Finite-Element Models for Electrical Machines by Henneberger

EPE-PEMC 2002 Dubrovnik & Cavtat P. 1

Finite-Element Models for Electrical Machines

T. Busch, G. HennebergerDEPARTMENT OF ELECTRICAL MACHINES (IEM)AACHEN INSTITUTE OF TECHNOLOGY (RWTH)

Schinkelstrasse 4D-52062 Aachen, Germany

Phone: +49 241 80 9 7636 / Fax: +49 241 80 9 2270e-mail: [email protected]

KeywordsElectrical Machines, Modelling, Permanent magnet motors, Machine tool drives, Transversal fluxmotors

AbstractAfter a brief introduction, several examples of the use of Finite-Element models for ElectricalMachines are described by means of research works carried out at the Department of ElectricalMachines (IEM), Aachen Institute of Technology (RWTH), Germany. Static torque calculations withlarge Finite-Element models are as well presented as transient calculations of eddy currents [1].Another topic is a calculation procedure to determine the mechanical and acoustic behaviour ofelectrical machines [2]. Finally a coupled simulation to calculate the dynamic behaviour is outlined,where two-dimensional Finite-Element calculations are coupled with physical machine models.

IntroductionThe development of electromagnetic devices as machines, transformers, heating devices and otherkinds of actuators confronts the engineers with several problems. For the design of an optimizedgeometry and the prediction of the operational behaviour an accurate knowledge of the dependenciesof the field quantities inside the magnetic circuit is necessary. The losses in the device have to becalculated for the construction of a suitable cooling system. If the noise has to be taken into account,the acoustic behaviour has to be predicted.

The physical correlations like the Maxwell equations are well known for many years, but theanalytical calculation methods forced a lot of neglect and simplifications. Corrections factorswere determined by practical experience to consider miscellaneous effects. Upcoming in theseventies of the last century, the Finite-Element Method (FEM) is today state-of-the-art forthe calculation of structural-dynamic, thermal and, of course, electromagnetic problems. Withthe improvements of the performance of personal computers and workstations the modelshave become three-dimensional with the number of elements increasing. The bandwidth ofpossible applications is advancing steadily and research projects are opening up newperspectives for the development of electrical machines.

Static and transient Finite-Element calculations of the electric and magnetic field enable the designersto optimize well known electro-magnetic devices with regard to the torque-to-mass ratio and thedynamic, thermal and acoustic behaviour. Furthermore, the Finite-Element method approves, thedevelopment and optimisation of new devices without the necessity of extensive prototyping.

In this paper the design of new machines is demonstrated for a spherical motor and a transverse fluxmachine as well as the optimisation of well known machines with new tools. Here, a claw-polealternator and a permanent-magnet synchronous machine are acoustically and electrically simulated.


Spherical motorThe spherical motor is a multi-coordinate direct drive with three degrees of freedom [3]. The sphericalrotor is able to rotate in three axes, a rotation ϕ around the vertical axis and a declination in γrot- andϑrot- direction (Fig. 1). Possible applications are machine tools and robotic devices, utilising theadvantages of this high-dynamic direct drive which contains no mechanical transmission elements likegears. The motor consists of a permanent-magnet rotor sphere and a stator hemisphere with a largenumber of stator poles. The guiding of the rotor is realised by a hydrostatic bearing to achieve highstiffness and low friction. The stator hemisphere and the stator poles are made of a soft-magneticcomposite to reduce eddy-current losses. The arrangement of the poles has a decisive influence on thetorque characteristic. The current-dependent torques are calculated with a combinednumerical/analytical method [4]. The static cogging torques have to be calculated with a Finite-Element-model of the complete motor geometry.

