IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS AND CONTROL INSTRUMENTATION, VOL. IECI-22, ;NO. 2, MAY 1975:

Optimum Design of Permanentic: Generatorsfor Control Applications

COLIN D. DICENZO, SENIOR MEMBER, IEEE, HESAMUDDIN AHMAD, ANDNARESH K. SINHA SENIOR MEMBER, IEEE

Abstract-Permanentic generators are particularly suited for time-optional control applications. The design equations for a permanenticgenerator are derived and then applied to an example of design indi-fcating all steps in the process. A number of simplng assumptionshave been made which can be modified according to the requirementsof the design specification.

NOMENCLATURE

1i Length of the magnet.Ig RRadial length of the airgap.A Armature mmf.A80' Steady-state short-circuit armature mmf.I t: Armature current.0I~,s Steady-state short-circuit armature current.D Diameter of the machine.E, EMF generated by the air-gap flux.-ENL excitation voltage; driving voltage in Blondel dia-

gram.,B AMagnetic flux density.Bm B when the energy product (BH) is maximum.Bo0 B when the stabilized mage is short-circuited

(by an infinite permanence).Bs B at the point of stabilization.N :-: ;Number of armature turns in series per phase.:H M:agnetizing force at magnet termnals.Q8J Armature mmf due to short-circuit current trans-

formed to the B-H plane.D8GL- Internal voltage drop due to short-circuit trans-

-formed to the B-H plane.P0 Permeance of air-gap.PO Permeance of equivalent magnetic source.,A- Permeability of free s.pace = 4ir X 1O-7.K,i D.C. resistance per unit length of conductor.P6 leakage permelance.Kf Ratio of effective pole width to magnet widths.KI Ratio of total permeance to that of path con-

sidered.K16 Winding constant, product of pitch-factor and

winding distribution factor.

Manuscript received November 11, 1974.C. D. diCenzo and N. K. Sinha are with MeMaster University,

Hamilton, Ont., Canada.'H. Ahmad is with the Generation Electrical Department, Ontarioydro, Toronto, Ont., Canada.

INTRODUCTIONMANYAUTHORS [1], E2], [3] have point-ed out that

small motors with permanent-magnet rotors havesome advantages over the electromagnetic version. Re-cently, it has been shown that such motors are ideallysuited for time-optimal control of angular position [4].Permanent-magnet generators find applications in mobilemilitarv units such as aircraft and missile generators,mainly due to their smaller weight and grater relability.Synchronous generators up to 75 kVA at 1714 rpm havebeen developed by the U.S. Signal Corps.

Hence, the need for optimum design of permanent-magnet generators is obvious. Such a design based onminimizing the D2l product for a specified kVA (whereD is the diameter and 1 the length of the machine) wasfirst proposed by Hanrahan and Toffolo [5], but they havenot given a practical design of such a machine.The object of this paper is to present an actual design,

procedure which makes the ealculation of an optimummachine possible. Simplifying assumptions have beenmade which can be easily modified according to theactual design problem, and yet the same design equationscan be utilized to achieve the designer s aims.

DESIGN EQUATIONSAll the design equations are derived in this section using

the MKSC systein of units. The magnetic quantities areon -per pole basis and the electric quantities on per phasebasis.

SpecificationsTypical specifications of a 3-phase ac generator are

I) kVA, 2) frequency, 3) speed, 4) voltage, and 5) power-factor. Thus the number of poles is also known. Inherentvoltage regulation or short-circuit current may also bespecified.

The B-H PlaneThe B-H curve of the magnet, material is shown in Fig.

1. Lines I, II, and III determine the operation. If themagnet is stabilized by bringing the generator up to ratedspeed under short-circuit, the operating point moves downto PS. Slopes Si, SD, and So are the unit permeances ofthe machine corresponding to Pi, P , and, Po. The normal

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DICENZO et a.: PERMANENTIC GENERATOR DESIGN

B g SI

Br I

BOB1 : !'Q

Bs P

: H2.. , ~~~~~~~~~~~~~H-::: ~H1 H'2 Hs N C

Fig. 1. The B-H plane.

operation would now occur along line I and the machineis said to be stabilized. For no lQad on the generator, theoperating point P1 is obtainedcs the intersection of lines:I and II. Point P2 represents the operating conditionsunder full-load conditions.;:A simple geometry of the machine considered is shownin Fig.: 22, Where b, h, and Im are the dimensions of themagrnet. Permeances P0and Pi-can now be transformedto the B-H plane as folows:

