Field analysis for rotating induction machines andits relationship to the equivalent-circuit method

Subscriptsr,n,qRSx,y,z

Superscriptsk,l,m -

S. Williamson, Ph.D., A.C.G.I., D.I.C., C.Eng., M.I.E.E., andA.C. Smith, B.Sc. (Eng.)

Indexing terms: Electromagnetics, Induction motors, Machines

Abstract: This paper presents a field analysis for rotating induction machines which includes the effects ofrotor skew, and accurately models the discrete nature of the rotor curent. It is verified by comparison withthe conventional equivalent-circuit method, and by using it to derive the equivalent-circuit for a balanced3-phase skewed-rotor induction machine.

normally accomplished by means of the phase-equivalentcircuit. This technique of analysis, which was introduced bySteinmetz,1 is well established and is developed to thestage that performance prediction for most types of balancedinduction motor is largely a matter of routine. The generalavailability of electronic calculating machines, in particular,has facilitated repeated complex arithmetic manipulation,so that analogue methods such as the circle diagram haveeffectively become obsolete. Symmetrical-componenttheory has been ingeniously used in conjunction with thecircuit-analysis method to enable the effects of unbalancedterminal voltages2 and general stator windings3 to betaken into account, although the visual and conceptualsimplicity of the phase-equivalent circuit is now lost.

Despite the dominance of equivalent-circuit techniques,a number of authors have proposed analyses based on adirect solution of Maxwell's field equations in an idealisedmodel of the machine.4"8 The motivation behind thisdevelopment was to gain further insight into the operationof the machine,4"6 or to examine some particular aspectof machine behaviour.7'8 Such an approach was adoptedby Greig and Freeman9 and developed by Freeman,10 whointroduced a versatile and convenient system of analysisbased on a stratified model of the machine. Broadly, thestator and rotor are represented by a series of smoothconcentric homogeneous regions having material propertieswhich depend on those of the parts of the machine theyreplace. In particular, the cage of the rotor and its circum-ferentially adjacent teeth become a homogeneous layerof uniform conductivity and thickness. Maxwell's fieldequations are solved in this idealised model, usually on aspace-harmonic basis, with appropriate matching at regionalboundaries. Although the eventual determination of thestator input current requires the explicit or implicit calcu-lation of terminal impedances, the method remainsessentially a field-solution analysis.

The field-solution method based on a multilayer modelhas found its widest application in the analysis of linearinduction machines, where it has been used by a numberof authors.11"15 There are several reasons for this choicefor such application. First, there is an obvious physicalsimilarity between the mathematical model and the linearmachine it represents, particularly as far as the rotor isconcerned. A linear-motor rotor invariably consists of aconducting sheet of aluminium lying on a smooth laminatediron block. Secondly, conventional circuit analysis hasproved to be incapable of modelling the asymmetric magneticand electric conditions which prevail in the linear motor,whereas the field analysis has been successful in this respect.

83

List of symbols

a,b,c,dB

cSn

EgHJKsKbs KbR

NS'NRPq

P'

ssTwZD

zu

a.7eXR[i

V

T

CJ

= transfer matrix elements= magnetic flux density= Mth stator-winding conductor density distri-

bution= induced e.m.f.= airgap length= magnetic field strength= x-directed component of current density= generalised skew factor= stator/rotor slot-breadth factor= number of stator/rotor slots= real machine specific slot permeance= model replacement machine specific slot

permeance= general layer depth= fraction slip= effective complex turns ratio= machine length= harmonic surface impedance looking down

from rth region= harmonic surface impedance looking up

from rth region= stator model coil pitch= rotor skew (mechanical degrees)= propagation constant= rotor slot pitch= absolute permeability= harmonic number= fundamental pole pitch for 2-pole wave= supply frequency (in rad/s)

region numberrotorstatorrectangular co-ordinates

= harmonic wavenumbers

1 Introduction

The analysis of balanced 3-phase induction motors, excitedfrom balanced 3-phase alternating-voltage sources, is

Paper S84B, first received 25th July and in revised form 19thNovember 1979Dr. Williamson and Mr. Smith are with the Department of Engineer-ing, University of Aberdeen, Marischal College, Aberdeen AB9 IAS,Scotland

IEE PROCEEDINGS, Vol. 127, Pt. B, No. 2, MARCH 1980

0143-7038/80/020083 + 08 $01-50/0

A third significant advantage is offered by the explicitdetermination of the magnetic field in the airgap, facilitatingstudies of 3-dimensional forces and stability.

