Dynamic modeling of a synchronous reluctance machine for transient simulation of vibrations under variable rotor magnetization

Abstract

This work introduces a new approach for the dynamic simulation of a permanent magnet-assisted synchronous reluctance machine with the ability to consider dynamic changes in the rotor magnetization. The aim is to comprehensively analyze the dynamics of a machine through transient simulations of the occurring magnetic and mechanical forces that influence the noise and vibration characteristics. A simplified magnetic model considering the effects of magnetic reluctances, leakage flux, and magnetic saturation is utilized to efficiently calculate the dynamically changing magnetic forces in the air gap. Unlike conventional designs employing rare earth magnets in the rotor, the design at hand utilizes non-rare earth magnets that enable adjustments of the magnets’ flux output. The novelty of the presented approach lies in its ability to consider these dynamic changes when calculating the air gap flux. The magnetic forces are then applied to an elastic multibody model of the motor, which includes the rotor, stator, bearings, and the housing, for the computation of the bearing forces and housing deformations. The presented multi-physical model allows for transient simulations of the forces acting on the bearings and the housing, capturing the dynamic response of the motor under varying rotor magnetization, air gaps, and loads. With the proposed approach, this study offers predictions regarding critical vibration characteristics that occur during dynamic operation, providing valuable insights for noise reduction efforts.

Introduction

Various electric machines have emerged as key components driving the advancement of modern mobility concepts. Most prominently, permanent-magnet synchronous motors (PMSMs) relying on rare earth magnets are used due to their remarkable power densities and high torque production. However, the mining of rare earth magnets comes with significant ecological detriments and their scarcity often leads to cost and availability issues. In contrast, synchronous reluctance machines (SynRMs) do not require rare earth magnets, as they generate torque through magnetic reluctance. The reluctance force arises from a special rotor geometry that enables the passage of magnetic flux in one direction while hindering its flow in another direction through the presence of flux barriers. Due to their straightforward rotor design, SynRMs are very efficient and capable of high speeds while providing low manufacturing and maintenance costs. However, the absence of rotor magnets in the SynRM leads to reduced power densities and power factors compared to conventional PMSM. To address these limitations, permanent magnet-assisted SynRMs were introduced. ¹ For this machine topology, non-rare earth permanent magnets are embedded within the flux barriers of the rotor, thereby enhancing torque production. This hybrid machine topology can mitigate the downsides of the pure PMSM or the pure SynRM. ² Commonly, non-rare earth permanent magnets exhibit lower remanent flux densities and lower coercive forces compared to their rare-earth magnet counterparts. However, this apparent drawback can be used favorably. The low coercive force facilitates the change of the magnets’ flux output during operation and thus provides an additional degree of freedom (DOF) for controlling the machine’s torque output. ³ This kind of motor is also referred to as variable flux SynRM.

The wide operating speed range of SynRMs renders them especially susceptible to undesired vibrations that may arise from magnetic or mechanical disturbances. While electromagnetic vibration and noise are mainly caused by higher space and time harmonics in the air gap flux, magnetic saturation, and slot openings, mechanical vibration can mostly be associated with the bearings, shaft misalignment, and rotor eccentricities. ⁴ For electric machines, the prevailing approach is to calculate the magnetic flux by the finite element method (FEM) for a steady-state scenario, and subsequently applying these forces to a finite element (FE) model of the stator.^5–8 In most of these approaches, the vibration and noise analysis is done in the frequency domain. Other methods generate look-up tables from FE data for subsequent transient simulations.^9,10 While FEM is well-suited for precise calculation of the magnetic flux, its computational cost remains high, sometimes even prohibitively high.

Lumped parameter models such as equivalent magnetic circuits (EMCs) bridge the gap between analytical and FE models by utilizing equivalent circuit elements to calculate the magnetic flux. ¹¹ In contrast to fundamental wave models, EMC facilitate the systematic incorporation of nonlinearities such as inhomogeneous air gaps and iron saturation. Models of significantly reduced complexity can already sufficiently well represent the motor behavior with respect to the forces, currents, and voltages, allowing for much faster computation times.^12,13 Hence, utilizing appropriately selected EMC networks as the foundation for transient dynamical simulations of noise and vibration phenomena are promising.

Based on earlier works,^14,15 in this article, a method for transient multi-physical modeling and simulation of a variable flux SynRM is introduced, with particular emphasis on accurate calculation of radial magnetic forces within the air gap. The mechanical motor model is given by an elastic multibody system (EMBS), thereby enabling the consideration of a variable air gap, elastic bearings, and rotor eccentricities. Leveraging this framework, the goal is to analyze the vibrations along the bearings and the stator and their impact on the housing under various transient scenarios. Furthermore, through the systematic and detailed incorporation of the rotor magnets in the EMC, the implications of a change in the flux output of the PMs during operation and the resulting forces are investigated.

When utilizing FEM for the simulation of the magnetic forces,^5–10 it is computationally expensive to incorporate dynamic magnetic properties such as a changing rotor magnetization. Also, when utilizing look-up tables, they need to be recomputed for each new set of parameters. In contrast, the presented approach makes use of the EMC method for the magnetic quantities, which comes with the advantage of being able to systematically incorporate the flux output of the permanent magnets. Other work also makes use of the EMC method,^13,12 but only for static analysis and without the consideration of dynamic changes in the rotor magnets. The presented model enables transient adjustments in the permanent magnets’ flux output while the electromagnetic forces can be simulated efficiently and with good quality. By incorporating this dynamic element, the model provides a more comprehensive understanding of the complex interactions between electromagnetic forces, mechanical vibrations of the motor housing, and resulting noise through efficient transient simulations.

Mechanical modeling

The fundamental mechanical components of the electric motor are given by the rotor, the stator, the bearings, and the housing. In order to analyze the dynamical excitations of the housing caused by the magnetic forces in the air gap or imbalances of the rotor, the mechanical parts of the motor are described within the framework of an EMBS. The deformation of the elastic bodies in EMBS is assumed to be small compared to their respective rigid body movements, allowing the separation of large nonlinear motions into a nonlinear rigid body motion and a linear elastic deformation with respect to a floating frame of reference.^16,17 In combination with model order reduction (MOR) methods, EMBS simulations achieve much faster computation times than classical FE analyzes while still being sufficiently accurate. Here, the housing, bearings, and the stator are represented by elastic bodies, as these components predominantly contribute to the vibration characteristics of the motor. Further, the influence of the coils in the stator slots is neglected. For simplicity, the shaft and rotor are assumed to be rigid. The elastic bodies are modeled separately, which allows for the consideration of different material properties of the respective components. The stator is composed of electrical steel while the bearings are made of regular steel. In contrast, the housing is made of aluminum. Due to its lower density and stiffness, the aluminum housing is more susceptible to vibrations, which can lead to an increased level of noise. The material properties for the elastic bodies are given in Table 1. Subsequently, the parts are discretized into FE meshes using the commercially available software Abaqus ¹⁸ and appropriately joined using tie constraints. With this kinematic constraint, two separate surfaces are tied together such that there is no relative motion between the corresponding surface nodes. The elastic parts are shown in Figure 1. Due to the high computational complexity of FE models, a dimensionality reduction is indispensable for transient simulations.

