Keywords

1 Introduction

The SRMs present many benefits for high-speed applications (e.g., electric compressor) compared with other types of machines. They can be operated at a very high-speed as they have no sliding contacts [1, 2], and can be operated in extreme temperature conditions. They are a competitor for the permanent-magnet synchronous machines in electric vehicle applications because of their simplicity and low cost, and their ability to operate at high speed with low maintenance [3,4,5].

In the literature, we find different methods of electromagnetic modeling of electric machines; semi-analytical modeling based on the magnetic equivalent circuit (or permeance network) [6, 7], subdomain method in linear conditions (i.e., infinite permeability of the iron parts) [8,9,10], and the exact subdomain method, which takes into account the iron permeability [11,12,13,14,15]. In addition, we find analytical methods based on multilayers [16,17,18,19] or elementary subdomains for the local saturation effect [20, 21].

The analytical methods cited previously give an accurate electromagnetic result compared with numerical calculations, with reduced computation time. In this paper, we will present a comparison study of 2-D electromagnetic performances between the developed linear model based on the subdomain method in linear conditions. In order to analyze the validity of the developed model, the results have been compared with those computed numerically using FEMM [22] in linear and non-linear conditions.

2 Analytical Model

The analytical model based on the subdomain method in linear conditions is given in [8,9,10]. In order to simplify the model, we have considered the following assumptions:

  • End-effects are neglected, i.e., A = {0;  0;  A z}.

  • Eddy-currents effects in all materials are neglected.

  • Current density in the stator slots has only one component along the z-axis, i.e., J = {0;  0;  J z}.

  • The slots have a radial sides.

  • The relative permeability is considered infinite for the iron parts (i.e., the saturation effect is neglected).

The schematic representation of the studied 8/6 conventional SRM is shown in Fig. 1.

Fig. 1
figure 1

Example of 8/6 conventional SRM

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In developing the 2D analytical model, a magnetic vector potential formulation is used in polar coordinates. It consists of solving the partial differential equation [23] due to

Maxwell’s equations in each domain of the studied machine. The equations to be solved in each region are given by

$$ \frac{\partial^2{A}_{zI}}{\partial {r}^2}+\frac{1}{r}\cdot \frac{\partial {A}_{zI}}{\partial r}+\frac{1}{r^2}\cdot \frac{\partial^2{A}_{zI}}{\partial {\theta}^2}=0\kern1em \left(\mathrm{Region}\ \mathrm{I}\right), \vspace*{-15pt}$$
(1a)
$$ \frac{\partial^2{A}_{zj}}{\partial {r}^2}+\frac{1}{r}\cdot \frac{\partial {A}_{zj}}{\partial r}+\frac{1}{r^2}\cdot \frac{\partial^2{A}_{zj}}{\partial {\theta}^2}=0\kern1em \left(\mathrm{Region}\ j\mathrm{th}\right).\vspace*{-15pt} $$
(1b)
$$ \frac{\partial^2{A}_{zi}}{\partial {r}^2}+\frac{1}{r}\cdot \frac{\partial {A}_{zi}}{\partial r}+\frac{1}{r^2}\cdot \frac{\partial^2{A}_{zi}}{\partial {\theta}^2}=-{\mu}_0\cdot {J}_{zi}\kern1em \left(\mathrm{Region}\ i\mathrm{th}\right). $$
(1c)

Solving of PDEs given by Eqs. (1a–1c) allows to obtain the general solution of A z in each domain [8,9,10]. The integration constants are determined by using a Fourier series expansion of A z in each region and the boundary conditions (BCs) [8,9,10,11,12,13,14,15,16,17,18,19,20,21]. The linear systeme can be written as

$$ \left[A\right]\cdot \left[X\right]=\left[B\right], $$
(2)

where:

  • [A] is the square matrix of the integration constants obtained by BCs of dimension Q × Q with Q = 4N + Q r ⋅ (1 + M) + Q s ⋅ (1 + K) in which Q s and Q r represent, respectively, number of the stator and rotor slots, and N, M, and K represent, respectively, the finite number of spatial harmonics terms in various regions.

