Automated Finite Element Aided Design of Skewed Rotor Induction Motors

Chris Riley, Alex Michaelides and Klaus Höffer Vector Fields Ltd, Kidlington, OX5 1JE

David Griffiths1 and Jan K. Sykulski2 1FR-HiTemp Ltd, Fareham, PO14 4QA 2Electrical Power Engineering University of Southampton, SO17 1BJ

Abstract This paper describes the way in which the commercial 2d electromagnetic Finite Element Analysis (FEA) package OPERA-2d has been used, in conjunction with the MatLab programming language and Microsoft Excel, to perform automated simulation and performance analysis of induction motors. The objective of this work is to develop a practical tool for design purposes. In this application full 3d transient solutions are not viable due to their extended solution times. It is a requirement however that significant 3d effects, in particular those due to rotor bar skew, are taken into account. OPERA-3d results are used to validate the multi-slice modelling procedure. 1. Introduction The computational effort required when carrying out full 3d transient solutions is known to be high and often of a level that is unacceptable within a design office environment. Alternatively 2d analysis can provide solutions within a reasonable time frame, the drawback to this approach being the inability to model significant 3d effects, such as skew. Techniques developed to represent skew in the 2d environment have been reported on since the early 1990’s [Ho et al. 1997], [Tenhunen et al. 2001], [Williamson et al. 1995]. These have varied in approach, often being driven by the level of available processing capability. It is evident from this work that the most popular method exploits a multi slice technique using a number of 2d slices in the Z (axial) direction. With the expanding capability of software and the increased processing power of PCs, the opportunity to harness FEA as part of the design process is realistic. In addition, modelling techniques such as multiple slices in the 2d environment, previously requiring bespoke routines, can also be accommodated by the commercially available software OPERA-2d. However, the use of FEA often remains the province of a specialist. Available commercial packages are often considered more an analysis tool rather than a design tool. The use of such software requires users to have a background in FEA as well as considerable machine design capability. This paper reports on one such development where a commercial 2d FEA package OPERA [Vector Fields. 2002] has been integrated, without any modifications, into a design system. The resulting package being so configured that it introduces the capability to model skew and removes, through the use of parametric input data together with charting and numerical output facilities, the requirement for the operator to have any FEA capability. 2. Multi Slice 2d Finite Element Model Model generation is achieved through multiple calls to OPERA-2d to create an op2 data file of the machine geometry and magnetic properties. The stator is generated first with the required number of slices, followed by the rotor, thus creating a model having sequentially numbered regions. The automated program takes its input from the basic physical geometry of the motor, such as the number and dimensions of slots and poles. Due to the combination of the present motor’s stator and rotor slots, the periodic conditions facility available within OPERA-2d is not exploited.

The construction of the model is based on a single half slot. This is an approach that aids the achievement of a balanced mesh. Whilst automated modelling can easily suffer from poor mesh generation, arising from a conflict in the geometry and the automated assignment of elements, the program overcomes such associated difficulties through matching the size of the model to the number of elements required. This modelling technique is particularly important in the most sensitive regions e.g. tooth tips. Figure 1a shows a segment of regions used in the model construction for a typical machine. Within OPERA-2d, the use of the automated facility ‘RMGap’ to create an airgap mesh at any rotor position, is precluded from application with multi slice models. A fully modelled air gap is therefore required. With the air gap between rotor and stator fully defined by the user it must also be sufficiently refined to permit repositioning of the rotor as demanded by the rotor skew. Ideally the air gap is most suitably modelled with an odd number of regions 3, 5 etc. In considering this, three layers were considered insufficient to achieve a smooth mesh in the radial direction, whilst five layers increased the number of regions used in the model, a condition preferably avoided as OPERA has a finite limit on the permitted number of regions that can be used in any one model. A balance was achieved by selecting four regions in the air gap (see Figure 1b).

(a)

(b)

Figure 1. Geometry of the analysed motor - (a) Overall (b) Air gap detail

3. Multi Slice 2d Solver The steady state AC solver was used for the analysis of a multi slice induction motor model operating at steady state. Minimising computational effort is an important feature and thus the minimum number of linear elements required to achieve a high accuracy solution was employed. Typically a model is set up with voltage driven coils and utilises the “External Circuit” option available in OPERA to link each slice of the model. Lumped resistance and inductance values (external to the finite element model) are included in the circuit definition to account for elements not catered for in the two dimensional model (e.g. end winding effects). The Motor Simulation Program (MSP) provides fully automated control of the pre- and post-processing of the Finite Element (FE) model. It calculates the performance and provides results in a clear manner unencumbered by the trappings of a typical FE solution. The program operation is shown simplistically in Figure 2, and a more detailed chart outlining its operation is given in Figure 3. With the FE solver invoked from within the main MatLab controlling routine, the model is analysed for multiple rotor positions, as derived from the slot ratio, over one rotor tooth pitch. For

each position of the rotor the OPERA-2d post-processor provides results for stator peak currents per phase, rotor power loss, force, torque and rotor conductor peak current.

