## 1. Introduction

Benefiting from the advantages of a simple mechanical structure—the rotor does not carry any windings, commutators, or permanent magnets (PMs)—and a robust, fault-tolerant nature, low-cost maintenance, high-thermal capability, and high-speed potential [

1,

2,

3], SRM is receiving renewed attention as a viable candidate for various adjustable-speed and high-torque applications such as in the automotive and traction fields [

4,

5,

6,

7,

8].

However, a major disadvantage of this machine is the undesirable electromagnetic vibration and acoustic noise, which are mainly excited by the radial UMF acting on the salient stator and rotor poles [

9,

10]. Moreover, it poses a drawback for SRM in noise-sensitive applications and still creates bottlenecks in vehicle propulsion. It is important to consider noise and vibration problems during the process of electrical machine design. Electrical machine noise and vibration are mainly of electromagnetic, aerodynamic, and mechanical origin, the most important of which are generated by electromagnetic sources [

11]. Also, in SRM, the attraction magnetic force can be divided into tangential and radial components relative to the rotor. The tangential magnetic force is converted into rotational torque, and the radial magnetic force converts into magnetic pressure equal to the radial magnetic force per unit area of the stator tooth and UMFs, which contributes to the radial vibration behaviour and therefore the motor noise [

9,

12]. A perfect machine with balanced stator windings should have net zero UMFs on the stator structure. However, UMFs can be present in machines having diametrically asymmetric disposition of slots and phase windings [

13,

14]. This magnetic force acts on the stator of these machine configurations due to an asymmetric magnetic field distribution in the air gap.

In the interest for design and optimization of electrical machines, there are various modelling methods; the first step in these is the magnetic field calculation. Some comprehensive reviews of the models of electrical machines for magnetic field prediction along with their (dis)advantages can be found in [

15,

16,

17,

18,

19,

20,

21,

22,

23,

24] and their references. Currently, the Maxwell‒Fourier method is one of the most used semi-analytical methods, and combines the very accurate electromagnetic performances calculation with a reduced computation time compared to numerical methods. In models from this method (viz., multi-layer models, eigenvalues model, and subdomain technique), the magnetic field solutions are based on the formal resolution of Maxwell’s equations by using the separation of variables method and the Fourier’s series. In electromagnetic devices, the major assumption is that an infinite permeability of iron parts has to be assumed [

25]. Therefore, the global and/or local saturation effect is neglected. It is interesting to note that an overview of the existing (semi-)analytical models in the Maxwell‒Fourier method with a global and/or local saturation effect has been realized in [

24], where some details and the (dis)advantages of these techniques can be found. To overcome that issue, Spranger et al. (2016) [

21] and Dubas et al. (2017) [

24,

26] have recently developed new techniques to account for finite soft-magnetic material permeabilities:

multi-layer models using the convolution theorem (i.e., Cauchy’s product theorem). The adjacent regions (e.g., rotor and/or stator slots/teeth) are assumed to be one homogeneous region with a relative permeability developed as a Fourier’s series expansion;

the subdomain technique using a superposition that allows for any non-periodic subdomain. The subdomain connection is performed directly in both directions. The general solutions of Maxwell’s equations are deduced by applying the principle of superposition by respecting the BCs on the various edges of subdomains.

For the same reason, another technique based on subdomain technique and Taylor polynomial has been developed and only applied in spoke-type PM synchronous machines (PMSM) [

27,

28]. Spranger’s approach has been extended and used in different machines with only the global saturation effect. It has been applied with the finite soft-magnetic material permeability in synchronous reluctance machine [

29], surface-mounted PMSM [

30], and many structures of PMSMs (i.e., for inset-/surface-/spoke-type PMSMs with different PM magnetization patterns and internal/external rotor) [

31], with the nonlinear

B(H) curve in switched reluctance machine [

32,

33]. The Dubas superposition technique has been implemented in radial-flux electrical machines with(out) PMs supplied by a direct or alternate current (with any waveforms) [

34]. This technique has been extended to: (i) the thermal modelling for the steady-state temperature distribution in rotating electrical machines [

35], and (ii) elementary subdomains in the rotor and stator regions for full prediction of magnetic field in rotating electrical machines with the local saturation effect solving by the Newton‒Raphson iterative algorithm [

36]. The Dubas superposition technique is very interesting since, like Spranger’s approach, it enables the magnetic field calculation in iron parts of slotted structures. Apart from its complexity, the main downfall of Spranger’s approach is that it suffers from the Gibb’s phenomenon at boundaries between slots and teeth. This introduces inaccuracies in the computation of the field and results in higher computational times [

37].

