundaries except the end-core BZR PER =maximum axial flux density at a k'=complementary modulus of eJlipticplane which is also eliminated by con- point at a distance r on the end-core integralssidering the mirror images of the current plane due to the rotor's peripheral Kd=distribution factorsheet components. It is also assumed current sheet, lines per square inch K=value of the complete normal elliptic

BzER RET = maximum axial flux density at a integral of the first kind, correspond-that the end-core plane is not saturated. point at a distance r on the end-core ing to a modulus kFor either stator or rotor, the effect is plane due to the rotor's return 1=1ength of distance

determined separately by algebraic addi- current, lines per square inch M=general constant of peripheral currenttion of the partial effects of each one of BZR RAD =maximum axial flux density at a of sinusoidal distributiontheir current sheet components. Finally, point at a distance r on the end-core n = number of zones of the stator peripheral

plane due to the rotor's radial cur- current sheetthe two effects of stator and rotor are rent sheet, lines per square inch p=fractional coil pitchcombined at each point on the basis of BERT=algebraic sum of the three rotor P t/the generator's vPector diagram. components, total rotor effect, lines Q = no. of armature slots

Experiments were conducted to verify per square inch QT =no. of rotor slotscalculations, conclusions of the corn- c=constant of elliptic integrals qi,=no. of conductors in a regular rotor slotparison are included. C=number of parallel circuits per phase Paper 62-l88, recommended by the AIBB Rotating

cr = ar-br, mf Machfinery Committee and approved by the AIBE=S a - b,, ni Technical Operations Department for presentation

Nomenclature E = value of the complete normal elliptic at the AIEE Winter General Meeting, New York,integral othseodhdcre- submitted October 9, 1981; made available for

a =auxiliary constant for computation of sponding to a modulus k printing December 8, 1961.elliptic integrals (has in every case F=-function of elliptic integral of the first 3. A. TEGOPOUJLOS Is with the Westinghousea value as indicated) kind Electric Corporation, East Pittsburgh, Pa.

700 Tegopoulos-Flux Imnpinging on End Plate of Turbine Generators FEBRUARY 1963