magnetic equivalent circuit modeling

8/11/2019 Magnetic Equivalent Circuit Modeling

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IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 22, NO. 2, JUNE 2007 259

Magnetic Equivalent Circuit Modeling of Induction Motors

Scott D. Sudhoff , Senior Member, IEEE , Brian T. Kuhn , Member, IEEE , Keith A. Corzine , Member, IEEE ,and Brian T. Branecky

Abstract Finite element models are invaluable for determiningexpected machine performance. However, nite element analysiscan be computationally intense; particularly if a large numbersof studies or high bandwidth studies are required. One method toavoid this difcultyis to extract machine parameters from theniteelement model and use the parameters in lumped parameter mod-els. While often useful, such an approach does not represent spaceharmonics or asymmetries in the motor. A methodology for con-structing a state-variable model, based on a magnetic equivalentcircuit of themotoris describedherein. In addition, theparametersfor this model are based solely on geometrical data. This approach

is an excellentcompromise between thespeed of lumpedparametermodels and the ability of nite element methods to capture spatialeffects. Experimental validation of the model is provided.

Index Terms Geometry-based modeling, induction machinemodeling, magnetic equivalent circuit (MEC), space harmonics.

I. INTRODUCTION

Finite element analysis (FEA) is a common method of study-ing spatial harmonics in electric machinery. This analysis canyield excellent accuracy, but is computationally intense. Other analysis approaches that include spatial effects are magneticallycoupled circuit modeling [1], [2]andmagnetic equivalent circuit(MEC) modeling [3][8]. This latter approach has the advantageof a close association with the physical eld distributions in themachine. It has been applied to a variety of machines includ-ing induction machines [3][6], synchronous machines [7], andswitched reluctance machines [8].

The MEC approach has the advantage that it includes spatialdependencies as in the case of FEA, but is computationally lessintense, thereby allowing it to be used in higher bandwidth stud-ies in which the interaction of power converter and machine isimportant, or in situations where a large number of studies arerequired (for example, population-based automated design tech-niques). Simply put, MEC modeling can provide a compromise

between nite element and qd -type models.

Manuscript receivedSeptember3, 2003; revised January 10, 2006. This workwas supported in part by A. O. Smith Corporation and in part by the Ofce of Naval Research through the ESRDC. Paper no. TEC-00228-2003.

S. D. Sudhoff is with the Department of Electrical and Computer Engi-neering, Purdue University, West Lafayette, IN 47907 USA (e-mail: [email protected]).

B. T. Kuhn is with SmartSpark Energy Systems, Champaign, IL 61820 USA(e-mail: [email protected]).

K. A. Corzine is with the Department of Electrical and Computer En-gineering, University of Missouri-Rolla, Rolla, MO 65409 USA (e-mail:[email protected]).

B. T. Branecky is with A. O. Smith Corporation, Milwaukee, WI 53224 USA(e-mail: [email protected]).

Digital Object Identier 10.1109/TEC.2006.875471

In this paper, the MEC approach to modeling the inductionmachine is examined in detail. A model is developed, whichis based solely on geometrical data, material parameters, andwinding distributions. The model takes into account local satu-ration of individual stator and rotor teeth aswell as the back-ironsections of the machine. Also, it takes rotor skewing, stator androtor tooth effects, space harmonics in the stator windings androtor bars, and end ring ux components into account. It is validfor asymmetrical excitation, broken end ring, and broken rotor bar conditions. One limitation of the proposed model is that theaccuracy during transient conditions is limited because it doesnot include distributed circuit effects in the rotor conductors. Anadditional factor that may limit performance during transientsis that stator end ring leakage inductance has not been included.

Of the prior work in the MEC area, the approach herein isperhaps most similar to [3] and [4]. However, this work featuresa more integrated treatment of the ux linkage and node poten-tial equations, and utilizes a qd framework that is benecial intreating leakage inductance. The use of a qd framework in or-der to simplify the calculation of the leakage inductance is alsoused in [5]; however, that model is restricted to the magneticallylinear condition. Unlike [6], this work sets forth expressionsfor permeance in terms of geometrical data rather than rely-ing on a secondary FEA analysis. Finally, this paper sets fortha much more detailed algorithm to calculate the permeancesthan [3][6]. As an aside, it has been recently shown [9] that abar-by-bar type representation can be reduced to a system withonly four states; however, this is restricted to symmetrical andmagnetically linear conditions.

