This application note presents the modelling and analysis of an axisymmetric linear motor using Opera-2D. The chosen design is a four-phase tubular Switched Reluctance Motor (SRM), designed and characterized in
Two types of analysis are presented using the axisymmetric model and results are compared with measured data from the actual prototype. The first analysis looks at the force versus position characteristic for each individual phase and will be referred to hereafter as the static characterization. The second analysis simulates the dynamic behavior of the device when the plunger is released from a position of maximum static magnetic force and moves to a fully aligned position, hereafter referred to as dynamic characterization.
The prototype motor presented in
 has a four-phase stator and a 17 segment plunger; each plunger segment comprises a tooth and slot of equal width.The geometry of the motor is shown in Figure 1, which depicts one stator phase and two plunger segments, with indicative dimensions.
The electrical parameters of the tubular SRM, as presented in  are:
- number of turns per phase: 560
- phase current: 0.5357 A
- phase resistance: 18 W
The material BH data provided in the referenced paper, were used in the simulations. However, the data are believed only to be approximate, and this is expected to have some impact on the accuracy of the results.
The insulating regions were defined using the default material characteristics of air.
Modelling with Opera-2d
The simulation of the tubular switched reluctance machine was performed using the Opera-2d Finite Element Software. The model was defined using the axisymmetric option and solved using the linear motion LM solver.
2D Axisymmetric Model
Due to the fact that the model is rotationally symmetric we can use the axisymmetry option in Opera-2d in order to reduce the computational effort required.
Since four stator phases are fed independently, each coil needs to have a different coil number assigned to it, as shown in Figure 2. The geometry of the first coil is created and copied three times, using the COPY command to create the geometry for the rest of the phases.
The four stator and 17 plunger segments were similarly created by building one segment and copying the geometry, with appropriate offset in the Z direction.
The detail of the meshing in and around the air gap is shown in Figure 3. Note the consistent meshing of the air gap in the Z direction; this maintains the quality of the new mesh at each time step, as the plunger moves through the MEDIUM group.
LM Analysis – static characterization
The model was set up to use the LM solver for both the static and the dynamic analysis. The static analysis could have been performed by running a series of static (ST) solver simulations, with the plunger at different positions. However, by running the LM solver with a slow plunger speed – such that the effects of any transients produced by the movement are negligible – the equivalent of a series of static solutions could be performed in a single run of the LM solver.
The geometry construction, material properties assignments, meshing data and analysis setup are all done by running a series of scripts that can be easily parametrized and modified. The drive is setup using the integrated Circuit Editor. The four coil regions defined in the model are linked to the external drive circuit, which allows for either voltage or current control. The circuit schematic and the electrical properties of the phase that is being fed (phase 1 in this case) are presented in Figure 4.
With a single phase excited, the static analysis calculates the distribution of the flux in the machine at each time step. The calculation is then repeated for each of the other phases in turn. Thus, the force on the plunger can be evaluated at multiple positions, and the differences between the effects produced by the inner and outer phases can be observed.
The movement is defined as a constant speed and the force on the plunger is evaluated at regular time steps, corresponding to a set of plunger positions. The speed is set to give a full mechanical step in one second, where a mechanical step is the distance between successive alignments of the plunger teeth with a particular stator phase, i.e. half of the length of a plunger section. At such a low speed, there should be no significant transient effects due to motion.
LM Analysis – dynamic characterization
The dynamic behavior of the tubular switched reluctance motor is also analyzed in
The dynamic analysis is performed in order to determine the stabilization time of the plunger when starting from an unaligned position. For this, a single phase is excited, with the plunger initially at the position of maximum static magnetic force, i.e. a half tooth-width offset from the excited stator phase. The movement is then determined by the equation of motion of the plunger. The equation takes into account the electromagnetic force, the friction, the viscous damping coefficient, which is proportional to speed and the plunger mass.
In this analysis, the plunger is released from rest and allowed to move freely based on the equation of motion. The total duration for the dynamic analysis is 0.8 seconds, during which the solver outputs 30 steps. Because of the high transients at the beginning of the analysis when the source is switched on and the plunger is released, the analysis is run using an adaptive time step. The plunger position, speed and acceleration are logged, as well as phase voltages and currents.
The same external drive circuit from the static analysis is used.
Results and discussion
The results from the static and dynamic simulations of the tubular motor are compared with the measured data below.
The static results are presented in Figure 5, where the force from each of the motor’s four phases obtained from the analysis is compared with the measured data.
As it can be seen, the agreement between simulated and measured results is excellent for all stator phases. Both sets of results show lower force from the two inner stator phases than from the outer ones. The reason for this can be understood from the difference in the flux distribution between the phases, as shown in Figure 6.
The FEM results for the dynamic case, showing the plunger motion from the unaligned position to stabilization, are presented in Figure 7 along with the experimental results, obtained in
 using the built prototype. Again, very good agreement is seen between the simulated and measured data.
The results presented here demonstrate the utility of the Opera-2D LM solver in the design and analysis of an axisymmetric linear switched reluctance motor. The simulations performed for both the static and dynamic characterizations produced results that are in excellent agreement with measurements.
The total number of elements in the model is approximately 38,000. The solving time for the series of static cases is 10 minutes (for all 25 positions), while the solving of the motion coupled simulation takes around one and a half hours.
 El Amraoui, Lilia; Conception Electromécanique d’une gamme d’actionneurs Linéaires Tubulaires a Réluctance Variable, Ph.D. Thesis, École Centrale de Lille, FRANCE, 2002