This application note describes the different methods that can be employed for modelling an Induction Machine using the Opera-2d and Opera-3d FEA software. Both the motor and generator operating modes are considered. The torque characteristic is obtained using full transient with motion analyses. The transient analysis also offers the information needed to further calculate the currents and losses vs. slip characteristics in the rotor bars.
Finally, the same machine is analysed using the Opera-3d package and the process of solving a slice of the full model, which significantly reduces the solving process, is presented.
- Induction machine – Modelling considerations
- Characterization using Opera-2d
- Characterization using Opera-3d
- Fast machine characterization using the dynamic solver (Opera-3d Dynamic)
- Reducing the problem size by using a 2d Slice
The Induction Machine (IM) is the most widely used type of electrical machine, by some estimates  accounting for 90% of all small size motors (10 – 750W) and 60 – 70% of medium size motors (0.75 – 375 kW). This is mainly due to their reliability, low cost and wide range of torque / speed characteristics. The IM is also used as a generator, especially for small and medium-size applications and where the requirement for the rotor to operate over a wide range of speeds is significant (e.g. wind turbines).
Although these machines have been in production and have been optimized for many decades, a stricter requirement for increased efficiency of electrical machines – both motors and generators – means that further significant improvements can only be achieved by an in-depth knowledge of the detailed electromagnetic effects in the machine at various operating points.
The Opera package includes a dedicated engineering environment for the modelling, analysis and characterization of Induction Machines. The Machines Environment allows the definition of the machine based on existing templates, offering a wide flexibility in terms of customization of models and analyses. The accuracy of Finite Element Modelling is coupled with a machine-specific post-processing stage in order to offer a quick and efficient design solution.
The induced rotor fields in an Induction Motor travel at a different frequency than the applied stator fields. This asynchronous effect produces torque and is behind the operating principle of the IM. The ratio between the stator and rotor frequencies is given by the slip, s, such that:
Similarly, in an Induction Generator, asynchronous fields are created in the rotor to give synchronous fields in the stator.
Because of this phenomenon, the behaviour of the Induction Machine cannot be captured by static simulations (or frozen flux paths). The different stator and rotor frequencies can be simulated either by including motion, and thus directly accounting for the effects of slip, or by assuming some simplifying hypothesis that allows modelling the dynamic effects at a single frequency for the entire machine. Being able to analyse the machine without the inclusion of motion helps reduce the simulation time, as is demonstrated later in this application note.
Characterization using Opera-2d
The model used in this example is a relatively small Induction Machine, with squirrel-cage rotor bars in the rotor and the overall dimensions presented in Table 1. Table 2 specifies the operational parameters of the IM.
Dimension Value Units Stator outer radius 125 [mm] Stator inner radius 70 [mm] Rotor outer radius 68 [mm] Shaft Radius 20 [mm] Stator tooth width 6.215 [mm] Stacklength 300 [mm] Number of stator slots 36 – Number of poles 4 – Number of rotor slots 28 – Number of stator phases 3 – Winding type Single layer –
Table 1. Geometric dimensions for the IM
Dimension Value Units Frequency 50 [Hz] Turns / coil 180 – Peak voltage 300 [v] Bar conductivity 3E+07 [S/m]
Table 2. Drive settings for the IM
The 2d model of the Induction Machine is presented in Figure 1. Opera-2d allows the minimum symmetry section of the machine (¼) to be modelled.
The machine is designed to be supplied from a 50Hz supply and has 4 poles. Hence, the synchronous speed of the machine – given by equation (3) – is 1500 rpm. As explained in the previous section, the speed at which the rotor is moving is dictated by the slip.
Full transient analysis with motion (Opera-2d RM) – Motor and Generator mode
Opera-2d RM is a transient eddy current solver, extended to include the effects of rigid body (rotating) motion. The solver also supports co-simulation with circuits and coupling to mechanical equations.