II. NOTATION AND OPERATORS

Throughout this work, scalar quantities will be designatedwith italics while matrix/vector quantities will appear inboldface. Two variables, i and j , will denote matrix/vector indexes when used in a subscript. In this context, i refers tostator quantities and takes on values [1, . . . , N ss ] where N ss isthe number of stator slots. The index j refers to rotor quantitiesand will take on values [1, . . . , N rs ] where N rs is the number of rotor slots. Integer operations on these indexes are ring-mappedback to the set. For example, if operating on the stator set indexwith N ss = 12 , then 12 + 1 maps to 1 and 1 1 maps to 12.This paper will make use of an element-by-element vector multiply. Given three vectors x, y,z R N where R denotes theset of real numbers and N is in the set of natural numbers, theelement-by-element multiply given by

zk = xk yk k [1,...,N ] (1)0885-8969/$25.00 2006 IEEE


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260 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 22, NO. 2, JUNE 2007

Fig. 1. Stator geometry.

will be denoted

z = x y. (2)The operator is communitive and associative and its impliedorder of operation is higher than multiplication.Throughout this paper, Heaviside notation p is used for the

time derivative operator, i.e.,

p = ddt

. (3)

III. MODEL STRUCTURE

A. Stator Denitions

The machine considered herein is assumed to possess a three-phase stator winding. Stator variables are denoted in a formf x where f may be a voltage v, current i (positive into themachine), or ux linkage , and x may be a , b, or c.

Stator quantities are also represented as vectors of the formf abc = [ f a f b f c ]T (4)

where the superscript T denotes the transpose operator. In termsof abc variables, the stator voltages, currents, and ux linkagesare related by Ohms and Faradays laws; in particular, the stator voltage equation may be expressed

v abc = r s i abc + p abc (5)

where r s is the stator winding resistance.The stator geometry is depicted in Fig. 1. The stator slots are

numbered 1 N ss . The ith slot contains N a,i turns of the a-phasewinding, N b,i turns of the b-phase winding, and N c,i turns of the c-phase winding, where positive turns are out of the page.The turns vector N abc,i is dened as

N abc,i = [ N a,i N b,i N c,i ]T . (6)

The number of spans describes the number of times eachphase winding links the radialeld density at a position s alongthe stator, where s is referenced from the center of stator tooth1 and measured in the counterclockwise direction as viewedfrom the front of the machine.

The winding function [10] is a measure of the number of times that a given winding set links the ux density at a position

s . Taking the positive direction of ux density to be from the

Fig. 2. Rotor geometry.

rotor to the stator, in a machine with a continuously distributedwinding of turns density N (s)), the winding function satises

dW (s)ds

= N (s). (7)The number of spans is determined by integrating the turn

density. The constant of integration is arbitrary since,by Gaussslaw, the integral of the ux density around the air gap is zero,but is normally determined such that the average value of thewinding function as s varies from 0 to 2 is zeroa practicethat results in MMF associated with a given winding to be equal

to the winding function for that winding times the windingcurrent.In a machine in which the windings are lumped into dis-

crete positions (slots), the number of spans is a discrete variablecalculated in accordance with

W i +1 = W i N i (8)where N i and W i are the turns in slot i and span between slot iand slot i + 1 , respectively. In (8), W 1 is selected so that

N ss

i =1

W i = 0. (9)

In the three-phase system discussed herein, the span vector W abc,i is dened as

W abc,i = [ W a,i W b,i W c,i ]T (10)

and is governed by

W abc,i +1 = W abc,i N abc,i (11)subject to the constraint that

N ss

i =1

W abc,i = [ 0 0 0 ]T . (12)

B. Rotor DenitionsThe rotor structure is depicted in Fig. 2. Therein, open rotor

bars are shown but this need not be the case. The squirrel cagerotor consists of N rs rotor bars separated by N rs rotor teeth. Thebars and teeth are indexed 1, . . . , N rs . The current owing inrotor bar j is denoted ib ,j .

The electrical connection of the bars is depicted in Fig. 3.Therein rb ,j , rfe,j , and rbe ,j denote the resistance of the j throtor bar, the resistance of the segment of the front end ring thatconnects bar j to bar j + 1 , and the resistance of the segment of the back-end ring that connects bar j to bar j + 1 , respectively.The provision to allow these to vary in a bar-by-bar fashion is

included in order to accommodate damaged rotor conditions.


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SUDHOFF et al. : MEC MODELING OF INDUCTION MOTORS 261

Fig. 3. Electrical rotor connections.

The loop current owing around tooth j (with a sign conven-tion such that positive current will cause a positive ux outwardin the radial direction) is denoted ir ,j . The loop currents arerelated to the bar currents by

ib ,j = ir ,j ir ,j +1 . (13)Finally, the loop current around the front and back-end rings

are denoted ife and ibe , respectively, and are dened with adirection such that the ux associated with these loops is out-ward in the axial direction. Although the set of loop currents[ir ,1 , . . . , i r ,N rs , i fe , ibe ] are independent, physically, the de-scription of the corresponding branch currents through the barsin segments is not unique, and so one of these may be assignedarbitrarily. In order to simplify the analysis to follow, the currentibe is set to zero.