One of the features which has prevented a more wide-spread application of the field-analysis method to rotatingmachines is the modelling of the slotted regions, and, inparticular, the modelling of the rotor. The replacement ofthe squirrel cage by a homogeneous conducting sheetcauses certain important features to be lost. Discrete barcurrents in the real machine produce slot-harmonic fields,which may constitute a significant part of the rotor leakagefield. Furthermore, skewing of the rotor constrains thebar currents to flow in paths which are strictly nonaxial,thus altering the stator/rotor coupling and axially redistri-buting the airgap flux.16 Both of these effects are lostwith a homogenous-sheet rotor model. In addition, therehas been some doubt concerning the ability of the field

The machine is regarded as a system of coupled circuits,and coupling impedances are used to account for the e.m.f.induced in any one circuit as a result of current flowing inany of the others. Each series-connected stator windingconstitutes one circuit. In the rotor, the induced e.m.f.sdiffer in frequency, amplitude and distribution withharmonic order, so that separate rotor coupling impedancesmust be determined for each harmonic. A single rotorbar is used to typify the whole of the rotor cage, as iscommon practice.

If the stator has N separate series-connected windings,and it is thought that M stator harmonics are sufficientto give the required accuracy, then the coupling impedancesmay be assembled into an N + M square matrix. Theterminal voltages are related to the winding and bar currentsby a matrix equation, as shown in eqn. 1 for the particularcase N = 3,M= 6:

>si~

^S2

^S3

0

0

0

0

0

0

=

7 7•^Sl.Sl ^ - S l ^ZS2,S1 ZS2,S2

^S3, SI ^S3, S2

ZRX.SX ZRLSI

ZR2.S1 ZR2.S2

7 7£jR3,Sl ZjR3,S2

Z~R4,S1 ZR4,S2

Z ZD

7 7^R6,S\ ^R6,S2

ZSX,S3

ZS2,S3

Zs3, S3

ZRl,S3

ZR2.S3

ZR3,S3

ZR4,S3

ZRS,S3

ZR6,S3

ZS1.RI

ZS2,RX

Z~S3,RX

ZRX.RX

0

0

0

0

0

ZS1,R2

ZS2,R2

ZS3,R2

0

ZR2.R7

0

0

0

0

ZSX,R3

Zs2,R3

ZS3,R3

0

0

ZR3,R3

0

0

0

Zsi.RA

ZS2,R4

Zs3, «4

0

0

oZR4,R4

0

0

ZS1,RS

Zs2,RS

ZS3,RS

0

0

0

0

ZRS.RS

0

Zsi,R6

Zs2,R6

Zs3,R6

0

0

0

0

0

ZR6,R6

hi

TS2

TS3

TRX

IR2

IR3

IR4

IRS

IR6

(1)

analysis to include the effects of slot leakage. These limi-tations have been overcome by techniques which borrowheavily from the equivalent-circuit method,17 bu. the modelhad to be simplified considerably to facilitate the modifi-cation.

Recently, a new form of field analysis, which allowsthe discrete nature of the rotor currents to be taken intoaccount, was proposed.18 This paper considerably extendsthe theory developed there to include the effects of rotorskew. The new theory' takes full account of all impedancesdue to flux entering the airgap, and in addition, includesslot leakage directly. It offers the advantages which arepresent in the homogeneous-rotor version used for linear-induction-motor analysis. Asymmetrical stator windingsand terminal voltages are accommodated with ease, and themagnetic fields in the various machine parts are determinedincidentally. The method is verified by comparison withthe conventional circuit-theory approach. The analysisis applicable to both cage and slip-ring machines, but willbe developed here in the form appropriate to a cage machine,for the sake of brevity.

