Finite Element Model in Electrical Machine Design

8/6/2019 Finite Element Model in Electrical Machine Design

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Finite element models in electrical machine design

K. Hameyer, F. Henrotte, Hans Vande Sande, Geoffrey Delige, Herbert De Gersem

Katholieke Universiteit LeuvenEE dept. ESAT / ELECTA

Kasteelpark Arenberg 10, B-3001 Leuven Heverlee

Abstract In the last thirty years, ongoing and

fast developments of numerical techniques for

the particular questions concerning electrical

machinery can be noticed. This development can

be seen in parallel with the efforts and success of

the soft- and hardware computer industry in

producing powerful tools. Computers as well as

the commercial software are now ready to be

applied to solve realistic numerical models of

technical relevance. Of course, there are enough

complicated problems and questions, which are

not yet answered to a satisfactory extend.

This paper is intended to focus on special

problems concerned to better solution of

electrical machine design and simulation

questions. Particular attention is given to the

topics of material modeling, winding and coil

models and to the dynamic simulation of

electrical machinery.

Today, efficient numerical solutions can be

obtained for a wide range of problems beyond

the scope of analytical methods. In particular

the limitations imposed by the analytical

methods, their restrictions to homogeneous,

linear and steady state problems can be

overcome using numerical methods.

INTRODUCTION

At the time, when not too many computers whereavailable yet, the design of electrical machines wasperformed in the classical way, by using one-dimensional models. Particular electro magneticparts of the machine are considered to form ahomogenous element in a magnetic circuitapproach. In this approach the knowledge ofparticular design factors is assumed. Such modelsenabled the calculation of specific stationaryworking points of the machine. Laplacetransformations applied to such models made itpossible to analyze the machine for the dynamicoperation. The refinements of the 1D models yieldthe technique of magnetic equivalent circuitmodels.

A very rough figure can be drawn in saying that thedevelopment of numerical methods for electricalmachines started with the finite difference methodwhich quickly was followed and further overtakenfrom the finite element method.With the first numerical models of electricalmachines, in the 70s, electromagnetic fieldsconsidering imposed current sources could besimulated by applying first order triangularelements [1]. Further developments of the finiteelement method lead via the definition of externalcircuits to problem formulations with imposedvoltage sources, which where more accuratemodels with respect to the realistic machine that isoperated by a voltage. Todays developments aredirected to all aspects of coupled fields. There arethe thermal/magnetic problems or structuredynamic field problems coupled to theelectromagnetic field; e.g. acoustic noise intransformers and rotating machines excited byelectromagnetic forces.Parallel with the developments of the finite elementmethod first attempts to numerically optimise thefinite element models where made. Optimisationsof realistic machine models where first performedin the late 80s and early 90s. Various deterministicand heuristic methods have been applied and can befound in the conference proceedings e.g.COMPUMAG and CEFC of that time.Research groups at universities have code availablethat goes further in modeling. For example, it canconsider external electrical circuits with powerelectronic components operating the machinemodel. Such models are consuming a lot ofcomputational time. However, research directed toaccelerate the numerical solvers is going on as well.Extensions of the overall machine models can bemade by applying more realistic material models.Hystersis effects and material anisotropy are still alack in many software packages. Special windingmodels allow to reduce the size of a numericalmodel significantly and therefore the computationtime as well. To avoid long lasting transientsimulations single-phase machines can be modelledby a time-harmonic approach by modelling separate


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rotor models for the rotary fields in reverserespectively rotation direction of the rotor.However, with all developments of numericaltechniques around the electrical machine analysis,

it must be noticed that the classical approach cannot be replaced by the numerical models. Enormousexpertise and knowledge to model the variouselectromagnetic effects appropriate and accuratelyis still required. The modeling/solving/post-processing of numerical motor models still requiressubstantial engineering time and can not replace thequick and correct answer of an exercised motordesign engineer. Both experts, probably in oneperson, numerical modeling and electromagneticmotor design engineer, are required for successfuland novel developments in this engineeringdiscipline.

