The purpose of this Application Note is to present the setup and analysis of a Finite Element model that shows the effects of induced eddy currents using the Opera-3d suite. Problem 24 is an electromagnetic benchmark proposed by the International Compumag Society and is a part of the Testing Electromagnetic Analysis Methods (T.E.A.M)  benchmarks, which cover a wide range of electromagnetic devices and phenomena.
The Finite Element Software suite Opera-3d offers the perfect tools for modelling transient phenomena in electromagnetic systems and the induced effects that such types of application cause. The advanced time-stepping controls and the use of hexahedral elements in the regions where eddy currents are induced are the key to the accurate simulation of the complex fields generated in this benchmark.
The detailed description of the experimental setup that is used for defining Problem 24 is given in . The device used is a simplified topology of a Switched Reluctance Motor with one pair of stator and rotor teeth. Unlike a real SRM, however, the stator and rotor are not laminated – allowing substantial eddy currents to be induced. The axial length of the machine is relatively short compared with the overall diameter, which makes the 3D end–effects significant. A pair of coils is wound around the stator poles and the rotor is locked at a position of partial misalignment with the stator poles. This increases the local saturation at the tip of the teeth and makes the overall distribution of the field in the machine (especially near the gap) more difficult to predict and compute.
The coils are powered with a step voltage and a series of measurements are performed in order to evaluate the response of the device. Values of current, flux in the rotor teeth, flux density at a certain point in the airgap and torque are all measured during the transient and are the basis for the evaluation of the simulation model.
The use of hexahedral elements in regions where eddy currents are induced is highly recommended for these provide the most accurate field evaluation. The geometry of the device is modelled such that it can be divided into regions that allow for hexahedral mesh. Layering is applied on the faces that would see the highest levels of induced eddy currents in order to adequately compute the skin effects produced by the step voltage. Also, in the regions where most of the saturation occurs (i.e. overlapped sections of the teeth tip) the mesh size is reduced so that the flux paths are modelled sufficiently well.
The magnetic properties of the EN9 steel used for the stator and rotor laminations have been measured and the values of the BH data presented in . The BH curve used in the FE model is shown in Figure 2.
The layering must produce a mesh that adequately resolves the field decay with depth in the material, i.e. it must resolve the skin depth (δ), given by the standard formula:
, where μ – absolute permeability [H/m]
σ – electrical conductivity [S/m]
f – frequency [Hz]
A transient excitation with a rise time of 0.2ms is applied, the resulting frequency f being 5000 Hz.
For the EN9 steel, assuming a value of permeability μ=1.25E-03 [H/m] and conductivity σ=4.54E+06 [S/m], the skin depth is approximately 0.1mm.
The mesh at the sides of the stator tooth, where most of the eddy current effects are expected is shown in Figure 3.
The rotor is locked at an angle of 22 degrees with respect to the centre line of the stator poles. The mesh in the airgap region where overlap occurs is denser in order to allow for the field lines to curve and model the saturation at the tip of the teeth accurately (Figure 4).
Since the effect of the axial field component is significant for this device, an appropriate level of discretisation is also needed in the air regions extending from the airgap in this direction (Z).
Due to the geometrical symmetry that exists in the model, the FE model can be reduced to a quarter of the device, by applying a positive 180 rotational symmetry about Z and a tangential magnetic symmetry condition in the XY plane. This reduces the number of elements in the model and the solving time.
The drive function for the TEAM 24 model consists of a step voltage of 23.1 V applied to the stator windings. The two coils are connected in series, each coil having 350 turns. The total phase resistance is 3.09 Ω.
The height of the coils has an important impact on the distribution of the fields induced in the stator teeth and the local saturation. This is why the choice was made to use meshed conductors, instead of the simplified, filamentary, representation.
The electrical circuit used to drive the model is presented in Figure 5. The phase resistance is modelled as a bulk resistor and the winding (which uses the symmetry defined in the model) is driven using a functional voltage. The voltage versus time information is provided in . This characteristic is loaded from a text file and used as an input for the functional voltage drive (Figure 6).
The initial transients in the model are significant and require the use of a small step at the start of the simulation. However, as the transients die down and the eddy currents are more evenly distributed, the electromagnetic time constant increases, allowing a larger time-step.
The time-transient solver used with Opera-3d allows for either a fixed or an adaptive time-stepping method. The size of the fixed time step as well as the minimum and maximum limits for the size of the adaptive time-step can be controlled during the simulation using a customized control file, in order to follow as closely as possible the simulation profile. The switch from a fixed time step to an adaptive one, and vice-verse, is controlled during the simulation.
