Magnetic Equivalent Circuit Modelling of Synchronous Reluctance Motors

Abstract

This paper proposes a modelling technique for Synchronous Reluctance Motors (SynRMs) based on a generalized Magnetic Equivalent Circuit (MEC). The proposed model can be used in the design of any number of stator teeth, rotor poles, and rotor barrier combinations. This technique allows elimination of infeasible machine solutions during the initial machine sizing stage, resulting in a lower cohort of feasible machine solutions that can be further optimized using finite element methods. Therefore, saturation effects, however, are not considered in the modelling. This paper focuses on modelling a generic structure of the SynRM in modular form and is then extended to a full SynRM model. The proposed model can be iteratively used for any symmetrical rotor pole and stator teeth combination. The developed technique is applied to model a 4-pole, 36 slot SynRM as an example, and the implemented model is executed following a time stepping strategy. The motor characteristics such as flux distribution and torque of the developed SynRM model is compared with finite elemental analysis (FEA) simulation results.

1. Introduction

Magnetic Equivalent Circuits (MECs) provide a clear upper hand in the modeling, analysis and optimization of electric machines since they offer a significant gain of accuracy in comparison to the empirical/analytical methods [1,2,3,4]. MECs have been widely used to analyse electromagnetic devices and saturation [5,6,7,8,9] and in the design of electric machines [10,11,12] such as induction machines [7,13,14] and wound-rotor synchronous machines [15,16], to name a few.

In the past decade, there have been considerable studies focusing on magnetic models, structures and solution techniques [17] to model electrical machines. The solution technique involves circuit analysis utilising Kirchoff’s Laws and establishing a system of equations. Generally, a system of equations is established by applying Kirchhoff’s Current Law (KCL) at every node in the MEC [8,17], where in magnetic permeance is used to express node potential in terms of the flux. An induction machine and its fault conditions have been studied in [18] using the concept of nodal MEC analysis. The magnetic and electric circuit analogy property has been utilised in [15,19] to model a synchronous machine, which is further solved using the nodal analysis. Refs. [20,21] use the MEC based analytical techniques for the study of three-phase SynRM and permanent-magnet assisted SynRM(PMaSynRM) with flux barriers of uniform width.

However, another way is a mesh-based MEC model which is built on Kirchhoff’s Voltage Law [10,22,23,24,25,26,27]. In such an algebraic system, magnetic reluctances define the relation between flux tubes and the various node potentials. In the induction motor model developed in [28] with a mesh-based MEC model, the accuracy is very low owing to the assumption that the air-gap reluctances and mesh do not depend on the rotor position. It is to be noted that very few researchers have incorporated rotation in models developed using the mesh-based MEC techniques. In such a loop/mesh based MEC model, the network equation is dynamic with change in rotor position, and the changes in the circuit topology need to be incorporated into the matrix formulation.

Traditionally, the MEC models of SynRMs are performed in the d-q axes for the calculation of the overall performance [29]. The authors of [30] employ a combined MEC-FE model for SynRMs which give flux linkages with good accuracy, but it involves two MEC models, namely with d and q-axes networks. Two separate d-q models were created for SynRM in [31] to produce an accurate saliency ratio; however, it is found that, even with two different models of the rotor, predicting a magnetic field in an air-gap is a challenge. Ref. [32] introduces a semi-analytical calculation method for SynRM modelling. In the majority of the existing research using MEC modelling of SynRMs, the model is fixed at the position of maximum magnetic linkage, or at two positions corresponding to the d-q axes alignment. However, Ref. [33] tries to overcome this limitation by incorporating the rotor position in a three-dimensional MEC model.

The authors of [34] report an analytical design technique applied to SynRMs with circular-shaped flux barriers that predicts the field distribution characteristics within the air-gap. Winding functions have also been commonly used to determine the stator MMF and reluctance network, which in turn would be instrumental in predicting rotor and air-gap flux densities. Flux barriers and the associated reluctances are determined by the use of tools such as conformal mapping. Ref. [35] proposes yet another analytical method that uses a combination of MEC and winding functions to predict the radial flux along the air-gap in the SynRM. Winding functions are used in the stator model whilst rotor was modelled using MEC taking into consideration the effect of stator slots on the rotor.