2 Theoretical development

2.1 Coupling impedance and the impedance matrix

Induction machines almost invariably operate from aconstant voltage source. The input currents are there-fore dependent quantities which must necessarily bedetermined as part of any analysis. For a general unbalancedsupply, this may be readily accomplished by means of thecoupling-impedance concept.

84

The present paper, in essence, is concerned with the use ofthe field-based analysis to determine the values of thecoupling impedances.18

2.2 MuItiregion model

Fig. 1 shows the relationship between the model (at C)and the machine it replaces (at A). The notional intermediatestage in the representation (B), in which the real slots arereplaced by parallel-sided slots of width equal to the slotmouth widths, is introduced as an aid to the determinationof the correct permeabilities for the slotted layers. Thispoint will be clarified in Section 3.1.

Each region of the stack replaces part of the developedmachine,9'10 and is characterised by its relative position,thickness, material constants and velocity relative to thestationary (stator) reference frame. The regions may beisotropic or anisotropic19'20, but are assumed to be hom-ogenous and linear. It is further assumed that the iron isinfinitely laminated, so that the conductivity of the ironregions is zero measured along any axis. The sinusoidalsteady state is assumed to prevail, although Freeman21

has used a layer model for transient problems.

2.3 General form of field analysis

Complex Fourier analysis is used to resolve the excitedwinding into double harmonic series in the axial (x) directionand the circumferential (y) direction (Sections 2.4and 2.5).The homogeneous nature of the regions in the modelensures that the axial and circumferential variations ofthe harmonic fields in the various regions will be the same

IEE PROCEEDINGS, Vol. 127, Pt. B, No. 2, MARCH 1980

as those of the parent harmonic excitation. This enables theanalysis to proceed on a harmonic-by-harmonic basis, andwe may, consequently, confine ourselves to the solutionof Maxwell's equations in the spe'cial case of a knownvariation along two axes and with time. Employing ageneral form of notation, all field quantities may be writtenin the following form:

a - (2)

in which Re denotes the real part, and the summationsrange over all positive and negative values of the wavenumber(/, m). sk is introduced here as a general parameter whichwill later be interpreted as fractional slip, where appropriate.

The solution of Maxwell's equations under these assump-tions produces, for the magnetic field strengths in a generallayer,

fjklm _ "' fitHx - -jHy

= C cosh ez + D sinh ez

rjklm _ ]_ (Jklm

*{Jilm + e(C sinh ez + D cosh ez)}

in which the propagation constant e is defined by

(3)

(4)

(5)

(6)

and C and D are constants of integration. Eqns. 3—6apply for any layer, with the appropriate values of nx,fly, fiz and Jxlm being inserted. A similar set of equationsmay be written for each layer in the model, giving a total oftwice as many constants of integration (per harmonic pair)as there are regions. Fortunately, these constants need notnecessarily be evaluated explicitly. In most instances, itis sufficient to determine the field values at the upper andlower surfaces of the excitation and to cross from regionto region by invoking the conditions of continuity.9'10

The values of Hylm at the upper and lower boundariesof the excited layer (assumed to be the nth region) are

denoted Hn and Hn.x, and obtained from eqns. 3—6 as

Hn = &n{u(an-\) + lcnZn-x) (7)

^n-l = - M"(an ~ 1) + fcn/^i) (8)-(/jU2co//)/n

an, bn and cn are transfer matrix elements whose valuesare listed in Appendix 6.1. Z^ . , and Z®-\ are harmonicsurface impedances5'10 looking up from and down fromthe nth layer; they are defined in Appendix 6.2.

