Low-Frequency Electromagnetic Fields and Finite Element Method

Abstract

Electromagnetic field analysis is the basis for solving the engineering coupled electromagnetic and thermal field problems. Based on the low-frequency Maxwell’s equations, some key problems concerning the formulations and numerical implementations of typical 3-D eddy current analysis methods, using different potential sets, such as A-V-A (or employing a reduced vector magnetic potential A_r to convert to A_r-V-A_r) and T-Ψ-Ψ, are briefly explained. Furthermore, the numerical solvers based on different potential sets have been developed by the author’s group and verified in the Testing Electromagnetic Analysis Methods (TEAM) benchmarking practices. In this chapter, the Galerkin weighted residual method, a key technique in numerical implementation, is elaborated, and the effectiveness of edge element, for example, in effectively reducing computational cost in industrial applications is discussed. Strengthening the theoretical basis of finite element analysis of electromagnetic fields and correctly understanding the significance of the combination of advanced numerical computation with accurate material property modeling will be more helpful in improving the effectiveness of modeling and simulation and further promoting the use of simulation in industrial applications.

Appendices

Appendix: Formulation of A-V-A and Galerkin Weighted Residual Processing

The Galerkin’s weighted residual processing is demonstrated based on the potential set of A-V-A using nodal element. Although the model shown in Fig. 2.3 looks simple, but it does not lose generality. For the more general case, there is no substantial difference or difficulty in the derivation process here. This is helpful for knowing basic derivation process.

Fig. 2.3

A-V-A model

Basic Model of A-V-A

In the A-V-A model, A (magnetic vector potential) and V (electric scalar potential) are used in the conductor—eddy current region, and only the magnetic vector potential A is used outside the conducting region.

Where σ and μ are the conductivity and permeability of the conducting material, respectively, and the magnetic anisotropy and nonlinearity needed to be considered. In the non-conducting region, the conductivity σ is zero, and the relative permeability μ_r is equal to 1.

The symbols in Fig. 2.3 are given the following meanings:

ω :

non-eddy current region, Ω: eddy current region

S :

interface between eddy current region and non-eddy current region

$\varvec{n}$ :

exterior normal unit vector of conductor surface in eddy current region

$\varvec{n}^{{\prime }}$ :

internal normal unit vector of conductor surface in eddy current region, where

$$ \varvec{n}^{{\prime }} = - \varvec{n} $$

(2.27)

Γ: The outer boundary of the whole solution domain, including generally the following two types:

$$ \Gamma _{h} {:}\,\varvec{H} \times \varvec{n} = 0 $$

(2.28)

$$ \Gamma _{b}{:}\, \varvec{B} \cdot \varvec{n} = 0 $$

(2.29)

A :

magnetic vector potential (Wb/m)

V :

integral quantity of electric scalar potential V′ to time (Vs)

$$ V = \int V^{{\prime }} {\text{d}}t $$

(2.30)

μ :

permeability of conductor (H/m)

σ :

Conductor conductivity (S/m), where the material is set to electric linear and σ is constant

Considering that the permeability of the material in the eddy current region of conductor is treated as anisotropic in any direction, its general tensor form is

$$ \bar{\mu } = \left[ {\begin{array}{*{20}c} {\mu_{xx} } & {\mu_{xy} } & {\mu_{xz} } \\ {\mu_{yx} } & {\mu_{yy} } & {\mu_{yz} } \\ {\mu_{zx} } & {\mu_{zy} } & {\mu_{zz} } \\ \end{array} } \right] $$

(2.31)

The reluctivity of the material should, therefore, be expressed as follows:

$$ \bar{v} = \left[ {\bar{\mu }} \right]^{ - 1} $$

It should be pointed out that only for the sake of simplicity of deduction, the anisotropic permeability in the following formula is simply written as μ, and the anisotropic reluctivity is simply written as $\frac{1}{\mu }$.

Definition: $\varvec{B} = \nabla \times {\varvec{A}}$.

Governing Equation of A-V-A

The governing equations of A-V-A are as follows:

3.1 Eddy Current Region

$$ \nabla \times \frac{1}{\mu }\nabla \times \varvec{A} - \text{[}\nabla \frac{1}{{\mu_{c} }}\nabla \cdot \varvec{A}\text{]} + \sigma \left( {\frac{{\partial \varvec{A}}}{\partial t} + \nabla \frac{\partial V}{\partial t}} \right) = 0 $$

(2.32)

In square brackets of Eq. (2.32) is a penalty function term to enforce the zero divergence condition, the same below, where μ_c needs to be isotropic to ensure the symmetry of the coefficient matrix. μ_c can be determined by the permeability of neighboring elements and updated during iteration.

$$ \nabla \cdot \sigma \left( { - \frac{{\partial \varvec{A}}}{\partial t} - \nabla \frac{\partial V}{\partial t}} \right) = 0 $$

(2.33)

3.2 Non-Eddy Current Region

$$ \nabla \times \frac{1}{{\mu_{0} }}\nabla \times \varvec{A} - \nabla \frac{1}{{\mu_{0} }}\nabla \cdot \varvec{A} = 0 $$

(2.34)

where μ₀: permeability in air.