Figure 1.

Sectional view of the elastic components.

Table 1.

Material properties of the elastic bodies given by Young’s modulus $E$ , Poisson ratio $\nu$ , and density $\rho$ .

Material	$E$ (GPa)	$\nu$	$\rho$ (kg/m3)
Aluminum	70	0.34	2700
Electric steel	190	0.3	7600
Steel	210	0.27	7800

Model order reduction

Each elastic body is described by a linear FE model with $N$ degrees of freedom. The state vector ${\mathit{q}}_e(t)\in {\mathbb{R}}^N$ describes the displacements of the FE nodes and thereby the body deformations over time. The equations of motion for the elastic model result in a linear second-order differential equation of the form

$${\mathit{M}}_e{\ddot{\mathit{q}}}_e(t)+{\mathit{D}}_e{\dot{\mathit{q}}}_e(t)+{\mathit{K}}_e{\mathit{q}}_e(t)={\mathit{B}}_e\mathit{u}(t)$$

(1)

There, the matrices ${\mathit{M}}_e$ , ${\mathit{D}}_e$ , ${\mathit{K}}_e\in {\mathbb{R}}^{N\times N}$ are called the mass, damping, and stiffness matrices, respectively, while ${\mathit{B}}_e$ denotes the system’s input matrix. Since $N$ is usually large, high-computational effort is needed in order to solve the system equation (1). As far as the dynamic analysis is concerned, the number of equations to be solved should preferably be small. Thus, a reduction of the elastic DOF is crucial. The number of DOF can be significantly reduced by utilizing MOR techniques. The goal is to find a reduced dynamical system representation of dimension $n$ with $n\ll N$ that approximates the dynamical behavior of the full system sufficiently accurate in a predefined region of interest. One widely used MOR technique is linear projection based MOR. ¹⁹ There, the idea is to find a projection matrix $\mathit{V}\in {\mathbb{R}}^{N\times n}$ that enables the approximation of the nodal displacements by

$${\mathit{q}}_e\approx \mathit{V}{\stackrel{\sim }{\mathit{q}}}_e$$

(2)

with the reduced coordinates ${\tilde{\mathit{q}}}_e$ . The reduced dynamical system of dimension $n$ emerges by inserting equation (2) into Equation (1) and left-multiplication with ${\mathit{V}}^{\mathrm{\top }}$ . An in depth description of different MOR methods for the calculation of the projection matrix $\mathit{V}$ can be found in the literature.^20,21 Within the scope of this work, the projection matrix is calculated using modal truncation. Thus, the nodal displacement vector is approximated by a linear combination of $n$ dominant eigenvectors ${\varphi }_i$ and corresponding eigenfrequencies ${\lambda }_i$ through

$${\mathit{q}}_e(t)=\sum _{i=1}^n{\mathit{\varphi }}_i{e}^{{\lambda }_{i}t}=\mathbf{\Phi }{\tilde{\mathit{q}}}_e$$

(3)

where is $\mathbf{\Phi }$ is also referred to as the modal matrix. For the system from Figure 1, six modal shapes and their respective eigenfrequencies are illustrated in Figure 2. In this regard, modes 4 and 18 are to be highlighted, as they represent the first two vibrational modes of the stator where the magnetic forces act upon.

Figure 2.

Selected mode shapes of the elastic body: mode 1, $f_1=1300$  Hz; mode 2, $f_2=1532$  Hz; mode 4, $f_4=2002$  Hz; mode 18, $f_{18}=4609$  Hz; mode 19, $f_{19}=4942$  Hz; and mode 20, $f_{20}=5334$  Hz.

Elastic multibody system

In this step, the rigid rotor together with the reduced elastic bodies from the “Model order reduction” section are assembled to form the EMBS. The rotor is modeled as a rigid body with mass $m$ and moment of inertia $\mathit{J}$ . It is connected to the housing through the elastic bearings. This is done by introducing stiff spring elements that connect the bearings with the rotor. The whole EMBS can then be written as

$$\begin{array}{cc}\mathit{M}(\mathit{q})\stackrel{\sim }{\ddot{\mathit{q}}}+\mathit{k}(\dot{\mathit{q}},\mathit{q},t)& =\mathit{g}(\dot{\mathit{q}},\mathit{q},\mathit{u},t)\\ \mathit{y}(t)& =\mathit{h}(\mathit{q}(t))\end{array}$$

(4)

with symmetric and positive definite mass matrix $\mathit{M}$ , gyroscopic forces $\mathit{k}$ , generalized coordinates $\mathit{q}$ that contain all elastic and rigid DOF, the applied forces $\mathit{g}$ with the input vector $\mathit{u}$ , and the output vector $\mathit{y}$ . Note, that $\mathit{u}$ embodies all external forces acting on the EMBS such as the magnetic forces in the air gap as well as the load torque, while the output $\mathit{y}$ contains the rotational position $\gamma$ and velocity $\dot{\gamma }$ of the rotor as well as the displacements of certain surface nodes of the housing.

Electromagnetic system

The electromagnetic model facilitates the computation of the magnetic flux within the motor, yielding both radial and tangential forces in the air gap. The radial forces are subsequently applied to the stator teeth, while the tangential forces are responsible for the torque generation. In order to enable transient simulations, the computational speed of the FEM for calculating air gap forces is insufficient. As a result of this limitation, the magnetic flux is modeled using an EMC that emerges from approximating the flux distribution through lumped elements such as reluctances and magnetomotive forces.

