# New Scientiﬁc Contribution on the 2-D Subdomain Technique in Cartesian Coordinates: Taking into Account of Iron Parts

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## Abstract

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## 1. Introduction

#### 1.1. Context of this Paper

- Graphical method of Lehmann (1909) [7];

#### 1.2. State-of-the-Art: Saturation in Maxwell-Fourier Methods

- Multi-layers models (only the global saturation):
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- Carter’s coefficient: The slotted machine is transformed into a slotless equivalent structure by applying the usual Carter’s coefficient [28]. Generally, the armature slotting is taken into account through the SC mapping method. The analytical magnetic field distribution is determined in five or six homogeneous layers (i.e., exterior, slotless stator, winding/air-gap, magnets, and rotor) [29,30,31]. In [29], the magnetic permeabilities in stator/rotor iron cores have a constant value corresponding to linear zone of the $B\left(H\right)$ curve. An iterative technique to include the nonlinear properties of core material has been developed in [30] (for a no-load operation) and [31] (for a load operation whose the source term in the slot caused by the armature currents is represented by a winding current region over the stator slot-isthmus). In this type of modeling, the local distribution of flux densities in the teeth and slots is neglected. However, by calculating the flux entering the stator surface from the air-gap magnetic field and thus assuming uniform distribution of flux, the flux density in middle of the stator teeth can be obtained.
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- Saturation coefficient: It represents the ratio between the total magnetomotive force (MMF) required for the entire magnetic circuit and the air-gap MMF [32]. The main magnetic saturation is included in the saturation factor, in an iterative way, by using the nonlinear $B\left(H\right)$ curve. The saturation effect is accounted for by modifying the air-gap length [32,33,34] or by changing the physical properties of magnets (in this case, the saturated load operation is calculated by considering an equivalent no-load operation with a fictitious magnet having a remanent flux density that creates the same MMF as the one created by both real magnet and stator MMF) [35]. The analytical magnetic field distribution is mainly determined in one or two regions (viz., air-gap or air-gap/magnets) of slotless machines by applying the Carter’s coefficient [32]. The slotting effect can be neglected [32,35] or taken into account through the SC mapping method [33,34]. The magnetic fluxes in the stator/rotor iron cores are obtained from the air-gap magnetic field [32,33,35] or/and with a simple MEC [34]. This technique has been applied to surface-mounted/-inset magnets machines [32,33,34,35], surface-inset magnet machines [33], and others electrical machines.
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- Concept wave impedance: They are based on a direct solution of Maxwell’s field equations in homogeneous multi-layers of magnetic material properties, viz., the magnetic permeability and the electrical conductivity. This approach, developed by Mishkin (1953) [36], was first applied to squirrel-cage induction machine in Cartesian coordinates with three-layers (i.e., stator slotting, air-gap, and rotor slotting). It was used and enhanced by many authors, viz.,
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- simplification of the electromagnetic theory [37];
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- extended with an infinite number of layers [38];
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- converted into equivalent circuits and terminal impedance [39];
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- included the curvature effect with the magnetizing current [40];
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- taking account of the slot-opening effect [45], i.e., the multi-layers model is combined with the subdomain technique for slotted structures by assuming infinitely permeable tooth-tips;
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- Convolution theorem: The electrical machine is divided into an infinite number of (in)homogeneous layers. The permeability in the stator and/or rotor slotting is represented by a complex Fourier series along the direction of permeability variation The permeability variation in the direction of the periodicity is directly included into the solution of the magnetic field equation. The resulting formulation, based on a direct solution of Maxwell’s field equations using the Cauchy’s product theorem (i.e., the discrete convolution of two infinite series), is completely defined in terms of complex Fourier series [47]. Recently, Djelloul et al. (2016) [48] extended this modeling taking into account the nonlinear $B\left(H\right)$ curve in each soft-magnetic section by an iterative procedure. For the moment, this technique has been applied to a switched reluctance machine [48] and a synchronous reluctance machine [49].

- Eigenvalues model (only the global saturation): The electromagnetic field can be solved directly by applying the method of Truncation Region Eigenfunction Expansions (TREE) [50]. The iron cores have finite magnetic permeability and finite height/width. The studied domains can be (non)conductive. The boundary value problem is formulated in terms of the magnetic vector potential, which is expanded in a series of appropriate eigenfunctions. The unknown coefficients of the series are computed by solving a matrix system (by using a standard method such as the lower upper decomposition), which is formed by applying the usual interface conditions. The corresponding eigenvalues are the real roots of a function with the geometrical and the magnetic permeability of the core as parameters. Nevertheless, an iterative numerical method (e.g., the bisection [50] and Newton-Raphson [51] method) is always adopted to compute the discrete eigenvalues in both the odd and even parity solutions. For the moment, this technique has been widely applied to the non-destructive testing of conductive materials (e.g., for the I-cored [50] and E-cored [52,53] probes, for a long coil with a slot in a conductive plate [51], etc.).
- Hybrid models (the local/global saturation): The analytical solution can be combined with numerical methods [54,55,56,57] or (non)linear MEC [58,59,60,61,62,63,64,65,66,67]. Usually, the analytical solution is established in uniform regions of very low permeability (e.g., air-gap, and magnets) and other methods are sought in regions where magnetic saturation cannot be neglected (i.e., the stator and/or rotor iron cores).