The ux linking the path of loop current ir ,j will be denoted r ,j . The area of this ux is dened to be from the center of bar j 1 to the center of bar j , and extend the length of the stackof the machine. By Ohms and Faradays laws, the loop currentsare related to the ux linkages by

0 = rb ,j (ir ,j ir ,j +1 ) + rb ,j 1(ir ,j ir ,j 1)+ rbe ,j ir ,j + r fe ,j (ir ,j ife ) + pr ,j (14)

where ibe has been taken to be zero. The ux linking the surfaceenclosed by the path of ibe and ife will be denoted be and fe ,respectively. From Fig. 3, and again taking ibe to be zero

0 = N rs

j =1

r be ,j ir ,j + pbe (15)

0 =N rs

j =1

r fe ,j (ife ir ,j ) + pfe . (16)

By Gauss law, the ux linkages r ,j , be , and fe are notindependent. Instead, they must satisfy

N rb

j =1

r ,j + be + fe = 0 . (17)

In (17), note that each ux linkage can be interpreted as a simple

ux since each ux linkage is associated with a single turn.

Finally, as a matter of denition, the rotor quantities will berepresented in vector form as

f r = [f 1 , . . . , f N rs ]T . (18)

C. Transformation of Stator Variables

Instead of analyzing the machine in terms of abc phase vari-ables, it is convenient to utilize qd 0 variables. The reasons for this are twofold. First, it leads to simplication of the analy-sis for the wye-connected case whereupon the zero sequencecomponents can be neglected (automatically eliminating cut-sets associated with the fact that the sum of the currents is zero).Second, it leads to a simplication in calculation of the leakageinductance. In particular, in terms of abc variables there canbe mutual leakage inductance between the stator windings. Interms of qd 0 variables, however, the mutual leakage inductancebetween windings, can, in general, be neglected.

The transformation between qd 0 variables and abc variablesmay readily be expressed in the form

f qd 0 = K s f abc (19)

where qd 0 variables are of the form

f qd 0 = [ f q f d f 0 ]T (20)

and the transformation K s is taken to be

K s = 23cos cos( 2/ 3) cos( + 2 / 3)sin sin( 2/ 3) sin( + 2 / 3)1/ 2 1/ 2 1/ 2

.

(21)

Note that this is not the standard qd -transformation as set forth

in, for example, [11]. This transformation was rst proposedin [12], though not in the context of magnetic circuit modeling.

D. Model Integration

The fundamental property of a time-domain model is theability to calculate the output variables and time derivatives of the state variables based on the input variables and the statevariables. The inputs to the electrical model will be the stator voltages vabc andthe electrical rotor position r . Theoutputs willbe the stator currents iabc and electromagnetic torque, T e . Thestate variables will be the stator ux linkage vector (expressedin terms of qd 0 variables) qd 0 , the rotor ux linkage vector r , and the front-end rotor ux fe [be is not used as a statevariable because it is algebraically related to r and be by(17)].

The algorithm to do this is, in theory, straightforward. Inparticular, given the rotor position and stator and rotor uxlinkages, the currents iqd 0 , i r , and i fe can be found. Once thisis accomplished, the time derivatives of the stator ux linkagesmay be expressed

p qd 0 = v qd 0 r s i qd 0 (22)where vqd 0 is calculated using (19) and the time derivatives of the rotor ux linkages are given by

pr ,j = r b ,j (ir ,j ir ,j +1 ) r be ,j (i r ,j ibe )


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Fig. 4. Structure of electrical model.

r b ,j 1(ir ,j ir ,j 1) r fe ,j (ir ,j ife) (23)

pfe = N rs

j =1 rfe ,j (ife ir ,j ) (24)

where (22) was obtained by applying the transformation (21) to(5), and (23) and (24) were derived by straightforward manipu-lation of (15) and (16), respectively.

Fig. 4 illustrates the structure of the electrical model.Therein,T e denotes the electromagnetic torque, which will be an alge-braic function of electrical states and an output to the mechan-ical model; it is calculated in a torque calculation block (TCB)described in Section IV-D. The angle r is the electrical rotor position that is a state of the mechanical model. The remainder of this paper is devoted to the MEC block, wherein the stator androtor currents, i qd 0 , i r , and ife are calculated from the stator uxlinkage vector (expressed in terms of qd 0 variables) qd 0 , therotor ux linkage vector r , the rotor end ux linkage fe , andthe rotor position r . Note that the current ibe does not appear in Fig. 4 since, as previously discussed, it can be set to zerowithout loss of generality.

IV. MEC M ODELING

The approach used to calculate the currents from the uxlinkages and position will be MEC modeling. The stator androtor equivalent circuits are depicted in Figs. 5 and 6, respec-tively. These gures also serve to dene the node potentials andprincipal permeances associated with the model. As can be seenin Fig. 6, all potentials are referenced to the center of the rotor.