In this paper, the authors propose applying the Dubas superposition technique in polar coordinates [

26] to SRM with sinusoidal current excitation, which has not yet been realized in the literature. The soft magnetic material permeability is constant corresponding to the linear zone of the

B(H) curve. Nevertheless, as in [

33,

36], it should be mentioned that the material properties could be updated iteratively to take the nonlinear

B(H) curve of the material into account. However, this is beyond the scope of the paper. In this investigation, the magnetic flux density distribution inside the machine, electromagnetic performances and non-intrinsic UMFs have been calculated for 6/4 SRM supplied by sinusoidal waveform of current with two various non-overlapping (or concentrated) windings. One of the case studies is a M1 with a non-overlapping all teeth wound winding (double-layer winding with left and right layer) and the other is a M2 with a non-overlapping alternate teeth wound winding (single-layer winding). All results obtained with the proposed semi-analytical model are verified by 2D FEM [

38] for different values of iron core relative permeability (viz., 100 and 800). The comparisons with FEM show good results.

## 2. Studied SRMs and Magnetic Field Solutions

#### 2.1. Machine Geometry and Assumptions

Figure 1 represents the studied SRMs having two various non-overlapping windings: (i) M1 with a non-overlapping all teeth wound winding (double-layer winding with left and right layer) (see

Figure 1a), and (ii) M2 with a non-overlapping alternate teeth wound winding (single-layer winding) (see

Figure 1b). The three-phase SRMs have six stator slots and four rotor slots, and do not contain any stator tooth tips. The main geometrical parameters of two studied SMRs are shown in

Figure 1 and are given in

Table 1 for the semi-analytical and numerical comparisons. These machines have been partitioned into nine regions as shown on

Figure 2, viz.,

Region I is the air gap;

Regions II and III are the rotor yoke (i.e., between rotor shaft and rotor slots/teeth) and the stator yoke, respectively;

Region IV is the rotor slots;

Region V is the rotor teeth;

Regions VI and VII are the stator slots of the first layer (i.e., right in the slot) and second layer (i.e., left in the slot), respectively;

Region VIII is the stator teeth;

Region XI is the non-periodic air gap (i.e., between the two-layer winding of the stator slots).

The semi-analytical model, based on the exact subdomain technique, is formulated in 2D, in polar coordinates, and in magnetic vector potential with the following assumptions:

The end-effects are neglected, i.e., $\mathit{A}=\left\{0;0;{A}_{z}\right\}$;

The eddy-current effects in the materials are neglected;

The current density in the stator slots has only one component along the z-axis, i.e., $\mathit{J}=\left\{0;0;{J}_{z}\right\}$;

The magnetic materials are considered as isotropic with constant magnetic permeability corresponding to linear zone of the B(H) curve;

The stator and rotor slots/teeth have radial sides (see

Figure 2).

However, it accounts for:

The internal/external rotor topology;

The saturation, slotting and curvature effect;

The (non-)overlapping winding distribution;

Any current waveform (i.e., sinusoidal or rectangular).

#### 2.2. General Solution with Non-Homogeneous Neumann BCs

Magnetic vector potential

$\mathit{A}$ is calculated analytically with solving the magnetostatic Maxwell’s equations with the separation of variables method, viz.,

where

${\mu}_{0}$ is the vacuum permeability.

According to [

24,

26], the solutions to

$\mathit{A}$ in all regions of conventional SRM are:

**Air gap subdomain (Region I)**: The solution of (1) in Region I,

$r\in \left[{R}_{3};{R}_{4}\right]$ &

$\forall \theta $, is defined by:

where

n is a positive integer, and

$\left\{{A}_{10};{A}_{20};{A}_{1n};{A}_{4n}\right\}$ are the integration constants of Region I.

**Stator and rotor yoke subdomain (Region II and III)**: In adding Dirichlet BC of

**A** at

$r={R}_{1}$ and

$r={R}_{ext}$, viz.,

${A}_{zII}\left({R}_{1},\theta \right)=0$ &

${A}_{zIII}\left({R}_{ext},\theta \right)=0$, the solution of (1) in Region II,

$r\in \left[{R}_{1};{R}_{2}\right]$ &

$\forall \theta $, can be written as:

where

$\left\{{A}_{50};{A}_{5n};{A}_{6n}\right\}$ are the integration constants of Region II.

The solution of Region III, $r\in \left[{R}_{5};{R}_{ext}\right]$ & $\forall \theta $, is similar to (4) by replacing $\left\{{A}_{50};{A}_{5n};{A}_{6n}\right\}$ with $\left\{{A}_{70};{A}_{7n};{A}_{8n}\right\}$ and ${R}_{1}$ with ${R}_{ext}$.

