SPEEDAM 2006 International Symposium on Power Electronics, Electrical Drives, Automation and Motion

1-4244-0194-1/06/$20.00 2006 IEEE

Abstract--This paper presents the analysis of a Permanent Magnet Induction Machine (PMIM) with the help of the Finite Element Method (FEM).

A PMIM consists of two rotors, one with permanent magnets freely rotating in the air gap and one cage rotor. This arrangement allows to built a direct drive wind power generator with characteristics of an induction machine.

To calculate this type of machine transient calculations are necessary. Due to computation time a 2-D model was chosen. Further on the moving mesh and eddy current method was used. As the wind power generator is connected directly to the grid, a circuit model which is coupled with the FEM model is included.

The aim of this work was so far not to optimize the machine but to develop a tool to calculate it. First results show a good compliance with measured results.

Index Terms--Finite element methods, Induction machines, Permanent magnet machines, Wind power generation

I. INTRODUCTIONAn as yet little-considered electrical machine has the

potential to the optimal generator for wind turbines: the permanent magnet-excited induction machine (PMIM). A freely suspended intermediate permanent magnet rotor rotates between stator and rotor cage (Fig. 1) and supports the excitation of the machine. This allows asynchronous function despite having a large air gap and small pole pitch. The principle combines the advantages of the induction machine (little maintenance, easy grid connection and stable operation at the supply network) with the advantages of the permanent magnet-excited synchronous machine (high torque at small pole pitch, good efficiency). When utilizing the PMIM as a generator for wind-turbines the generator can be directly coupled to the turbine and the grid. Thus the gearbox and the frequency converter, which contribute significantly to wind turbine losses, can be abandoned.

Fig. 1. Idealized cross-section of the PMIM.

The idea of an internally excited induction machine - in those days realized with electromagnets - was first discussed by Punga and Schn [1] in 1926 for application as a one phase locomotive drive. Supply of power to the electromagnets via slip-rings, however, reduces the reliability of the system. A logical alternative is to replace the electromagnets by permanent magnets, which was done by Douglas [2] in 1959 and Sedivy [3] in 1967. The main objective of those projects was to improve the power factor of a conventional induction machine. Due to the fact that the energy product of available permanent magnets was relatively low, the machine was described as rather impractical.

Some years later in 1992, further investigations of the PMIM - now using rare earth magnets - were carried out by Low and Schofield [4]. The chosen PMIM configura-tion was different in that an internal magnet rotor and a squirrel-cage ring were utilized. Despite a rather good performance of the small prototype, the project was stopped because of financial considerations.

The latest investigations have been carried out at Darmstadt University of Technology by Hagenkort [5], Gail [6] and Troester [7]. There the aim was not only to improve the power factor, but to implement a directly driven induction machine with a large diameter for wind power applications.

Continuing with this work, this paper describes the finite element analysis of a PMIM. The aim is to show, how the PMIM can be modeled and how the characteristics (e.g. torque, efficiency, power factor) depend on the magnet height of the machine.

II. MODELLING A PMIM USING FINITE ELEMENTSAs the PMIM has a squirrel cage and permanent

magnets, the calculation method has to be examined closely. Some FEM-Programs are able to calculate PM-machines or induction machines but not both at a time. Those programs can do steady-state or harmonic calculations. To calculate the PMIM transient calculations are necessary. Thus a program had to be taken, which is able to do transient calculations, which lead to the general purpose software ANSYS.

Further decisions about the calculation method had to be made. According [8], FEM-calculations of induction machines have been applied basically with three different classifications:

2-D or 3-D Fixed mesh or moving mesh Eddy current model or circuit model

Finite Element Analysis of a Permanent Magnet Induction Machine

E. Trster, M. Sperling, Th. Hartkopf Darmstadt University of Technology, Institute of Electrical Power Systems, Field of Renewable Energies

Landgraf-Georg Strae 4, D-64289 Darmstadt, Germany

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A. 2-D or 3-D Model Three-dimensional calculations consume a

considerable calculation effort of the computers. So for field solution two-dimensional models are still preferred, although some attributes of three-dimensional nature (e.g. rotor end-ring, stator overhang windings, rotor skew, radial ventilation ducts, inter-bar currents) have to be neglected.

B. Fixed Mesh or Moving Mesh Model The fixed mesh model is characterized by a rotor mesh

lying stationary in relation to the stator mesh. The stator winding is fed with a current with slip frequency. This method has the advantage of being less time-consuming but has also some disadvantages. These are mainly that variable speed and effects of variable air-gap permeability, caused by the passing of the stator and rotor teeth by each other, can not be considered. Further on the non-linear iron saturation effect is reduced with the assumption of a time-averaged reluctance, instead of instantaneous reluctances.

