Finite-element Models For Electrical Machines By Henneberger

Finite-Element Models for Electrical Machines T. Busch, G. Henneberger DEPARTMENT OF ELECTRICAL MACHINES (IEM) AACHEN INSTITUTE OF TECHNOLOGY (RWTH) Schinkelstrasse 4 D-52062 Aachen, Germany Phone: +49 241 80 9 7636 / Fax: +49 241 80 9 2270 e-mail: [email protected]

Keywords Electrical Machines, Modelling, Permanent magnet motors, Machine tool drives, Transversal flux motors

Abstract After a brief introduction, several examples of the use of Finite-Element models for Electrical Machines are described by means of research works carried out at the Department of Electrical Machines (IEM), Aachen Institute of Technology (RWTH), Germany. Static torque calculations with large 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 of electrical 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.

Introduction The development of electromagnetic devices as machines, transformers, heating devices and other kinds of actuators confronts the engineers with several problems. For the design of an optimized geometry and the prediction of the operational behaviour an accurate knowledge of the dependencies of the field quantities inside the magnetic circuit is necessary. The losses in the device have to be calculated 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 the analytical calculation methods forced a lot of neglect and simplifications. Corrections factors were determined by practical experience to consider miscellaneous effects. Upcoming in the seventies of the last century, the Finite-Element Method (FEM) is today state-of-the-art for the calculation of structural-dynamic, thermal and, of course, electromagnetic problems. With the improvements of the performance of personal computers and workstations the models have become three-dimensional with the number of elements increasing. The bandwidth of possible applications is advancing steadily and research projects are opening up new perspectives for the development of electrical machines. Static and transient Finite-Element calculations of the electric and magnetic field enable the designers to optimize well known electro-magnetic devices with regard to the torque-to-mass ratio and the dynamic, thermal and acoustic behaviour. Furthermore, the Finite-Element method approves, the development 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 flux machine as well as the optimisation of well known machines with new tools. Here, a claw-pole alternator and a permanent-magnet synchronous machine are acoustically and electrically simulated.

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Spherical motor The spherical motor is a multi-coordinate direct drive with three degrees of freedom [3]. The spherical rotor is able to rotate in three axes, a rotation ϕ around the vertical axis and a declination in γrot- and ϑ rot- direction (Fig. 1). Possible applications are machine tools and robotic devices, utilising the advantages of this high-dynamic direct drive which contains no mechanical transmission elements like gears. The motor consists of a permanent-magnet rotor sphere and a stator hemisphere with a large number of stator poles. The guiding of the rotor is realised by a hydrostatic bearing to achieve high stiffness and low friction. The stator hemisphere and the stator poles are made of a soft-magnetic composite to reduce eddy-current losses. The arrangement of the poles has a decisive influence on the torque characteristic. The current-dependent torques are calculated with a combined numerical/analytical method [4]. The static cogging torques have to be calculated with a FiniteElement-model of the complete motor geometry.

γrot ϑrot rotor sphere

ϕ

stator poles

permanent magnets

stator

hydrostatic bearing

Figure 1: Basic structure of the spherical motor

Combined numerical/analytical calculation method Looking at Fig. 1, one can imagine, that the motor geometry causes large Finite-Element models with high element numbers. As a result the meshing and computational time is very high. Therefore it is not reasonable to calculate the torques with a Finite-Element model of the whole geometry for different cases of current supply in order to optimize the stator pole arrangements. Otherwise the geometry is too complicated to calculate the torques in an analytical way. Therefore a combined numerical/analytical calculation method has been developed for the calculation of the total currentdependent 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 and appropriate 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 by a current injection to one pole in the model. The models consist of 7 up to 9 poles and differ in the positions of the neighbouring poles, which surround the pole carrying the current. Fig. 2 shows the model in case that this pole is located at the border of the stator sphere. EPE-PEMC 2002 Dubrovnik & Cavtat

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pole (carrying the current) stator pole

stator yoke

permanent magnet

ϑrel

ϕrel

rotor yoke

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. 3 shows the calculated ϕ- and ϑ-components of the thrust force of the spherical motor. They result from the difference of a calculation at a current of 4 A and a calculation at 0 A, so they do not include the cogging 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 this pole above the permanent magnets in the rotor and not on the positions of the neighbouring poles. The total torques of the spherical motor are calculated by multiplying the thrust-force contributions of each pole with the corresponding distance between the pole and the pivot axle. Therefore, the thrust-force characteristics, which were calculated with the FE-models are approximated with trend functions depending 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 current of 3 A. Using this calculation method, various stator-pole arrangements were investigated concerning the achievable torques.

