Simplified modeling of electromagnets for dynamic simulation of transient effects for a synchronous electric motor

This study aims to show an approach for the dynamic simulation of a synchronous machine. The magnetic forces in the air gap are calculated efficiently using simplified approaches without neglecting important effects. For the modeling of the magnetic forces, an equivalent magnetic circuit is constructed in which the magnetic saturation and the leakage flux are taken into account and coupled with the electrical circuit at the end. The calculated magnetic forces are then passed to a mechanical model of the motor. Together with a predefinable load torque, the resulting motor rotation and the forces in the bearings are identified. The presented model is then investigated in a small example. This novel approach is intended to provide a method of calculating dynamically the forces transmitted from the shaft to the motor housing and to create the basis for evaluating electric motors for vibrations, noise, and harshness under varying loads and input voltages.

characteristics and flux distributions, which is time-consuming and offers limited flexibility for design modifications. Other modeling approaches like the equivalent magnetic circuit (EMC) 8 provide information on static working points only and neglect effects resulting from magnetic saturation, which become significant for higher loads and speeds. The identification of parameters in simplified magnet models based on surrogate parameters is usually a challenging task and these models are often only valid in a small frequency range.
This article discusses the possibilities to model the magnetic forces transiently while taking into consideration the magnetic saturation and leakage flux and transfer the calculated forces into a mechanic model to investigate the forces that occur in the bearings and the housing. In Section 2, the simulation framework is described with the inputs and outputs for each subsystem. In Section 2.1 the modeling of the magnetomotive forces is depicted in detail. The used detailed EMC surrogate model, including magnetic reluctance, leakage flux, magnetic saturation, and fringing is described and the challenging part of transforming the implicit differential equations (IDEs) into a differential algebraic system is explained. Beginning with a flux-based model, taking the mentioned effects into account, the EMC is coupled with the electric circuit which results in a semi-implicit differential equation. The calculated magnetomotive forces are then transferred to the mechanical model, which is described in Section 2.2. Using Neweul-M 2 , 9 the system containing the rotor and the shaft is formulated as a rigid multibody system, 10 with force application points as input and with the mechanical states as output.
Afterward, in Section 3, a small example simulation is presented with a user-defined load torque and input voltage. In Section 4, a short conclusion is drawn with respect to the current capabilities of the model and an outlook on how the presented system can be used and improved.

| MODELING
The motor considered in this study is a three-phase synchronous motor with eight poles, so the stator consists of 48 teeth. The number of slots per pole and phase is q = 2. In synchronous machines, the spatially distributed windings in the stator are fed by sinusoidally alternating voltages with a phase shift of 2/3 π relative to each other. These input voltages cause currents in the winding thus creating a rotating magnetic field. The rotor speed is synchronous to the magnetic field in the stator granting the machine its name. As depicted in Figure 1, the simulation model is divided into two parts. The system input is a voltage U input , which is used to calculate the magnetomotive forces in the air gap in the magnetic model. The calculated magnetic forces F mag rad,tan are then transferred to the mechanical model, together with userdefined load torque. As a result, the bearing forces, the generated motor torque, and the mechanical states α and its derivative are calculated. This modular and generic structure allows quick adjustments in the implementation if the motor geometry, number of pole pairs, or inputs are changed.
In Figure 2, the stator with the three-phase windings around the teeth and the rotor are shown. In Ref. 11 a detailed approach of modeling the magnets in a long stator motor is described. The modeling approach of the synchronous motor in this study is quite similar regarding the mathematical formulation and transformation of the differential equation, so details can be found there. In the following, it is only briefly described to convey the underlying idea. Important for the behavior of the magnets are the air gap s between rotor and stator, the length of the flux lines in the iron l Fe , the permeability of the iron μ r with its variation by the current I, and the movement of the flux lines along the stator and rotor. Also, the magnetic hysteresis, the occurring coupling capacities in high frequencies, and the change of the air gap due to an elastic shaft or an eccentricity of the rotor have an influence on the magnetomotive force but are neglected in this simplified approach. The calculated forces of the electromagnetic model are the inputs for the mechanical model. This can be used to calculate, for example, the bearing forces. Note that here a transient dynamic simulation must be done and so methods like the finite element method (FEM) are not feasible due to their enormous computation times.

| Modeling of the magnetic circuit
The idea is to replace the magnetic currents and effects with equivalent electrical currents and components. An extract of the EMC is shown in Figure 2 and is described in the following.
or rearranged in matrix form with a matrix T Note that the entries in the upper right corner of the matrix are connecting the last loop between the last and first stator teeth. The reluctance in the iron components depends on the cross-sectional area and the relative magnetic permeability μ r . This can be calcu- here A Fe is the cross-sectional area of the iron component, B is the magnetic flux density, H is the magnetic field strength, and α α , 1 2 , and α 3 are positive constants. The BH curve gives a correlation between the magnetic field H and the flux density B. This curve is material-specific and, therefore, known for the iron used. In the applied model, the parameters α α , 1 2 , and α 3 are optimized so that the function for μ ϕ ( ) r matches the material properties. While the hysteresis is neglected in this case due to a lack of data, the model is nonetheless able to consider magnetic hysteresis.
Applying Kirchhoff's second circuit law and Ohm's law for the stator and rotor yields and Using the earlier imposed definitions, Equations (5) and (6) can be given in matrix form where n is the number of windings per slot and I is the input current, the matrix for the system under consideration is of form In the analyzed standard motor, the number of windings per slot is n = 20. Combining Equations (7) and (9) leads to second circuit law for the electric circuit, and Faraday's law of induction, the currents in Equation (10) can be replaced, which then leads to with n n n [ , , ] 1 2 3 T representing the three columns of N. After reordering, this results in an IDE in semi-implicit form Handling IDEs is not an easy task. Therefore, using a transformation presented in Schmid et al. 11 this IDE is transformed into a differential  To take further advantage of the symmetry, it is only necessary to simulate the electric period T el over one pole pair, which is equal to Once the simulation is complete, the radial and tangential tensile stress along the air gap can be calculated according to The time-dependent forces for a certain area element followed by multiplying the increment of the discretized air gap circumference x Δ with respective to line load component is given as The primary effort in computing the air gap forces using this approach is the computational time consumed by the FEA simulation. Figure 4 shows the result of an FEA simulation containing the tangential and radial forces. As mentioned above, there is a relationship between these forces. On the right side of Figure 4, the tangential force is linearly scaled to fit the peaks of the radial forces.
The decay behavior can be approximated by a PT1 element but is neglected in this study. As a first approximation, the relationship is thus assumed to be linear and the scaling factor determined in this way is used to estimate the tangential forces from the calculated radial forces.