The motor investigated in this work is part of the research project “Effective reluctance machine for emission-free mobility without rare earths” funded by the Ministry of Science, Research, and Arts of the Federal State of Baden-Württemberg in the framework of “Innovationscampus Mobilität der Zukunft”. In the previous project phase, a new machine concept for increasing the efficiency of hybrid SynRMs has been developed. The new concept is based on solving the conflicting goals of pure SynRMs between optimal electromagnetic utilization of the rotor and mechanical load capacity by introducing a fiber-reinforced polymer material into the flux barriers and using magnets without rare earths. In this way, the efficiency can be improved, and the power density can be increased. By substituting air with fiber-reinforced polymers, the rotor gains increased strength and resistance to mechanical stresses, thereby ensuring reliable performance and minimizing the risk of structural deformations or failures during operation and thus allowing for higher maximum speeds. ²² For the second project phase, the design and manufacturing of the motor prototype is done. The considered motor is a three-phase $(m=3)$ synchronous motor with $n_{\mathrm{s}}=36$ stator slots and a distributed stator coil winding, connected to result in six poles, that is, $p=3$ pole pairs. The number of windings per slot is $n_{\mathrm{w}}=4$ . For each pole, there are three radially magnetized AlNiCo magnets embedded into the rotor as depicted in Figure 3.

Figure 3.

Sectional view of one motor pole including the rotor with its permanent magnets and the stator yoke, teeth and copper windings.

Due to the magnetic symmetry inherent to each rotor pole pair and when assuming a constant air gap, it is usually possible to consider only one pole for magnetic simulations. This effectively reduces the computational burden associated with the simulation. However, in the current investigation, the magnetic forces depend on the air gap width $\mathit{s}$ , which can vary along the rotor’s circumference due to the elastic housing and bearings. Therefore, only considering one pole is not sufficient here.

Magnetic circuit

For the calculation of the magnetic flux, the stator, the rotor, and the air gap are approximated by areas of constant cross-section and modeled with appropriate reluctances. Figure 4 shows the EMC of one pole. Note, that compared to earlier work, ¹⁴ here the reluctance network has been extended to incorporate permanent magnets in the rotor, enabling the consideration of a varying rotor flux. The magnetomotive forces caused by the coils around the stator teeth are given by $\mathbf{\Theta }=[{\mathrm{\Theta }}_1,\dots ,{\mathrm{\Theta }}_{36}]$ , while the magnetomotive forces of the permanent magnets in the rotor are approximated by ${\mathbf{\Theta }}_{\mathrm{PM}}=[{\mathrm{\Theta }}_{\mathrm{PM},1},\dots ,{\mathrm{\Theta }}_{\mathrm{PM},36}]$ . The fluxes occurring in each branch of the reluctance network are represented by

$$\mathit{\varphi }=[{\varphi }_1,{\varphi }_{\sigma ,1},\dots ,{\varphi }_{36},{\varphi }_{\sigma ,36}{]}^{\mathrm{\top }}\in {\mathbb{R}}^{n_{\mathrm{s}}}$$

(5)

where ${\varphi }_{\sigma ,i},i\in \{1,\dots ,n_{\mathrm{s}}\}$ denotes the leakage fluxes between the stator teeth. Moreover, the teeth fluxes ${\mathit{\varphi }}_{\mathbf{T}}$ can be expressed as

$$\begin{array}{cc}{\mathit{\varphi }}_{\mathbf{T}}& =[{\varphi }_1+{\varphi }_{\sigma ,1}+{\varphi }_{36}+{\varphi }_{\sigma ,36},\dots \\ & {\varphi }_{35}+{\varphi }_{\sigma ,35}+{\varphi }_{36}+{\varphi }_{\sigma ,36}{]}^{\mathrm{\top }}\\ & =\underset{\mathit{T}}{\underbrace{\left[\begin{array}{cccccccc}1& 1& 0& \dots & \dots & 0& 1& 1\\ 1& 1& 1& 1& 0& \dots & \dots & 0\\ ⋮& \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & ⋮\\ 0& \dots & 1& 1& 1& 1& 0& 0\\ 0& \dots & \dots & 0& 1& 1& 1& 1\end{array}\right]}}\mathit{\varphi }\end{array}$$

(6)

with the transformation matrix $\mathit{T}\in {\mathbb{R}}^{36\times 72}$ . Similarly, ${\mathit{\varphi }}_{\mathbf{F}}={\mathit{T}}_{\mathrm{F}}\mathit{\varphi }$ denotes the fluxes in the air gap causing the attractive forces between stator and rotor, ${\mathit{\varphi }}_{\mathbf{S}\mathbf{t}}={\mathit{T}}_{\mathrm{St}}\mathit{\varphi }$ the fluxes in the stator yoke, and ${\mathit{\varphi }}_{\mathbf{R}}={\mathit{T}}_{\mathrm{R}}\mathit{\varphi }$ the fluxes through the rotor.

Figure 4.

The magnetic equivalent circuit for one pole with the considered reluctances and magnetomotive forces.

The reluctances within the given reluctance network can be classified into two distinct groups. First, the nonlinear reluctances of the iron components in the stator and rotor. Second, the linear reluctances in the air gap. The stator yoke reluctances $R_{\mathrm{Fe},\text{St},i}$ , the stator teeth reluctances $R_{\mathrm{Fe},\text{T},i}$ , and the rotor reluctances $R_{\mathrm{Fe},\text{R},i}$ are described by

$$R_{\mathrm{Fe},x,i}=\frac{l_{\mathrm{Fe},x}}{{\mu }_0{\mu }_r({\varphi }_{x,i})A_{\mathrm{Fe},x}}$$

(7)

with the permeability of free space ${\mu }_0$ , the length of the flux lines $l_{\mathrm{Fe},x}$ , the cross-section area $A_{\mathrm{Fe},x}$ of each component $x\in \{\text{St, T, R}\}$ , and the relative permeability ${\mu }_r$ of the iron. To account for saturation effects in the iron components, the relative permeability ${\mu }_r$ depends on the respective flux ${\varphi }_{x,i}$ passing through the component. While ${\mu }_r$ may be constant in a small range, the iron saturation can not be neglected for increasing flux densities that can occur in high load scenarios. The relation between flux density $B$ and relative permeability ${\mu }_r$ is reflected within the material specific BH curve, where $H$ denotes the magnetic field strength and $B$ the corresponding flux density. In practical scenarios, the BH curve is commonly provided as a data set generated by FE simulations or measurements. However, for the purpose of analytical tractability, it can also be approximated using a function of the form

$$H(B)={\alpha }_{1}B+{\alpha }_2{B}^{{\alpha }_3}$$

(8)

with positive constants ${\alpha }_1,{\alpha }_2$ , and ${\alpha }_3$ that can be obtained by least-squares fitting to available data. Utilizing (8) and the relation $B=\varphi /A$ allows to express the flux dependent relative permeabilities of the iron by

$$\begin{array}{cc}{\mu }_r({\varphi }_{x,i})& =\frac{B}{{\mu }_{0}H(B)}=\frac{B}{{\mu }_0({\alpha }_{1}B+{\alpha }_2{B}^{{\alpha }_3})}\\ & =\frac{1}{{\mu }_0({\alpha }_1+{\alpha }_2{\left(\frac{{\varphi }_{x,i}}{A_{\mathrm{Fe},x}}\right)}^{{\alpha }_3-1})}\end{array}$$