The main disadvantage of a fixed mesh model is, that it only works with pure induction machines without permanent magnet, because the moving field of the magnets can not be taken into account. Thus the moving mesh method had to be taken.

In the moving mesh model a new position of the rotor is calculated in each time-step. By this method the disadvantages of the fixed mesh model can be overwhelmed, unfortunately the time required for simulations is much higher. A modern computer takes a couple of hours to calculate a solution with an adequate meshing and high number of time steps using two-dimensional elements, while for fixed mesh calculation with similar conditions this would take a couple of minutes.

This large time needed is in part due to the fact that rotor bar currents are not known. Therefore they start from zero, causing a long transient that must be calculated. To overcome this problem, the values of the previous load step are taken as start values for the next load step which brings some improvement in calculation time.

C. Eddy Current or Circuit Model The third classification is how the results are

evaluated. In the eddy current model everything is calculated in the field solution. Currents flow naturally in the conductive materials according the variation of the magnetic field and the electrical resistivity of the conductor. Skin effect is automatically included, as well as leakage inductance effects. In the circuit model method the currents are calculated outside the field solution in an equivalent circuit diagram. The field solution is then used to estimate circuit parameters (specially the coupling impedances), checking compatibility with the results. One advantage of this method is that harmonic currents can be treated separately, specially at machine starting when the rotor may run in the synchronous area of some harmonics. Another advantage is that skewed slots can be represented better by using the skew factor for harmonics.

To handle skewed machines with the eddy-current model, a multi-sliced model would be necessary, what means making parallel field solutions for many sections of the rotor taken in different axial positions. Nevertheless the advantages of the eddy-current model seem to overwhelm the disadvantages leading to the choice of this method.

III. THE FEM-MODELThe basic parameters of the modeled generator are:

TABLE IBASIC MACHINE PARAMETERS

Parameter Symbol Value

Diameter D 5 m Air gap 2x 5 mm Pole pairs p 150

Phases m 3Slots per pole and phase q 1

Stator voltage Vs 690 V Frequency f 50 Hz

By taking advantage of the symmetry, the number of elements and thus computation time can be reduced dramatically, thus only one pole of the machine was modeled as shown in figure 2.

Fig. 2. Meshed model with circuit elements

The generator is connected directly to the grid. This is taken into account by a circuit connected to the FE model (Fig. 2). Besides the voltage sources representing the grid, leakage inductances representing the overhang winding are included. The elements of the FE model are connected to the circuit via stranded coil elements on the right.

After each time step, the connections between the stator, intermediate rotor and the cage rotor had to be renewed. Figure 3 shows the constraint equations between the stator and the intermediate rotor. Vertical

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lines symbolize even connections, diagonal lines odd connections.

Fig. 3. Constraint equations in the air gap

Figure 4 shows the results for the torque derived from the transient calculations. The aim is to find the steady-state operation point of one load step. At steady-state operation the torque within the intermediate permanent magnet rotor is equal to zero. To achieve this a PI controller was programmed, which starts to change the load angle at 0.4 s. At 0.8 s the steady-state operation is reached. The final results have been calculated by taking a mean value of the last 400 ms.

Fig. 4. Transient progress of the torque

In the following table some characteristics of the model are presented.

TABLE IIMODEL PROPERTIES

Number of elements 787

Typical time step 0.5 ms

Type of processor used AMD Athlon 64 X2 Dual-Core 4400+ Computation time per load step approx. 40 min

The accuracy of the calculated results depends very much on the time step and the number of periods calculated to find the steady-state operation. The following figure shows PM rotor torque, power and load angle in dependence on the number of steps per period. A reasonable number seems to be 40, which is equivalent to a time step of 0.5 ms. A higher number would bring more accuracy but also increases the computation time.

Fig. 5. Accuracy in dependence on the number of steps per period

An interesting phenomenon concerning the accuracy can be found when plotting the current locus. In the area of pure active current (power factor = 1) the results are obviously wrong as shown in the following figure. This effect can not be recognized in the torque characteristic but only in the current locus. A possibility to overcome this problem is by reducing the time step and increasing the number of periods calculated.

Fig. 6 Current locus with calculation fault

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IV. RESULTSTo get an idea, how the PMIM works, the flux

distribution at no load (Fig. 7) and at a slip of 3% (Fig 8) is presented.

Fig. 7 Flux lines at no load

Fig. 8 Flux lines for a slip of 3%

In figure 9 the effect of the additional magnet ring is shown. Calculations have been carried out for the arrangement with and without magnets. Obviously already a magnet height of only 5 mm per magnet has a great influence on the achievable torque, the efficiency and the power factor.

Fig. 9. Comparison of the arrangement with (PMIM) and without (IM) permanent magnets

Figure 9 shows very clearly, that an induction machine without permanent magnets can not be used as a direct drive wind power generator. Torque, efficiency and power factor would be far too poor.