2.4 Stator coupling impedances

Neglecting end effects, the fields produced by an excitedstator winding are invariant in the axial (x) direction; thisis obtained in eqns. 2—6 by inserting the value m = 0.

If the rth stator slot contains cSn r conductors of then th stator winding and is positioned at y = yr (measured inthe stator reference frame), the conductor density distri-bution for the nth stator winding is given by

(10)

where

KlQn.rexp(/*[yr) (11)

S r-1

k = vkn/T is a wavenumber. K%s is the t»feth-harmonicstator slot breadth factor, defined by

k _ sin(kbs/2)b (12)

When excited with current ISn, the nth stator windingproduces a yfeth harmonic of current density, given by

Jk = IsnCL (13)

IF^

A B

Fig. 1 Development of the multilayer model

IEEPROCEEDINGS, Vol. 127, Pt. B, No. 2, MARCH 1980 85

This value may be inserted into eqs. 7—9 to determine thefields and hence the coupling impedances.

2.4.1 Stator—stator coupling impedances'. The stator-drivenharmonic fields rotate synchronously in the stator referenceframe, so that sk = 1 in eqns. 2—5. If the wth statorwinding is excited by a current ISn (in A), the e.m.f. itinduces in the m th stator winding is given by

Sm,Sn = EkSm,Sn

(14)

in which it has been assumed that the mth and nth windingsboth occupy the qth layer, and Jk is given by eqn. 13. Ifthe nth. winding does not lie in the same layer as the mthwinding, the first term inside the brackets of eqn. 14 shouldbe set to zero.

2.4.2 Rotor—stator coupling impedances: In the rotorframe, the stator-driven fields rotate at slip speed, so thatsk is the fractional slip of the yfeth stator harmonic field,defined in the usual manner, with l = k (eqns. 2-6) . Thee.m.f. induced in the skewed rotor bar by the unskewedstator-driven field may be obtained via the flux-cutting ruleas follows:

Rk,Sn =

2.5 Ro tor coupling impedances

(15)

The skewed nature of the rotor cage results in axial vari-ations in the fields, as well as introducing axially directedfield components. The physical limitations introduced bythe finite length of the rotor stack now assume someimportance, and a suitable means of introducing their effectmust be incorporated in the model. The method employedis based on a technique due to Preston and Reece.22

The rotor is regarded as being one of an infinite array ofidentical rotors arranged on a common shaft, with a single,infinitely long, stator. The rotors are given alternate skew,so that a typical instantaneous current-flow pattern maylook as shown in Fig. la, where the lengths of the arrowsare intended to represent current amplitudes. The back ironis continuous between rotors, so the model has a continuousstack of rotor laminations with alternately skewed cagesembedded at regular intervals. By symmetry, no flux willcross the planes normal to the common axes (shown by thedashed lines XX' and YY' on Fig. 2a. In the real machine,the bar currents produce little axial leakage flux at the coreends, a situation which can be simulated by arranging forXX' and YY' to correspond with the core ends. To do this,the cages in the infinite array are moved closed togetheruntil their end rings overlap and the end-ring currentscancel. The resulting rotor arrangements appear asillustrated in Fig. 2b. The end-ring currents are now com-pletely absent, and their effect must be reintroduced intothe coupling impedences as an external component.

The sources of the rotor currents are the stator-drivenfields, so that the frequency and distribution of the barcurrents are determined by the stator harmonics and therotor speed. The discrete nature of the rotor bars results inthe production of rotor-slot harmonics. If the i>ftth har-monic bar currents are of amplitude IRk and frequency

86

sku), then 2-dimensional Fourier analysis reveals that theresulting current-density distributions are of the followingform:

rk _ y y rkim

where

Jklm =

and Klsm is a generalised skew factor defined by

ffife /tan 7

dR XR I tan 7 + m

Kim =sin {(w/2)(l tan 7 - m)}

(w/2)(/tan7 —m)

(16)

(17)

(18)