The analysis is run at a constant speed (calculated based on the value of the slip), for 4 cycles of the armature drive with a fixed time-step of 0.05 ms, hence 400 steps per cycle. The stator windings are driven using the electric circuit presented in Figure 2. The rotor bars (represented in the bottom loop) are short-circuited to allow the induced eddy currents to circulate. In order to limit the starting transients the three phase supply voltage is ramped from 0 to peak value over the first cycle. The next three cycles allow the currents and voltages to completely stabilize and the machine to be considered as operating at steady-state.
The transient analysis creates a text file containing the circuit values (currents and voltages in any component) at each time-step along with information related to the motion: rotor angle, speed and acceleration. The torque produced by the device is also logged.
In the following examples, the IM is analysed at various slip speeds, both as a motor and as a generator. The list of considered cases is presented in Table 3.
Slip Rotor speed [rpm] Mode 0.95 75 Motor 0.4 900 Motor 0 1500 – -0.25 1875 Generator -0.65 2475 Generator
Table 3: List of simulations
For slip values between 0 and 1 the IM acts as a motor – current is applied to the stator to cause the rotor to turn. At a slip value of 0 (rotor speed equal to the synchronous speed) the machine will not produce any useful torque. This is the point of crossing between the motor and generator modes. Once the machine works as a generator, power will start to flow back into the supply network and the torque will become negative reflecting the mechanical power needed to rotate the rotor above the synchronous speed.
The torque obtained for the 5 different slip values is presented in Figure 3. As can be seen, the average torque value varies as a function of the slip frequency at which the machine is operating.
At a slip value of 0 (1500 rpm), the rotor fields are rotating with the same frequency as the stator fields, which results in zero average torque. Decreasing the rotor speed to 900 rpm, the equivalent rotor slip is 0.4 and the average torque is 1.5 Nm. By further reducing the rotor speed to 75 rpm, we obtain the maximum average torque of around 1.8 Nm for a slip value of 0.95, but with large torque pulsations.
For the generator mode two different rotor speeds are chosen: 1875 rpm (equivalent to a slip value of – 0.25) and 2475 rpm (equivalent to a slip value of – 0.65). The average values of the torque for these two cases are -1.1 Nm and -1.95 Nm respectively.
The change in operating mode, from motor to generator, can also be seen in the phase shift between the supply voltage and the current. In Figure 4 the current in phase one for two slip values (0.95 and –0.65) is plotted using the normalized phase voltage as a reference for the phase shift.
As would be expected, in the case of the motor operation the phase current is lagging the voltage by less than 90 electrical degrees, whereas for the generator case, the current is lagging by more than 90 electrical degrees.
This type of analysis is useful to investigate the transient behaviour of the machines, especially the start-up characteristics and load variation as well as cogging torque. However, an initial machine characterization is usually done by looking at the steady-state behaviour of the machine. The average steady-state torque vs. slip characteristic gives a good idea of the machine behaviour, both as a motor or a generator.
The time taken for a full rotating solution running for 4 cycles of the armature field (0.08s), for one slip value, is around 25 minutes on an Intel i7 2.8GHz workstation. Hence, although this solution provides the most comprehensive characterization of the machine, running a full transient analysis might not be time-efficient when looking at the initial estimation of the machine performance. For this, the Opera-2d AC solver offers a much quicker way of obtaining the average steady-state torque for various slip values.
Fast machine characterization using the dynamic (Opera-2d AC) solver
The dynamic solver (Opera-2d AC / Opera-3d Dynamic EM) solves eddy current models where the driving currents or voltages are varying sinusoidally in time. This allows the calculation of the steady-state torque of an Induction Machine by running a non-motional simulation, considerably reducing the computation time.
The main point to be considered when modelling Induction Machines at a single frequency is that the currents induced in the rotor bars will vary at the slip frequency rather than the armature frequency. This is important because the setup of the dynamic analysis needs to account for the difference between the frequencies in the stator windings and the rotor bars / conductors.