FINITE ELEMENT MODELS

It all starts with Maxwells equations (Table I).Every electromagnetic phenomenon can beattributed to the seven basic equations, the fourMaxwell equations of the electrodynamic andthose equations of the materials. The latter can beisotropic or an-isotropic, linear or non-linear,homogenous or non-homogenous.

Table I. Maxwells equations.

differential form integral form(i)

= E Bt dt

dd

C

= rE(ii) JH = Id

C

= rH(iii) =B 0 B S = d

S

0

(iv) =D D S = d Q

S

(v) B H=

(vi) ED =(vii) EJJJJ +=+=

00 c

The Maxwell equations are linked by interfaceconditions. Together with the material equationsthey form the complete set of equations describingthe fields completely. is the magnetic flux, I theconducted current, Q the charge, C indicates thecontour integral and S the surface integral.The seven equations describe the behavior of theelectromagnetic field in every point of a fielddomain. All electric and magnetic field vectors E,

D, B , H, and J and the space charge density arein general functions of time and space. Theconducting current density can be distinguished bya material/field dependent part Jc and by an

impressed and given value J0 . It is assumed thatthe physical properties of the materials permitivity, permeability and conductivity areindependent of the time. Furthermore it is assumedthat those quantities are piecewise homogenous.The Maxwell equations represent the physicalproperties of the fields. To solve them, mainly thedifferential form of the equations and mathematicalfunctions, the potentials, satisfying the Maxwellequations, are used. The proper choice of apotential depends on the type of field problem.The electric vector potential for the displacementcurrent density will not be introduced here, because

it is only important for the calculation of fields incharge-free and current-free regions such as hollowwave-guides or in surrounding fields of antennas.Various potential formulations are possible for thedifferent field types. Their appropriate definitionensures the accurate transition of the field problembetween continuous and discrete space. Using theseartificial field quantities reduces the number ofdifferential equations. Considering a problemdescribed by n differential equations, a potential ischosen in such a way that one of the differentialequations is fulfilled. This potential is substituted inall other differential equations, the resulting system

of differential equations reduces to n-1 equations.

Flux

Movement

Coil

Fig. 1. (top) Geometry, direction of movement, expectedflux path and (below) the associated 3D FEM

discretisation of a linear transversal flux (TF) motor. (thesurrounding discretized air is not plotted)


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To build up a FEM model, in a pre-process thedevices geometrical shape is approximated byfinite elements. In two-dimensional models thestandard elements are commonly triangles and in

the three-dimensional case tetrahedron elementscan be used. This type of elements is very flexibleand capable to approximate very well evencomplicated geometrical shapes (Fig. 1). In thisstep of modelling the device, material and boundaryconditions have to be defined.Outgoing from the computed potential solution, theinteresting field values, such as field strength andmagnetic flux density can be calculated for allplaces inside the model. From the obtained fieldvalues e.g. the forces and other physical quantities,such as the induced currents and voltages can bederived.

Computer models enable a thorough study ofactuators without the requirement of expensiveprototyping. Prototyping is of course required at theend of the design procedure to verify simulatedresults or if required to improve the devicesnumerical model. Various different geometricalvariants of an actuator concept can be studied andthe particular properties can be compared. Figure 2shows two different variants of a concept of a linearTF actuator with their associated forcecharacteristic.

-10

-5

0

5

10

0 0,2 0,4 0,6 0,8 1

Position (in # pole pitches)

Netforce

(in

N)

Fig. 2. 3D FEM models of variants of a TF linear motorwith the associated force vs. position characteristic.

SPECIAL MATERIAL MODELLING

The ongoing trend towards the miniaturisation ofsystems and the trend of system integration forcethe development engineers to construct smaller andsmaller devices. Key here is the appropriate

utilisation of the materials used. The recentdevelopments and ongoing research in the field ofhard- and soft-magnetic material deliveredmaterials with improved material properties

(Fig. 3). Therefore, extremely small electric motorswith very compact electromagnetic circuits can bedeveloped (Figs. 4 and 5) by applying appropriateand accurate material models.