Analytical methods including the d,q model [36,37,38] and co-energy method for characterization of SynRMs [39] have been studied extensively, but, due to its limited accuracy, these models are mostly preferred as a pre-design step for the FEA based analysis. Ref. [40] proposed an analytical method for the Reluctance type motor topologies where the MEC model was developed considering the rotor alone. This utilises a Fourier series expansion of the stator magnetic potential to obtain the magnetic potential of the rotor under consideration. Furthermore, the torque was calculated from the radial magnetic component, which in turn was extracted from the differential of potentials of the stator and rotor. Refs. [41,42,43] considers the saturation effect in SynRM analytical modelling. Thus, although studies have successfully demonstrated that analytical models exhibit low computational cost, it has major limitations in terms of accuracy when the studied geometries become complex. Recent studies [41,44] with a focus on synchronous machine topologies, using the MEC principles yet again to model and further optimise the machine performances. However, the fact remains that all these models require redesigning from scratch if any of the dimensions or stator/pole configurations changes are to be incorporated.

In this paper, an MEC model for a SynRM is presented and is derived with the ability to change rotor position, number of stator teeth and rotor pole numbers, and also with the ability to change the number of rotor barriers. The paper presents the detailed modeling of the air-gap flux with rotor position. The model is then used to analyse the torque characteristics of an example SynRM.

The proposed MEC modelling technique for SynRMs can be ideally used to analyze a large cohort of machines with different dimensions, rotor barriers, slot pole combinations and to develop feasible preliminary designs for further FEA analysis. Compared to the d-q based models, the model presented in this paper relies on a single dynamic network allowing the potential to add asymmetric features to the rotor. Section 2 discusses MEC Model development for SynRM, and the variable lumped elements of the air-gap permeance. Section 3 outlines the Mathematical Model developed for solving the MEC. Simulation strategy used to determine the torque along with the machine characteristics and validation with the FEA has been presented in Section 4. Conclusions and prospects of future work are given in Section 5.

SynRM Fundamental Unit MEC Model

Figure 1 shows a generalized model of the fundamental unit of a SynRM, namely a 2-pole rotor and an idealised 2-pole stator consisting of single barrier in the rotor structure.

The main components include the MMF sources denoted by $F_w$, the stator permeances associated with the back-iron $P_{sb}$, the tooth $P_{st}$, the rotor position dependent air-gap permeances $P_{ag1}$ and $P_{ag2}$ and the rotor permeances $P_{1ab}$, $P_{2ab}$, respectively.

Table 1. Nomenclature and definition of variables.

${\overline{F}}_u$	Magnetic Potential vector at nodes ‘u’
${\overline{F}}_s$	Magnetic Potential vector at nodes ‘s’
${\overline{F}}_w$	Magneto motive force vector
${\overline{F}}_{ra}$	Magnetic Potential vector at nodes ‘ra’
${\overline{F}}_{rb}$	Magnetic Potential vector at nodes ‘rb’
${\overline{F}}_{rc}$	Magnetic Potential vector at nodes ‘rc’
$P_{st}$	Stator tooth permeance
$P_{sb}$	Stator back iron permeance
$P_{leak}$	Stator outer leakage permeance
MEC	Magnetic Equivalent Circuit

provides a list of variables used in this MEC model along with their definitions. These elements of the magnetic circuit can be modelled based on their geometry, configuration and material characteristics. Permeances can also be implemented in nonlinear form to account for saturation effects. This permeance network in Figure 1 allows the model to be extended to model any slot/pole combination and number of rotor barriers. The next section demonstrates the extension of the basic model to a 4-pole, 36-slot SynRM.

2. MEC Model Development for a 4-Pole 36-Slot SynRM

The basic configuration described in the previous section is utilised to expand the model to a 4-pole, 36 slot SynRM model with three rotor barriers.