(9)

Note that using this simplification, magnetic hysteresis is not considered. The air gap is modeled with two separate reluctances. While $R_{\sigma ,i}$ incorporates the leakage flux between two neighboring teeth, $R_{\mathrm{L},i}(s_i)$ denotes the air gap reluctances between stator and rotor, depending on the time-varying air gap width $s_i(t)$ . They are given by

$$R_{\sigma ,i}=\frac{l_{\sigma }}{{\mu }_0{A}_{\sigma }}\text{and}{R}_{\mathrm{L},i}(s_i)=\frac{s_i}{{\mu }_0{A}_{\mathrm{L}}}$$

(10)

where $l_{\sigma }$ , $A_{\sigma }$ , and $A_{\mathrm{L}}$ are the respective lengths and cross-section areas of the modeled air gap sections.

The equations for the magnetic network can now be formulated utilizing Kirchhoff’s second circuit law and Ohm’s law for magnetic circuits yielding

$$\begin{array}{rl}{\mathrm{\Theta }}_1+{\mathrm{\Theta }}_2-{\mathrm{\Theta }}_{\mathrm{PM},1}+{\mathrm{\Theta }}_{\mathrm{PM},2}& =R_{\mathrm{Fe},\mathrm{St},1}({\varphi }_1+{\varphi }_{\sigma ,1})+R_{\mathrm{L},1}({\varphi }_{36}+{\varphi }_1)\\ & +R_{\mathrm{L},2}({\varphi }_1+{\varphi }_2)+R_{\mathrm{Fe},\mathrm{R},1}{\varphi }_1\\ & +R_{\mathrm{Fe},\mathrm{T},1}{\varphi }_{\mathrm{T},1}+R_{\mathrm{Fe},\mathrm{T},2}{\varphi }_{\mathrm{T},2}\end{array}$$

(11)

for the first mesh including the stator yoke and the rotor, and

$$\begin{array}{rl}{\mathrm{\Theta }}_1+{\mathrm{\Theta }}_2& =R_{\mathrm{Fe},\mathrm{St},1}({\varphi }_1+{\varphi }_{\sigma ,1})+R_{\sigma ,1}{\varphi }_{\sigma ,1}\\ & +R_{\mathrm{Fe},\mathrm{T},1}{\varphi }_{\mathrm{T},1}+R_{\mathrm{Fe},\mathrm{T},2}{\varphi }_{\mathrm{T},2}\end{array}$$

(12)

for the second mesh including the stator yoke and the leakage flux. By continuing for all meshes of the magnetic network, the relations can be stated in matrix form as

$${\mathit{T}}^{\mathrm{\top }}\mathbf{\Theta }+{\mathit{T}}_{\mathrm{PM}}{\mathbf{\Theta }}_{\mathrm{PM}}={\mathit{A}}_{\mathbf{m}\mathbf{a}\mathbf{g}}(\mathit{\varphi },\mathit{s})\mathit{\varphi }$$

(13)

where $\mathbf{\Theta }$ and ${\mathbf{\Theta }}_{\mathrm{PM}}$ denote the magnetomotive forces caused by the stator coils and the permanent magnets, respectively, and ${\mathit{A}}_{\mathbf{m}\mathbf{a}\mathbf{g}}(\mathit{\varphi },\mathit{s}))\in {\mathbb{R}}^{72\times 72}$ contains all magnetic reluctances. Note, that in the interest of enhancing readability, the dependence of ${\mathit{A}}_{\mathbf{m}\mathbf{a}\mathbf{g}}$ on the fluxes $\mathit{\varphi }$ and the air gaps $\mathit{s}$ are subsequently omitted. The transformation matrix ${\mathit{T}}_{\mathrm{PM}}$ allows to consider the magnetomotive forces of the permanent magnets in the rotor and is given by

$${\mathit{T}}_{\mathrm{PM}}=\left[\begin{array}{ccccc}-1& 1& 0& \dots & 0\\ 0& & \dots & & 0\\ 0& 1& -1& \dots & 0\\ ⋮& \ddots & \ddots & \ddots & ⋮\\ 0& & \dots & -1& 1\\ 0& & \dots & & 0\\ -1& 0& \dots & 0& 1\\ 0& & \dots & & 0\end{array}\right]\in {\mathbb{R}}^{72\times 36}$$

(14)

Applying Ampère’s law and considering the distributed winding of the stator coils establishes a relation between the electric phase currents and the magnetomotive forces of the stator teeth through

$$\begin{array}{cc}\mathbf{\Theta }& ={\left[\begin{array}{cccccc}\tilde{\mathit{N}}& -\tilde{\mathit{N}}& \tilde{\mathit{N}}& -\tilde{\mathit{N}}& \tilde{\mathit{N}}& -\tilde{\mathit{N}}\end{array}\right]}^{\mathrm{\top }}\left[\begin{array}{c}{I}_u\\ I_v\\ I_w\end{array}\right]\\ & =\mathit{N}\mathit{I}\end{array}$$

(15)

where the winding matrix $\tilde{\mathit{N}}$ is given by

$$\tilde{\mathit{N}}=n_{\mathrm{w}}\left[\begin{array}{ccc}0& -1& 1\\ 1& 1& -1\\ -1& 0& 1\\ 1& -1& -1\\ -1& 1& 0\\ 1& -1& 1\end{array}\right]$$

(16)

with $n_{\mathrm{w}}$ denoting the number of turns per coil. Note, that the winding topology of the stator results in six magnetic stator poles. While in similar works mostly concentrated windings have been considered,^13,14 here the stator has a distributed winding topology.

