8/7/2019 Modelling Rotor Skin Effect

http://slidepdf.com/reader/full/modelling-rotor-skin-effect 1/6

Modeling Three Phase Induction Machines with Rotor Skin-Effect

H.J. Coelinght, P.C. Breedveld 1 and J. van Dijk 2

Drebbel Institute for Systems Engineering'Dept. of Electrical Eng. 2Dept. of Mechanical Eng., University of Twente, P.O.Box 217, 7500 AE Enschede, The Netherlands

phone: +31-534892707, fax: +31-534892223, e-mail: mechatronics@rt.el.utwente.nl www: http://www.rt.el.utwente.nllmechatronics

ABSTRACT

Coordinate transformations can be applied to a bond graph model of a three phase induction machine, such that its structure becomessimilar to that of a simple DC machine. This transformation is applicable to every n-phase machine model. The resulting model enlargesthe insight in the operating principle of the induction machine. The model is also extendible to include non-linear effects such as rotorskin-effect. It is a competent model for position and velocity loop control design.

1. INTRODUCTION

The three phase induction motor is an electric motor with arelatively simple, robust construction and a low price, whichexplains its frequent applicat ion. The stator contains a numberof coils that conduct three alternating currents with a phaseshift of 120". These currents generate a rotating magnetic fie ldin the air gap between the stator and the rotor with an angularfrequency that equals the sta tor elec trical frequency. The rotorcoils (rotor cage) are exposed to this varying field and thusrotor currents are induced. The frequency of the inducedcurrents equals the difference between the stator currentfrequency and the mechanica l angular frequency, i.e. the slip-frequency. The induced currents are directed perpendicular tothe radial magnetic field and Lorentz forces are generated.

Initial bond graph modelThe theory of the generalized machine in [10] is taken as astarting point for modeling electric machines. This generalizedmachine is a model of a two phase machine used to describe

the principle of electro-mechanical energy conversion. Themost important assumptions are:

The air gap between the stator magnetic structure and therotor magnetic structure is uniform. All magneticvariations due to slots are represented by an effectivesmooth airgap.The stator variables are expressed in a stationaryreference frame and the rotor variables in a referenceframe fixed to the rotor.The magnetic material of both the stator and the rotorstructure is magnetically linear, it does not exhibitsaturation.Instead of current car rying conductors in slots the current

density is represented by a sinusoidal current sheet.Only the radial component of the magnetic field in the airgap is considered, the tangential component is neglected.

I: JM

Figure 1: The initial bond graph model of a three phaseinduction motor.

This mathematical model of the generalized machine is alsoused in [3] to obtain a bond graph model of electricalmachines. The core of this model (Figure I) is an IC-elementcontaining an inductance matrix and an expression for thetorque . These results will be used and developed fur ther in thispaper to describe a three phase induction motor.The induction matrix [10] is a 6><6matrix, that relates the

fluxes A and the currents i in a motor :

MabCde!(e)] [i~bCl L .L; . ir = abc,de! "abc,de!def de!

(1)

where the mutual-induction matrix depends on the position ofthe rotor e . The stator and rotor variables are expressed in the

coordinate axis abc respectively de! The mutual inductancebetween the three stator or rotor coil s, is approximated by [6]:

".27t Is'm: = l' cos- = --I3 2

(2)

where r' is the stator self inductance and m S is the mutualinductance between two stator coils. A formulation for thetorque T can be obtained by differentiat ing the co-energy Eco

to the rotor posit ion e (Lagrange-approach):

E I · T L .co = Z'abc,def abc.def+abc.def

aEco (iabc,def' e)T=

a e( 3 )

Note that T is the torque needed to put mechanical energy intothe t ransducer (Figure I) .

Star-node connectionsCoils of the stator or the rotor are connected in a star-node,

which is either insulated or connected to neutral. It isimportant to consider this difference, as in a three phase motorwith an insulated star-node only two independent currents canflow. Transformations, as applied in the next section, thenrequire a partial inversion of induction matrices due to thisrestriction. This can be shown by a causal analysis of the bondgraph model, where split causali ty occurs at the transformers.In case a motor with an insulated star-node is modeled, thematrix inversion can be avoided by approximating the isolationby a large resistor. This is in line with the practical situationwhere ideal insulation is impossible. In this case threeindependent currents can flow to the star-node and splitcausality and thus par tial matr ix inversion, does not occur.