Galerkin Weighted Residual Processing

4.1 Galerkin Residuals

The Galerkin weighted residual technique is applied to (2.32)–(2.34), and the continuity condition of the boundary field quantity (B, H) is considered.

$$ \left( {\varvec{B}_{1} - \varvec{B}_{2} } \right) \cdot \varvec{n} = 0 $$

(2.35)

$$ \left( {\varvec{H}_{1} - \varvec{H}_{2} } \right) \times \varvec{n} = 0 $$

(2.36)

The discretization equation is derived. The subscripts 1 and 2 indicate both sides of the interface.

The Galerkin residuals corresponding to (2.32) and (2.33), using scalar weight function, are, respectively, as follows:

$$ \varvec{R}_{\mathbf{1}} = \mathop \int \limits_{\Omega } N_{i} \left( {\nabla \times \frac{1}{\mu }\nabla \times \varvec{A} - \nabla \frac{1}{{\mu_{c} }}\nabla \cdot \varvec{A} + \sigma \left( {\frac{{\partial \varvec{A}}}{\partial t} + \nabla \frac{\partial V}{\partial t}} \right)} \right)\varvec{ }{\text{d}}v $$

(2.37)

$$ R_{0} = \mathop \int \limits_{{\varvec{\Omega}}} N_{i} \nabla \cdot \sigma \left( { - \frac{{\partial \varvec{A}}}{\partial t} - \nabla \frac{\partial V}{\partial t}} \right)\varvec{ }{\text{d}}v $$

(2.38)

where N_i is a scalar weight function. For ease of derivation, R₁ is rewritten separately as three terms

$$ R_{11} = \mathop \int \limits_{\Omega } N_{i} \nabla \times \frac{1}{\mu }\nabla \times \varvec{A }\,{\text{d}}v $$

(2.39)

$$ R_{12} = - \mathop \int \limits_{\Omega } N_{i} \nabla \frac{1}{{\mu_{c} }}\nabla \cdot \varvec{A }\,{\text{d}}v $$

(2.40)

$$ R_{13} = \mathop \int \limits_{\Omega } N_{i} \sigma \left( {\frac{{\partial \varvec{A}}}{\partial t} + \nabla \frac{\partial V}{\partial t}} \right)\varvec{ }{\text{d}}v $$

(2.41)

4.2 Residuals Processing

4.2.1 Eddy Current Region

For (2.39), the vector formulation is

$$ \nabla \times \left( {N_{i} \frac{1}{\mu }\nabla \times \varvec{A}} \right) = \nabla N_{i} \times \frac{1}{\mu }\nabla \times \varvec{A} + N_{i} \nabla \times \frac{1}{\mu }\nabla \times \varvec{A } $$

(2.42)

(2.42) is sorted out to:

$$ N_{i} \nabla \times \frac{1}{\mu }\nabla \times \varvec{A} = \nabla \times \left( {N_{i} \frac{1}{\mu }\nabla \times \varvec{A}} \right) - \nabla N_{i} \times \frac{1}{\mu }\nabla \times \varvec{A } $$

(2.43)

Substituting (2.43) into (2.39)

$$ R_{11} = \mathop \int \limits_{{\Omega } } \nabla \times \left( {N_{i} \frac{1}{\mu }\nabla \times \varvec{A}} \right)\varvec{ }{\text{d}}v - \mathop \int \limits_{{\Omega } } \nabla N_{i} \times \frac{1}{\mu }\nabla \times \varvec{A}\,\varvec{ }{\text{d}}v $$

(2.44)

The vector formulation is

$$ \mathop \int \limits_{{\Omega } } \nabla \times \varvec{a} \,{\text{d}}v = \mathop {\oint }\limits_{s} \varvec{n} \times \varvec{a }\,{\text{d}}s $$