2.1. The Magnetic Equivalent Circuit Model

MEC modeling is based on the representation of the electromagnetic behavior in the form of a network composed of concentrated permeance elements. Each part is then approximated by geometric shapes. Each permeance can be determined by:

$$\begin{array}{c}\hfill P=\frac{{\mu }_0{\mu }_{r}A}{l_t}\end{array}$$

(1)

where ${\mu }_0$ represents the permeability of free space, and A and $l_t$ are the equivalent cross-sectional area perpendicular to the flux path and length of the flux path, respectively.

The permeance network can be sectioned into three regions, namely the stator, rotor and air-gap. The MEC model also represents electromagnetic interaction between a single reference stator tooth and any section of the rotor. The remaining permeances between individual rotor pole sections and other stator teeth are modelled iteratively by introducing an appropriate shift to the reference stator tooth to represent the positions of other stator teeth. This allows the model to be easily adopted for any stator tooth–rotor pole combination.

Stator and rotor MEC sections have constant geometry and permeability throughout the rotational motion of the motor, and hence these can be represented with constant permeance units. Due to the magnetic saturation property of the ferromagnetic material involved in the machine modelling, these elements are generally represented by Inherently Nonlinear Permeance units.

2.2. Magnetic Equivalent Circuit Model of the Stator

Figure 2 shows the MEC model with lumped permeances for a section of SynRM where the variables are defined as follows.

The permeances $P_{st}$ and $P_{sb}$ model the stator tooth and stator back iron, respectively. The magnetomotive force (MMF) source in each tooth represents equivalent MMF produced by the stator coils under load conditions where

$$\begin{array}{c}\hfill F_{coil}=NI\end{array}$$

(2)

The variable N is the number of turns in the coil and I is the current in the coil. In addition, the slot leakage permeance unit is added between adjacent teeth to calculate leakage flux flowing through the stator slot. Application of KCL to the stator nodes yield the mathematical model for the stator [5,8,45].

At node $u_{N_s}$ in Figure 2:

$$(F_{u,Ns-1}-F_{u,Ns})P_{sb}+(F_{u,1}-F_{u,Ns})P_{sb}+(F_{s,Ns}-F_{u,Ns}-F_{w,Ns})P_{st}=0$$

(3)

This yields $N_s$ equations which can be formulated as a matrix equation:

$$\left[K_{sb}\right]{\overline{F}}_u+\left[P_{st}\right]{\overline{F}}_s-\left[P_{st}\right]{\overline{F}}_w=0$$

(4)

where $\left[K_{sb}\right]$ is a $\left[N_s\times N_s\right]$ matrix with stator permeances. The diagonal elements of $\left[K_{sb}\right]$ are the sum of all permeances connected to the corresponding node, e.g., the value at the ith row and ith column represent the sum of elements connected to node i of the stator network. The off-diagonal elements correspond to element between the respective nodes, e.g., the value at the ith row and jth column represent permeance between nodes i and j. The variables ${\overline{F}}_u$ and ${\overline{F}}_s$ represent the node MMF vectors of the corresponding nodes.

2.3. Magnetic Equivalent Circuit Model of the Rotor

In order to establish an accurate model, the rotor segments are split into flux carrier and flux barriers. As shown in the rotor permeance network in Figure 2, the thick nodes show the interconnection nodes. The nodes denoted with subscript ‘ra’ represent the upper layer while the nodes ‘rb’ and ‘rc’ represent the second and the third layers. The dotted lines connecting any two nodes represent the Permeance of that region. The flux barriers are represented by the permeances interconnecting nodes between each layer e.g., between $1_{rb}$ and $2_{rb}$, $2_{rb}$ and $3_{rb}$. The flux carriers are represented by interconnecting nodes between each layer, e.g., $2_{ra}$ and $2_{rb}$, $2_{rb}$ and $2_{rc}$, etc. To develop a mathematical representation of this permeance network, KCL is applied at each node obtain a network of equations. The Layer-a of both halves of the pole can be represented by the equation:

$$\left[P_{ag,diag}+P_a^{}\right]{\overline{F}}_{ra}+\left[P_{ab}^{}\right]{\overline{F}}_{rb}+\left[P_{ag}\right]{\overline{F}}_{\mathrm{s}}=0_{14\times 1}$$

(5)

The matrix $\left[P_a^{}\right]$ represents Layer-a of the rotor permeance network. The diagonal elements of $\left[P_a^{}\right]$ are equal to the negated sum of the permeances connected to each node in Layer-a and adjacent nodes. The off-diagonal elements correspond to element between the nodes, e.g., ith row and jth columns represent the element between the ith and jth nodes. The matrix $\left[P_{ab}^{}\right]$ represents the interconnecting permeance network between the nodes a and b. The matrix $\left[P_{ag}\right]$ represents the air-gap permeance network where the elements within the matrix at the $N_x^{th}$ row and ith column represents the air-gap permeance value between the $N_x^{th}$ stator tooth and the ith node on the rotor surface.