In contrast, the magnetomotive forces ${\mathbf{\Theta }}_{\mathrm{PM}}$ of the permanent magnets in the rotor do not directly depend on the phase currents. Instead, they are approximately constant depending on the remanent flux density $B_r$ . However, if the phase currents are high enough, the magnetic field $H$ generated by the stator coils may exceed the critical field strength $H_{\mathrm{crit}}$ at the flexion point of the permanent magnets' BH curve for a short period of time, that is, $|H|>|H_{\mathrm{crit}}|$ , resulting in the magnet returning to a working point of different flux density compared to the starting point. This effect may be controlled and utilized to either increase or decrease the magnet’s flux output. In order to reproduce this physical behavior, the magnetomotive forces ${\mathbf{\Theta }}_{\mathrm{PM}}(t)$ are modeled as time dependent parameters. As illustrated in Figure 3, there are three radially placed permanent magnets per rotor pole. However, for the sake of simplicity, they are summarized to one magnet per rotor pole. Since there are six magnetomotive forces per pole in the reluctance network, the corresponding values of ${\mathbf{\Theta }}_{\mathrm{PM}}$ need to be chosen appropriately in order to approximate the flux distribution caused by one summarized magnet. The magnetomotive force of each permanent magnet ${\mathrm{\Theta }}_{\mathrm{PM},i}$ in the reluctance network is thus calculated through

$${\mathrm{\Theta }}_{\mathrm{PM},i}(\gamma ,{\hat{\mathrm{\Theta }}}_{\mathrm{PM}})={\hat{\mathrm{\Theta }}}_{\mathrm{PM}}\cos (p\gamma +\frac{2p}{n_s}(i-1)\pi )$$

(17)

with the desired magnetomotive force ${\hat{\mathrm{\Theta }}}_{\mathrm{PM}}$ of the summarized magnet. Note, that ${\mathrm{\Theta }}_{\mathrm{PM},i}$ depends on the rotor angle $\gamma$ , since the actual position of the rotor magnets changes with the rotor’s rotational position, while for the reluctance network the lumped elements are assumed to be stationary. Consequently, it becomes necessary to adapt the parameter ${\mathrm{\Theta }}_{\mathrm{PM},i}$ depending on the specific rotor angle. This setup allows to directly impose arbitrary magnetization values and thus to analyze the system behavior for varying magnetization. Consequently, combining (13) and (15) results in the static algebraic equation of the magnetic network

$${\mathit{T}}^{\mathrm{\top }}\mathit{N}\mathit{I}+{\mathit{T}}_{\mathrm{PM}}{\mathbf{\Theta }}_{\mathrm{PM}}={\mathit{A}}_{\mathbf{m}\mathbf{a}\mathbf{g}}\mathit{\varphi }.$$

(18)

With this equation, the relation between the input currents $\mathit{I}$ and the resulting magnetic fluxes $\mathit{\varphi }$ is established.

Next, the coupling between the electric circuit and the magnetic network is realized through Kirchhoff’s second circuit law and Faraday’s law of induction for the stator coils. The introduction of the phase voltages $\mathit{U}=[U_u,U_v,U_w{]}^{\mathrm{\top }}$ and the Ohmic resistances of the three phases ${\mathit{R}}_{\mathrm{el}}=\text{diag}([R_1,R_2,R_3])$ then yields

$$\mathit{U}={\mathit{R}}_{\mathrm{el}}\mathit{I}+{\mathit{N}}^{\mathrm{\top }}\dot{{\mathit{\varphi }}_{\mathrm{T}}}$$

(19)

Solving (19) for the currents $\mathit{I}$ and inserting it into (18) gives

$${\mathit{T}}^{\mathrm{\top }}\mathit{N}{\mathit{R}}_{\mathrm{el}}^{-1}(\mathit{U}-{\mathit{N}}^{\mathrm{\top }}\dot{{\mathit{\varphi }}_{\text{T}}})+{\mathit{T}}_{\mathrm{PM}}{\mathbf{\Theta }}_{\mathrm{PM}}={\mathit{A}}_{\mathbf{m}\mathbf{a}\mathbf{g}}\mathit{\varphi }$$

(20)

which can be rearranged into

$$\begin{array}{rl}& {\mathit{T}}^{\mathrm{\top }}\mathit{N}{\mathit{R}}_{\mathrm{el}}^{-1}{\mathit{N}}^{\mathrm{\top }}\mathit{T}\dot{\mathit{\varphi }}\\ & ={\mathit{T}}^{\mathrm{\top }}\mathit{N}{\mathit{R}}_{\mathrm{el}}^{-1}\mathit{U}+{\mathit{T}}_{\mathrm{PM}}{\mathbf{\Theta }}_{\mathrm{PM}}-{\mathit{A}}_{\mathbf{m}\mathbf{a}\mathbf{g}}\mathit{\varphi }\end{array}$$

(21)

with respect to the fluxes $\mathit{\varphi }$ using (6), where the phase voltages $\mathit{U}$ and the magnetomotive forces of the permanent magnets ${\mathbf{\Theta }}_{\text{PM}}$ are the inputs. Following previous work, ¹⁵ (21) can be transformed into an explicit differential algebraic equation (DAE) by introducing the auxiliary variable

$$\mathit{\eta }=\frac{1}{n_{\text{w}}}{\mathit{N}}^{\mathrm{\top }}{\mathit{\varphi }}_{\text{T}}$$

(22)

which reflects the sum of magnetic flux per phase. Solving (20) for $\mathit{\varphi }$ and using (6) yields

$$\begin{array}{rl}{\mathit{\varphi }}_{\mathrm{T}}& =\mathit{T}{\mathit{A}}_{\mathrm{mag}}^{-1}{\mathit{T}}^{\mathrm{\top }}\mathit{N}{\mathit{R}}_{\mathrm{el}}^{-1}(\mathit{U}-{\mathit{N}}^{\mathrm{\top }}\dot{{\mathit{\varphi }}_{\mathrm{T}}})\\ & +\mathit{T}{\mathit{A}}_{\mathrm{mag}}^{-1}{\mathit{T}}_{\mathrm{PM}}{\mathbf{\Theta }}_{\mathrm{PM}}\end{array}$$