0 ~Pg = ;07b(-1)*- ~~~~~p- AO D~~~~~~~~~~~~~~~~~~1;where

b' effective pole-width of the machine;=bkf. (2)

Hence

lmbkfPg

and

.0 .Pb j (3)Sg=-P bokfI (3)-: ~~~blmn --

The leakage permeance P1 is the result of contributionsfrom many paths and is given by

P-Ki,l[1 srD-Pb-2i rh 7rD-Pb 1

215

Fig. 2. Machine geometrY.

where') nz

K idjotI0IIIKa *ir

No-Load VoltageAt no-load the magnet operates at point Pl in Fig I

at which point

EN-L KeN40 (6)

where

Ke 4.44 fKw. (7)

But

g= blmBgi (8),and

BgI= SIgH (9)-Thus,

ENL KeN (blm) SgH. (o10)Substituting for S0 from (3) we have

- - ENL = 1KK1 (bhlmNHi)

Short-Circuit CurrentFor steady-state short-circuit the generator will operate-

at point P, in Fig. j. At short-circulit all the voltage isutilized as a drop in internal impedance. The linejoiiing0 to P1 divides any air-gap: voltage into impedance dropand -terminal voltage. Hence, the- generatd voltage atshort-circuit equals the armature drop represented by the

(.q)

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS AND CONTROL INSTRUMENTATION, MAY 1975

flux-density D. in the B-H plane; the corresponding de-magnetizing-mmf is QS-.From the geometry of the B-H curve

Qsc= nQ2where n is the per unit short-circuit current. Also,

,~~~~~q IB, --2B81 B12 + yR1,

where -y is per unit voltage regulationFrom similar triangles P1S1O and P2S20

91_ QSO n

Bt2 QSc- Q2 n-i

Using the relationship between B,, and Bt2

(12)

Thus, if Q2 is defined such that- F2- F2 + Ad = F1 + Q2

then it follows that(16)

(17)P

2-=PAd2

(13)

(14)

Thus the point of operation (F2, 2) is found simply byadding Q to the no-load mmf F1.

Tran$sforming to the B-H plane the effect of short-cir-cuit demagnetization can be written as

Q5CS1h (18)

~15) Since short-circuit current is almost completely direct-:aXis eureth

Short-Circuit Armature Re6action

When the generator is operating on load the armaturecurrent gives rise to an mmf which can be either in a mag-

netizing direction or demagnetizing direction dependingon a load power factor. As in most cases the generator-loadis inductive, the armature reaction almost always has a

demagnetizing effect. The magnetic characteristic of a

permanent-magnet- generator is shown in Fig. 3. With no-

load on the generator the point of operation is given by(F1, ). Let the operating poinlt -because of the load beat (F2,4)2). The leakage flux at this point is 4)2 and the air-

gap flux 4)2 = )2 -412- This value of air-gap flux requiresa certain mmf drop across the gap - F1 = Rg2g Sub-tracting this value from F2 will glve the demagnetizingmmf Ad2 causing operationl at (F2,42), i.e.,ed2 = IA-

F829g2

o0

A s = KaNIscwhere K. is given by

Ka

Thus,

QSo

Using eq. (3),

Q8c

where

(19)

6 kw7r p

SgK0aNIs1sth

N- Kq15

(20)

-(21)

(22)

Kq = loKfKaIsc. (23)Short-Circuit Voltage DropThe air-gap: voltage on short-circuit equals the drop due

to internal impedane. This voltage drop is represented bythe line OPS :in the B-H plane in Fig. 1. A simple calcula-tion gives

ls'4KRIS 22 122

- Ke Ke+D2which can be reduced to

QKDs' C.-R -Qsc-Q. Sk~X

(24)

(25)

where

4KRIS2t K-~ eb JJRsIs \2Kz = el

(26)

(27)

Fig. 3. Magnetic characteristic of permanent-magnet generator, and

1 +- '

nry

216

..DI S P, EI TOR DESIGN 17

X QBC/N. (28)

Ge-metr-in th TfhH Planee first geometrical relation s osed at point P.