In Opera-2d AC solutions these effects can be accounted for in two different ways:
- the model is analysed at the drive frequency of the armature (usually 50 or 60 Hz) and the conductivity of the rotor bars or windings is scaled by the slip value:
- the model is run at the slip frequency rather than the true AC frequency. The use of the slip frequency to drive the model results in the value of variables such as the back-emf to be reduced by the ratio of the slip frequency to synchronous frequency. This can be remedied by scaling the length of the machine with the ratio defined below. Also, the phase resistance per unit length and the end inductance of the winding will need to be divided by the same ratio. However, in this example the phase resistance is defined as a bulk resistor rather than as a per unit length resistance, in which case this doesn’t need to be scaled.
The second method is used in the following to characterize the Induction Machine in both the generator and motor modes. The aim of this analysis is to produce the torque vs. slip characteristic over the entire simulation range.
The machine is characterized over the slip range from -1 to 1, which is equivalent to a rotor frequency from 100 to 0 Hz. The resolution in slip points is 0.1, or 5 Hz. The average torque over the range of slip values is shown in Figure 5.
The average values obtained for the 5 cases that were analysed with the full transient analysis are also shown on the plot for comparison. As can be seen, the results obtained from the two analyses show a very good agreement, the AC dynamic analysis making a good estimate of the machine characteristic without including motion.
The time taken for one case in the dynamic analysis (for one slip value) is around 5 seconds. This is much more convenient when doing an initial estimation of the machine overall performance or when coupling the analysis to the integrated Optimizer.
Enhancing machine performance by including skew
In order to reduce the ripple torque produced by the Induction Machine, the rotor or the stator of the machine is often skewed. Opera-2d allows for skew in all the relevant electromagnetic solvers. The user specifies the skew angle and the number of representative slices, with each slice considered to be offset by a different angle, defined by the skew angle and the number of slices.
Using the example model, a skew angle of 10 degrees is applied to the stator (equivalent to the angle of a slot) and the skewing is represented by 5 slices. The torque characteristics for the motor case at a slip value of 0.4 (or 900 rpm) and for the generator case at slip -0.65 (or 2475 rpm) for the models with and without skew are displayed in Figure 6.
The fields on the five slices for the motoring operating case with a slip of 0.4, for a particular rotor position are shown in Figure 7. The rotation angle between the first and the last slice is 10 mechanical degrees. The torque at each position is calculated as an average of the torque in the five slices.
The slice option can be used in dynamic solutions (Opera-2d AC) as well as in fully transient ones (Opera-2d RM). The comparison between the skewed and the un-skewed models solved using the dynamic solver is presented in Figure 8.
It is worth noting the slightly higher average torque predicted by the dynamic solution for the skewed model. In order to investigate the cause of this discrepancy, we need to look at the drive waveforms computed using the full transient with motion analysis. We can see in Figure 9, where half of cycle of the phase current is plotted, that the waveform for the un-skewed model has a much higher harmonic content. The fundamental calculated from the Fourier analysis with a peak of 0.949 A is also plotted for reference. Because of the simplifying assumption that the dynamic solver does by considering an ideal sinusoid drive, the results obtained from the skewed model will be closer to the full transient with motion results where the harmonic distortion due to slots and bars is also included. The rotor position for which the dynamic analysis is performed is also likely to influence the value of the torque obtained.
Characterization using Opera-3d
Fast machine characterization using the dynamic (Opera-3d Dynamic) solver
The Opera-3d package allows for a complete model characterization, both as a fully transient analysis or as a steady-state dynamic case and includes all the 3d field effects. The same example model is built in Opera-3d and solved using the 3d Dynamic solver in order to see the impact that the end-effects have on the machine performance. The model is built and solved using the 3d Machines Environments.
A radial-axial cross-section of the full 3d model is presented in Figure 10. The solved model is reduced to a quarter of the full model by using both a rotational and an axial symmetry.
The method used to account for the different frequencies in the stator and the rotor in the absence of movement is to change the conductivity of the rotor bars, as described earlier, while driving the model at the true 50 Hz frequency.
The flux density and the flux distribution in the machine, running as a motor, can be seen in Figure 11.