Figure 2. Architecture of the Motor Simulation Program (MSP)

Figure 3. Essentials of MSP operation

Performance Results

Post ProcessFE Results

FE Solver

FE Model Construction

ParametricInput Data

Programs & Libraries

Analytical & Control Software MatLab

OPERA

MS Excel

START

Loop Variable Slip

END

Stator Design

Rotor Design

Process Results and Output the MS Excel

Loop Variable Slices

Input Data

OPERA FEA

OPERA AC Solver

OPERA Output Results

Increment

Slip

Increment

Slice

With the completion of the analysis, control is returned to the core MatLab function where the output data files created by OPERA-2d are processed using MatLab. Field plots can be viewed on the screen as the study takes place, or subsequent to the run, but are not necessary for an efficient execution of the analysis. The results produced by OPERA-2d are averaged according to the number of slices in the model and number of rotational steps. The overall result is combined with calculated losses for drag and exported to an MS Excel spread sheet. Results can then be processed using the full range of charting functions available within MS Excel. 4. Three Dimensional Finite Element Analysis In order to validate the trends obtained in multi slice 2d modelling, a three dimensional analysis of the induction motor has been undertaken. Two model configurations are being examined, namely the ‘straight bar’ (no skew) and ‘skewed rotor’ configurations. The three dimensional models were built using the OPERA-3d Solid Modeller. The OPERA-2d model was imported into the Solid Modeller, with the regions automatically translated into volumes but retaining their individual material identity (label). Material labels included the stator iron, rotor iron and copper cage. Electromagnetic material characteristics (BH curves and conductivity) were subsequently assigned to each label. The stator and rotor slot combination (29 stator & 32 rotor slots) meant that the symmetry boundary conditions available in OPERA-3d ELEKTRA-SS could not be exploited. Therefore the full 360 degree geometry needed to be defined. In the ‘straight bar’ rotor configuration, the imported OPERA-2d rotor surfaces were simply extruded into the third (Z) dimension. The OPERA-3d formulation allows for the stator coils not to form part of the mesh. The double layer distributed winding was defined by defining the two dimensional plan view of the coil and the angular span between slots, for the software to automatically generate the three dimensional coil structure. The OPERA-3d model of the ‘straight rotor’ machine is shown in Figure 4.

Figure 4. OPERA-3d model of the ‘straight bar’ IM. In the ‘skewed rotor’ configuration, the extruded rotor volume was skewed by one rotor slot pitch along its length, by selecting the volume and applying a twist of 360/32 = 11.25 degrees. The full

axial length of the machine was modelled due to the variation of the fields at the two ends of the machine. This resulted in a large FEA model, with a total number of elements reaching1,800,000. The skewed rotor geometry is shown in Figure 5.

Figure 5. Flux density in the skewed rotor IM defined in OPERA-3d. In summary, the main features of the model included:

• Skewed rotor geometry and parametric non-meshed conductor geometries • Non-linear magnetic properties of materials • Assignment of rotor conductivity for computation of eddy currents in the rotor cage.

Each operating point was successfully solved using the ELEKTRA Steady State AC solver on a 2.4MHz machine with 1.5Gb RAM in 23 hrs. 6. Multi Slice 2d Results Figure 6 shows for a typical model how torque varies with rotor angular position. This highlights the need for averaging torque results over the rotor tooth pitch and illustrates the importance of selecting a suitable elemental step of the rotor in order to avoid any detrimental impact on results.

Figure 6. Torque variation as a function of rotor position

Torque

0.7800

0.7810

0.7820

0.7830

0.7840

0.7850

0.7860

0.7870

0.7880

0.7890

0 10 20

Mechanical Position (Degrees)

Tor

que

(Nm

)

A family of performance curves for the same machine indicates that as the number of slices increases, the variation between the results produced decreases (see Figure 7).

Figure 7. Comparison of torque - speed characteristics for 2 – 5 slice 2d models This is demonstrated here with the estimation of torque. The greatest difference exists between a 2 slice and 3 slice model. Variation between 3 and 4 slices is significantly less whilst variation between 4 and 5 slices is reduced further. It is doubtful, due to the marginal difference in results obtained for 4 and 5 slices, if performing an analysis with 5 slices is computationally economic. Discrepancies between test and simulated results can be attributed to temperature and end winding effects, which can have an impact on the impedance of the stator and rotor electrical circuits. In addition, a deviation in magnetic properties of the stator and rotor laminations and manufacturer supplied material data can also have an impact on the torque predicted by the program. The MSP software was installed and run on a 2.4 GHz machine with 1GB of RAM. Preparation, solution and post processing times vary with the model complexity and do so approximately linearly with the number of slices modelled. Those obtained for the test case are given in Table 1.