Figure 1: Basic structure of the spherical motor

Combined numerical/analytical calculation methodLooking at Fig. 1, one can imagine, that the motor geometry causes large Finite-Element models withhigh element numbers. As a result the meshing and computational time is very high. Therefore it is notreasonable to calculate the torques with a Finite-Element model of the whole geometry for differentcases of current supply in order to optimize the stator pole arrangements. Otherwise the geometry istoo complicated to calculate the torques in an analytical way. Therefore a combinednumerical/analytical calculation method has been developed for the calculation of the total current-dependent torques of the spherical motor. The four most important steps of the method are:

• Preparation of a Finite-Element model of one stator pole with its nearest neighbours andappropriate rotor magnets

• Numerical calculation of the current-dependent thrust forces of this stator pole

• Approximation of the thrust-force characteristic using trend functions

• Analytical calculation of the total torques using the trend functions.

Five different spherical Finite-Element models were created to investigate the thrust forces caused bya current injection to one pole in the model. The models consist of 7 up to 9 poles and differ in thepositions of the neighbouring poles, which surround the pole carrying the current. Fig. 2 shows themodel in case that this pole is located at the border of the stator sphere.

rotor sphere

stator hydrostatic bearing

stator poles permanent magnets

ϕ

γrot

ϑrot


Figure 2: Finite-Element model of a motor section (mesh not displayed)

The thrust forces have been computed with a solver package developed at the institute [5]. Fig. 3shows the calculated ϕ- and ϑ-components of the thrust force of the spherical motor. They result fromthe difference of a calculation at a current of 4 A and a calculation at 0 A, so they do not include thecogging forces.

Figure 3: Calculated thrust forces Fϕ and Fϑ

It is assumed, that the generated thrust force of one stator pole only depends on the position of thispole above the permanent magnets in the rotor and not on the positions of the neighbouring poles. Thetotal torques of the spherical motor are calculated by multiplying the thrust-force contributions of eachpole with the corresponding distance between the pole and the pivot axle. Therefore, the thrust-forcecharacteristics, which were calculated with the FE-models are approximated with trend functionsdepending on the pole position and the current. Fig. 4 shows the total torque around the normal axis Tϕ

depending on the rotation about the normal axis ϕ and the declination of the normal axis ϑ at a currentof 3 A. Using this calculation method, various stator-pole arrangements were investigated concerningthe achievable torques.

Figure 4: Total torque Tϕ around the normal axis

ϕrelpermanent magnet

stator yokepole (carrying the current)

stator pole

rotor yokeϑrel

80 Nm70 Nm60 Nm50 Nm40 Nm


Calculation of the cogging torquesIt has to be taken into account, that the cogging torques depend on the complete geometry of stator androtor. For the calculation of these torques an overall Finite-Element model of the spherical motor hasbeen built. This model consists of round about 900000 elements (Fig. 5). The cogging torques have tobe calculated at various different rotor positions. For this reason the meshed stator and rotor model aretwisted against each other, glued in the air gap and then the air gap is meshed.

Figure 5: Finite-Element model and flux distribution

Transverse flux machineThe transverse flux machine is a direct drive with a high torque-to-volume ratio. Different topologiesof transverse flux machines have been developed at the Department of Electrical Machines in Aachen[6,7,8]. The calculation of additional eddy-current losses is very important for the prediction of themachine performance. Therefore, a 3-dimensional FEM solver with a time stepping algorithm is used,which is also capable of simulating the rotor movement.

Design of the machineThe geometry of the magnetic circuit and the complete machine layout are presented in Fig. 6. For abetter view the left figure shows only one pole of one phase in a linear arrangement.

Figure 6: Geometry of the magnetic circuit and complete machine layout

winding

yoke

air-gaps

flux concentrator

permanent magnets

Flux density

windings

yokes


The armature winding in the inner stator is surrounded by U-shaped soft-iron parts, which are arrangedcircumferentially in a distance of a double pole pitch. The limbs of the U-yoke are shifted by anelectrical angle of 180° against each other. The U-yoke lamination is stacked in circumferentialdirection. The geometrical shift is achieved by bending the complete stack before bonding. In order toget the utilisation of the magnetic circuit as high as possible, the cross sections of the U-cores getbigger at the air gap by using pole shoes. In the external rotor a flux concentrating design with soft-magnetic pieces between the rare earth permanent magnets, which are magnetised with alternatingpolarity in circumferential direction, is applied. The magnetic flux is of three dimensional nature in theflux concentrating parts in the rotor. Therefore they have to be made of a soft magnetic composite(powder iron) material instead of laminated iron.