The switching between fixed and adaptive time stepping is done using a special control comi file which needs to exist in the current project folder. The controls implemented for this simulation are presented below:
$constant #tol 1E-7
// Switch to adaptive time step and reduce the time step; output only at the specified output points
$STRING TIMESTEP_METHOD ADAPTIVE
$STRING TIMESTEP_MINMAX OVERRIDE
$CONSTANT #TIMESTEP_MIN 1E-05
$CONSTANT #TIMESTEP_MAX 1E-03
$STRING TIMESTEP_OUTPUT YES
// After the current transient decreases switch to simple time step
$STRING TIMESTEP_METHOD SIMPLE
$STRING TIMESTEP_DELTAT OVERRIDE
$CONSTANT #TIMESTEP_DELTAT 1E-02
$STRING TIMESTEP_OUTPUT YES
// When current stabilises increase the time step
$STRING TIMESTEP_DELTAT OVERRIDE
$CONSTANT #TIMESTEP_DELTAT 2E-02
$STRING TIMESTEP_OUTPUT YES
The simulation starts with a simple time-step, which is switched to an adaptive time-step when the step voltage is applied. The time-stepping algorithm will reduce the adaptive time step towards the minimum value defined by the user (#TIMESTEP_MIN = 1E-05) so that the errors due to the highly transient effects are minimized. As the simulation continues and the transients begin to decrease, the solver will slowly increase the time-step until it reaches the maximum value defined in the control.comi (#TIMESTEP_MAX = 1E-03).
When the circuit time constants are high enough and the effects produced by the eddy currents become less significant, the time-stepping method is switched back to fixed and the time step is adjusted accordingly, in two stages.
When comparing a system variable (TTime in this case) with a constant, a user defined tolerance is used, rather than using a strict equality, in order to avoid floating-point rounding errors that might otherwise occur.
Measurements and results
Four parameters are measured during the test: phase current, flux in the rotor tooth, flux density at a particular point in the airgap and torque. All of the measured data are presented in .
The current in the coils is measured using a current shunt connected in series with the coils. Since the electric circuit is voltage driven, the current is an unknown, reflecting the system’s impedance.
A search coil, placed around the rotor pole at 8.7 mm from the tooth’s tip, is used to measure the flux that passes through the rotor.
The flux density is measured using a Hall probe, placed in the airgap, close to one of the rotor’s pole corners (exact coordinates are given in ). The evaluation of field quantities at this point is influenced by the local saturation in the tip of the teeth and the singularities that appear at the corners.
The fourth quantity that is measured is the electromagnetic torque which is produced by the tendency of the rotor to align with the stator teeth.
As it can be seen from Figure 7, the predicted current in the device overlaps almost perfectly the experimental results. This suggests that the circuit inductance has been well represented by the FEA model and that the time-stepping algorithm gives the right discretisation in order to correctly solve the circuit equations.
The eddy current effects can be observed in Figures 8 – 12, where the evolution of the induced current density through the cross-section of the stator and rotor is shown, at times 2E-03, 1E-02, 2E-02, 1E-01 and 3E-01 respectively. The values are in A/mm².
The flux density distribution for the same cases as before, through the model cross-section, is shown in Figures 13-17.
From both eddy-current and flux density distribution plots it can be seen that an accurate mesh layering close to the surface of the device is essential for the calculation of the stored magnetic energy. From the images it can also be seen that even if the saturation at the tip of the stator tooth occurs about the time when the current reaches the ‘steady-state’ value (0.1 s), the flux in the device continues to re-distribute even after this point, making the assumption that the field is uniformly distributed incorrect.
The correlation between the measured field in the stator tooth and the fields on a patch at the same coordinates in the FEA model is presented in Figure 18. After the initial transient, the error between the measured and simulated flux linkages is less than 2%.
The flux density in the airgap near the stator tooth tip where most of the saturation occurs is particularly difficult to model because of the inherent errors associated with the singularity of the tooth corner. The mesh elements in these regions need to be very well defined in terms of aspect ratio and discretisation in order to allow for the correct curving of the field lines near this point.
From the point of view of the accuracy of the measured data, the fact that the Hall probe used to measure the field is not infinitely small and that the accuracy of its positioning may not be precise needs to be considered. Both of these might also introduce some uncertainty to the measurements.
These being said, after an initial error of around 15% when the field is first applied, the error between measured and simulated flux density values remains below 5%. This error can be further reduced to below 1% if a smaller element size is used in the gap region. This, however, would significantly increase the computational time.
Finally, the torque calculations reflect the accuracy of the field evaluation by means of the magnetic energy stored in the device. The measured vs. modelled values of torque are presented in Figure 20.
The modelling of induced eddy currents using the time-transient (ELEKTRA TR) solver is applied to the TEAM 24 benchmark problem. The steps necessary for the model building and analysis setup are presented, with emphasis on the particular features raised by this type of analysis. The mesh layering needed for capturing the correct effects of the induced eddy-currents as well as the functional drive source and adaptive time-step control are all discussed in the first part of the application note.
The results obtained from the FE simulation show a good agreement with the available measured data. The local field distribution is evaluated by measuring the flux in the rotor teeth and the airgap flux density during the transient analysis. The correct calculation of the overall energy in the device is assessed by way of the torque measured at the shaft. The torque results also show a good agreement with measured data, validating the mesh definition and the time-transient simulation control.
 N. Allen and D. Rodger. “Description of team workshop problem 24: Nonlinear time transient rotational test rig” [Online] http://www.compumag.org/jsite/images/stories/TEAM/problem24.pdf