:whereline louts the f:41 axis. Hence,

B. SzI

AdJ

QE (S+e~ S+,O1S

(29)

(30)

(A + SoQ)(.1)

w' nhbedue.r4 tgive

Bo S S- +Sf H

S SF1 +.( * e~~~~~~H 4-

Sut etnq. (3 ) frm .4

ILS+H+ff J1th+X,t

S. +(HUh.S0S1

(32)

(33),

(34)

(30)d iangeqJO),

Sw - + (So+ Sz)Q. (37)tqA tio (9)9 (30, (>, n (37)ar the four i

dependent equation defin the geometry of- t curve

P OCEP tO'FOR DESIGX.,~~~ ~~~it..s .>In the hove designeqatnitis h that two.difr ut s of ndepedenteqi tio xi . ations

(2), (30) (2), d (37) relate toth g;eintr the1 plae ^whilt Q (I,t (I),1 (22), d (24) dep

o thm chine parammert and eA triI quantitiesuk n load vltge artcnit c .nt armat er ation u~d oIt dr aue t 6sh ni ntoth acine. XX2 et' u ^ai:

Theequatities Q a i4,l ob 4 frm-the. iparameters fix the6point ,P.which ie he curve^ tsatisf all equation TI other wods, the tw o uti are led for a Q-D fit. A sip approach to thispoehmi nis

Select a renahlediameter for th machin. For-thidiametr assuime a value of b and A. and S, c-an then heCalculated froim eq (3) and (4) oti tha (hl)P

i% ikg.an thenheeasily ohtainA'froe qs (29) (3), and (32). Fo

11ecurve/as ½ i F' 4 the variationd of Q, anidisplWoted soa nFg.5.

IAT/Inch

g -.M ete efI. .~I

Now n so eb leulated' frmeq. (2)jwicpent on th tor ahle of theachineeEqua,tion,,(25)') i-the quatio a srgt l ati aD12, he slopef-ti lhne 4hconstt/athedeohbaecalul ted. Tus,the intreetion of traWhlnwith the curve eiwse point w , ensnd henlgi the require Q tThs, 1t d B hbei known, ican hea i

rom e. (3) n te number of gimeni hyq"(28). The load voltage and the regulutio hei known,i-the no-loadv nd. hen e:tlent of the machine can caulated m eq. (II.Hence,fior a tul daiametera- width,of tIhpth vlues of lingtiof themh ff v ofh are calculated and plott own i g . Promcurve, theoptim tin leng h-orchd ianet-r i thugskThesamproc uealbcne repeated for differ ntdiveters and.-the optimummacine fr minimu 2 can heobined Thfo g example the produ"re.

EXAM?PLE OP DESIGNTConsidr a' mainewNith theflowng peeificitions:

2 kYA,3200 H 115 vots/A onneced- 240 gwVtri Here

' 3200n2

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS AND CONTROL INSTRUMETATION, MA 197

Hence,

120 X3200ofplsp 20X0 = 16= number of poles.

\ ~~~~~ ~ ~~~~~~~~~24000D=2.5", b=O.25"

2000

I'phase 5.65 amperes'.3 X 115

Assume a diameter of 2.5 inches. The resulting polepitch is 0.49 inches. Assumingratio of length to pole pitchas 2.5 gives I 1.25 inche approximately.- .Taking the pole are as~ 75% of the pole pitch, the pole-shoe = 0.37 inches.Now if Kf -1.2 to 1.6, then the approximate width of

the pole b = O.2 to 0.3 inches.I \ \ \ \ /Assuming an inherent roltage regulation of 25%

ENL = 115 +4115/4 144 volts.:\

\/With a sinusoidal wave-form of air-gap flux, we haENL 4.44fKN4,010- where,, g Bg. bl, Since Bo

/liesbetween 40 to 55 kilolines per square inch, letBg=45 kilolines/in2. Let N 80. Total numbe'r of conductos =

480.Selecting one slot per pole per phase, the number of

slots N8= 48, and the number of conductors per slot,10. Full-pitch winding gives KU =-1.0: - - \ \>-Selecting 2O Gwifre-of -diameter 0.037 inces,the-dimensions of the slot may now be ealculated, assumingsinge-layer winding.

Q;/105 (0.037 X 5 + .005) 0.190 inches Let d2 eFig.S. PlotofQ1 and D802. .002 inches. 1 = (0.037+ .005) X 2+ .005 0.087

inches. Thus, theslot dimesionsareknownfr the calcula-1 (in ch) - - t i O n o f tion of X i.

3¢ : { 0 2.5", bw 0.25'" r Itm ay be mentioned here that ratio fthe height of

the slot to the wdth is fairly constant for a particularmachine, and does not vary very much if the winding is

changed.KR forSWG2O wire 12.3ohms per 1000ft. at

;C.The short-circuit current- can be calculte from eq.

.(15)

1+ r' _ 1+0.2521" n ~~~~~~~~~~~~~~~5.- --;i-r- -0 25

Hence I, X 5.65' 28,.2 amperes.From thesevalues, the minimum len-gthhas been cal--

culated as explained in theprous sectin, andthe result- : : , is plotted inFig. 6. The minimum) length is 1.2inches for

D -2.5 inchies.