7 being the rotor axial skew, in mechanical degrees. Thesummation for the circumferential harmonic number (yl ineqn. 16) reflects the production of the rotor-slot harmonics,thus

= (yh+vlNR)-T

(19)

The summation for axial harmonic number arises from theuse of the Preston and Reece model,22 so that

m — vw

(20)

2.5.1 Rotor-rotor coupling impedances: The rotor-slotharmonics induce e.m.f.s in the rotor bars of the samedistribution and frequency as the bar currents which setthem up. To determine the induced e.m.f. per bar offrequency sfew and distribution appropriate to the i>feth

32 ic

ic

[• V1

Fig. 2 Development of axial representation for skewed rotor bars

1EEPROCEEDINGS, Vol. 127, Pt. B, No. 2, MARCH 1980

harmonic, it is necesssary to sum slot and axial harmoniccomponents as follows:

Rk = £ Egmexp(jsko>t) (21)

The values of ERm are obtained using the flux-cutting

principle:

_ joJHzskwKl

bRKIsm l(l + mtany)

l2dR X l2+m2

{JMmdR+HR-HR_x} (22)

2.5.2 Stator-rotor coupling impedances'. The rotor-slotharmonics can, under certain circumstances, have a pro-found effect on the stator windings. The present analysiscan be extended to include coupling impedances betweenthe stator and the rotor-slot harmonics, but it will beassumed here that such effects are negligible.

As there are no loss-producing mechanisms in thepresent machine model, since the winding and bar resist-ances have not yet been incorporated into the couplingimpedances, the stator—rotor coupling impedances may beobtained from

_fife ~

(23)

The bracketed term is a phase shift, which reflects the factthat there is a spatial phase angle between the referencerotor bar and the n th stator winding.

2.6 Ex ternal coupling impedances

The stator coupling impedances calculated in Section 2.4.1make no allowance for the winding resistance or the end-turn leakage reactances, which must be added in as separateitems before the winding currents are determined. Exter-nally connected impedances such as starting capacitorsmust also be included at this stage. The rotor harmoniccoupling impedances are likewise increased by adding in theeffective resistance and end-turn leakage reactance per bar.In representing the bar currents by Fourier series (Section2.5), the assumption has been made that the currentdensity is uniform over the bar cross-section. It is necessary,therefore, to modify the effective bar resistance to takeskin effect into account. A simple multiplicative factor foraccomplishing this is given by liwschitz-Garik.23

2.7 Calculation of input power, torque and fieldcomponents

The input power is readily obtained from the windingcurrents and applied voltages in the usual way. The torquemay be calculated using either Maxwell's stresses, once thefield components are evaluated, or by a straightforwardconsideration of the power balance within the machine.The latter is probably the simplest approach. If ironsaturation plays an important role in the operation of themachine being examined, it is possible at this point to usethe machine as a basis for updating the relative permeabilitiesof the saturating regions.

IEEPROCEEDINGS, Vol. 127, Pt. B, No. 2, MARCH 1980

3 Relationship between the field analysis andequivalent-circuit methods

3.1 Impedance components

The validity of the impedance components calculated usingthe present method will be established with reference to thesimplest possible form of induction machine — a single-phase motor with a single concentrated coil on the stator.This is done for simplicity, but also because more compli-cated stator-winding systems can be derived from this basicbuilding block by superposition. The motor is representedin developed form by Fig. 1, and the jV-turn stator coil hasits sides in slots positioned at y = 0 and y = a in the statorreference frame.

3.1.1 Stator coupling impedances: Fourier analysis of thestator coil (eqn. 11) gives

Co —^

sin (ka/2) exp \ j(kot — n) (24)

The usual assumption of infinitely permeable iron is made,to allow comparison with the equivalent-circuit method.This assumption produces \xz — juy = °° in the rotor andstator back-iron regions, and \iz = °° in the slotted regions.