80 Nm 70 Nm 60 Nm 50 Nm 40 Nm

Figure 4: Total torque Tϕ around the normal axis EPE-PEMC 2002 Dubrovnik & Cavtat

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

Figure 5: Finite-Element model and flux distribution

Transverse flux machine The transverse flux machine is a direct drive with a high torque-to-volume ratio. Different topologies of transverse flux machines have been developed at the Department of Ele ctrical Machines in Aachen [6,7,8]. The calculation of additional eddy-current losses is very important for the prediction of the machine 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 machine The geometry of the magnetic circuit and the complete machine layout are presented in Fig. 6. For a better view the left figure shows only one pole of one phase in a linear arrangement. winding yoke windings yokes

air-gaps

flux concentrator permanent magnets Figure 6: Geometry of the magnetic circuit and complete machine layout

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The armature winding in the inner stator is surrounded by U-shaped soft-iron parts, which are arranged circumferentially in a distance of a double pole pitch. The limbs of the U-yoke are shifted by an electrical angle of 180° against each other. The U-yoke lamination is stacked in circumferential direction. The geometrical shift is achieved by bending the complete stack before bonding. In order to get the utilisation of the magnetic circuit as high as possible, the cross sections of the U-cores get bigger at the air gap by using pole shoes. In the external rotor a flux concentrating design with softmagnetic pieces between the rare earth permanent magnets, which are magnetised with alternating polarity in circumferential direction, is applied. The magnetic flux is of three dimensional nature in the flux 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 no common rotating field in the three-phase design of a transverse flux machine, but only three independent alternating fields which are electrically shifted by 120°. The necessary mechanical shift is done in the rotor by shifting the complete rotor rings, consisting of the magnets and the powder-iron parts, 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-magnetic material to prevent high stray fluxes from one soft-magnetic piece to the other. The electric conductivity of this material is high for using the ring as a damper to displace magnetic flux from the carrier adjacent to the ring. The carrier itself is also made of a non-magnetic material with a good electric conductivity.

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

yoke

non-magnetic ring soft-magnetic pieces and permanent magnets

Figure 7: FE mesh of the transverse flux machine

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

δ=

1 . π⋅ f ⋅ σ⋅µ

(1)

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

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

Calculation of the eddy currents The calculation of the eddy currents is based on a 3D Finite-Element method with a time stepping procedure. The potential formulation is using two vector potentials for the magnetic and electric field, r the magnetic vector potential A in the regions without eddy currents and both the magnetic and r electric vector potential T in the eddy-current regions. With this approach the investigation of the influence of the sinusoidal stator current and partic ularly the movement of the permanent magnet excited rotor on the eddy current losses is possible. Fig. 8 exemplifies the eddy current distribution in the 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 suitable materials, especially for the passive rotor parts, e.g. carriers and fixations. The convergence of the transient calculation is strongly influenced by the resistivity of the eddy-current regions with the presented approach. A steady state solution is reached after only a few periods because the transverse flux machine is not a rotating-field but an alternating-field machine with a simple operation principle. EPE-PEMC 2002 Dubrovnik & Cavtat

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

Calculation procedure The calculation procedure has been presented in [9] for star-connected alternators and in [10] for deltaconnection. This procedure can be split into three blocks: the magneto-static computations, leading to the magnetic forces on the stator metal, the structural-dynamic calculation of the relevant harmonics and 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 various speeds are executed on models of one pole pitch as shown in Fig. 9. Since the modelling of the winding head for machines with the number of stator slots per phase winding q > 1 is very complicated, the model is simplified in these regions. All other magnetically relevant regions are modelled precisely. In each of the five models the rotor is rotated by an angle of:

∆α = 2° ⋅ m

with

m ∈ {0;1; 2;3; 4} ,

(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 driven by a constant direct current in the rotor-excitation coil. A three-phase current is impressed into the stator coils. The amplitude of the current and the load angle are functions of the generator speed. In order to take saturation effects into account, they are determined in a model with q = 1 and compared to measurements [12]. The magnetic forces are calculated in each static FE calculation based on the flux-density distributions and the material data. Interpolation of different stator tooth positions leads to 30 time steps or 14 spectral modes for two stator-tooth pitches.