Leakage permeances, which would appear in parallel withthe MMF sources in Figs. 5 and 6, are not shown in thesediagrams; they will be considered when formulating the uxlinkage equations. The stator MMF source may be representedas being either in the yoke or the tooth; it is placed in the yokeherein so that its value (is,i ) directly corresponds to the currentin the corresponding slot. From Fig. 5, and noting that the slotcurrent in the ith stator slot is given by

is,i = N Tqd 0,i i qd 0 (25)

Fig. 5. Stator MEC.

Fig. 6. Rotor MEC.

where N qd 0 ,i is related to N abc,i by (18), the stator nodal equa-tions may be readily expressed as

P st ,i (M sb ,i M st ,i )+ P sb ,i 1(M sb ,i M sb ,i 1 N Tqd 0,i 1 i qd 0)+ P sb ,i (M sb ,i M sb ,i +1 + N Tqd 0,i i qd 0) = 0 (26)

and

P st ,i (M st ,i M sb ,i ) + P stst ,i 1(M st ,i M st ,i 1)+ P stst ,i (M st ,i M st ,i +1 )+

N rt

j =1

P rtst: i,j (M st ,i M rt ,j ) = 0 . (27)

Equations (26) and (27) may also be written as

A sb sb M sb + A sbst M st + A sb si i qd 0 = 0 (28)A st sb M sb + A st st M st + A st rt M rt = 0 . (29)

Consideration of Fig. 6 leads to the rotor nodal equations

P rt ,j (M rt ,j M rb ,j ir ,j )+ P rtrt ,j 1(M rt ,j M rt ,j 1)


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Fig. 7. Stator ux linkage ux paths.

+ P rtrt ,j (M rt ,j M rt ,j +1 )

+N ss

i =1

P rtst: i,j (M rt ,i M st ,j ) = 0 (30)

P rt ,j (M rb ,j M rt ,j + ir ,j )+ P rb ,j 1(M rb ,j M rb ,j 1)+ P rb ,j (M rb ,j M rb ,j +1 )+ P rtc ,j M rb ,j = 0 (31)

which may also be expressed in matrix-vector form as

A rt st M st + A rt rt M rt + A rt sb M rb + A rt ri i r = 0

(32)

A rb rt M rt + A rb rb M rb + A rb ri i r = 0 (33)

where the submatrices in (32) and (33) may be established bycomparison to (30) and (31).

A. Stator Flux Linkage Equations

Fig. 7 depicts the ux paths used in the derivation of thestator ux linkage equations. Therein, note that all radial uxentering sector i links the q -, d-, and 0-axis circuits W q,i , W d,i ,and W 0,i times, respectively. The ux entering this sector hastwo sources. First, there is the ux that enters the stator tooth.From Fig. 5, this ux may be expressed

st ,i = P st ,i (M st ,i M sb ,i ). (34)This component of the ux includes magnetizing ux as com-ponents of the leakage ux that travel through the MEC.

Adding up the number of times each of these ux componentslink each of the axis windings yields

qd 0 =N ss

i =1

W qd 0,i st ,i [ 1 1 1 ]T + L liqd 0 . (35)

In (35), L l represents a leakage inductance matrix that takesinto account the fact that there will be components of the uxlinkage not included in the MEC. The calculation of Ll will beset forth in Section VI.

In matrix vector form (35) may be rewritten as

A s sb M sb + A s st M st + L l i qd 0 = qd 0 . (36)

B. Rotor Flux Linkage Equations

The rotor end ux linkages are given by

fe =N rs

j =1

P fe ,j (ife ir ,j ) (37)

and

be = N rs

j =1

P be ,j ir ,j (38)

wherein P fe,j and P be ,j denote the permeances of the magneticpath around the front and back rotor end ring segments, re-spectively, and ibe is taken to be zero in accordance with thecomments of Section III.

The ux linking each current loop may be expressed as

r ,j = P b ,j (ir ,j ir ,j +1 ) + P b ,j 1(ir ,j ir ,j 1)+ P fe ,j (ir ,j ife ) + P be ,j ir ,j + mr ,j (39)

where P b ,j is the permeance of the magnetic path inside the j thbar and mr ,j is the ux through the rotor tooth that may beexpressed

mr ,j = P rt ,j (M rb ,j M rt ,j + ir ,j ). (40)Substitution of (37), (38), and (40) into (39) yields

r ,j = P rt ,j M rb ,j P rt ,j M rt ,j P b ,j 1 ir ,j 1 P b ,j ir ,j +1+ ( P b ,j + P b ,j 1 + P fe ,j + P be ,j + P rt ,j )ir ,j

P fe,jP feN rs

k =1

P fe ,k ir ,k (41)

where

r ,j = r ,j + P fe ,jP fe

fe (42)

and

P fe =N rs

j =1

P fe ,j . (43)