The stator-coil self impedance is obtained as

(25)

Eqn. 25 has been written to separate the airgap and slot-leakage components. The airgap impedance is exactly thatwhich would be obtained using classical methods, whereasthe slot leakage reactance is normally given by

Xslot — 3 cs cs-

(26)

Comparision between eqns. 25 and 26 reveals how theappropriate choice for the circumferential permeability intoothed regions should be made. For a typical toothedregion (say region q), the appropriate choice is

(27)

where Pg is the specific slot permeance of that part of theslot that the ^th region models in the real machine (Fig.1A), and P'q is the same for the 4 th region in the parallel-sided-slot replacement machine (Fig. IB). When the per-meabilities /iy3, My4 and juyS are chosen in accordance witheqn. 27, the slot-leakage-reactance component in eqn. 25becomes identical with eqn. 26.

The yfeth harmonic coupling impedance between thestator coil and the rotor bar is obtained in like fashion; it isgiven by

7k -£R,S — sm(28)

87

This expression may be verified using classical methods.The rotor and stator slot mouth widths are accounted forby the factors K^R and K%s, and the exponential term is aphase constant which reflects the relative positions of thestator winding and the rotor bar in their respective referenceframes. If point conductors are assumed, K%R = K*s = 1 •

3.1.2 Rotor coupling impedances: The determination ofthe yfeth-harmonic rotor-bar self-coupling impedancerequires summation both for axial harmonic variation andfor the slot harmonics. The resulting expression may bewritten, for purposes of comparison, as the sum of magnet-ising, slot leakage and differential harmonic leakage compo-nents.

yk —7^RR £

yk, gap I, slot

rk'K, diff

where

(29)

(30)

which is the usual expression multiplied by the factor(KbR)2 which takes into account the rotor slot mouthwidth. If slots having negligible mouth widths are assumed,

tan27)

eR

OR

tan2 7)

1 dR- 7 " ( M y 9 + ju*9 tan27)5 bR

(31)

If the eddy-current effects in the teeth are small, classicalmethods give

Xslot = +eR

+ — —. sec2 7•5 CR

(32)

A comparision of eqns. 31 and 32 reveals that the choicefor nyQ given by eqn. 27 is still appropriate, together with

yk

cosec2 \ k\R —2K bR 2b j

(33)

which compares well with the equivalent-circuit component

I 2 ,cosec 5 k\R{

- U (34)

The differences between eqns. 33 and 34 once again arisefrom the allowance for finite slot mouth widths. Zero slotmouth width gives bR = 0,K*R = 1, and the exprs. 33 and34 become identical.

3.2 Equivalent circuits

In Section 3.1, the validity of the coupling impedances hasbeen established. In this Section, the relationship betweenthe coupling impedances and the impedances commonlyfound in an equivalent-circuit model is derived.

Consider a balanced 3-phase induction motor, in whichall harmonic fields produce negligible rotor—stator coupling,so that only the fundamental forward-travelling rotorharmonic term need be considered. Eqn. 1 thereforebecomes

31 yi

Z\R

Z2R

Z3R

ZRR

h

hTR

(35)

in which numeral subscripts refer to the stator windings andthe letter R refers to the fundamental rotor term.

Let us assume that the phase sequence is 1-2-3, so thatthe stator voltages and currents become

Vx = V V2 = a2V

L = a2l

V3 = aV

L = of

where a = (— 1 +/\ /3)/2. Substituting these values intoeqn. 35 and eliminating the rotor current, one obtains,from the top row,

V = 1 UZn+a2Zl2+aZi3)-

'RR(36)

Z u is comprised of an airgap term, plus a leakage term; asonly the fundamental field is to be considered, that portionof the airgap reactance due to all the other harmonics mustbe regarded as leakage. The airgap component in eqn. 25includes this harmonic leakage, but the method used toderive eqn. 25 can be used without harmonic summation togive the appropriate magnetising component for one phase.Thus

£*mn0 icjr (37)

It is straightforward to show that this expression is equal toone-third the value calculated for magnetising reactance in abalanced machine, with a small correction to allow forfinite slot mouth widths. Therefore, when the stator-windingresistance is incorporated,

Zn = (38)

The stator coupling impedance Zl2 is given by Z12 = aZ, asthe fundamental field is produced _by an assumed phasesequence 1-2—3. Similarly Z13 =a2Z.