The complete machine consists of three phases. In contrast to conventional machines there is nocommon rotating field in the three-phase design of a transverse flux machine, but only threeindependent alternating fields which are electrically shifted by 120°. The necessary mechanical shift isdone in the rotor by shifting the complete rotor rings, consisting of the magnets and the powder-ironparts, from one phase to the next. Accordingly, the stator cores in all phases can be arranged in line.

The complete arrangement of the active parts in the rotor is framed by a ring made of a non-magneticmaterial to prevent high stray fluxes from one soft-magnetic piece to the other. The electricconductivity of this material is high for using the ring as a damper to displace magnetic flux from thecarrier adjacent to the ring. The carrier itself is also made of a non-magnetic material with a goodelectric conductivity.

Finite-Element model and meshThe Finite-Element model of the transverse flux machine only consists of a cutting of one phaseincluding a double pole pitch in circumferential direction, which is the smallest symmetry unit of themachine. Comparative investigations between linear and rotationally symmetric Finite-Elementmodels have shown, that the consideration of a geometrically linear model is sufficient, because of thesmall pole pitch. The mesh for the non-air parts of the Finite-Element model is shown in Fig. 7. Thetotal mesh including the air regions consists of 250000 tetrahedral elements and 50000 nodes.

Figure 7: FE mesh of the transverse flux machine

yoke

non-magnetic ring

soft-magnetic piecesand permanent magnets


The mesh for the eddy current regions in the rotor, which consist of the permanent magnets and thecarrier parts, has to be influenced by the penetration depth δ of the electromagnetic field:

. f µ⋅σ⋅⋅π

=δ1

(1)

Special attention has to be paid to the periodic boundary conditions of the Finite-Element model incircumferential direction. The mesh in both edge layers has to be exactly the same to apply a periodicboundary condition. Therefore, one of the edge layers is meshed with triangle elements and thencopied on the other edge layer. After this operation the three-dimensional meshing of the model withtetrahedral elements is done. The chosen approach for the mesh generation is therefore a combinationof two- and three-dimensional meshing.

Rotor movementAnother important point for the meshing of the model is the simulation of the rotor movement. Theapplied technique permits the use of only one mesh for the complete transient calculation. This isrealised with a layer in the airgap between stator and rotor, which has exactly the same mesh inequidistant spacing ∆x in the direction of movement. This equidistant spacing is depending on thedesired geometric step width from one time step to the next. The simplest strategy of producing thismeshed layer is to mesh only a part of the layer with the width ∆x and then copy this mesh in the di-rection of movement. After every rotor position change of n x⋅ ∆ the positions of the nodes in the layerare congruent again. Therefore, stator and rotor mesh are completely independent and they are shiftedagainst each other but it is not necessary to mesh the airgap region again. Only the constraints have tobe defined anew after each transient step.

Calculation of the eddy currentsThe calculation of the eddy currents is based on a 3D Finite-Element method with a time steppingprocedure. The potential formulation is using two vector potentials for the magnetic and electric field,the magnetic vector potential A

r in the regions without eddy currents and both the magnetic and

electric vector potential Tr

in the eddy-current regions. With this approach the investigation of theinfluence of the sinusoidal stator current and particularly the movement of the permanent magnetexcited rotor on the eddy current losses is possible. Fig. 8 exemplifies the eddy current distribution inthe permanent magnets and in the rotor ring with moved rotor.