In section 2 coordinate transformations are applied to the bondgraph model in order to reduce the number of relevant phases

535

8/7/2019 Modelling Rotor Skin Effect

http://slidepdf.com/reader/full/modelling-rotor-skin-effect 2/6

and to introduce a unique set of coordinates. Section 3discusses skin-effect in induction machines and describes howthe bond graph model should be extended to incorporate thisphenomenon. In section 4 simulation exper iments are shown ofthe bond graph models and section 5 ends this paper with adiscussion of the presented models.

2. COORDINATE TRANSFORMATIONS

The model of a three phase induction machine can be

simplified by the applica tion of coordinate transformations.The first transformation will reduce the number of relevantcoordinate axis from three to two and the secondtransformation will express stator and rotor variable in aunique set of coordinates. In [8] both transformations areapplied simultaneously, but in this paper the twotransformations are applied sequentially, to show applicabili tyof the coordinate transformations to every n-phase machinemodel. For these machines the first transformation (A), from nto two, is different and the second (B) is identica l.

Transformation A: Reduction of relevant phasesTo reduce the number of relevant phases, and coordinate axis,the vectors are decomposed. This procedure is described forthe stator current vector. For other vectors an identicalprocedure is followed. The stator current vector in the newreference frame is:

The elements of matr ix A are determined by the decompositionshown in Figure 2, where the three current vectors areexpressed both in the old reference frame abc and the newreference frame a f J O .(Rotor variables are transformed from

reference frame defto reference frame beO, using matrix A.)

fJ

a

.sI c{I is

c

Figure 2: Decomposition of the stator current vector usingtransformation A.

The transformation matrix A equals:

27r 27rcos¢ cos(¢+3) cos(¢-3)

A =[ - f .sin¢. 27r . 27r ( 5 )sm(¢+3) sm(¢-3)

L/2 }_ J i. }_ J i.2 2 2

This t ransformation matrix is a 3x3 matrix, so it seems that the

number of phases is not reduced. However the thirdtransformed phase, denoted with subscript 0, is the so-calledzero phase needed for power-invariant transformations as

( 4 )

explained in [8]. In the Appendix I a derivation of the powerconserving transformation matrix A is given.The zero phase does not playa role in the energy conversion,therefore the number of relevant coordinate axis is reducedfrom three to two.

Transformation B: Unique coordinatesThe second transformation will express the stator variables interms of rotor coordinates. The rotor coordinates 0 and EO are

fixed on the rotor. The new stator variables are obtained bymultiplication with the transformation matrix B«(}). This

procedure will be shown for the stator current vector.

i& O= B(B)· i ~ p o ( 6 )

In Figure 3 the relation between the two coordinate axis isvisualized, the afJ-axis are fixed and the be-axis rotate with aspeed Q) equal to the derivative of the angle ().

fJ

·R 'RIp· .....~\I

Figure 3: Visualization of the transformation B.

The zero phase is identical in both coordinate sets, theorthogonal transformation matrix B«(}) (cf. (45) ) equals:

[

cosB sinB 0 ]B(B) = - sinB cosB 0

o 0 1

(7)

Application of transformation AWithout loss of generality transformation A can be appliedwith the displacement < />equal to zero. This transformation will

effect the variables in the constitutive relations. The generalprocedure to transform the relating matrices will be shownusing the stator self-inductance matrix as an example:

[I'mS

']s s «s s t' ". i: ( 8 )A:abe = Labe . ' abc = m m . abc

mS mS IS

Both the flux vector and the current vector will betransformed, using AT = A·I • Because ia{3o= A· iabc , t.: can be

written as AT.ia{3o and also X'abc can be written as A \~.'41o.

Substituting these results in (8) results in:

AT 1s LS AT.s ,s A-T L' AT.s (9 )" " a p o= abc ' . ' a PO=> " a PO= . 'abc ' . ' a PO

The expression for the stator se lf- inductance matr ix in the newcoordinates equals:

o(10)

o

The self inductance per re levant phase is 1.5 times its previousvalue, which expresses the reduction from 3 to 2 phases.