Similarly, Layer-b and Layer-c of the pole can be represented by Equations (6) and (7), respectively:

$$\begin{array}{c}\hfill \left[P_{ab}^{}\right]{\overline{F}}_{ra}+\left[P_b^{}\right]{\overline{F}}_{rb}+\left[P_{bc}^{}\right]{\overline{F}}_{rc}=0_{14\times 1}\end{array}$$

(6)

$$\begin{array}{c}\hfill \left[P_{bc}^{}\right]{\overline{F}}_{rb}+\left[P_c^{}\right]{\overline{F}}_{rc}=0_{14\times 1}\end{array}$$

(7)

The variables ${\overline{F}}_{ra}$, ${\overline{F}}_{rb}$ and ${\overline{F}}_{rc}$ represent the MMF vectors at nodes $ra$, $rb$ and $rc$, respectively.

The modeling strategy of the air-gap permeance modelling with respect to the varying rotor position will be discussed in detail in the section below.

2.4. SynRM Air-Gap Permeance Model

The flux path associated with air-gap have variable dimensions due to a rotational movement of the rotor. Hence, each pole is connected to each tooth via a piecewise continuous permeance varying with rotor position. In this analysis, the rotor surface is divided into multiple regions defined by the flux barrier and their relative positions as shown in Figure 3.

The air-gap permeance linking the $N_x^{th}$ stator tooth with the ith node on the rotor pole is modelled by $P_{AG,N_x,i}$. At each time step, the rotor position changes and the air-gap network has to be recomputed. The analysis of the air-gap permeance at various rotor positions requires the arc length of the overlapping regions between the stator teeth and regions of the rotor that are adjacent to the air-gap [46]. For each stator interconnection node, a logic function described below determines the existence of a connection to one or more rotor surface nodes based on the rotor position.

In Figure 3, “Tooth1” represents any one of the teeth in the stator. The regions 1, 2 and 3 represent three distinct nodes in the rotor. If the angle corresponding to “Tooth1” is [${\theta }_1$, ${\theta }_2$] and the angle corresponding to the rotor region “i” is [${\alpha }_i$,${\alpha }_{i+1}$] for i = 1, 2, 3, the arc length of the overlapping region $l_i$ between “Tooth1” and the rotor region “i” can be calculated by:

$$\begin{array}{c}\hfill l_i=\left\{\begin{array}{ccc}0& \mathrm{if}& {\alpha }_{i+1}<{\theta }_1\mathrm{or}{\alpha }_i>{\theta }_2\\ r^{\prime }\left({\alpha }_{i+1}-{\alpha }_i\right)& \mathrm{if}& {\alpha }_i>{\theta }_1\mathrm{and}{\alpha }_{i+1}<{\theta }_2\\ r^{\prime }\left({\alpha }_{i+1}-{\theta }_1\right)& \mathrm{if}& {\alpha }_i<{\theta }_1\mathrm{and}{\theta }_1\le {\alpha }_{i+1}\le {\theta }_2\\ r^{\prime }\left({\theta }_2-{\alpha }_i\right)& \mathrm{if}& {\theta }_1\le {\alpha }_i\le {\theta }_2\mathrm{and}{\alpha }_{i+1}>{\theta }_2\\ r^{\prime }\left({\theta }_2-{\theta }_1\right)& \mathrm{if}& {\alpha }_i<{\theta }_1\mathrm{and}{\alpha }_{i+1}>{\theta }_2\end{array}\right.\end{array}$$

(8)

where r${}^{\prime }$ is the mean air-gap radius.