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2.) Structural-dynamic computation The forces in spectral mode are transformed into a mechanical FE-model [13]. Since all mechanically relevant components have to be modelled, there is no symmetry and a full 360° model as shown in Fig. 10 has to be generated. This model is used to determine the deformation and oscillation of the relevant harmonics, based on the material data. Here, transversely isotropic materials are used to represent the stator metal sheets. In order to reduce modelling and calculation expenses and since the rotor contributes barely to the acoustic outcome, the rotor region is modelled as a solid cylinder with the 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. 11 shows the deformation of the claw pole alternator. In the case of the claw-pole alternator the relevant harmonics are acoustic orders 5 and 6 or mechanical orders 30 and 36. Since these two orders lead almost to the complete noise output, all other orders are neglected.

Figure 11: Scaled deformation of the claw-pole alternator

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3.) Acoustic simulation In 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 are meshed and the stator and rotor region is merged in order to reduce numerical errors in the acoustic BE method caused by small air gaps. Again a 360° model is used. Onto this model, the surface velocities of the structural-dynamic calculations are interpolated. These velocities are used to drive the acoustic BE solver. On a half spherical boundary area the emitted sound distribution and the total sound power are evaluated. A sound-reflection plane represents the lower half space. Repeating this calculation chain for various generator speeds and the relevant harmonics leads to the sound-power characteristic of the machine.

Simulation of a PMSM with SIMPLORER-FLUX2D-Coupling The coupling of a simulation tool like SIMPLORER with FEM calculations allows to simulate the behaviour of more complex geometries like that of a conventional electric machine. The simulation parameters depending on the geometry of the machine can be adjusted. As an example for a coupling of SIMPLORER and FLUX2D a start-up of an electronic commutated permanent-magnet synchronous machine (PMSM) has been simulated at the Institute for Electrical Machines [14]. The complete controlling and the differential-equation system of the machine are implemented in SIMPLORER. The machine is operated in field-orientated coordinates. The three-phase currents and voltages of the machine 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 to this the estimation of the torque is replaced by a 2D Finite-Element calculation with FLUX2D. In order to bring the two simulations into agreement, the inductance of the machine has to be recalculated. For this the phasor diagram is approximatively calculated. Finally the simulations are repeated. In the case of a PMSM as an injection-pump drive a minimal start-up-time of 100 ms from 0 to 3500 rpm is required. The machine shown as a FE-model in Fig. 12 has an outer diameter of the stator of 120 mm. The rotor diameter is 34.5 mm. The length of the machine is 60 mm. As permanent magnet material ferrit is used with a remanence of 0.35 T. The stator has a copper winding with 24 slots and the 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 stack factor is kCu = 0.35.

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

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Coupling of SIMPLORER and FLUX2D The idea behind the coupling of the SIMPLORER-simulation with the FEM program FLUX2D is to use the exactness of the FEM calculation to determine the torque of the machine at every time step considering the geometry of the machine. A model of the machine is built with FLUX2D. This is coupled by an electrical circuit in FLUX2D to the coupling module in SIMPLORER. The coupling module is added to the simulation sheet of the machine as shown in Fig. 13. The SIMPLORER simulation 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 the speed, 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 any longer smooth but undulating as shown in Fig. 14, an effect depending on the stator slots. The reluctivity of the motor depends on the position of the rotor. If the reluctivity is lower the torque is higher than at the point of higher reluctivity.

Figure 14: Calculated torque of the coupled simulation

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Using the coupling of SIMPLORER and FLUX2D or similar products it is possible to calculate dynamic processes of complex geometries. This is of advantage if it is not easily possible to derive an analytical equation for the torque of an electrical machine. If the coupling is used, the calculation time increases dramatically. But it is a possibility to verify the results of a simulation done without FEM and adjust it. In a next step the FEM calculation with FLUX2D is coupled to the simulation with SIMPLORER in order to substitute the equation for the estimation of the torque by the output of FLUX2D. The results are compared and because of the great difference the inductance of the machine is recalculated using the coupling. The SIMPLORER parameters are adjusted and the simulations with and 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 the simulation without coupling.

Conclusion This paper presents four different examples of the way Finite-Element models are being used at the Institute for Electrical Machines at Aachen Institute of Technology. The bandwidth of the applications covers static torque and force calculations for new types of electrical machines, transient calculation of eddy current losses, a procedure to determine the mechanical and acoustic behaviour of electrical machines 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 less extensive. Regarding the ongoing improvements of the performance of personal computers and workstations these examples are showing, that the complexity of the applications of Finite-Element models will increase further on.

References [1] [2]

[3] [4] [5] [6] [7]

[8] [9]

[10]

[11]

[12] [13]

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[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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