0

100

200

300

400

500

600

700

800

900

1000

1880 1900 1920 1940 1960 1980 2000 2020 2040 2060 2080

Year

kJ / m3

Steel

AlNiCo

Ferrit

SmCo5

Sm2Co17

NdFeB

(BH)max/ 2

(BH)max.theor. = 960 kJ / m theoretical

(BH)max. 75 % = 720 kJ / m technically possible

Fig. 3. Development of permanent magnet material.(source: VACUUMSCHMELZE , Germany)

statorpermanent magnet system

rotorwinding

Fig. 4. Magnetic field solution of a small electromagneticstepper motor excited by novel rare earth permanent

magnet material.


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Fig. 5. Three members of the Penny-Motor family: a) HEtype for high torque; b) YL type for low price; c) EC type

for electronic commutation. (source: mymotors &actuators gmbh, Germany)

A. EDDY CURRENTS IN LAMINATEDFERRO-MAGNETIC ELECTRIC STEEL

High permeable steel laminations are usually usedfor the design of the magnetic core of electricalmachines. This is because of its magnetically goodproperties. However, the material is electrically

very good conducting as well. As a consequeceeddy current losses have to be considered in themodel of steel lamination when time varying fieldshave to be assumed. To lower the eddy currentlosses in AC fields several iron laminates withcoating material on both surfaces are applied. Thecoating material is less conductive and lesspermeable when compared to the iron. Thisprevents excessive eddy current losses but at theexpense of a higher reluctance of the globalmagnetic flux path [2] (Fig. 6).

( )11,

( )22 ,

z

J

zl

xl

Fig. 6. One sheet of ferromagnetic lamination withcoated surfaces.

For the analysis of electrical machines usually a2D-FEM electrodynamic model is built which liesin the plane of the flux. For this case, no eddycurrents in the x-plane of the lamination (Fig.6) can

be considered. To cope with this problem the modelhas to be built in the xy-plane, where the flux thenis perpendicular to it. To realize a solution for thismodel, an electrodynamic in-plane formulationusing the electric vector potential T is chosen:

JT = , (1)

this yields the time-harmonic formulation

z

mz

z VjTj

x

T

x l =+

. (2)

is the resistivity, the permeability, j is theangular frequency, Vm the magnetomotive forceand lz the length of the problem in z-direction.With this model it possible to determine the lossesfor single sheets and for multiple layerarrangements of ferromagnetic lamination (Fig.7).For the simulation of multiple layers an FEMapproach with external magnetic circuits for theflux in z-direction has to be chosen [3].

mR

Fig. 7. 2D electrodynamic model of a laminated material

of various layers combined with a magnetic equivalentcircuit [3].

Semi-analytical simulations consider the losses in asingle laminate and neglect the conductivity andpermeability of the coating material. The modelpresented here deals with coating material with afinite resistivity. As a consequence, the closing pathof the current may cross the coating layers. Theeddy current losses are completely different fromthe simplified analytical model. As an externalcondition, the total magnetic flux through themodel has to equal the applied flux. The magnetic


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equivalent circuit represents the parallel connectionof all domains in the electrodynamic model, excitedby a flux source (Fig.7).By using this hybrid modelization technique, the

coupled FEM and magnetic equivalent circuitmodel, dependencies of the eddy current losses canbe studied with respect to the frequency and thenumber of iron layers (Fig. 8a, b, c). The results canbe employed to simulate the materials behavior inelectrical machines for various frequencies. Toconsider the complicated shape of e.g. an inductionmachine these simulated results have to betransformed in such a way that they can be used inthe 2D model of a realistic electrical machine.

a) b)

c)

Fig. 8. Eddy current density distribution for a field at a)50 Hz, b) at 500 Hz and c) voor comparison, multiple

laminations at a frequency of 50 Hz [3].

B. FERROMAGNETIC ANISOTROPY

Grain-oriented ferromagnetic materials exhibitanisotropy at both the microscopic and themacroscopic level. In order to assess themacroscopic anisotropy of these materials, singlesheet testers are often used. This is done bymeasuring the magnetization curve ( )measmeas HB of a

series of small strips, cut out of a metal sheet undervarious angles. As single sheet testers can onlyperform unidirectional measurements, thesemagnetization curves represent a relation betweenthe components of the magnetic flux density B [T]and the magnetic field strength H[A/m] along

some fixed directions. The reluctivitymeas[Am/Vs]

meas

measmeas

B

H= , (3)

depends non-linearly on Bmeas. Fig. 9 shows themeasured reluctivity curves for a conventionalgrain-oriented (CGO) steel, along several directionswith respect to the rolling direction [4, 5].