For stator tooth $N_s$ at node $S_{Ns}$ in Figure 2, KCL can be applied as:

$$\begin{array}{cc}& ({\overline{F}}_{s,N_s}-{\overline{F}}_{u,N_s}-{\overline{F}}_{w,N_s})P_{st}+({\overline{F}}_{s,N_s}-{\overline{F}}_{s,1})P_{leak}+({\overline{F}}_{s,N_s}-{\overline{F}}_{s,N_s-1})P_{leak}\hfill \\ & +\sum _{i=-7}^7({\overline{F}}_{s,N_s}-{\overline{F}}_{i,ra})P_{AG,N_s,i}=0\hfill \end{array}$$

(9)

The nodal equations for nodes $S_1$ to $S_{Ns}$ can be represented as:

$$\begin{array}{c}\hfill \left[P_{st}\right]{\overline{F}}_s+P_{leak}\left[K\right]{\overline{F}}_s-\left[P_{st}\right]{\overline{F}}_u-\left[P_{st}\right]{\overline{F}}_w+\mathrm{rowsum}\left(P_{ag}^T\right){\overline{F}}_s-P_{ag}^T{\overline{F}}_{ra}=0\end{array}$$

(10)

where

$$\begin{array}{c}\hfill \left[K\right]={\left[\begin{array}{cccccc}2& -1& 0& 0& \cdots & -1\\ -1& 2& -1& 0& \cdots & 0\\ 0& -1& 2& -1& \cdots & 0\\ ⋮& ⋮& ⋮& ⋮& \ddots & 0\\ -1& 0& 0& -1& \cdots & 2\end{array}\right]}_{Ns\times Ns}\end{array}$$

(11)

3. Matrix Formulation of the MEC

Following the nodal analysis, the MEC can be described by simultaneous matrix equations formulated as a larger matrix equation:

$$\begin{array}{c}\hfill \left(\begin{array}{ccccc}\left[K_{sb}\right]& \left[P_{st}\right]& 0& 0& 0\\ 0& \left[P_{ag}\right]& \left[P_{ag,diag}+P_a\right]& \left[P_{ab}\right]& 0\\ 0& 0& \left[P_{ab}\right]& \left[P_b\right]& \left[P_{bc}\right]\\ 0& 0& 0& \left[P_{bc}\right]& \left[P_c\right]\\ \left[P_{st}\right]& \left(\begin{array}{c}\left[P_{st}\right]+P_{leak}\left[K\right]+\hfill \\ rowsum{\left[P_{ag}\right]}^T\hfill \end{array}\right)& -{\left[P_{ag}\right]}^T& 0& 0\end{array}\right)\left(\begin{array}{c}{\overline{F}}_u\\ {\overline{F}}_s\\ {\overline{F}}_{ra}\\ {\overline{F}}_{rb}\\ {\overline{F}}_{rc}\end{array}\right)=\left(\begin{array}{c}\left[P_{st}\right]\\ 0\\ 0\\ 0\\ \left[P_{st}\right]\end{array}\right){\overline{F}}_w\end{array}$$

(12)

Torque Calculation

The electromagnetic torque produced by the rotor can be written in terms of co-energy as:

$$\begin{array}{c}\hfill T_{em}=-{\left.\frac{\partial W_c}{\partial \theta }\right|}_{constant-flux}\end{array}$$

(13)

Co-energy is expressed as:

$$\begin{array}{c}\hfill W_c=\int F_{a}d{\varphi }_a\end{array}$$

(14)

$F_a$ and ${\varphi }_a$ are the air-gap MMF and flux. The air-gap flux $F_a$ can be written as a function of permeance ${\varphi }_a$:

$$\begin{array}{c}\hfill {\varphi }_a=F_a{P}_a\end{array}$$

(15)

Assuming that the permeances are linear and constant for a given rotor position, co-energy (14) can be expressed as:

$$\begin{array}{c}\hfill W_c=\frac{1}{2}{F}_a^2{P}_a\end{array}$$

(16)

Substitution of (16) in (13) and the resolution of the partial derivative yields:

$$\begin{array}{c}\hfill T_{em}=-\frac{1}{2}{F}_a^2\frac{d{P}_a}{d\theta }\end{array}$$

(17)

Using the MEC model $T_{em}$ can be calculated as the summation of all the torque components produced by each of the air-gap permeances. This is achieved by application of (17) to each of the air-gap elements with rotor position and summation.