(2.44) can be rewritten as follows:

$$ R_{11} = - \mathop {\oint }\limits_{s} N_{i} \varvec{n} \times \frac{1}{\mu }\nabla \times \varvec{A }\,{\text{d}}s - \mathop \int \limits_{{\Omega } } \nabla N_{i} \times \frac{1}{\mu }\nabla \times \varvec{A }\,{\text{d}}v $$

(2.45)

Apparently, $\varvec{n} \times \frac{1}{\mu }\nabla \times \varvec{A} = \varvec{n} \times \varvec{H}$ is the tangential component of the magnetic field intensity at the interface. According to the continuity condition of the tangential component of the magnetic field intensity at the interface, the surface integral in (2.45) will be canceled from each other with the corresponding surface integral in the non-eddy current region ω derived from (2.34), and then (2.45) contains only the volume integral term, i.e.,

$$ R_{11} = - \mathop \int \limits_{{\Omega } } \nabla N_{i} \times \frac{1}{\mu }\nabla \times \varvec{A}\,\varvec{ }{\text{d}}v $$

(2.46)

The vector formulation is

$$ \nabla N_{i} = \frac{{\partial N_{i} }}{\partial x}\varvec{ i} + \frac{{\partial N_{i} }}{\partial y}\varvec{ j} + \frac{{\partial N_{i} }}{\partial z}\varvec{ k} $$

(2.47)

$$ \frac{1}{\mu }\nabla \times \varvec{A} = \frac{1}{{\mu_{x} }}\left( {\frac{{\partial A_{z} }}{\partial y} - \frac{{\partial A_{y} }}{\partial z}} \right)\varvec{i} + \frac{1}{{\mu_{y} }}\left( {\frac{{\partial A_{x} }}{\partial z} - \frac{{\partial A_{z} }}{\partial x}} \right)\varvec{j} + \frac{1}{{\mu_{z} }}\left( {\frac{{\partial A_{y} }}{\partial x} - \frac{{\partial A_{x} }}{\partial y}} \right)\varvec{k} $$

(2.48)

With the result of (2.47) and (2.48),

$$ \begin{aligned} R_{11} & = - \mathop \int \limits_{\Omega } \nabla N_{i} \times \frac{1}{\mu }\nabla \times \varvec{A} {\text{d}}v \\ & = - \left[ {\mathop \int \limits_{\Omega } \left( {\frac{1}{{\mu_{z} }}\frac{{\partial N_{i} }}{\partial y}\left( {\frac{{\partial A_{y} }}{\partial x} - \frac{{\partial A_{x} }}{\partial y}} \right) - \frac{1}{{\mu_{y} }}\frac{{\partial N_{i} }}{\partial z}\left( {\frac{{\partial A_{x} }}{\partial z} - \frac{{\partial A_{z} }}{\partial x}} \right)} \right){\text{d}}v} \right] \varvec{i} \\ & \quad - \left[ {\mathop \int \limits_{\Omega } \left( {\frac{1}{{\mu_{x} }}\frac{{\partial N_{i} }}{\partial z}\left( {\frac{{\partial A_{z} }}{\partial y} - \frac{{\partial A_{y} }}{\partial z}} \right) - \frac{1}{{\mu_{z} }}\frac{{\partial N_{i} }}{\partial x}\left( {\frac{{\partial A_{y} }}{\partial x} - \frac{{\partial A_{x} }}{\partial y}} \right)} \right){\text{d}}v} \right] \varvec{j} \\ & \quad - \left[ {\mathop \int \limits_{\Omega } \left( {\frac{1}{{\mu_{y} }}\frac{{\partial N_{i} }}{\partial x}\left( {\frac{{\partial A_{x} }}{\partial z} - \frac{{\partial A_{z} }}{\partial x}} \right) - \frac{1}{{\mu_{x} }}\frac{{\partial N_{i} }}{\partial y}\left( {\frac{{\partial A_{z} }}{\partial y} - \frac{{\partial A_{y} }}{\partial z}} \right)} \right){\text{d}}v} \right] \varvec{k} \\ \end{aligned} $$

(2.49)

For (2.40), the vector formulation is

$$ \nabla \left( {cd} \right) = d\nabla c + c\nabla d $$

which leads to

$$ \nabla \left( {N_{i} \frac{1}{{\mu_{c} }}\nabla \cdot \varvec{A}} \right) = \nabla N_{i} \frac{1}{{\mu_{c} }}\nabla \cdot \varvec{A} + N_{i} \nabla \left( {\frac{1}{{\mu_{c} }}\nabla \cdot \varvec{A}} \right) $$

(2.50)