(23)

where ${\mathit{A}}_{\mathrm{mag}}^{-1}$ is invertible due to its structure and $R_{\mathrm{Fe},x,i},R_{\mathrm{L},i},R_{\sigma ,i}>0,\mathrm{\forall }i\in \{1,\dots ,36\}$ . By left multiplication with ${\mathit{N}}^{\mathrm{\top }}$ , (23) can be expressed with respect to $\mathit{\eta }$ utilizing (22), resulting in

$$n_{\mathrm{w}}\mathit{\eta }={\tilde{A}}_{\mathrm{mag}}^{-1}\underset{=\mathit{I}}{\underbrace{{\mathit{R}}_{\mathrm{el}}^{-1}(\mathit{U}-n_{\mathrm{w}}\dot{\mathit{\eta }})}}+{\mathit{N}}^{\mathrm{\top }}\mathit{T}{\mathit{A}}_{\mathrm{mag}}^{-1}{\mathit{T}}_{\mathrm{PM}}{\mathbf{\Theta }}_{\mathrm{PM}}$$

(24)

with ${\tilde{A}}_{\mathrm{mag}}^{-1}={\mathit{N}}^{\mathrm{\top }}\mathit{T}{\mathit{A}}_{\mathrm{mag}}^{-1}{\mathit{T}}^{\mathrm{\top }}\mathit{N}$ . Finally, solving (24) for the time derivatives results in the differential equation for the auxiliary variable

$$\dot{\mathit{\eta }}=\frac{1}{n_{\text{w}}}(\mathit{U}+{\mathit{R}}_{\mathrm{el}}{\mathit{K}}_{\mathrm{PM}}{\mathbf{\Theta }}_{\mathrm{PM}})-{\mathit{R}}_{\mathrm{el}}{\tilde{A}}_{\mathrm{mag}}\mathit{\eta }$$

(25)

with ${\mathit{K}}_{\mathrm{PM}}={\tilde{A}}_{\mathrm{mag}}{\mathit{N}}^{\mathrm{\top }}\mathit{T}{\mathit{A}}_{\mathrm{mag}}^{-1}{\mathit{T}}_{\mathrm{PM}}$ .

Note, that ${\tilde{\mathit{A}}}_{\mathrm{mag}}$ and ${\mathit{K}}_{\mathrm{PM}}$ still depend on the fluxes $\mathit{\varphi }$ instead of the auxiliary variable $\mathit{\eta }$ . Since the mapping (22) is not unique, there exists no explicit way to write $\mathit{\varphi }$ as a function of $\mathit{\eta }$ . However, an implicit relation can be established by solving (24) for $\mathit{I}$ and inserting into (18) yielding the algebraic equation

$$\begin{array}{rl}\mathit{\varphi }& ={\mathit{A}}_{\mathrm{mag}}^{-1}{\mathit{T}}^{\mathrm{\top }}\mathit{N}{\tilde{A}}_{\mathrm{mag}}{n}_{\mathrm{w}}\mathit{\eta }\\ & +{\mathit{A}}_{\mathrm{mag}}^{-1}({\mathit{T}}^{\mathrm{\top }}\mathit{N}{\mathit{K}}_{\mathrm{PM}}-{\mathit{T}}_{\mathrm{PM}}){\mathbf{\Theta }}_{\mathrm{PM}}\end{array}$$

(26)

and thus completing the DAE formulation, which is utilized for calculating the fluxes $\mathit{\varphi }$ of the magnetic circuit. Note, that the resulting phase currents $\mathit{I}$ may be calculated with the help of (24).

Magnetic forces

In a last step, the magnetic forces resulting from the air gap fluxes need to be calculated. As depicted in Figure 4, the radial forces perpendicular to the rotor surface can directly be attributed to the air gap fluxes ${\mathit{\varphi }}_{\mathrm{F}}$ . They can then be approximated by

$$F_{\mathrm{rad},i}({\varphi }_{\mathrm{F},i})=\frac{{\varphi }_{\mathrm{F},i}^2}{2{\mu }_0{A}_l}$$

(27)

These radial forces are acting perpendicular on the stator teeth and thus contribute significantly to the overall excitation of the stator and housing. In contrast, however, the tangential forces in the air gap are responsible for the generated torque and thus the rotation of the rotor shaft. They can be related to the radial forces through

$$F_{\tan ,i}(\delta ,{\varphi }_{\mathrm{F},i})=\sin (\delta )F_{\mathrm{rad},i}({\varphi }_{\mathrm{F},i})$$

(28)

with the angle $\delta ={\gamma }_{\mathrm{rmf}}-\gamma$ between the rotating magnetic field and the rotor’s rotational position $\gamma$ . For synchronous machines, it holds that ${\dot{\gamma }}_{\mathrm{rmf}}=p\dot{\gamma }$ with the number of pole pairs $p$ . By investigating (28), it can be seen that the tangential forces are zero for $\delta =0$ , which means that no torque is produced when the rotating magnetic field and the rotor are perfectly synchronous. However, because of internal losses and the applied load torque, usually $\delta >0$ holds, resulting in a torque that keeps the motor synchronous. Note, that the highest amount of torque is produced when $\delta =\pi /2$ . For $\delta >\pi /2$ , the motor falls out of synchronism. Further, the case of generator operation with $\delta <0$ is not considered in this study.

Dynamic equations for the complete magnetic system

In summary, the input–output description of the magnetic model is defined by the DAE

$$\begin{array}{rl}\dot{\mathit{\eta }}& =\frac{1}{n_{\text{w}}}(\mathit{U}+{\mathit{R}}_{\mathrm{el}}{\mathit{K}}_{\mathrm{PM}}{\mathbf{\Theta }}_{\mathrm{PM}})-{\mathit{R}}_{\mathrm{el}}{\tilde{A}}_{\mathrm{mag}}\mathit{\eta }\\ \mathit{\varphi }& ={\mathit{A}}_{\mathrm{mag}}^{-1}{\mathit{T}}^{\mathrm{\top }}\mathit{N}{\tilde{A}}_{\mathrm{mag}}{n}_{\mathrm{w}}\mathit{\eta }\end{array}$$

(29\rm a)

$$+{\mathit{A}}_{\mathrm{mag}}^{-1}({\mathit{T}}^{\mathrm{\top }}\mathit{N}{\mathit{K}}_{\mathrm{PM}}-{\mathit{T}}_{\mathrm{PM}}){\mathbf{\Theta }}_{\mathrm{PM}}$$

(29\rm b)

with the differential state $\mathit{\eta }$ and algebraic variable $\mathit{\varphi }$ . The model inputs are the phase voltages $\mathit{U}$ , the air gap $\mathit{s}$ , the magnetomotive force ${\hat{\mathrm{\Theta }}}_{\mathrm{PM}}$ and the rotor angle $\gamma$ . Further, the outputs are defined by the forces in the air gap

$${\mathit{F}}_{\mathrm{mag}}={\left[\begin{array}{c}{F}_{\mathrm{rad},1},F_{\tan ,1}\dots F_{\mathrm{rad},36},F_{\tan ,36}\end{array}\right]}^{\mathrm{\top }}$$