**i****-th Stator slot subdomain (Region VI and VII)**: The solution of (2) in Region VI,

$r\in \left[{R}_{4};{R}_{5}\right]$ &

$\theta \in \left[{\gamma}_{1i}-f/2;{\gamma}_{1i}+f/2\right]$, is defined by:

where

$m$ and

$k$ are positive integers,

${\gamma}_{1i}={\gamma}_{i}-\left(e+f\right)/2$ and

$f$ are respectively the position and opening width of first layer winding in the

i-th stator slot,

$\left\{{C}_{1i0};{C}_{2i0};{C}_{1im};{C}_{2im};{C}_{3ik};{C}_{4ik}\right\}$ are the integration constants of Region VI,

${v}_{mf}=m\pi /f$ and

${\lambda}_{ks}=k\pi /\mathrm{ln}\left({R}_{5}/{R}_{4}\right)$ are respectively the periodicity of

${A}_{zVIi}$ in

θ-and

r-edges.

The solution of Region VII, $r\in \left[{R}_{4};{R}_{5}\right]$ & $\theta \in \left[{\gamma}_{2i}-f/2;{\gamma}_{2i}+f/2\right]$, is similar to (5) by replacing $\left\{{C}_{1i0};{C}_{2i0};{C}_{1im};{C}_{2im};{C}_{3ik};{C}_{4ik}\right\}$ with $\left\{{C}_{5i0};{C}_{6i0};{C}_{5im};{C}_{6im};{C}_{7ik};{C}_{8ik}\right\}$, $J1{\left(i\right)}_{z}$ with $J2{\left(i\right)}_{z}$, and ${\gamma}_{1i}$ with ${\gamma}_{2i}={\gamma}_{i}+\left(e+f\right)/2$.

**i****-th Non-periodic air gap and i-th stator tooth subdomain (Region XI and VIII)**: The solution of (1) in Region VIII, $r\in \left[{R}_{4};{R}_{5}\right]$ & $\theta \in \left[{\gamma}_{i}-e/2;{\gamma}_{i}+e/2\right]$, and in Region XI, $r\in \left[{R}_{4};{R}_{5}\right]$ & $\theta \in \left[{\delta}_{i}-d/2;{\delta}_{i}+d/2\right]$, can be obtained directly from (5) with $J1{\left(i\right)}_{z}=0$.

For Region VIII, $\left\{{C}_{1i0};{C}_{2i0};{C}_{1im};{C}_{2im};{C}_{3ik};{C}_{4ik}\right\}$ is replaced by $\left\{{D}_{1i0};{D}_{2i0};{D}_{1im};{D}_{2im};{D}_{3ik};{D}_{4ik}\right\}$, ${\gamma}_{1i}$ by ${\gamma}_{i}$, $f$ by $e$, and ${v}_{mf}$ by ${v}_{me}=m\pi /e$.

For Region IX, $\left\{{C}_{1i0};{C}_{2i0};{C}_{1im};{C}_{2im};{C}_{3ik};{C}_{4ik}\right\}$ is replaced by $\left\{{D}_{1i0};{D}_{2i0};{D}_{1im};{D}_{2im};{D}_{3ik};{D}_{4ik}\right\}$, ${\gamma}_{1i}$ by ${\delta}_{i}$, $f$ by $d$, and ${v}_{mf}$ by ${v}_{md}=m\pi /d$.

**j****-th Rotor slot and j-th rotor tooth subdomain (Region IV and V)**: The solution of (1) in Region IV,

$r\in \left[{R}_{2};{R}_{3}\right]$ &

$\theta \in \left[{\alpha}_{j}-a/2;{\alpha}_{j}+a/2\right]$, is defined by:

where

${\alpha}_{j}$ and

$a$ are respectively the position and opening width of

j-th rotor slot,

$\left\{{B}_{1j0};{B}_{2j0};{B}_{1jm};{B}_{2jm};{B}_{3jk};{B}_{4jk}\right\}$ are the integration constants of Region IV,

${v}_{ma}=m\pi /a$ and

${\lambda}_{kr}=k\pi /\mathrm{ln}\left({R}_{3}/{R}_{2}\right)$ are respectively the periodicity of

${A}_{zIVj}$ in

θ-and

r-edges.