F I G U R E 3 Simulated section in the finite element analysis
The calculated magnetomotive forces F mag,tan and F mag,rad are now fed forward to the mechanical model of the motor. Later, additional mechanical degrees of freedom for rotation around the other axes will be considered for a detailed analysis of bearing forces.

| Mechanical modeling of the rotor
Given the geometry and mass of the rotor and shaft the moments of inertia are calculated. Using this, the equation of motion for this simple system is defined by with J as the moment of inertia about the center of mass, the acceleration α , that is the second derivative of the generalized coordinate, and f the generalized forces. The input vector u 49×1   consists of the stacked load torque, and radial and tangential forces.
The vector B transforms the input forces to the defined application frames of the forces. The mechanical rigid body model is derived and simulated using the software Neweul-M 2 . 9 In Figure 5, the mechanical model is illustrated. The discretely positioned force application frames per stator pole are shown on the sketched cross-section through the rotor as an example. A tangential and radial force F i mag,rad/tan  acts on each frame around the motor. Also, the body-fixed coordinate system and the acting load torque can be seen.
Since the objective is to study the vibrations that the motor transmits to the housing, the forces that occur in the rotor bearing are calculated in the simulation.
In further investigation, the next step is to replace the rigid body with elastic elements, such as an elastic shaft. An elastic shaft would in turn have an effect on the size of the air gap, which affects the magnetic forces in the gap. Feedback to the magnetic model would then be necessary, but the magnetic model, as described, already uses an air gap that can vary over time, although is assumed to be constant in this first investigation. In addition, it must be considered how the housing is coupled to the motor via the bearings, but this is not a major obstacle using the described force elements. This is close to the goal of dynamically investigating the noise vibration harshness behavior of the engine.

| EXAMPLE SIMULATION
For the simulations, a CPU with 12 cores and 3.20 GHz is used.
For 10 s simulation time, the computational time is approximately 20-30 min. In this example, a constant three-phase alternating current U inp is the input of the system, with a peak of 400 V and a F I G U R E 4 Relationship between radial and tangential forces from magnetic finite element method simulation (left) and the reasonable scaling for the tangential forces (right) F I G U R E 5 Sketch of the mechanical model with the defined force application points, the body-fixed coordinate system, and the acting load moment BECHLER ET AL. forces are in that range. The generated motor torque M motor , without the acting load torque, can be seen on the right side of Figure 6. This figure also shows that torque ripples occur, which is a characteristic effect in many electric machines.
The user-defined load torque is varying over time. As shown in All other forces on the rotor cancel each other out. To show a nonconstant force curve in the bearing, a static eccentricity is now modeled in the rotor. Therefore, the center of gravity of the rotor is shifted by 0.1 mm from the axis of rotation. Since the axis of rotation remains the same, the air gap between rotor and stator also remains constant. Figure 8 shows the resulting force in x-direction on the bearing. As expected, the resulting force is highest when the angular velocity of the rotor is also highest and oscillates at the speed of rotation.
F I G U R E 6 Absolute values of the first four tangential magnetomotive forces F mag,tan,1−4 (left) and the generated torque on the motor M motor by all tangential forces without a load torque (right) F I G U R E 7 Applied load torque M load (left) and resulting mechanical states α and α (right) F I G U R E 8 Bearing force in the x-direction that occurs due to the modeled eccentricity

| CONCLUSION AND OUTLOOK
In this paper, a method was presented to model in a simplified way the electromagnets of a synchronous motor. Starting with an EMC model of the magnets, it is possible to couple the calculated forces that arise in the air gap to a mechanical model. Important effects, such as the magnetic saturation and leakage flux, are not neglected in the process. As result, the magnetomotive forces in the air gap of the motor are the input for a mechanical model. The coupled model allows a very efficient transient simulation and makes an evaluation of the acting forces possible over long times. This type of modeling will by no means replace the detailed magnetic finite element simulation, but for the application of vibration analysis at load changes or changes of the input voltage during operation, this approach could be suitable.
To improve the accuracy of the model, parameter identification is mandatory to validate the model for the specific motor. Therefore, an experiment with the presented motor or a detailed magnetic FEM simulation could be used. The simulation model is modular, which makes it possible to add mechanical or magnetic effects or change the motor design quickly. For example, the magnetic hysteresis, which causes a weakening of the magnetic field during the pole reversal processes, should be implemented during the next steps. In addition, the influence of coupling capacities that occur in the highfrequency range on the magnetic field 14 should be investigated.
On the mechanical side, the next steps are the connection to the housing via force elements representing the bearings and the replacement of the rigid shaft by an elastic shaft. For this step, it is then necessary to implement the feedback to the magnetic model to account for the change in the air gap. Also, other mechanical effects causing vibrations, for example, eccentricity and bearing stiffness can be implemented to investigate and improve the emerging vibrations on the motor.