(30)

and the phase currents

$$\mathit{I}={\tilde{\mathit{A}}}_{\mathrm{mag}}(n_{\text{w}}\mathit{\eta }-{\mathit{N}}^{\mathrm{\top }}\mathit{T}{\mathit{A}}_{\mathrm{mag}}^{-1}{\mathit{T}}_{\mathrm{PM}}{\mathbf{\Theta }}_{\mathrm{PM}})$$

(31)

Parametrization

For the simulations in this contribution, the magnet model described in the “Magnetic circuit” section is parametrized according to the design presented in our previous work. ²² The characterizing parameters of the magnetic model are the lengths and cross sectional areas of the flux tubes represented by $A_l,A_{\sigma },l_{\sigma },A_{\mathrm{Fe},x}$ , and $l_{\mathrm{Fe},x},x\in \{\text{St, T, R}\}$ , the number of turns per stator coil $n_{\mathrm{w}}$ , the constants ${\alpha }_1,{\alpha }_2$ , and ${\alpha }_3$ of the nonlinear BH curve for ${\mu }_r$ as well as the electric phase resistances $R_1,R_2$ , and $R_3$ .

The geometric parameters for the lumped network elements are chosen based on available computer-aided design data. Thereby, the flux paths through the stator yoke, the stator teeth, the air gap, and the rotor are approximated by the flux tubes illustrated in Figure 4. Further, the number of turns per stator coil $n_{\mathrm{w}}$ and the electric phase resistances $R_1,R_2$ , and $R_3$ are given by the design of the prototype. The parameters for the nonlinear BH curve for ${\mu }_r$ are determined by fitting the analytic model (8) to the available FE data in a least-squares sense, as illustrated in Figure 5. This parameter set can be seen as a starting point for the dynamic simulations, but requires further tuning based on the final prototype design by parameter identification and model validation.

Figure 5.

Analytical approximation of the magnetic field intensity and flux density (BH) curve for the nonlinear permeability ${\mu }_r$ of stator and rotor material.

Dynamic simulation

The presented multi-physical model of the SynRM is now employed to conduct numerical exemplary simulations. For this, the elastic bodies of the housing, stator, and bearings are discretized resulting in a total of $N\approx$ 440,000 elastic DOF, while there are six DOF for the rigid rotor. The elastic DOF are reduced to $n=20$ modal coordinates through modal truncation as described in the “Model order reduction” section, thereby covering a frequency range of 0–6 kHz. The modal reduction of the elastic parts is performed using the software MatMorembs. ²³ To facilitate the application of magnetic forces ${\mathit{F}}_{\text{mag}}$ to the stator, one interface node per stator tooth is selected, where the concentrated magnetic force is exerted onto the flexible body of the stator. Furthermore, several output nodes on the housing surface are randomly chosen for the analysis of the housing vibration. As these nodes are located on the outer surface of the housing, they can be utilized for the calculation of the sound pressure level and thus the noise generated by the machine. The overall system setup is illustrated in Figure 6, showing the coupling between the electromagnetic and mechanical model. The mechanical model, that is, the EMBS (1), is set up with the software Neweul- ${\text{M}}^2$ , ²⁴ joined with the electromagnetic model and solved in Simulink using standard numerical solvers. For each time step of the numerical integration, the magnetic fluxes in the air gap ${\mathit{\varphi }}_{\mathrm{F}}$ are obtained by solving (29), utilizing the air gap width $\mathit{s}$ from the previous integration step. Subsequently, the magnetic forces are calculated with the help of (27) and applied to the mechanical model. The mechanical output $\mathit{y}$ contains the rotational position $\gamma$ and velocity $\dot{\gamma }$ of the rotor as well as the displacements of certain surface nodes of the housing. Furthermore, the current air gap width $\mathit{s}$ is provided as feedback for the next integration step. Due to the small changes in the air gap width from one integration step to the next step, that is, a weak coupling from the mechanical to the magnetic model, this sequential computation approach is reasonable.

Figure 6.

Overall structure of the machine model with the three-phase input voltages $\mathit{U}$ and the mechanical output $\mathit{y}$ containing the rotational angle $\gamma$ and velocity $\dot{\gamma }$ as well as the bearing forces and the nodal displacements of the surface nodes.

In the first simulation, the dynamical system behavior is analyzed during a run-up of the machine with no load. The machine is supplied with a three-phase input voltage with a peak of 100 V yielding an increasing rotational velocity. In this case, the magnetomotive forces of the permanent magnets in the rotor are kept constant. Subsequently, the resulting bearing forces and vibration of the surface nodes of the housing are investigated. In the second simulation, the change in magnetization of the rotor magnets is considered. In this scenario, magnetization of the rotor magnets is changed during the simulation and the occurring forces and housing vibrations are analyzed.

Run-up

The machine is run up to a maximum rotational velocity of 10,000  $1/min$ , while the magnetomotive force of the rotor magnets is kept constant. Further, a static rotor eccentricity is considered by shifting the center of gravity of the rotor by 0.1 mm from the axis of rotation. This creates an imbalanced rotation and thus high forces on the bearings and the stator. As illustrated in Figure 7, the resulting displacements and forces peak at the maximum velocity and oscillate at the speed of the rotation. Evidently, even minor unbalances can lead to substantial increases in bearing forces and vibrations of the housing.

Figure 7.

Vibrations and forces during run-up from 0 to 10.000  $1/min$ : (a) vibration of the surface nodes on the housing during run-up; and (b) bearing forces during run-up.