54.7 RD

TD

Fig. 9. Definition of the Goss-tecture with rollingdirection (RD) and transverse direction (TD).

0

300

600

900

1200

1500

0.0 0.4 0.8 1.2 1.6 2.0

Flux densityB [Vs/m2]

Reluctivi

ty[Am/Vs

10 0

203090

80

4070

6050

Fig. 10. Reluctivity curves for grain-oriented steel undervarious angles referencing to the RD.

However, for anisotropic ferromagnetic materials,the vectors representing the magnetic field and the

flux density are not parallel with each other. It isimpossible to deduce the angle between B and H,by exclusively considering the measuredmagnetization curves. For ferromagnetic materialshaving a Goss-texture, like most transformer steels,the paper demonstrates a way to compute this anglea posteriori, by combining the measurements froma single sheet tester with a physical anisotropymodel.

The magnetizations

Mr

[A/m] within a domain

tends to align with one of the easy axes of thecrystal. Each deviation from this equilibrium state


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corresponds to an increase of energy, which is dueto the intrinsic anisotropy of the crystal. For cubiccrystals, the anisotropy energyEa [J/m

3] is given b:

( ) ( )232221221232322222110a KKK ++++=E (4)

with 1,2 and 3 the direction cosines of sMr

in the

crystallographic coordinate system [4]. K0 is anarbitrary constant. K1 and K2 are the anisotropyconstants. In case of a cubic iron crystal, these areK1 = 0.48105 J/m3 and K2 = 0.05105 J/m3 [5]. Onthe other hand, if an external field H

r is applied,

sMr

tends to align with it. The corresponding

energyEh [J/m3] is the so-called field energy

HME

rr

= s0h . (5)

The process of coherent rotation of the domains canbe considered as a competition between the

anisotropy energy (2) and the field energy (3):s

Mr

stabilizes in a direction for which the total energy isminimal.

For silicon iron having a Goss-texture, (2) and (3)simplify into

( )221

a sin34sin4 =

K

E (6)and

( ) = coss0hHME (7)

with the angle between the field Hr

and the RD,

and the angle betweens

Mr

and the RD. After

evaluating (6) obviously, = 54.7 corresponds toa maximum ofEa.

Hybrid Model. The hybrid method works in twosteps. The direction of the magnetizationM is firstdetermined for a fixed direction ofH, by means ofthe previous discussion. If < 54.7 , the minimumofEa+Ehat low values of is determined. Else, theminimum for high values of is determined. Thesecond step consists in extracting the amplitude ofM, knowing its direction, from the single strip testermeasurements (Fig. 11). This is done as follows.The flux densityB is related to the applied field Hand the magnetizationMby:

MHBrrv

+= 0 (8)The measured and applied field strength are equal.By projecting (8) onto the direction of the applied

field, an expression for the amplitude ofM isobtained:

( ) ( ) = cos//0meas

HBM . (7)

M B / 0

H Bmeas / 0

a

?

RD

Fig. 11. Definitions within the measurement set-up.

Subsequently, the components ofB are obtained byprojecting (6) on the RD and the TD:

+=

+=

sinsinsin/

coscoscos/

0

0

MHB

MHB, (8)

Using the previously presented hybrid approach, itis possible to compute the two components ofBand hence the reluctivity tensor. If this symmetricalsecond order tensor is considered in its principal

coordinate system, it has zero off-diagonal entries[6]. For this application, the RD and the TD are theprincipal axes of the tensor, asB and Hare parallelin these directions. As a consequence

=

TD

RD

TD

RD

TD

RD

0

0

B

B

H

H

, (9)

with

( ) ( )

( ) ( )

sin/sin

and

cos/cos

TD

RD

BH

BH

=

=

(10)

for the specified measurement point with theangle betweenB and the RD.The hybrid model is analyzed for the data setplotted in Fig. 12. It can be shown, that for lowfieldsH, the flux density vector stays close to theRD or the TD, irrespective of the field direction .On the other hand, for very higher fields (> 1,7 T),Hand B tend to align. This is in correspondencewith the theory of magnetization [7,8,9]. Fig. 13reveals how the field behaves, if the flux density


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rotates in space with constant amplitude. The result,for |B| = 1.7 T is plotted in Fig. 14.