The torque vs. slip characteristic of the full 3d model is presented in Figure 12. The results from the 2d analysis are also shown for comparison. As can be seen, the average steady-state torque in slightly reduced in the 3d model, although the overall shape of the characteristic remains the same. This is because the 2d model did not include the end ring resistance and inductance. Of course, an estimate of this could be added into the circuit for the rotor in the 2d model.
Reducing the problem size by using a 2d Slice
The models that are built using the 3d Machines Environments can also be solved by Opera-3d dynamics using the 2d Slice feature. This allows for a slice at the middle of the machine to be solved under the assumptions that apply to 2d models (i.e. only including end effects through a circuit component). The benefit of using this option is that a model built using the 3d software can be quickly characterized as a 2d case, before moving on to a full 3d one. This option is applied as a symmetry bounding box with appropriate boundary conditions, so that the details referring to the 3d model are never lost and the user can quickly change between the two models just by changing the type of symmetry applied. The 2d slice representation of the IM used in this example is shown in Figure 13. The model is extruded in the Z direction and the mesh is automatically mapped from one side of the model to the other creating a prismatic mesh. This ensures a reduced solving time since the number of elements is basically the same as in an equivalent Opera-2d model.
In order to include the effects of the induced eddy currents circulating in the rotor bars, a feature has been implemented in Opera-3d which allows cell entities as bulk conductors in a circuit. The bulk conductors are regions that have conductivity assigned to them and can be included as elements of the electric circuit. Hence, current redistribution due to skin and proximity effects can be easily modelled using standard cells.
Each bulk conductor needs to have an ‘entry’ face and an ‘exit’ face to define the path that the current will take inside the volume. These faces represent the terminals of the bulk conductors and need to have a polarity assigned to them (positive or negative). (The option of using a single face as both positive and negative terminal is also available for closed conductive regions).
The rotor bar conductor faces at one end of the 2d slice model make up the positive polarity of the bulk conductors, while the ones at the other end are linked to the negative terminal of the bulk conductor. Since the rotor bars are connected in parallel, all of the faces with the same polarity can use the same boundary label. In Figure 14 two of the faces that make up the positive terminals of the rotor bulk conductors are shown along with the respective setting in the Boundary Conditions dialog.
The bulk conductors are linked in the external circuit in the same way as previously described in the 2d models. The loop defining the rotor winding is shown in Figure 15; the stator loops remain the same as in the full 3d model. The axial length symmetry implied by the slice option is used to scale the conductors in the circuit to give the correct impedance.
The torque vs. slip characteristic obtained from the 2d slice representation of the model is shown in Figure 16. This is compared to the one obtained from the full 3d model and the one from the 2d model.
As expected, the torque values obtained from the full 3d model are lower than in the case of the other two models. This is because of the inclusion of the ends effects which have been omitted in both Opera-2d (rotor only) and Opera-3d slice (stator and rotor).
The solution time for the 2d Slice models is almost the same as the time taken by the Opera-2d Dynamic models and significantly lower than a full 3d model. The timing comparison is presented in Figure 17.
This application note describes the different methods that can be employed for modelling an Induction Machine using the Opera-2d and Opera-3d FEA software. Both the motor and generator operating modes are considered. The torque characteristic obtained using the full transient with motion analysis is presented. The transient analysis also offers the information needed to further calculate the currents and losses vs. slip characteristics in the rotor bars.
After describing the principles that govern the torque production in the IM, a simplifying hypothesis that can be employed to characterize a machine rapidly is presented. Based on this, a time-efficient method of analysing the IM with Opera-2d Dynamics AC analysis without the inclusion of motion is shown.
The model is also analysed using the Opera-3d software and the characteristics obtained from the full 3d models are presented and compared to the 2d results. Finally, the option of solving for a 2d Slice of the full 3d model is employed. This allows a very efficient solution comparable in performance time to Opera-2d dynamics solution but, at the same time, preserves the characteristics of the full 3d model.
 IEA report “Walking the torque: Proposed work plan for energy-efficient policy opportunities for electric motor-driven systems”, May 2011