The solution of Region V, $r\in \left[{R}_{2};{R}_{3}\right]$ & $\theta \in \left[{\beta}_{j}-b/2;{\beta}_{j}+b/2\right]$, is similar to (6) by replacing $\left\{{B}_{1j0};{B}_{2j0};{B}_{1jm};{B}_{2jm};{B}_{3jk};{B}_{4jk}\right\}$ with $\left\{{B}_{5j0};{B}_{6j0};{B}_{5jm};{B}_{6jm};{B}_{7jk};{B}_{8jk}\right\}$, $a$ with $b$, and ${\alpha}_{j}$ with ${\beta}_{j}$.

#### 2.3. Magnetic Flux Density

The field vectors

$\mathit{B}=\left\{{B}_{r};{B}_{\theta};0\right\}$ and

$\mathit{H}=\left\{{H}_{r};{H}_{\theta};0\right\}$ are coupled by:

where

${\mu}_{rc}$ is the relative recoil permeability of iron parts.

Using

$\mathit{B}=\nabla \times \mathit{A}$, the components of

$\mathit{B}=\nabla \times \mathit{A}$ can be deduced by

#### 2.4. Stator Current Density Source

The stator current densities in the stator slots for double-layer concentrated winding are defined as [

33]:

where

${i}_{g}=\left[\begin{array}{lll}{i}_{a}& {i}_{b}& {i}_{c}\end{array}\right]$ is the vector of phase currents whose currents’ waveform is sinusoidal with a phase shift of

$2\pi /3$ electric,

$S=f\cdot \left({R}_{5}{}^{2}-{R}_{4}{}^{2}\right)/2$ is the surface of the stator slot coil, and

${C}_{\left(1\right)}^{T}$ &

${C}_{\left(2\right)}^{T}$ are the transpose of the connection matrix between the three phases and the stator slots that represent the distribution of stator windings in the slots of the M1 with all teeth wound (double-layer winding with left and right layer) (see

Figure 1a) is given by [

33]:

For the M2 with alternate teeth wound (single-layer winding) (see

Figure 1b), the same model is used with few modifications:

These connection matrices can be generated automatically by using the ANFRACTUS TOOL developed in [

39].

#### 2.5. Boundary Conditions

The ICs in this semi-analytical model can be divided into two types, viz.,

**θ-edges ICs:** over angle interval for given radius value $\left\{{R}_{2};{R}_{3};{R}_{4};{R}_{5}\right\}$;

**r-edges ICs:** over radius interval for given angle $\left\{{\alpha}_{j}\pm a/2;{\beta}_{j}\pm b/2;{\gamma}_{i}\pm c/2;{\delta}_{i}\pm d/2;{\gamma}_{i}\pm e/2\right\}$.

Therefore, we obtain on the:

- -
The ICs between Region II, IV and V at

$r={R}_{2}$ as:

- -
The ICs between Region I, IV and V at $r={R}_{3}$ are similar to (13)–(16) by replacing II with I and ${R}_{2}$ with ${R}_{3}$.

- -
The ICs between Region I, VI, VII, VIII and XI at

$r={R}_{4}$ as:

- -
The ICs between Region III, VI, VII, VIII and XI at $r={R}_{5}$ are similar to (17)–(24) by replacing I with III and ${R}_{4}$ with ${R}_{5}$.

- -
The ICs between Region IV and V at

${\alpha}_{j}+a/2={\beta}_{j}-b/2$ and

${\alpha}_{j+1}-a/2={\beta}_{j}+b/2$ for

$r\in \left[{R}_{2};{R}_{3}\right]$:

- -
The ICs between Region VII and VIII at

${\gamma}_{i}+c/2={\delta}_{i}-d/2$ and between Region VI and VIII at

${\gamma}_{i+1}-c/2={\delta}_{i}+d/2$ for

$r\in \left[{R}_{4};{R}_{5}\right]$:

- -
The ICs between Region VI and XI at

${\gamma}_{i}-e/2={\gamma}_{i}-c/2+f$ and between Region VII and XI at

${\gamma}_{i}+e/2={\gamma}_{i}+c/2-f$ for

$r\in \left[{R}_{4};{R}_{5}\right]$:

The system of the 36 BCs matrix (Equations (13)–(36)) is used to determine the coefficients of **A** in nine regions.

Figure 3 briefly represents a flowchart of the subdomain technique.

To solve the Cramer’s system, the number of integration constants is equal to $2\cdot \left(4N+2\right)+2{Q}_{r}\cdot \left(2+2M+2K\right)+4{Q}_{s}\cdot \left(2+2M+2K\right)$ where $N$, $M$ and $K$ are the finite numbers of spatial harmonic terms in the various regions.