Model Type Nodes Elements Solution Time per Step Rotation

Solution Time per Performance Point (18 steps)

No. Off No. Off h:m:s h:m:s

2 Slice 35592 70964 0:11:47 3:32:06

3 Slice 53388 106446 0:17:36 5:16:48

4 Slice 71184 141928 0:24:00 7:12:00

5 Slice 88980 177410 0:28:24 8:31:12

5 Slice Zero Skew 88980 177410 0:21:50 6:33:00

Table 1. Preparation, solution and post processing times 7. 3d Results The 400 Hz 6 pole motor has a synchronous speed of 8000 rpm. The machine was modelled at 2 operating points, namely at 7600 and 7800 rpm. Figure 8 shows the rotor cage current distribution

T o r q u e V s S p e e d

0

0 .5

1

1 .5

2

2 .5

6 5 0 0 7 0 0 0 7 5 0 0 8 0 0 0

S p e e d (r p m )

Tor

que

(Nm

) T e s t

5 S lic e

4 S lic e

3 S lic e

2 S lic e

M o d e l T y p e

in the skewed machine at 7800 rpm. Current driven circuits were used in order to assess the torque production capability of the straight and skewed rotor machines at equal current. In addition, at the 7600 and 7800 rpm operating points, the voltage driven OPERA-2d model results were compared with the corresponding OPERA-3d result (using the currents predicted in OPERA-2d).

Figure 8. Rotor cage current distribution in the skewed machine at 7800 rpm.

Figures 9(a,b) show the results from OPERA-2d voltage driven analysis on the ‘straight-bar’ and ‘skewed’ rotor alongside the corresponding results from OPERA-3d. Two and three-dimensional results are in good agreement and demonstrate the high quality results obtained from the multi-slice method implemented in OPERA-2d to account for skewing. The skewed configuration consistently produces marginally lower torque.

OPERA-3d simulations of the straight bar and skewed geometries at equal current confirmed that skewing, employed to smoothen the torque profile in a machine, comes at the expense of reduced torque production capability.

Torque comparison (OPERA 2d)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

7600 7800 8000Speed (rpm)

To

rqu

e (N

m)

2d Straight

2d Skew (5 slices)

Figure 9a. Comparison of torque figures for the skewed and straight rotor machines in OPERA-2d.

Torque Comparison (OPERA-3d)

00.20.40.60.8

11.21.41.61.8

7600 7800 8000Speed (rpm)

To

rqu

e (N

m)

3d Straight

3d Skew

Figure 9b. Comparison of torque figures for the skewed and straight rotor machines in OPERA-3d. The time required to solve a single step rotation in OPERA-2d is 28mins:24sec (177,410 elements) for the 5-slice model, compared to 23hrs:25mins for a single OPERA-3d solution (1,800,000 elements). This demonstrates the advantage of the method proposed in the paper. It must be mentioned however, that additional OPERA-3d cases at different frequencies or excitations can be solved in approximately 13hrs per case, by employing a coil field scaling technique. Further work is currently underway using voltage driven coils in OPERA-3d. This would help to validate trends observed in OPERA-2d relating to the change in impedance ‘seen’ by the stator phases when a skew is applied to the rotor. The effect of mesh size on the results also needs to be quantified. Conclusions The paper describes an induction motor FEA based design tool that is suitable for design office use and offers improved accuracy over equivalent empirical design programs. The program can provide solutions within acceptable timescales and removes the need for expertise in FEA. The approach makes use of the commercial software package OPERA-2d to achieve this and adopts a multiple slice technique to model skew in the 3-phase induction motors, without recourse to bespoke FEA software. The results demonstrate that the use of a 4-slice model achieves a good balance between accuracy in the modelling of skew and speed of solution. Acknowledgement The work is supported by FR-HiTEMP Limited, Titcfield, UK. References [1] S.L. Ho and W. N. Fu . A Comprehensive Approach to the Solution of Direct –Coupled Multislice Model of Skewed Rotor Induction Motors Using Time-Stepping Eddy-Current Finite Element Method. IEEE Transactions on Magnetics Vol. 33, No 3 May 1997, p 2265 - 2273. [2] A Tenhunen and A Arkkio Modelling of induction machines with skewed rotor slots. IEE Procedures Electrical Power Applications Vol. 148. No 1, January 2001. [3] PC-OPERA reference manual, Vector Fields Ltd, Oxford, UK. 2002. [4] Stephen Williamson, Timothy J Flack, Albertus F Volschenk Representation of skew in time-stepped two-dimensional finite-element models of electrical machines. IEEE Transactions Industrial Applications Vol 31, No 5, September 1995, p1009 - 1015.

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