  • [X] is the vector of unknowns (integration constants to determine) with dimension Q × 1.

  • [B] is the vector of electromagnetic sources terms with dimension Q × 1.

The vector [X] can divided into three parts as follows:

  • Part 1 is the air-gap (i.e., Region I) with dimension 4N × 1.

  • Part 2 is the rotor slots (i.e., Region jth) with dimension Q r ⋅ (1 + M) × 1.

  • Part 3 is the stator slots (i.e., Region ith) with dimension Q s ⋅ (1 + K) × 1.

Using the geometrical and physical parameters given in Table 1, we have calculated the computing time necessary for obtaining the vector [X] (viz., t = 58.85 s for 31 position) and the computing time necessary for making the mesh and analyze on FEMM (viz., t = 151.33 s for 31 positions. The auto-mesh is used).

Table 1 Parameters of 8/6 conventional SRM
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3 Simulation Results

The analytical expression of electromagnetic torque and the method for calculating the flux are given in [9, 10, 13,14,15,16,17,18,19,20,21]. We have used the parameters of SRM given in [24] for our comparison study.

Figure 2 shows the equipotential lines of A z in the machine due to phase A (I A = 20 A) at 180° rotor position obtained by numerical model. Figure 3 shows the radial and tangential magnetic flux density in the middle of the air-gap by feeding only phase A (I = 10 A) for rotor position 45°.

Fig. 2
figure 2

Equipotential lines of A z due to phase A for I A = 20 A and 180° rotor position

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Fig. 3
figure 3

Radial (a) and tangential (b) components of magnetic flux density in the air-gap for I A = 10 A and rotor position 45°

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Figure 4 shows a comparison between the numerical and analytical result of flux per phase due to phase A. In linear (I = 10 A), the numerical and analytical results are well in agreement. In non-linear, the relative error between analytical and numerical results is 46.56%. The mutual flux between phase A and others phases is shown in Fig. 5. In non-linear, the analytical and numerical results present 49.12% relative error. The mutual flux between phases A–C is null because the opening between this phases is π/2.

Fig. 4
figure 4

Waveform of flux per phase due to phase A

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Fig. 5
figure 5

Waveform of mutual flux obtained by feeding phase A in: (a) linear (I A = 10 A) and (b) non-linear (I A = 50 A)

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Figure 6 shows the comparison between analytical and numerical results of static electromagnetic torque due to phase A. It can be seen that the analytical model gives the accurate results in linear. In non-linear, we have a 42.15% relative error between the analytical and numerical results. This important relative error due to no considering the relative permeability of iron parts in the analytical model.

Fig. 6
figure 6

Waveform of the static electromagnetic torque due to phase A in: (a) linear (I A = 10 A) and (b) non-linear (I A = 50 A)

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Figure 7 shows the comparison between analytical model and non-linear numerical results of the maximum electromagnetic torque due to phase A for different values of current (0–100 A). The results of the max self-inductance due to phase A for different values of current (0–100 A) obtained by analytical model and non-linear numerical model are shown in Fig. 8. It can be seen that the analytical model gives a good result and the non-linear numerical results in limited current rang (I ≤ 30 A for max torque, and I ≤ 20 A for max self-inductance).

Fig. 7
figure 7

The maximum electromagnetic torque due to phase A for different values of current

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Fig. 8
figure 8

The maximum self-inductance due to phase A for different values of current

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4 Conclusion

In this paper, we have presented an analytical model based on the subdomain method of 8/6 conventional SRM in linear conditions. The analytical results are in excellent agreement with numerical ones. However, in non-linear conditions (I = 50 A), the developed model overestimates the electromagnetic performances with a maximum error of about 46%. However, by taking this error into account, this model can be effectively used in the optimization procedure, in which saving of computation time is required. In order to obtain more accurate performances in non-linear conditions, this model can be extended to the saturation case (taking into account the characteristic of relative permeability of iron parts). It will be the object of our future works.