536

8/7/2019 Modelling Rotor Skin Effect

http://slidepdf.com/reader/full/modelling-rotor-skin-effect 3/6

Application of transformation BTransformation B has a side effect as the transformationmatrix depends on the rotor position and thus on time. Thisside effect will be discussed in the sequel and is expressed inbond graphs terms as the gyristor. This is similar to the sideeffect of transformations of rigid-bodies in 3-dimensionalmechanics.

Introduction of a gyristor The IC-element in Figure 1 isdepicted in Figure 4a, where the rotor and stator multi-bondsare taken together as one 6-dimensional multi-bond. From nowon matrix A is a 6x6 block-diagonal matrix with twice (5) as

the two sub-matrices on the diagonal. Matrix B(O) is now a6x6 block-diagonal matrix with (7) and a 3x3 zero matrix on

the diagonal. Transformation B affects the variables in theelectrical domain. The variables and parameters in themechanical domain are unaffected.

eH e T

1f (j)

L(O)

( a )

e e] e2

~ )cT

1 MTF wf

~ e-e]1; B(O) 1 ;

GRGr=?

(b)

Figure 4: Transformation of the induction matrix.

The relation between the efforts e and flowsfin Figure 4a is:

d dL(B) .e=-(L(B)· 1)=--' I +L(B)· I

dt dt(11)

This should equal the corresponding relation between effor ts eand flows I in Figure 4b. The constitutive equations of thet ransformer in Figure 4b are :

Iz =B(6)· II

el =BT(B)·ez(12)

Due to transformation B the inductance matrix L in Figure 4bis no longer dependent of O . Therefore the relat ion between

efforts e, and flowsl, at the IC-element are:

d .ez=-(L· fz) = L· fz

dt(13)

Substitution of the constitutive equations of the transformer(B) in (13) results in the following re lation between the effortse, and flows I,

T' .el = B (B)· L· (B(B)· II + B(B)· III (14)

Comparison of (11) with (14) shows that:

L(B) = BT(B).L· B(B) (15)

Comparison of e1 in Figure 4b and e in Figure 4a shows that(14) is missing a term at the right hand side:

T .e-el =B (B)·L·B(B) (16)

because differentiating (15) with respect to time t, results in:

dL(B) = BT(B). L. iJ(B) + iJT(B). L· B(B)dt

(17)

The missing term (16) has to be add to the bond graph modeland can be represented by the gyristor Gr at the l-junction atthe left side of the transformer in Figure 4b. The expression forthe gyristor is:

Gr= iJT(B)· L· B(B) (18)

The gyristor can also be expressed in rotor coordinates, i.e.moved to the other side of the transformer. The expression forthe gyristor, using orthogonality of B(O), changes into:

The resulting bond graph model is depicted in Figure 5.

.. .C~·--l

=MTF=l =I l : L

B(O)

Figure 5: The bond graph model after a time dependenttransformation of an Ie-element.

The gyristor-C combination rewritten as a gyrator.Within the gyristor and G::-element combination the conversionof e lectr ic and mechanical energy takes place. This conversioncan be represented by the gyrator that is shown in Figure 6.

rL.beO .s.s

II~/ U beO T

~=l=~TF=l i M.qY~

I I A'B(O) ~ beO G

[Star 1node.

fRM

1

I

Figure 6: Transformed model of the induction motor.

The relations for the gyrator in Figure 6 are:

(20)

(21)

From the matr ix Gr, the sub-matrix G T of the gyrator can beobtained, as Gr, gives a relation between the angular speed ofthe rotor Q) and the vol tage vector coming out of the gyristor.

(22)

thus the sub-matr ix of the gyrator equals:

(23)

The equation for the torque can be obtained using the co-energy, as stated in (3):

T JEco 1 ·T dL (B) .=---=--, ,a o "-0'- ., ,a o "-0dB 2 cqiv.oe dB al' ,~(24)

Using the transformation to unique coordinates (24) can berewritten as:

537

8/7/2019 Modelling Rotor Skin Effect

http://slidepdf.com/reader/full/modelling-rotor-skin-effect 4/6

T - I.T B-T (B) d(BT(B)·L· B(B)) BT(B) .