Equation (41) is readily formulated as

A r rt M rt + A r rb M rb + A r ri i r = Tr (44)

C. MEC Solution Algorithm

The relationships of (28), (29), (32), (33), (36), and (44) forma set of 2N ss + 3 N rs + 3 equations and the same number of unknowns. They may be expressed as

Ax = b (45)

where A , x , and b are partitioned matrices of the form

A =


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A sbsb A sb st 0 0 A sbsi 0A st sb A st st A st rt 0 0 0

0 A rt st A rt rt A rt rb 0 A rt ri0 0 A rb rt A rb rb 0 A rb ri

A s sb A s st 0 0 L l 00 0 A r rt A r rb 0 A r ri

,

(46)

x = M Tsb MTst M

Trt M

Trb i

Tqd 0 i

Tr

T(47)

and

b = 0 0 0 0 Tqd 0 Tr . (48)

In this system of equations, each 0 represents a null matrixof appropriate dimensions, and subscripts sb, st, rt, rb,si, ri, s , and r denote association (either that this ele-ment came from that equation, or is multiplied by that variable)

with the potential of stator base, potential of stator tooth, poten-tial of rotor tooth, potential of rotor base, stator current, rotor current, stator ux linkage, and rotor ux linkage, respectively.

The system (45) is bilinear since many of the elements of A are a function of permeabilities that are in turn a functionof x . The system can be solved using any nonlinear solutionalgorithm. For this work, a GaussSeidel iteration was used(starting fromthe previous solution). In particular, the algorithmproceeds as follows.

1) Calculate constant permeances and leakage inductances.This step is only conducted once at the beginning of a sim-ulation. Expressions for permeances are given in SectionsV and VI.

2) Based on the rotor position, calculate the stator to rotor permeances as discussed in Section V-C.

3) Based on node potentials and currents, calculate eld in-tensity in each branch, andthen thecorresponding valueof permeability. Then calculate permeances or iron estimatedened in Sections V-A and V-B.

4) Using the results of Steps 13 form system (45) and solve.If

|x k e x k 1e |


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Fig. 9. Rotor permeance geometry.

Flux seeks the path of least reluctance; selecting the value of

r dbn that minimizes R yields

r bdn = 12 (r od 2r sd )r twN rb (55)

whereupon the rectangular approximation of the rotor toothpermeance is

P rt ,j = Fe ()Lr tw

r sd + rbdn(56)

and the rotor back-iron permeance is

P rb ,j = 2Fe , lin Lr dbn N rb(r od

2r sd

2r bdn )

. (57)

The permeance from the base node to the center may beexpressed

P rtc ,j = 2 Fe , lin L

N rb ln((r od 2r sd 2r dbn )/r id ). (58)

In (57) and (58), Fe , lin is the permeability of the iron in thelinear region since these regions do not saturate. Finally, thetooth tip to tooth tip permeance may be expressed

P rtrt ,j = 0N rs r tft L

(r id r tft ) N rs r tfw. (59)

C. StatorRotor PermeancesThe statorrotor permeance is computed for every stator

rotor tooth combination and is denoted by P rt ,st : i,j . In order to compute the permeance between a particular statorrotor tooth pair, the area of overlap between the teeth is computedwhereupon

P rt ,st: i,j = ai,j 0

g (60)

where g is the air gap length and ai,j is the area of overlapbetween the ith stator tooth and the j th rotor tooth. The area a i,jis calculated based on the machine geometry (taking skewing

into account) and the instantaneous position of the rotor. The

Fig. 10. Stator slot geometry.

details of the calculation of the area of tooth overlap are omittedbutare straightforwardto developgiven the machines geometry.

VI. LEAKAGE INDUCTANCE A. Stator Leakage Calculation

The objective of this section is the calculation of the stator leakage matrix L l . The rst point that should be made is that theL liqd 0 component of the stator leakage ux represents the com-ponent of leakage ux that ows in a path not entirely containedin the MEC. The other components of the stator leakage uxthat crossing from stator tooth tip to the adjacent stator tooth tipand the component that travels from the stator, across the air gapto the rotor, and again back to the stator without coupling therotor current is computed as part of the MEC and is not includedin this term. Thus, even though this term of the stator leakageux will be treated as being linear, it does not mean that theoverall stator leakage inductance is linear.

The premise of the method used to calculate L l is the a priori assumption that this matrix is diagonal. This assumptionis motivated by the formulation of the problem in terms of qd 0 variables, wherein the windings are orthogonal, rather thanabc variables, in which the windings are not orthogonal. Uponmaking this assumption, the calculation of each of the nonzeroelements of L l can be made independently.

Given that L l is diagonal, it may be expressed as

L l =

L lq 0 0

0 Lld 00 0 Ll0 . (61)

In order to calculate L ly , where y may represent q , d, or 0, consider Fig. 10, which shows the cross-sectional view of the ith trapezoidal-shaped stator slot. The width of the base andtop of the slot are denoted by wb and wt , respectively and theheight of the slot is denoted by h.