Fig. 12: The magnetization curves applied to explain themodel.

H

H

Fig. 13. B and as a function of H (data set fig. 12).

H

B

Fig. 14. H-locus when B rotates at 1,7 T (data set fig. 12).

SIMULATION OF THE TRANSIENT SYSTEM-BEHAVIOR BY STAGED MODELING

This section presents a methodology to achieve an

overall model of a power system that consists of abattery, an inverter, a permanent magnet servo-motor and a turbine. Particular attention is paid tothe fact that one single finite element model wouldnot be able to provide a satisfactory representationof the behavior of the overall system. A stagedmodeling is proposed instead, which succeeds inproviding a complete picture of the system andwhich relies on numerous finite elementcomputations.The key point in the realization of a staged model isto define carefully the interface quantities betweenthe different components of the system. Between

the inverter and the motor, the interfacing quantitiesare the electrical quantities associated with thethree stator phases. In particular, to represent theback-emfof the phase, we will pay attention to theflux distribution:

dSnIbI

phS

=r

),(),( (11)

which gives the total flux embraced by one statorphase as a function of the angular position of therotor and the current in that phaseI. If the motor

works in steady state operation, the flux distributionof one phase is sufficient, as the flux plot for theother phases are obtained by a simple angle shift. Ifthe machine works in a non-saturated state, thedependence on I can be neglected. The flux plotgathers in one table all the required informationconcerning the motor, seen from the inverter.At high motor-speeds respectively high supplyfrequencies, iron core losses may override copperlosses and have therefore to be estimated carefullyover a wide range of frequencies. A difficulty in thecalculation of core losses is that the magnetic fluxdensity not only varies in time but also varies

widely in space. Usually one distinguishes hystere-sis losses, which are proportional to the frequencyand eddy current losses, which are approximatelyproportional to the square of the frequency. Onewill therefore assume that core losses can beexpressed by the formula

221

)()(),( ICICIQcore

+= (12)

where the frequency independent constants Cl(I)and C2(I) are expressed by:


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=

=

core

eddy

core

dIb

C

IC

dIbhIC

2

0

2

2

2

0

1

),(2)(

,),()(

(13)

This assumption is one of the simplifications of thestaged modeling. It can be seen that the fluxdistribution (11) as well as the core lossesparameters (13) depend on ),( Ib and ),( Ib .They can therefore be computed beforehand for agiven geometry of the motor, by post-processingadequately a series of 2D static FE computations,and then storing the results into look-up tables.Because of the large difference in scale betweenone electrical period and the time required for the

motor to reach its steady state operation, a classicaltransient analysis would require several ten-thousands of time steps. The transient analysis istherefore split-up into two parts. The first partconsists in calculating 'over one period' thestationary current wave shapes for a fixed speed , a fixed commutation angle and fixedoperation temperature. The second part is thetransient dynamic and thermal analysis. Thissplitting is one of the simplifications of the stagedmodeling.During the 'over one period stationary' analysis, theinductances , resistances and back-emf's of the

phases, the extra voltages and resistances due to theelectronic components and the switching strategy ofthe inverter bridge are all considered for thecomputation of the wave shapes of the phasecurrents and voltages. One starts from the voltageequation of one stator phase:

tIRV += , with index CBA ,,= (14)

in which the flux coupling is given by (11). The

total resistance of the phase is computedanalytically and can be augmented by an extra

resistance due to the power electronic components.The total phase voltage V is equal to the power

electronic inverters dc-link voltagedc

V distributed

over the different phases, according to theparticular topology at each instant of time of theinverter and to the winding connection type, detarespectively Y-connection. In this example weconsider a Y-connection and it can be written:

dcVV = (15)

is the coefficient, which can be taken from the

table II, which considers the various switchingstates of the inverter.

Table II. Coefficient for the distribution of the dc-link

voltage to the three winding phases A, B, C.