Figure 8: Eddy-current density distribution in the permanent magnets and in the rotor ring

The extracted perceptions of the different loss mechanisms are used for the choice of suitablematerials, especially for the passive rotor parts, e.g. carriers and fixations. The convergence of thetransient calculation is strongly influenced by the resistivity of the eddy-current regions with thepresented approach. A steady state solution is reached after only a few periods because the transverseflux machine is not a rotating-field but an alternating-field machine with a simple operation principle.


The Claw-pole alternatorSynchronous claw-pole alternators are used in automobiles for generation of electricity. They arefairly efficient over a wide speed range and are inexpensive when built in high numbered seriesproduction. Another optimisation aspect is the audible noise of these alternators. Especially in the lowspeed range (n = 2000-4000 RPM), when thermo-dynamical noise of the engine and the alternator fannoise are still relatively low, the noise caused by the magnetic excited forces of the generator has to beinvestigated.

Calculation procedureThe calculation procedure has been presented in [9] for star-connected alternators and in [10] for delta-connection. This procedure can be split into three blocks: the magneto-static computations, leading tothe magnetic forces on the stator metal, the structural-dynamic calculation of the relevant harmonicsand the acoustic simulation of the generator.

1.) Magneto-static calculation

First, Finite-Element computations of the magneto-static field in the claw pole alternator at variousspeeds are executed on models of one pole pitch as shown in Fig. 9. Since the modelling of thewinding head for machines with the number of stator slots per phase winding q > 1 is verycomplicated, the model is simplified in these regions. All other magnetically relevant regions aremodelled precisely. In each of the five models the rotor is rotated by an angle of:

m⋅°=α∆ 2 with }4;3;2;1;0{∈m , (2)leading to five time steps.

Figure 9: Magnetic model, one pole pitch, simplified winding head

An edge-based static FE solver as described in [11] is utilised for each time step. The solver is drivenby a constant direct current in the rotor-excitation coil. A three-phase current is impressed into thestator coils. The amplitude of the current and the load angle are functions of the generator speed. Inorder to take saturation effects into account, they are determined in a model with q = 1 and comparedto measurements [12]. The magnetic forces are calculated in each static FE calculation based on theflux-density distributions and the material data. Interpolation of different stator tooth positions leads to30 time steps or 14 spectral modes for two stator-tooth pitches.


2.) Structural-dynamic computation

The forces in spectral mode are transformed into a mechanical FE-model [13]. Since all mechanicallyrelevant components have to be modelled, there is no symmetry and a full 360° model as shown inFig. 10 has to be generated. This model is used to determine the deformation and oscillation of therelevant harmonics, based on the material data. Here, transversely isotropic materials are used torepresent the stator metal sheets. In order to reduce modelling and calculation expenses and since therotor contributes barely to the acoustic outcome, the rotor region is modelled as a solid cylinder withthe same mass as the claw-formed real-life motor.

Figure 10: Mechanical full 360° model

A node-based structural-dynamic FE solver is utilised for each relevant harmonic and alternator speed.To increase the numerical accuracy, second order elements are used in the displacement solver. Fig. 11shows the deformation of the claw pole alternator. In the case of the claw-pole alternator the relevantharmonics are acoustic orders 5 and 6 or mechanical orders 30 and 36. Since these two orders leadalmost to the complete noise output, all other orders are neglected.

Figure 11: Scaled deformation of the claw-pole alternator


3.) Acoustic simulationIn the last step, the sound power level is calculated using an acoustic boundary-element model (BE).This model differs geometrically from the mechanical model. Here, only the surface boundaries aremeshed and the stator and rotor region is merged in order to reduce numerical errors in the acousticBE method caused by small air gaps. Again a 360° model is used. Onto this model, the surfacevelocities of the structural-dynamic calculations are interpolated. These velocities are used to drive theacoustic BE solver. On a half spherical boundary area the emitted sound distribution and the totalsound power are evaluated. A sound-reflection plane represents the lower half space. Repeating thiscalculation chain for various generator speeds and the relevant harmonics leads to the sound-powercharacteristic of the machine.