CONCLUSIONThe design example calculated indicates that the values

of Qc?and D2 are quit se sitive to thegeometry of theB-Hcurve of the materl. Heence, the choiceofmaterialis an important factor. Alnico V' and VIare6 the best

permanent mnagniet mater'ia'ls for Machines oif integralhorse-power capaeity. Forgsmaller machines ramic mag-

h (Inch)_ _netLs are preferred because of their lowrcst, but they have2 .3 .4 .-5 .. . ....... . - -.-.........much lower values of -residual inducti.-

Fig. Variation of machinelength height pole. example calculated herefor the, given gegerator

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IEEE TRANSACTIONS. ON INDUSTRIAL ELECTRONICS AND CONTROL INSTRUMENTATION, VOL. IECI-22, No. 2, MAY 1975

Specifications demonstrates the method of optimum designfor minimum D21.

REFERENCES[11 F. W. Merril, "Permanent magnet synchronous motors," AIEE

Trans., Vol. 73, Part III, pp. 1754-1759, 1954.[21 A. T. Puder and F. Strauss, "Salient-pole permanent magnet

alternators for high speed drive," AIEE Trans., vol. 78, Part 11,

pp. 333-338, November 1957.[3] B. Szabados, N. K. Sinha and C. D. diCenzo, "Stepping motors

versus permanentic motors for control applications, ElectroiEngineering, vol. 45, no. 544, pp. 61-63 1973.

[41 B. Szabados, N. K. Sinha and C. D. diCenzo, "A time-optimaldigital position controller using a permanent-magnet de motor,"IEEE19 Tran. Ind. Elect. Contr. Instrum. io IECI-19, pp.74-77, August 1972.

151 D. J. Hanrahan and D. S. Toffolo, "Permanent agnet gen-erators," AIEE Trans., Vol. 76, Part III, pp. 1098-1103, 1957.

Ana ysis o the Frequency Characteristics of a

Three-Phase Induction Motor

KOKICHI NITTA, SENIOR MEMBER, IiEEai, H. OKITSU, T. SUZUKI, AND IKURO MORITA

Abstract-In this paper, we discuss the frequency characteristicsof a three-phase induction motor theoretically and take up the trans-fer functions on. the speed variation to the small variation of supplyvoltage, supply frequency, and load torque. The frequency responsesare affected considerably by the electrical time constant at the rangeof,high signal frequency, but these transfer functions may be approxi-mated by the form of a first-order lag element as the value of me-chanical: time constant or the slip of a tested induction motor in-creases.

INTRODUCTIONTrHE three-hase induction motors -are usually operated

at a fixed speed or near synchronous speed. However,along with a recent remarkable development and wide-spread use of semiconductor devices, the induction motorshave been often used in the speed control system. Ac-coordingly it is absolutely necessary to study the dyna-mic performances of them. In the analysis of the dynamccharacteristics of electrical machines in the control sys-tem, we usually have two methods, one is the time domainanalysis and the other is the frequency domain analysis.Firs,t in the former situation the dynamic performance ofthree-hase induction motor has been analyzed by meansof analog simulation or state variable analysis, [1]-[3E,and second, we derive an approximate transfer functionof them on the basis of the equivalent circuit of the in-duction motor in the latter situation [4]. When themechanical time constant, which is the ratio of a momentof 'inertia to: a differential coefficient derived from torque-speed characteristics, Is much larger than the electrical

Manuscript received December 9, 1974.The authors are with the University of Tokushima, Tokushima,

Japan.,

time constant, we can successfully use the above transferfunction. But when an induction motor has a relativelysmall mechanical time constant, this trans function can-not express exactly the dynamic performances of it.And so, in this paper, we -discuss theoretically the fre-

quency response of an induction motor in considerationof all electrical time constants. Especially we expand the.theorem of ac carrier control system [5] to the induc-tion motor and analyze the dynarmc characteristics, of itby the perturbation method. As for the dynamic charac-teristics, we take up the transfer functions on the speed:variation to the small variation of supply voltage, supplyfrequeney, and load torque. From the results of the analy-sis, we see that the frequency responses are affectedconsiderably by each parameter of the induction motor.

TRANSFER FUNCTIONSThe diagram of a three-phase induction motor with

the primary winding on the stator is show,n in Fig. 1. Asthe d-q co-ordinate axes rotate at an electrical angular

as

d, T

Fig. 1. Three-phase induction motor.

219