--"'2'&0' . ae . "&0 (25)

I.T { dBT(B) dB(B) T }.T=--,&o' B(B)·--·L+L·--B (B) "&0(26)

2 dB dB

where L = C is used. Substituting (23) into (26) and using thefact that all flows are equal at a l-junction results in:

(27)

Using the fact that i1o·G T =i&;o·G the torque can be

rewritten as:

T =-ilo ·a T

The sub-matrix GT of the gyrator equals:

0 t 0 0 0 0 r .il-r 0 0 0 0 0 l" ·S- '1 0

GT=IL

0 0 0 0 0 0 . 1 1 0"&0 = _. m·il2 0 m 0 0 0 0 2

0 0 0 0 0 ·S-m -m'l o0 0 0 0 0 0 0

(28)

(29)

The expression for the torque (28), then results in:

T 1 1 (.S.R .s .R )= -m· I~'I -I '/~2 u e e u(30)

This expression is similar to the expression of the torque of atwo-phase induction motor [10].

3. ROTOR SKIN·EFFECT

Next we will show that the previously obtained model caneasily be extended to include skin-effect. Skin-effect is thetendency of a current to flow in the region with the lowestimpedance. It causes the current to flow at the outer-side of therotor. The impedance of a rotor bar is characterized by aresistance and an inductance, which increases respectivelydecreases with the frequency of the rotor current 0/. Thehighest slip-frequency and thus the highest frequency of therotor currents occur during start-up, as the angular frequencyof the rotor w is small. The increase of rotor resistance leads toan increase of the genera ted (starting) torque [7].A problem is that the skin-effect varies with the shape of therotor and rotor bars, but a general skin-effect model, valid for

all rotors and easy to connect to the existing induction motormodel, is preferred. Therefore the rotor is considered ahomogeneous rolled up sheet of conducting material. Thissheet has a conductivity and a permeability, which is anaverage of the conductivities and permeabilities of theconducting rotor bars and the laminated iron between thesebars.

Admittance of a conducting sheetThe rotor is assumed to be a rolled-up sheet of isotropicmaterial with given resistance and permeability. Theadmittance of this sheet is well known from skin-effect theory[4,5] and equals:

. R 4hrG"kin(J (J) ) = kttanh (kb ) (31)

where k is a constant, 2b is the thickness of the sheet, l thelength in axial direction, 2 h the length in radia l direc tion and ythe electric conductivity. During formulation of this expressionit is assumed that the system is in quasi-stationary state, inorder to neglect the displacement current. The magnetic fieldintensity is assumed to be an harmonic function of time. Thiscomplex admittance is generally approximated by a networkconsisting of resistances and inductances. Several methodsexist to find values for the components in the network.

In [9] a standard simplex optimization routine is used, whichminimizes a cost criterion containing the square root of theerror between the complex admittance and its approximation.In [7] two other methods are formulated. A Nyquist diagramshows that the accuracy of these two formulations is identica l.The first method uses continued iterative decomposition of thecomplex admittance, resulting in. a chain of resistances andinductances.The second method [7] determines the values of thecomponents in the approximate network on basis of partialfraction expansion, using the resistance and inductance of therotor. This approximate network consists of para llel branches.This method combines the advantages of the previous twomethods, i.e. it does not require an optimization for everypart icular rotor configuration, no additional parameter valuesare required and the parallel network facili tates an easy changeof the order of the approximation. This method will be used toapproximate the complex admittance (31).The complex admittance, rewritten according to [7] is:

(32)

(33)

(34)

where t is the rotor self inductance, m is the mutualinductance and It is the e lectr ical 'resistance of the rotor coil s.This admittance equals the complex admit tance formula ted by[4], as in [9] is stated that:

12 RR(J) =--- = ---

c /.q4b2 [R _ m(35)

I 4hJbG"kin(0) = RR = -[ - (36)

The admittance of the approximate network, consisting ofparallel branches tha t contain a resistance and an inductance is:

(37)

The expressions for the resistances and inductances in thisadmittance are [7]:

R 1C2

2R"kin,n= R g(2n -1 ) (38)

(39)

538

8/7/2019 Modelling Rotor Skin Effect

http://slidepdf.com/reader/full/modelling-rotor-skin-effect 5/6

An additional branch contains the remainder R ,b " ,Oand L,,,",o ofthe series approximation. The expressions for thesecomponents are [7]:

I I n o I--=R-L,--Rskin,O R n=! Rskin,n

(40)

L. _ I 32 . 3(IR _m). Rskin,Oskm,O 3 ,,4 ( 2no _1)2 RR

(41)

In the Nyquist diagram of Figure 7 both the complexadmittance of (32) and its approximation (37) are shown.Properties of the fictitious motor, used as an example in thispaper, can be found in Appendix II. The values of theparameters in the approximate network are given in Table 1.