Winding y with N y,i turns is in the top portion of the slot asshown by the cross-hatch lines and has a height of h1 . It shouldbe observed that the other windings also occupy this same area.As stated previously, it is not necessary to include the effects of these windings since there will not be any net mutual leakage

inductance through this path. However, it may strike the reader


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as odd to assume that all the windings are located in the samephysical space. To some extent, one may suppose that effectsof this localization may somewhat average out. Further thanthis, though, it should be remembered that the windings beingconsidered are the q, d, and 0 equivalent windings, not the a, b,and c. This has the result of effectively resulting in a more

uniform distribution. For the purposes of analysis, the windingwill be assumed to be perfectly uniform.Given these assumptions, it can be shown that

L ly = P lN ss

i =1

N 2y,i (62)

where

P l = 0L K

wt h21(2K + h1)2K 4 ln

K + h1K

K 3h1 + K 2h21

2 + Kh 31 +

h414

+ 0Lhwb wt

ln wb h

(wb wt )h1 + wt h (63)

In (63)

h1 = wt hwt wb

w2t h2

(wb wt )2 +

wb + wtwb wt

h2s sf

(64)

where ssf is the slot ll and

K = hwtw

b w

t

(65)

wt =s id2

+ ssd 2N ss s tw (66)

wb =s id2

+ stft 2N ss s tw (67)

h = ssd s tft . (68)Observe that the stator end turn permeance has not been in-

cluded, even though it could be in the proposed framework. Thisis because it is assumed that the included terms are dominant an assumption that will be supportedby theexperimental results.

B. Rotor Leakage CalculationThe rotor leakage permeances P b ,j , P fe ,j , and P be ,j are set

forth in this section. The calculation of the rotor bar leakagepermeance is based on Fig. 11. Therein, the trapezoidal area de-picted represents the bar conductor within the slot. The width of the base and top of the slot are denoted wb and wt , respectively,and the height of the conductor area as h. The variable x denotesdistance from the bottom of the slot and w(x) denotes the widthof the slot at a distance x from the bottom of the slot. Current isassumed to be owing out of the conductor. In order to establisha formula for the leakage permeance, the eld intensity will becalculated based on the path shown, and it will be assumed that

along the air portion of this path the eld intensity is uniform

Fig. 11. Rotor bar leakage permeance.

Fig. 12. End-bar leakage permeance.

and has a value H (x) and everywhere else along the path issmall enough to be neglected.

In order to calculate the leakage permeance P b ,j , note thatthe amount of this component of the leakage ux shown, which

will cross the ux plane through the tooth, is the same as theux passing the midline of the stator slot starting at the heightof the ux plane and stopping at the top of the conductor (anadditional component of leakage ux will ow from tooth totooth across the tooth-to-tooth permeance, but this componenthas already been incorporated through the permeance P rtrt ,j ).Based on these assumptions, it can be shown that

P b = 0Lhwb + wt

38

+ wb

(wt wb )2wt wb

2

+ wb lnwb + wt

2wt. (69)

The basis for the calculation of the end-bar leakage perme-ances is depicted in Fig. 12. Therein the shaded area depicts across-sectional view of the end-bar conductor, with the machineitself folded into a developed diagram. Assuming the eld in-tensity is only a function of x at its endpoint and follows thepath shown, and that the eld intensity in the rotor is zero

P fe ,j = P re ,j = 0Lseg

1 +

4r erth 1

ln 4r ert

4r ert + h 1

+ ln4r ert + h 24r ert + h 1

(70)


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where

h1 = r sd r tft (71)h2 = r sd + r tft (72)

and

Lseg =

N rs (r od r tft r sd ). (73)

VII. RESISTANCE CALCULATIONS

Herein, an expression for the stator winding resistance is notgiven since it is normally straightforward to derive as a functionof conguration. The rotor bar resistance, assuming no damage,may be readily expressed as

r b ,j = LN rs rc

((r od r tft r sd )) r tw N rs )( r sd r tft ) (74)

where rc denotes the resistivity of the rotor conductor material

(typically aluminum or copper). Assuming that the end ring hasthe same radial dimension as the bar and a thickness r ert , theend ring resistances may be expressed as

r fe,j = rbe ,j = (r od r tft r sd )rc

N rs (r sd r tft )r ert. (75)

VIII. MATERIAL CHARACTERISTICS

The permeability of the steel, Fe , is computed as a functionof the magnetic eld intensity H in accordance with

fe(H ) = K 2 ln( K 1 |H |+1)

|H | H = 0K 1K 2 H = 0 (76)

where

H = M

l (77)

and where M is the magnetic potential across the iron sectionand l is the length of the iron section. A variety of alternatives to(76) and (77) exist. The only requirement is that the expressionreturns the permeability given the magnetic eld intensity.