The following figure shows the breakdown power in dependence of the magnet height.

Fig. 10 Breakdown power in dependence on magnet height

By increasing the magnet height, the output power can also be increased to a certain extend. Above a certain value a saturation effect happens. This is very well known from PM synchronous machines. A magnet height of 15 to 20 mm seems to be reasonable. With a breakdown reserve of 60 % this would correspond a rated power of approx. 1.3 MW.

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Figure 11 shows the efficiency and the power factor versus the output power from no load to breakdown in dependence of the magnet height.

Fig. 11. Efficiency and power factor vs. power for a stator voltage of 690 V and different magnet heights (hm)

The diagram shows, that there is an optimum concerning efficiency and power factor. In this special case it is at a magnet height of 7 mm. This is true for a certain winding arrangement and stator voltage. By adopting one of those parameters, the optimum can be shifted to other magnet heights as shown in the following diagram.

Fig. 12. Efficiency and power factor vs. power for a stator voltage of 800 V and different magnet heights (hm)

The efficiency is rather high as only ohmic losses are considered. Further on the machine is not optimized and not very well utilized, as the power output is rather low for such a big machine.

Measured results of a 3 kW PMIM [6] are presented in figure 13. With increasing stator voltage the relation between stator voltage an internal voltage is reduced and thus the machine behaves more like a normal induction machine. In figure 14 FEM calculations of a wind power generator are presented, the stator voltage is increased and a similar behaviour to the measured results can be recognized. The current loci are plotted from no load up to breakdown slip.

Fig. 13 Current loci for different stator voltages (VS)Measurement results of a 3 kW PMIM [6]

Fig. 14 Current loci for different stator voltages (VS)FEM calculations of a wind power generator

V. CONCLUSIONSA model for calculating the permanent magnet

induction machine using finite elements has been developed. A survey showed, that transient calculations are necessary. Due to computation time a 2-D model was chosen. Further on the moving mesh and eddy current method was used.

A model of a direct drive wind power generator was built. As the transient calculation consumes a lot of

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calculation time, it was favorable that symmetry properties could be used and the geometry of the modeled pole was not complicated. By this computation time could be kept low. Other more complicated models showed to be much more time consuming. To reduce the computation time, the results of the preceding load step are used as start values for the next load step leading to a slight reduction in computation time. Further strategies to reduce computation time have to be developed.

Due to the transient calculation it is of course possible to analyse the transient behaviour of the machine. In this work, the primary aim was to find the steady-state operation and by this to calculate the torque characteristic and other important characteristics like efficiency and power factor. First results of a wind power generator are presented. They show already a small magnet can improve the achievable torque, the efficiency and the power factor in comparison to an induction machine of the same geometry.

The efficiency characteristic of the PMIM is favorable for wind power generators, as the efficiency is higher at partial load in comparison to rated load. At partial load the most wind energy is yield.

Comparisons to measurements [6] show the same behaviour of the calculated and the measured results.

Further calculations and optimisations have to show whether a competitive PMIM wind power generator can be built.

REFERENCES

[1] Punga, F.; Schn, L.: Der neue kollektorlose Einphasenmotor der Firma Krupp, Elektrotechnische Zeitschrift (1926), part I in Heft 29, pp 842-848, part II in Heft 30, pp 877-881.

[2] Douglas, J.F.H: Characteristics of Induction Motors With Permanent-Magnet Excitation, Trans AIEE (PAS), Vol 78, June 1959, pp 221-225.

[3] Sedivy, J.K.: Induction Motor with Free-Rotating DC Excitation, Trans AIEE (PAS), Vol 86, No 4 , 1967, pp 463-469.

[4] Low, W. F.; Schofield, N.: Design of a Permanent Magnet Excited Induction Generator, Proc. ICEM 1992 Manchester University, Vol. 3, pp 1077-1081.

[5] Hagenkort, B.; Hartkopf, T.; Binder, A.; Jckel, S.: Modelling a Direct Drive Permanent Magnet Induction Machine, Proc. ICEM 2000, Helsinki University of Technology, Vol. 3, pp 1495-1499.

[6] Gail, G.; Hartkopf, T.; Trster, E.; Hffling, M.: Static and Dynamic Measurements of a Permanent Magnet Induction Generator: Test Results of a New Wind Generator Concept, ICEM 2004, Cracow.

[7] Trster, E.; Gail, G.; Hartkopf, T.: Analysis of the Equivalent Circuit Diagram of a Permanent Magnet Induction Machine, ICEM 2004, Cracow.

[8] Williamson, S.: Induction Motor Modelling Using Finite Elements, ICEM 1994, Paris

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