Resistances Inductances

R,d(in,O 0.0543 c.: ge-4

«: 0.489 L skird ge-4

R.fkin ,2 1.36 L.fkin,2 ge-4R, 0.657 L, 12e-5

Table 1: Values oj the parameters in the approximate network.

Simulations will be performed with maximum rotor currentfrequency of 50 Hz. The approximation error increases withthis frequency. For a third order network the error at 50 Hzequals about 2%.

0

J-2"I

2'",-4

-,~ : JHZIm(G,,,"(jwR» -6

-,-8 ./~

-100 10 IS 20 25

Figure 7: Nyquist diagram of skin-effect admittance (1) andapproximation (2).

It can be seen from Figure 7, that in case the rotor currentfrequency equals zero, the rotor conductance is about 22 Qol,

which equals the rotor resistance of 44 mQ.

Bond graph modelingFigure 8 shows a bond graph model of the approximatenetwork. The lower branch represents the n o te rms of the seriesapproximation and the upper branch represents the remainder.This model has to be connected to the left l-junction of themotor bond graph model of Figure 6, only at the rotor current.

=! = g _ :R.,kin.o

n: Lsu«):«

Figure 8: Bond graph model of the approximation of the skin-effect admittance.

The skin-effect admittance replaces the rotor resistance andthe rotor leakage inductance in the motor bond graph model.This is done without affecting the structure of the motor bondgraph model. Only two elements in the model have to bemodified. The elec trical rotor resistance has to be removed andthe rotor leakage inductance will be removed from theinductance matrix. The leakage inductance equals thedifference between the self inductance and the mutualinductance, the fluxes can be expressed in terms of the leakage

inductance:

(42)

oR MT (.S .R) (LR MT).R1\.&0 = &0' '&0 +'&0 + &0 - &0 "&0 (43)

This results in new inductance matrix used for the bond graphmodel with skin-effect impedance

M&O] . [ i i co ]= LSR* . [ i& o ]MT ·R &O.R

&0 '&0 '&0

(44)

4. SIMULATION EXPERIMENTS

Simulations are performed of the start-up of the inductionmotor model with and without skin-effect. Parameter valuesare given in Appendix II. The input of the motor is a threephase voltage with a frequency of 50 Hz and an amplitude of220V. Simulation are performed in 20-sim [I] and the resultsare shown in Figure 9 and Figure 10.

4 0 0 A , .- - -- - -- -- - -- -- -

5008

A

·100 A

·5008 ~~~-~~---,tio:!:me:--~~-~~nO-,-'.5

Figure 9: Simulation of the rotor torque and angularfrequency of the motor model without skin-effect.

4 0 0 A , , -- - -- - -- - -- - -- - -,

5008

A

·10 0 A

·5008 ~~~-~~---;ti,;;:'me;;--~~-~~'O,..-J.5

Figure 10: Simulation of the rotor torque and angularfrequency of the motor model with skin-effect.

The simulations show:

A. The angular frequency of the rotor.B. The torque on the rotor iner tia.

539

8/7/2019 Modelling Rotor Skin Effect

http://slidepdf.com/reader/full/modelling-rotor-skin-effect 6/6

The motor model with skin-effect has indeed a higher starting-torque and the rotor reaches its constant angular velocity fasterdue to the skin-effect.