IX. EXPERIMENTAL RESULTS

Several studies were conducted to evaluate the performanceof the MEC model. The parameters of the wye-connected, 230V l-l rms, 60 Hz, 5-hp test machine are listed in Table I, and thewinding distribution is set forth in Table II.The apparatus for thelaboratory tests was a Magtrol HD-815 dynamometer operatingin constant speed mode. For each of the studies, the test motor wasline fedfrom a 240-V three-phase source. In order to removepower quality issues from the simulation, the measured voltagewaveforms were used as inputs to the simulation.

In conducting the studies, it was determined that the esti-mate of the rotor resistance based strictly on the resistivity of aluminum at 20 C and the expressions (64)(66) were low byapproximately 27%. This can be explained by several factors,including porosity and impurities in the aluminum, skin effect,neglecting skewing when deriving (64) and (65), and tempera-

ture rise during startup. To correct this, the expressions for rotor

TABLE IMACHINE PARAMETERS

TABLE IITurns Distribution

resistance (64) and (65) were multiplied by the factor r r ,mult inall of the studies.

Five studies were used to test the model; Study 1: balanced(source voltage) cold no load; Study 2: balanced cold full load;Study 3: balanced hot full load; Study 4: unbalanced cold fullload; Study 5: balanced cold full load with broken rotor endring. In the cold studies, the machine wasat room temperature(20 C),started, andthe data taken asquickly aspossible (thoughclearly the start-up transient would increase the rotor conductor temperature). For the hot study, the machine was allowedto run under load for approximately 1 h. The stator and rotor resistances used in the simulation were increased 25.7% and15.6% from the cold values, respectively, to approximate theincrease in resistance with temperature. This was based on thetemperature coefcient of copper and aluminum, and assuminga 100 C temperature rise. For the unbalanced study, a 2- resistance was placed in series with the a-phase. An overviewof the results from each study is shown in Table III. As canbe seen, the MEC approach provides a reasonable estimate of machine performance. The no-load studies are at 1799 r/mindue to the frictional and windage losses.

Fig. 13 illustrates the predicted and measured a-phase currentwaveforms, as well as the magnitude of the discrete Fourier transform (DFT) of the a -phase current waveforms for Study1 (balanced cold no load). This DFT, as well as those for theremainder of the studies, was based on a 0.3-s data windowsampled at 10 s. As can be seen, the simulation provides areasonable estimate of the magnitude and waveshape of theno-load current and current spectrum. In general, the spectralfeatures of the measured data are broader than those predicted,and include minor peaks not predicted by the MEC model.Thesediscrepanciesare notunexpected sincesmallasymmetriesthat physically exist will tend to broaden the spectral features

and add additional spectral components. This can be veried


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TABLE IIITEST RESULT OVERVIEW

Fig. 13. Study 1 (balanced cold no load) results.

by incorporating asymmetries in the MEC model as will bediscussed in Study 5.

Fig. 14 depicts Study 2 (balanced cold full load) results.Herein, it can be seen thatwhile the estimate of the current wave-form shape is reasonable, some spatial harmonics are somewhatoverestimated, most likely due to the method of computing thestator toothrotor tooth permeance.

The DFT of the simulated performance in Study 2 containsfeatures not seen in Study 1 due to the movement of the rotor teeth/bars relative to the stator MMF. The same effect can alsobe seen by comparing the measured DFTs, though in this casethe effect is more subtle because of spectral features caused by

bar-by-bar asymmetries.

Fig. 14. Study 2 (balanced cold full load) results.

The simulated and measured time-domain waveforms andfrequency spectrums for Study 3 (balanced hot full load) areshown in Fig. 15. The results are generally similar to those inStudy 2 (Fig. 14).

The results of Study 4 (unbalanced cold full load) appear inFig. 16. The simulated results are again in reasonable agree-ment with measurements. In this case, the number of low-levelspectral features is reduced from the other loaded studies (bothpredicted and measured). This may be associated with increaseddamping because of the added a-phase resistance.

Finally Fig. 17 illustrates the performance for Study 5 (bal-anced cold full load with broken end ring). A feature of par-ticular interest in this study is the broadening of the spectralfeatures in the simulated DFT. This is suggestive that one rea-son that the measured DFT spectrum has consistently broader features than the measured spectrums could be bar-by-bar andend-ring-segment by end-ring-segment variation in resistancedue to variation in porosity.

From a computational perspective, for the wye-connectedmachine there are 2 + N rs state variables (ux linkages) andthe number of unknowns in the MEC model (45) is 2N ss +3N rs + 3 . Thus, for the sample machine, there are 31 electricalstates and the system of algebraic equations associated with theMEC portion of the model is 159. This is a fraction of the order

of equations that would be required by a FEM-based model.