#### 1.3. Objectives of this Paper

## 2. A 2-D Subdomain Technique of Magnetic Field

#### 2.1. Problem Description and Assumptions

- The end-effects are neglected (i.e., that the magnetic variables are independent of z);
- The electrical conductivities of materials are assumed to be null (i.e., the eddy-currents induced in the copper/iron are neglected);
- The magnetic materials are considered as isotropic (i.e., the permeability can be assumed the same in the two directions);
- The effect of global saturation is taken into account with a constant magnetic permeability corresponding to linear zone of the $B\left(H\right)$ curve (i.e., the initial magnetization curve).

#### 2.2. Problem Discretization in Subdomains

- Region 1 $\left\{\forall x\wedge y\in \left[{y}_{1},\phantom{\rule{0.277778em}{0ex}}{y}_{2}\right]\right\}$ with ${\mu}_{1}={\mu}_{v}$;
- Region 2 $\left\{\forall x\wedge y\in \left[{y}_{3},\phantom{\rule{0.277778em}{0ex}}{y}_{4}\right]\right\}$ with ${\mu}_{2}={\mu}_{v}$;
- Region 3 $\left\{x\in \left[{x}_{1},\phantom{\rule{0.277778em}{0ex}}{x}_{2}\right]\wedge y\in \left[{y}_{2},\phantom{\rule{0.277778em}{0ex}}{y}_{3}\right]\right\}$ with ${\mu}_{3}={\mu}_{v}$;
- Region 4 $\left\{x\in \left[{x}_{5},\phantom{\rule{0.277778em}{0ex}}{x}_{6}\right]\wedge y\in \left[{y}_{2},\phantom{\rule{0.277778em}{0ex}}{y}_{3}\right]\right\}$ with ${\mu}_{4}={\mu}_{v}$.

- Region 6 $\left\{x\in \left[{x}_{2},\phantom{\rule{0.277778em}{0ex}}{x}_{3}\right]\wedge y\in \left[{y}_{2},\phantom{\rule{0.277778em}{0ex}}{y}_{3}\right]\right\}$ with ${\mu}_{6}={\mu}_{c}$;
- Region 7 $\left\{x\in \left[{x}_{4},\phantom{\rule{0.277778em}{0ex}}{x}_{5}\right]\wedge y\in \left[{y}_{2},\phantom{\rule{0.277778em}{0ex}}{y}_{3}\right]\right\}$ with ${\mu}_{7}={\mu}_{c}$.

#### 2.3. Governing Partial Differential Equations in Cartesian Coordinates

#### 2.4. Boundary Conditions

#### 2.4.1. Reminder on the Boundary Conditions at the Interface of Two Surfaces

#### 2.4.2. Application to the Air- or Iron-Cored Coil

#### 2.5. General Solutions

#### 2.5.1. Region 1

#### 2.5.2. Region 2

#### 2.5.3. Region 3

#### 2.5.4. Region 4

#### 2.5.5. Region 5

#### 2.5.6. Region 6

#### 2.5.7. Region 7

#### 2.6. Solving of Linear System

#### 2.7. Numerical Problems: Harmonics and Ill-Conditioned System

## 3. Comparison of the Semi-Analytic and Finite-Element Calculations

#### 3.1. Introduction

#### 3.2. Results Discussion

## 4. Conclusions

## Author Contributions

## Conflicts of Interest

## Appendix A The 2-D Magnetostatic General Solution in Cartesian Coordinates

#### Appendix A.1 Governing Partial Differential Equations (EDPs)

#### Appendix A.2 General Solution

## Appendix B Simplification of Laplace’s Equations According to Imposed Boundary Conditions

#### Appendix B.1 Case-Study No 1: A_{z} imposed on all edges of a region

**Figure B.1.**${A}_{z}$ imposed on all edges of a region: (

**a**) General and (

**b**) Principle of superposition.

#### Appendix B.2 Case-Study No 2: B_{y} and A_{z} are respectively imposed on x- and y-edges of a region

**Figure B.3.**${B}_{y}$ imposed on x-edges and ${A}_{z}$ imposed on y-edges of a region: (

**a**) General and (

**b**) Principle of superposition.