5. DISCUSSION

Coordinate transformation can lead to significant modelsimplifications. The first transformation reduces the number ofrelevant phases to two and the second transformation expressesthe model in one reference frame. It appeared that the bond

graph model of a three phase induction machine can berewritten in a configuration comparable with the model of asimple DC motor. This enlarges the insight in the principle ofoperation of the three phase induction motor and it makes thecorresponding bond graph model easier accessible. Thedeveloped basic model also supports the insight to augmenteasily non-linear effects such as the skin-effect.In [8] the transformations are applied in one step. In this paperthe transformation are applied in two separate steps, to showthat the first transformation is different for every n-phasemachine and the second transformation is identical for everymachine . Every n-phase induction machine can be representedby the bond graph model of Figure 6.The skin-effec t is normal ly expressed in di ffusion equat ions,but a lumped parameter model of the skin-effect in a thinconducting sheet, the impedance in (31) , is a competent modelto represent skin-effect in induction machines.This impedance is approximated to avoid too many (unknown)parameters in the model. Different approximations of theimpedance were mentioned. The approximation most suitedfor bond graph models is a series approximation, which onlyrequires two parameters, that were also needed for the bondgraph model without skin-effect and that needs no timeconsuming optimizations. If different parameter values orhigher order approximations are required, then this can easilybe realized in 20-sim [1].Simulations in 20-sim confirm that addition of the skin-effectmodel to the motor model increases the starting torque.

6. REFERENCES

[1] Controllab Products Inc. (1997), 20-Sim, ReferenceManual, University of Twente, Enschede, The Netherlands,http://www.rt.el.utwente.nIl20sirnlclp.htm.[2] Goldstein, H. (1981), Classical Mechanics, Addison-Wesley Publishing Company Inc.[3] Karnopp, D. (1991), State Functions and Bond GraphDynamic Models for Rotary, Multi-winding ElectricalMachines, 1. of the Franklin Inst., Vol . 328, No.1, pp. 45-54.[4] Kneller, E. (1962), Ferromagnetismus, Springer-Verlag,Berlin, Germany.[5] Lammeraner 1 . and M. Stafl (1966), Eddy Currents, IlifeBooks Ltd., London, UK.[6] Messerle, H.K. (1965), Dynamic Circuit Theory, PergamonPress, Oxford, U.K.[7] Paszek, W. (1981), Transientes Verhalten derInduktionsmaschine mit Hochstabliiufer, Archiv fiirElektrotechnik 63, pp.77-86.[8] Sahm, D. (1979), A Two-Axis Bond Graph Model of theDynamics of Synchronous Electrical Machines, 1. of theFranklin Inst. , Vol. 308, No.3, pp. 205-218.[9] Veltman, A. (1994), The Fish Method, PhD thesis, DelftUniversity of Technology, Delft, The Netherlands.[10] White , D.C. and H.H. Woodson (1959), ElectromechanicalEnergy Conversion, John Wiley & Sons Inc., New York, U.S.A.

I. APPENDIX: POWER INVARIANCE OFTRANSFORMATION MATRIX A

For power-conserving coordinate transformations thefollowing condition for the t ransformation matr ix A can beformulated [2,8]:

(45)

where I i s the identity matr ix. This condition requires A to beorthogonal. Therefore A is a n x n matrix, resulting in anadditional zero-phase component. The first two rows of Afollow from Figure 2.

cos¢J 2" 2"cos(¢J--) cos(¢J+-)3 3

A= - sin¢J . 2" . ¢ J 2" (46)-sm(¢J--) -sm( +-)3 . 3

a3l a32 a33

The elements in the third row follow from the orthogonalitycondition of the matrix. The multiplication of thetransformation matrix with its transposed is:

1 1-'2+a31a32 -'2+a31a33

1-'2+a32a33

(47)

This matrix cannot be equal to the identity matrix for anyvalue of a'l' a" and a"., as it should be according (45). It canbe made equal to a matrix with only elements on the diagonal.The terms off the diagonal are equal to zero when:

a'll = a'l2 = a33 =L.fi- - - 2(48)

When the transformation matrix is multiplied with a constantfactor c , it is also possible to make the terms on the diagonalequal to one. The transposed of the transformation matrix isalso multiplied with this factor.

30 0

2

AT.A =c 2 03

0 (49)2

0 03

2

When c' is chosen 2/3, the identity matrix is obtained. Theresulting transformation is shown in (5).

II. APPENDIX: FICTITIOUS MOTOR

Pro.Q~I!L_. unit._.. ._.._----_._---Stator resistance 39 [mQ]

Rotor resistance 44 [mQ]

Stator self induction 20 [mH]Rotor self induction 20 [mH]Mutual inductance 19.4 [mH]Rotor inertia 0.07 [kg m']Mechanical friction 0.1 [Nms/rad]

Table 2: The properties of the fictitious motor of [9].

540