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Fig. 15. Study 3 (balanced hot full load) results.

Fig. 16. Study 4 (unbalanced cold full load) results.

Fig. 17. Study 5 (broken end ring) results.

X. SUMMARY

An MEC representation of an induction machine is set forthand shown to be reasonably accurate in predicting machine per-formance over a range of operating points, including no-loadconditions, loaded conditions, unbalancedexcitation conditions,and broken end ring conditions. This model is notable in that allof the parameters are physical quantitiesthe dimensions of themotor, the lamination magnetic properties, and the winding con-guration. The approach is an excellent compromise betweenthe standard lumped-parameter models and time-stepped FEAin terms of computation time and accuracy.

REFERENCES

[1] X. Lou, Y. Liao, H. A. Toliyat, A. El-Antably, and T. A. Lipo, Multiple

coupled circuit modelingof induction machines, IEEE Trans. Ind.Appl. ,vol. 31, no. 2, pp. 311318, Mar.Apr. 1995.[2] H. A. Toliyat and T. A. Lipo, Transient analysis of cage induction ma-

chines under stator, rotor bar, and end ring faults, IEEE Trans. EnergyConvers. , vol. 10, no. 2, pp. 241247, Jun. 1995.

[3] V. Ostovic, A method for evaluation of transient and steady state per-formance in saturated squirrel cage induction machines, IEEE Trans. Energy Convers. , vol. 1, no. 3, pp. 190197, Sep. 1986.

[4] , A simplied approach to magnetic equivalent-circuit modeling of induction machines, IEEE Trans. Ind. Appl. , vol. 24, no. 2, pp. 308316,Mar.Apr. 1988.

[5] A. K. Wallace and A. Wright, Novel simulation of cage windings basedon mesh circuit model, IEEE Trans. Power App. Syst. , vol. PAS-93,pp. 377382, Jan.Feb. 1974.

[6] C. Delforge and B. Lemaire-Semail, Induction machine modeling usingnite element and permeance network methods, IEEE Trans. Magn. ,vol. 31, no. 3, pp. 20922095, May 1995.

[7] G. R. Slemon, An equivalent circuit approach to analysis of synchronous


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machines with saliency and saturation, IEEE Trans. Energy Convers. ,vol. 5, no. 3, pp. 538545, Sep. 1990.

[8] M. Moallem andG. E. Dawson, Animproved magnetic equivalent circuitmethod for prediction the characteristics of highly saturated electromag-netic devices, IEEE Trans. Magn. , vol. 34, no. 5, pt. 1, pp. 36323635,Sep. 1998.

[9] A. R. Munoz and T. A. Lipo, Complex vector model of the squirrel-cageinduction machine including instantaneous rotor bar currents, IEEETrans. Ind. Appl. , vol. 35, no. 6, pp. 13321340, Nov.Dec. 1999.

[10] N. L. Schmitz andD. W. Novotny, IntroductoryElectromechanics . NewYork, Ronald, 1965.

[11] P. C. Krause, O. Wasynczuk, and S. D. Sudhoff, Analysis of Electric Machinery . Piscataway, NJ: IEEE Press, 1996.

[12] W. A. Lewis, A basic analysis of synchronous machinespart I, AIEETrans., Part III. Power App. Syst. , vol. 77, no. 37, pp. 436456, Aug. 1958.

[13] R. Fiser, S. Ferkolj, and H. Solinc, Steady state analysis of inductionmotor withbrokenrotor bars, in Conf. Rec. 7thInt. Conf. Electr. Machines Drives , 1995, pp. 4246.

Scott D. Sudhoff (S87M91SM01) received the B.S., M.S., and Ph.D.degrees, from Purdue University, West Lafayette, IN, in 1988, 1989, and 1991,respectively, all in electrical engineering.

Currently, he is a Full Professor at theDepartmentof Electrical andComputer Engineering, Purdue University.

Brian T. Kuhn (M93) received the B.S. and M.S. degrees, from the Universityof Missouri-Rolla, Rolla, MO, in 1996 and 1997, respectively, both in electricalengineering.

Currently, he is a Senior Engineer at SmartSpark Energy Systems,Champaign, IL.

Keith A. Corzine (S92M97) received the B.S., M.S., and Ph.D. degrees,from the University of MissouriRolla, Rolla, MO, in 1992, 1994, and 1997,respectively, all in electrical engineering.

Currently, he is an Associate Professor at the Department of Electrical andComputer Engineering, University of Missouri-Rolla.

Brian T. Branecky received the B.S. and M.S. degrees, from the Texas TechUniversity, Lubbock, TX, in 1986, and the University of Wisconsin-Milwaukee,Milwaukee, WI, in 1998, both in electrical engineering.

Currently, he is a PrincipalEngineerat the A. O. SmithCorporate TechnologyCenter, Milwaukee, WI.

magnetic equivalent circuit modeling

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