Simulation of a PMSM with SIMPLORER-FLUX2D-CouplingThe coupling of a simulation tool like SIMPLORER with FEM calculations allows to simulate thebehaviour of more complex geometries like that of a conventional electric machine. The simulationparameters depending on the geometry of the machine can be adjusted. As an example for a couplingof SIMPLORER and FLUX2D a start-up of an electronic commutated permanent-magnet synchronousmachine (PMSM) has been simulated at the Institute for Electrical Machines [14]. The completecontrolling and the differential-equation system of the machine are implemented in SIMPLORER. Themachine is operated in field-orientated coordinates. The three-phase currents and voltages of themachine are transformed into a two-phase system with quadrature (index q) and direct axis (index d).

In a first step the torque of the machine is calculated analytically in SIMPLORER. In comparison tothis the estimation of the torque is replaced by a 2D Finite-Element calculation with FLUX2D. Inorder to bring the two simulations into agreement, the inductance of the machine has to berecalculated. For this the phasor diagram is approximatively calculated. Finally the simulations arerepeated.

In the case of a PMSM as an injection-pump drive a minimal start-up-time of 100 ms from 0 to 3500rpm is required. The machine shown as a FE-model in Fig. 12 has an outer diameter of the stator of120 mm. The rotor diameter is 34.5 mm. The length of the machine is 60 mm. As permanent magnetmaterial ferrit is used with a remanence of 0.35 T. The stator has a copper winding with 24 slots andthe rotor is symmetric so that there is no difference in the inductances of direct and quadrature axis.The pole pair number is p = 2. The inertia of the rotor is calculated to J=1.1032 mWs3. The stackfactor is kCu = 0.35.

Figure 12: FEM model of the PMSM, one pole pitch


Coupling of SIMPLORER and FLUX2D

The idea behind the coupling of the SIMPLORER-simulation with the FEM program FLUX2D is touse the exactness of the FEM calculation to determine the torque of the machine at every time stepconsidering the geometry of the machine. A model of the machine is built with FLUX2D. This iscoupled by an electrical circuit in FLUX2D to the coupling module in SIMPLORER. The couplingmodule is added to the simulation sheet of the machine as shown in Fig. 13. The SIMPLORERsimulation parameters must be adjusted. The time step is now set constant and the memory size higher.The simulation is started and every time step a FEM calculation in FLUX2D is conducted using thespeed, the time step, and the currents estimated in SIMPLORER.

Figure 13: SIMPLORER simulation sheet with coupling module to FLUX2D

Although the currents are about the same as without coupling the torque in the simulation is not anylonger smooth but undulating as shown in Fig. 14, an effect depending on the stator slots. Thereluctivity of the motor depends on the position of the rotor. If the reluctivity is lower the torque ishigher than at the point of higher reluctivity.

Figure 14: Calculated torque of the coupled simulation


Using the coupling of SIMPLORER and FLUX2D or similar products it is possible to calculatedynamic processes of complex geometries. This is of advantage if it is not easily possible to derive ananalytical equation for the torque of an electrical machine. If the coupling is used, the calculation timeincreases dramatically. But it is a possibility to verify the results of a simulation done without FEMand adjust it. In a next step the FEM calculation with FLUX2D is coupled to the simulation withSIMPLORER in order to substitute the equation for the estimation of the torque by the output ofFLUX2D. The results are compared and because of the great difference the inductance of the machineis recalculated using the coupling. The SIMPLORER parameters are adjusted and the simulations withand without the coupling are repeated. The simulations then show almost exactly the same behaviour.The coupling is not any longer necessary and the calculation time is reduced again when using thesimulation without coupling.