+a2Zl2 +aZ13 = + jXn (39)

The rotor self-coupling impedance ZRR also consists of anairgap component plus a leakage term ZR, which may bewritten as ZR = R2 + jskX2 (Section 3.1.2). The airgap

88 IEE PROCEEDINGS, Vol. 127, Pt. B, No. 2, MARCH 1980

component may be related to X, by comparing eqns. 30 and 37, to give

where T k is an effective complex turns ratio, defmed by

Consideration of the phase sequence reveals ZR2 = d R l , ZR3 = a2ZR1 . ZR1 is obtained as a more general version of eqn. 28, to allow for distributed and chorded windings:

from which

Eqn. 23 gives, finally

Substituting eqns. 39, 40, 43 and 44 into eqn. 35 and rearranging,

in which

Eqn. 46 will be recognised as defining the rotor/stator transformation ratio. Eqn. 45 may be represented by the equivalent circuit shown in Fig. 3. This equivalent circuit is exactly that derived by Butler and ~ i r c h ~ in their investi- gation of skew-effect parameters.

Fig. 3 Phase-equivalen t circuit for balanced polyphase induction motorz4

4 Conclusions

A field analysis has been proposed which is shown to be capable of accurately modelling the induction motor, including slot leakage and the effects of rotor skew. As such it represents a considerable advance on existing field analyses. The coupling-impedance approach makes the method extremely versatile, being formulated for an arbi- trary number of stator windings and any supply configur- ation. In addition, it offers the advantage that for a little extra computational effort it is capable of determining the field in any part of the machine.

5 References 1 STEINMETZ, C.P.: 'The alternating current induction motor',

Trans. Am. Inst. Elect. Eng., 1897, 14, pp. 185-217 2 BROWN, J.E., and BUTLER, 0.1.: 'A general method of analysis

of three-phase induction motors with asymmetrical primary con- nections', Proc. IEE, 1953, 100, Pt 11, pp. 25 -34

3 BROWN, J.E., and JHA, C.S.: 'Generalised rotating field theory for polyphase induction motors, and its relationship to sym- metrical component theory', ibid., 1962, 109A, pp. 59-69

4 MISHKIN, E.: 'Theory of the squirrel-cage induction motor derived directly from Maxwell's field equations' Q. J. Mech. Appl. Math., 1954,7, Pt. 4, pp. 472-487

5 CULLEN, A.L., and BARTON, T.H.: 'A simplified electro- magnetic theory of the induction motor using the concept of wave impedance', Proc. IEE, 1958, 105C, pp. 331 -336

6 PIGGOTT, L.S.: 'A theory of the operation of cylindrical induction motors with squirrel-cage rotors', ibid., 1962, 109C, pp. 270-282

7 HESMONDHALGH, D.E., and LAITHWAITE, E.R.: 'Method of analysing the properties of 2-phase servo-motors and a.c. tacho- meters', ibid., 1963, 110, ( l l ) , pp. 2039-2054

8 LAWRENSON, P.J., and RALPH, M.C.: 'Tooth-ripple losses in solid poles', ibid., 1966, 113, (4), pp. 657-662

9 GREIG, J., and FREEMAN, E.M.: 'Travelling-wave problem in electrical machines', ibid., 1967, 114, ( l l ) , pp. 1681-1683

10 FREEMAN, E.M.: 'Travelling waves in induction machines: input impedance and equivalent circuits', ibid. 1968, 115, (12), pp. 1772-1776

11 FREEMAN, E.M., and LOWTHER, D.A.: 'Normal force in singlesided linear induction motors', ibid., 1973, 120, (1 2), pp. 1499-1505