4. Simulation Results

The SynRM machine in

Table 2. SynRM parameters.

Parameter	Value
Flux-barrier/pole	3
Rotor shaft diameter	35 mm
Air-gap length	0.3 mm
Axial length	140 mm
Rated Current Density	10 A/m${}^2$
Number of turns	100
Number of phases	3
Rated frequency	200 Hz
Number of stator slots/poles	36/4
Stator outer/inner diameter	180/110 mm
Rotor outer diameter	109.4 mm
Material type	iron (ideal)
Material relative permeability	4000

has been simulated using a time stepped approach, and the results are validated by 2D-FEA using the Ansys-Maxwell Electromagnetic analysis software package. The block diagram representation of the time stepped simulation strategy is depicted in Figure 4.

Owing to the geometry, the air-gap permeance units are symmetrical between the other stator tooth and rotor poles. Thus, air-gap permeances for all possible combinations of stator tooth–rotor poles can be calculated from a single unit model represented by one rotor pole–stator teeth combination. In the simulation strategy shown in Figure 4, the ith rotor pole is used to calculate the air-gap permeances for each relative position with respect to the jth stator tooth. An iterating loop calculates the air-gap permeance parameters for all combinations of rotor pole and stator tooth within each time step. The air-gap permeance matrix hence formed is utilised to solve the full SynRM model for each time step.

FE simulation of the SynRM has been performed in an ANSYS Maxwell Electromagnetic analysis software package. The simulations were run on a 2.5 GHz Intel^® Core-i5 machine with a 64-bit operating system and 16 GB of RAM. The MEC method recorded a run time of 52 s with a step size of 0.1 for four cycles of simulation, whereas the same model using FEA took around 9 min for one cycle of simulation. Thus, the developed model could play a vital role in the design stage, where several alternative designs need to be quickly evaluated. The Flux Density distribution in the various stator tooth represented by points A, B, C and D is shown in Figure 5. The flux density obtained from FEA and MEC are compared in

Table 3. Stator tooth flux density comparison.

	$\mathit{\theta }$ = 7.2°		$\mathit{\theta }$ = 18°		$\mathit{\theta }$ = 25°
	FEA	MEC	FEA	MEC	FEA	MEC
A	0.421	0.448	0.457	0.406	0.211	0.37
B	0.383	0.373	0.4	0.378	0.58	0.42
C	0.774	0.80	0.833	0.832	0.791	0.81
D	0.56	0.496	0.59	0.51	0.65	0.51

. It can be observed that the flux density predicted using an MEC model are within an acceptable range. Figure 6 shows the flux density distribution in the stator and rotor at rotor position of $\theta$ = 61.2${}^{\circ }$ for excitation Current = 1 A, 2 A, 3 A, 4 A and 5 A, respectively. This shows that the machine core is magnetically utilised to its fullest. However, due to the unmodelled saturation effects, an operating flux-density beyond the knee point of the core material needs additional consideration.

Figure 7 shows the SynRM Torque obtained with the MEC model compared to FEA. The torque in the SynRM model for various excitation currents ranging from 1 A to 5 A have been plotted. The result demonstrates that the MEC model is able to predict the instantaneous torque waveform fairly accurately and is a sufficient form to determine the feasibility of a given machine design. The maximum torque developed at different excitation levels predicted with the MEC and FEA is compared in Figure 8, and is also found to achieve a very close match.

5. Conclusions

In this paper, an MEC model for SynRM has been proposed. The main motivation for the proposed modeling approach is to obtain a model that can be applied to any stator teeth/ rotor pole combination. The MEC modelling technique developed in this research is then applied in a time-stepped simulation strategy, and the results are compared with finite element simulation results. Comparisons show that the proposed MEC approach offers good accuracy and is able to predict the characteristics of SynRM motors with reasonable accuracy. The model exhibits promising results to be utilised as a rapid design tool to assess the feasibility of a SynRM design in the machine design process.