Change in magnetization

Now, the rotational velocity of the rotor is set to be constant at 10,000  $1/min$ with a peak input voltage of 100 V. In this simulation, the focus lies on the change in magnetization of the rotor magnets. No static rotor eccentricity is considered in this case. Such a change of the electromagnetic properties during operation might be beneficial for many practical applications, for example, when for a car the electric motors can be switched to a different behavior for low-speed usage in city traffic and high-speed usage on a highway. An exemplary change in magnetomotive force of the permanent magnets is simulated by means of the FEM and is depicted in Figure 8. By applying a high current to the stator coils for a short period of time, the resulting magnetic field is strong enough to change the flux output, that is, the magnetomotive force of the permanent magnets until the next high current is applied. Due to its rapid occurrence within milliseconds, the change is approximated as a step in the magnetomotive force for the transient simulation. Thus, the magnetomotive force of the permanent magnets steps from ${\hat{\mathrm{\Theta }}}_{\mathrm{PM}}=310\text{A}$ to ${\hat{\mathrm{\Theta }}}_{\mathrm{PM}}=237\text{A}$ after $t_{\mathrm{s}}=100\text{ms}$ . Figure 9 shows the transient behavior during the change in rotor magnetization. In Figure 9(a), the air gap flux density between the rotor and one stator tooth is depicted. As expected, the flux in the air gap decreases following the reduced flux output of the permanent magnets, resulting in smaller radial forces acting on the stator tooth, as shown in Figure 9(c). By looking at the frequency domain of the air gap flux in Figure 9(b), it is evident that after the change in magnetization also the higher harmonics have been significantly reduced. The same can be observed for the radial forces in Figure 9(c). Note that the radial force has twice the frequency of the air gap flux since the radial force is always acting in both directions, independent of the pole polarity. This behavior is incorporated into the model through (27), where the squared air gap flux is used to calculate the resulting radial force. Since the rotational frequency of the magnetic field at 10,000  $1/min$ is 500 Hz, the main excitation frequency on the stator teeth through the radial force has a frequency of ${\hat{f}}_0=1000\text{Hz}$ . Figure 9(d) shows the main excitation frequency as well as the occurring higher-order harmonics before and after the change in magnetization. The mean displacement of the surface nodes is illustrated in Figure 9(e), with the corresponding frequency spectrum in Figure 9(f). While the first peak corresponds to the excitation frequency ${\hat{f}}_0$ , the remaining peaks correspond to the eigenfrequencies of the elastic housing. Furthermore, the dominant modes and eigenfrequencies from Figure 2 are highlighted. Notably, the fourth eigenfrequency $f_4$ of the elastic housing is close to the first-order harmonic ${\hat{f}}_1$ of the radial force. The resulting forces on the bearings are depicted in Figure 9(g). Here, the bearing forces are only caused by unbalanced air gap forces caused by the change in rotor magnetization and are thus comparably small. Figure 10 shows the frequency spectrum of the housing surface nodes for different magnetizations and rotational velocities. The values for the magnetomotive force of the permanent magnets was chosen based on the possible range of flux output for the given geometry and material. Evidently, higher-order harmonics of the radial forces are visible in the frequency spectrum depending on the flux and, thereby, the magnetization of the permanent magnets. For instance, the second-order harmonic ${\hat{f}}_2$ is not visible in the frequency spectrum in Figure 10(a) when the permanent magnets’ flux output is small, but starts to appear when the flux output is increased, thereby altering the vibrational behavior of the housing. Figure 10(b) shows the frequency spectrum at 15,000  $1/min$ . There, the excitation frequency ${\hat{f}}_0$ is close to the second eigenfrequency $f_2$ , resulting in a higher peak in the frequency spectrum.

Figure 8.

Approximation of the change in magnetization of the permanent magnets through a step at $t_{\mathrm{s}}=100$  ms.

Figure 9.

Transient behavior of characteristic quantities before and after the change in magnetization at $t_{\mathrm{s}}=100$  ms. (a) Transient of the flux density in the air gap between the rotor and one stator tooth; (b) frequency spectrum of the flux density in the air gap at a stator tooth before and after the change in magnetization; (c) radial force acting on a stator tooth generated by the air gap flux; (d) frequency spectrum of the radial force acting on a stator tooth before and after the change in magnetization; (e) displacement of one exemplary surface node on the housing; (f) frequency spectrum of the mean nodal displacement of the surface nodes before and after the change in magnetization; and (g) bearing forces of bearing A and bearing B during the change in magnetization.

Figure 10.

Frequency spectrum of the housing surface nodes for magnetization values of ${\hat{\mathrm{\Theta }}}_{\mathrm{PM}}=100\text{A}$ (), ${\hat{\mathrm{\Theta }}}_{\mathrm{PM}}=500\text{A}$ () and ${\hat{\mathrm{\Theta }}}_{\mathrm{PM}}=1000\text{A}$ (): (a) spectrum at 10,000  $1/min$ ; and (b) spectrum at 15,000  $1/min$ .

Conclusion and outlook

A dynamic model of a variable flux reluctance machine has been introduced, covering the mechanical and electromagnetic properties of the machine. The utilization of an EMC model allows for the consideration of effects such as iron saturation and leakage flux, while being much more computationally efficient than FEMs. The outputs of the magnetic model are the radial and tangential forces in the air gap. These are the main source of vibration in such machines. They are also the input for the mechanical model, which is realized as an EMBS. To ensure fast computation speeds, the dimensionality of the elastic parts is reduced using MOR techniques. The ability to consider time-varying rotor magnetizations constitutes the novel contribution of this study. It facilitates transient simulations of the interaction between the housing and the magnetic forces contributing to the noise and vibration for time-varying rotor magnetizations by considering the most important magnetic and mechanical quantities.

In this study, two different scenarios were analyzed. During the run-up phase, the machine was subjected to increasing rotational velocities. The presence of a static rotor eccentricity added imbalance to the rotation, resulting in high forces on the bearings and stator. The simulation enabled the investigation of bearing forces and housing vibrations, providing information about the machine’s strain at high speeds. In the second scenario, the change in magnetization of the rotor magnets was simulated. Further, the vibrational behavior of the housing was analyzed for different magnetizations of the permanent magnets. The results show that the change in magnetic flux output of the permanent magnet can change the machine’s vibrational behavior significantly. This strongly depends on the actual flux output of the permanent magnets. For the considered change in magnetization, the effects on the bearings and the vibration of the housing are limited. However, if the flux output is further increased, the vibrational behavior is altered by introducing higher order harmonics in the air gap flux. This may be the case for alternate working points of the machine. However, the calculation of optimal magnetizations depending on the working point is beyond the scope of this study.

Overall, the dynamic simulation provided a deeper understanding of the variable flux SynRM’s behavior under varying conditions. This multi-physical approach facilitates the transient simulation and analysis of the most important magnetic and mechanical quantities. The insights gained from this study can contribute to the design and improvement of SynRMs, ultimately enhancing their efficiency and vibrational behavior by providing an accurate and fast simulation framework. Among the next steps is the investigation of vibrations of the housing and their influence on the noise radiation. Since the electromagnetic forces can now be simulated efficiently and with good quality, the noise radiation is a problem which can nicely be tackled without additional physical effects by the derived flexible multibody model.