12 EASTHAM, J.F., and BALCHIN, M.J.: 'Pole-change windings for linear induction motors', ibid., 1975, 122, (2), pp. 154-160

13 BALCHIN, M.J., and EASTHAM, J.F.: 'Performance of linear induction motors with airgap windings', ibid., 1975, 122, (12), pp. 1382-1390

14 BOLDEA, I., and BABESCU, M.: 'Multilayer approach to the analysis of single-sided linear induction motors', ibid., 1978, 125, (4), pp. 283-287

15 GIERAS, J.: 'General equations of electromagnetic field distri- bution in composite multi-layer structures for one-sided pene- tration', Acta Tech. CSA V, 1977, 22, pp. 361 -386

16 BINNS, J.J., HINDMARSH, R., and SHORT, B.P.: 'Effect of skewing slots on flux distribution in induction machines', Proc. IEE, 1971, 118, (3/4), pp. 543-549

17 EASTHAM, J.F., and WILLIAMSON, S.: 'Generalised theory of induction motors with asymmetrical airgaps and primary wind- ings', ibid., 1973, 120, (7)' pp. 767-775 '

18 WILLIAMSON, S., and SMITH, A.C.: 'Layer theory analysis for integral-bar induction devices'. Second Compumag Conference, Grenoble, September, 1978

19 WILLIAMSON, S.: T h e anisotropic layer theory of induction machines and induction devices', J. Inst. Math. & Appl., 1976, 17, pp. 69-84

20 WILLIAMSON, S.: 'Induction motor analysis and field calcu- lation using anisotropic layer theory'. First Compumag Confer- ence, Oxford, April, 1976

21 FREEMAN, E.M.: 'Computer-aided steady-state and transient solutions of field problems in induction devices', Proc. IEE, 1977, 124, (1 I), pp. 1057-1061

22 PRESTON, T.W., and REECE, A.B.J.: Transverse edge effect in linear induction motors', ibid., 1969, 116, pp. 973 -979

IEE PROCEEDINGS, Vol. 12 7, Pt. B, No. 2, MARCH 1 980

23 LIWSCHITZ-GARIK, M.M.: 'Computation of skin effect in barsof squirrel-cage rotors', Trans. Am. Inst. Electr. Eng., 1955, 74,pp. 768-771

24 BUTLER, O.I., and BIRCH, T.S.: 'Comparison of alternativeskew effect parameters of cage induction motors', Proc. IEE,1971, 118, (7), pp. 879-883

6 Appendixes

In this Section, Hn and Bn will be used to denote the valuesof Hylm and Bglm, respectively, at the uppermost boundaryof the n th region.

6.1 Transfer matrix elements

The values of Hylm and B^lm at the upper and lowerboundary surfaces of the rth. region may be related via atransfer matrix equation derived directly from eqn. 3—6:

r Jr ar br

cr dr

where

'r-l , Jr

Hr-!

from the (n + l)th layer) is defined as

In region 1, all field quantities must vanish at z = — °°, sothat D = C (eqns. 4—6) and Zf* is given by

By manipulation of the transfer matrix for region 2, oneobtains

'P +Z2 tanh e2SiZ? tanhe252 + Z2

where

Similar equations can be written for Z% ,Z% etc., until theexcitation layer is reached.

The surface impedance looking up into the nth layerfrom the (n — l)th layer is defined as

_ _<^Kn ~ iR

ar = dr = cosh eSr,

, 7M2e .br — ••—-— sinn

sinh eSr

6.2 Surface impedances and impedance cascading10

The surface impedance (looking down into the nth layer

In the uppermost region (say the TVth), the requirementthat all field quantities must vanish at z = °° produces thefollowing expression:

U

The cascading process down to the excitation layer thenfollows from

LN-I —7U i 7

tanh

90 IEE PROCEEDINGS, Vol. 127, Pt. B, No. 2, MARCH 1980