ConclusionThis paper presents four different examples of the way Finite-Element models are being used at theInstitute for Electrical Machines at Aachen Institute of Technology. The bandwidth of the applicationscovers static torque and force calculations for new types of electrical machines, transient calculation ofeddy current losses, a procedure to determine the mechanical and acoustic behaviour of electricalmachines and coupled simulations to calculate the dynamic behaviour of electrical machines.Depending on the requirements the Finite-Element models are either 2D or 3D and more or lessextensive. Regarding the ongoing improvements of the performance of personal computers andworkstations these examples are showing, that the complexity of the applications of Finite-Elementmodels will increase further on.

References[1] Blissenbach, R.; Henneberger, G.: Numerical calculation of 3D eddy current fields in transverse flux

machines with time stepping procedures, COMPEL, Vol. 20, Number 1, 2001, pp. 152-166[2] Kaehler, C.; Henneberger, G.: Calculation of the mechanical and acoustic behaviour of a clow pole

alternator in double and single star connection, 4th International symposium on advancedelectromechanical motion systems (Electromotion proceedings), Vol. 2, 2001, pp.553-558

[3] Weck, M.; Reinartz, T.; Henneberger, G.; De Doncker, R.: Design of a spherical motor with threedegrees of freedom, Annals of the CIRP, Vol. 49, 2000, pp. 289-294

[4] Busch, T.; Henneberger, G.: Designing methods for multi-coordinate drives, Linear Drives for IndustryApplications (LDIA Proceedings), 2001, pp.74-77

[5] Albertz, D.: Entwicklung numerischer Verfahren zur Berechnung und Auslegung elektromagnetischerSchienenbremssysteme , PhD thesis, Institute for Electrical Machines, ISBN 3-8265-6301-8, 1999

[6] Bork, M.; Henneberger, G.: New transverse flux concept for an electric vehicle drive system, Proc. Int.Conference on Electrical Machines (ICEM), Vol.2, 1996, pp. 308-313

[7] Blissenbach, R.; Henneberger, G.: Transverse flux motor with high specific torque and efficiency for adirect drive of an electric vehicle, Proc. ISATA, Clean Power Sources and Environmental Implicationsin the Automotive Industry, 1999, pp. 429-436

[8] Blissenbach, R.; Henneberger, G.: New design of a transverse flux machine for a wheel hub motor in atram, Proc. PCIM, Intelligent Motion, 1999, pp. 189-194

[9] Ramesohl, I.; Bauer, T.; Henneberger, G.: Calculation procedure of the sound fields caused bymagnetic excitations of the claw-pole alternator, 1st International Seminar on Vibrations and AcousticNoise of Electric Machinery (VANEM Proceedings), 1998, pp. 75-79

[10] Kaehler, C.; Henneberger, G.: Calculation of the differences in the acoustical behaviour of a claw-polealternatorwhen connected in delta and star, 2nd 1st International Seminar on Vibrations and AcousticNoise of Electric Machinery (VANEM Proceedings), 2000, pp. 127-131

[11] Albertz, D.; Henneberger, G.: On the use of the new edge based TAArrr

−, formulation for the calculationof time-harmonic, stationary and transient current field problems, IEEE Transactions on Magnetics,Vol. 36, 2000, pp. 818-822

[12] Küppers, S.: Numerische Verfahren zur Beredchnung und Auslegung von Drehstrom-Klauenpolgeneratoren, Dissertation, Institut für Elektrische Maschinen, RWTH Aachen, 1996

[13] Ramesohl, I.; Kaehler, C.; Henneberger, G.: Influencing factors on acoustical simulations includingmanufacturing tolerances and numerical strategies, 9th International Conference on Electrical Machinesand Drives IEE EMD(Canterbury, England), 1999, pp. 142-146


[14] Schlensok, C.; Henneberger, G.: Simulation of a PMSM with SIMPLORER-FLUX2D-Coupling, Proc.Int. Conference on Electrical Machines (ICEM), in press, 2002

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Finite-Element Models for Electrical Machines by Henneberger