Phase A Phase B Phase C

11 ++ 01 10 0 OFF 0

1/2 OFF -1/2-1/3 2/3 -1/31/3 1/3 -2/3

01+ 10 + 11 OFF 0 0OFF 1/2 -1/2-2/3 1/3 1/3-1/3 2/3 -1/3

10 11 ++ 010 0 OFF-1/2 1/2 OFF-1/3 -1/3 2/3-2/3 1/3 1/3

11 01+ 10 +0 OFF 0

-1/2 OFF 1/21/3 -2/3 1/3-1/3 -1/3 2/3

01 10 11 ++OFF 0 0OFF -1/2 1/22/3 -1/3 -1/3

1/3 -2/3 1/310 + 10 11 ++

0 0 OFF1/2 -1/2 OFF1/3 1/3 -2/32/3 -1/3 -1/3

The table consists of six sections, which correspondto the six periods between the switching-on of twosuccessive phase windings. For instance, let usconsider the last group of four lines in the table. Inthe considered period, the phase A has to beswitched on ( 10 + ) and the phase C has to beswitched off ( 01+ ), the phase B remains in thesame state as before ( 11 ). Immediately aftersending the switch-on signal to phase A, all threephases are magnetized by its correspondingcurrents. The inverter topologies correspondingwith the last two lines of the group of the table areused then. In both states, the voltage is positive inthe switched-on phase (i.e. phase A), the voltage isnegative in the switched-off phase (i.e. phase C),and the voltage is either positive or negative inphaseB, which allows to control the current in thatphase. After a while, the current in phase Creaches


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zero and the phase ceases to be conducting. Fromthat instant on, the first two lines of the group areapplied, the first one to let the current decrease(freewheeling) and the second one to make it

increase in the loop formed by the phasesA and B.Outgoing from the voltage equation (14) theexplicit time-stepping scheme for the phase-currents can be derived:

+=+

I

IRVII

)()()(

(16)The given scheme can be applied until the steadystate of the transient behavior, the steady statewave-shapes of the currents )(I over a period

is reached. With this, the electromagnetic torque

can be evaluated:

=

2

0

),()()(

2

3),(

IQdIT

core

(17)In a following step the dynamic differentialequation of motion is applied to form the transientanalysis of the overall drive system.

),()( TfJt

=+ , (18)

whereJ is the inertia of the rotating part, )(f is

the reaction torque exerted by the load and frictionforces. When temperature-sensible devices, such ase.g. permanent magnet excited systems are studied,at this point of analysis a thermal model has to beapplied as well [10]. With the described modelingboth, transient and steady state behavior of thedrive can be analyzed for e.g. arbitrary values ofthe commutation angle .However, to conclude this approach of combinedanalytical parameter model and numerical finiteelement model, building the staged model hasimplied assuming particulatr simplifications. On theother hand the staged model has the big advantage

of providing at a reasonable computational cost afaithful dynamic and steady state description of anoverall drive system. As the development goesalong, the staged model is open to gradualenrichment and improvements, thanks to further FEinvestigations aiming at determining its differentparameters or at estimating the influence of thesimplifications that have been done.Fig. 16 illustrates some results obtained from aservo-motor. Flux weakening consists here inselecting, at each speed , the commutation angle that maximizes the torque in that point ofoperation.

a)

b)

Fig. 15. Steady state wave-shape form of the phase

current at a speed of a) srad/1000= with O0=

and b) at srad/2500= atO

55= .

Fig. 16. Steady state torque of the speed andcommutation angle including the field weakeningrange of a permanent magnet excited servo-motor.

NUMERICAL OPTIMIZATION OF FINITEELEMENT MODELS

The design process of electromagnetic devicesreflects an optimisation procedure. Theconstruction and step by step optimisation oftechnical systems in practice is a trial and error-


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optimization of a small dc motor applying theevolution strategy [11].The objective is to minimise the overall materialexpenditure, determined by permanent magnet-,

copper- and iron volume subject to a given torqueof the example motor.

.min10)x( maxmax

cos

cos)(cos

+=

penaltyZt

txt

(19)

The use of penalty term in the form:

Finite Element Model in Electrical Machine Design

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