(2.50) is sorted out to:

$$ N_{i} \nabla \left( {\frac{1}{{\mu_{c} }}\nabla \cdot \varvec{A}} \right) = \nabla \left( {N_{i} \frac{1}{{\mu_{c} }}\nabla \cdot \varvec{A}} \right) - \nabla N_{i} \frac{1}{{\mu_{c} }}\nabla \cdot \varvec{A} $$

(2.51)

Substituting (2.51) into (2.40):

$$ R_{12} = - \mathop \int \limits_{\Omega } \nabla \left( {N_{i} \frac{1}{{\mu_{c} }}\nabla \cdot \varvec{A}} \right)\varvec{ }{\text{d}}v + \mathop \int \limits_{\Omega } \nabla N_{i} \frac{1}{{\mu_{c} }}\nabla \cdot \varvec{A }\,{\text{d}}v $$

(2.52)

The vector formulation is

$$ \mathop \int \limits_{\Omega } \nabla c\, {\text{d}}v = \mathop {\oint }\limits_{s} c\varvec{n} \,{\text{d}}s $$

The following form can be derived from (2.40):

$$ R_{12} = - \mathop {\oint }\limits_{s} N_{i} \left( {\frac{1}{{\mu_{c} }}\nabla \cdot \varvec{A}} \right)\varvec{n} \,{\text{d}}s + \mathop \int \limits_{\varOmega } \nabla N_{i} \cdot \frac{1}{{\mu_{c} }}\nabla \cdot \varvec{A }\,{\text{d}}v $$

(2.53)

The continuity at the interface ($\frac{1}{\mu }\nabla \cdot {\varvec{A}}$) becomes a natural interface condition. The surface integral of (2.53) will be canceled with the surface integral of (2.34), so only the volume integral term of

$$ R_{12} = \mathop \int \limits_{\Omega } \nabla N_{i} \frac{1}{{\mu_{c} }}\nabla \cdot \varvec{A} \,{\text{d}}v $$

(2.54)

The vector formulation is

$$ \frac{1}{{\mu_{c} }}\nabla \cdot \varvec{A} = \frac{1}{{\mu_{c} }}\left( {\frac{{\partial A_{x} }}{\partial x} + \frac{{\partial A_{y} }}{\partial y} + \frac{{\partial A_{z} }}{\partial z}} \right) $$

(2.55)

$$ \nabla N_{i} = \frac{{\partial N_{i} }}{\partial x}\varvec{ i} + \frac{{\partial N_{i} }}{\partial y}\varvec{ j} + \frac{{\partial N_{i} }}{\partial z}\varvec{ k } $$

(2.56)

Substituting (2.55) and (2.56) into (2.54):

$$ \begin{aligned} R_{12} & = \mathop \int \limits_{\Omega } \nabla N_{i} \frac{1}{{\mu_{c} }}\nabla \cdot \varvec{A}\, {\text{d}}v \\ & = \mathop \int \limits_{\Omega } \frac{1}{{\mu_{c} }}\frac{{\partial N_{i} }}{\partial x}\left( {\frac{{\partial A_{x} }}{\partial x} + \frac{{\partial A_{y} }}{\partial y} + \frac{{\partial A_{z} }}{\partial z}} \right){\text{d}}v\,\varvec{i} \\ & \quad + \mathop \int \limits_{\Omega } \frac{1}{{\mu_{c} }}\frac{{\partial N_{i} }}{\partial y}\left( {\frac{{\partial A_{x} }}{\partial x} + \frac{{\partial A_{y} }}{\partial y} + \frac{{\partial A_{z} }}{\partial z}} \right){\text{d}}v\,\varvec{j} \\ & \quad + \mathop \int \limits_{\Omega } \frac{1}{{\mu_{c} }}\frac{{\partial N_{i} }}{\partial z}\left( {\frac{{\partial A_{x} }}{\partial x} + \frac{{\partial A_{y} }}{\partial y} + \frac{{\partial A_{z} }}{\partial z}} \right){\text{d}}v\,\varvec{k } \\ \end{aligned} $$

(2.57)

For (2.41), we get

$$ \begin{aligned} R_{13} & = \mathop \int \limits_{\Omega } \sigma N_{i} \left( {\frac{{\partial A_{x} }}{\partial t} + \frac{\partial }{\partial x}\left( {\frac{\partial V}{\partial t}} \right)} \right){\text{d}}v\,\varvec{i} \\ & \quad + \mathop \int \limits_{\Omega } \sigma N_{i} \left( {\frac{{\partial A_{y} }}{\partial t} + \frac{\partial }{\partial y}\left( {\frac{\partial V}{\partial t}} \right)} \right){\text{d}}v\,\varvec{j} \\ & \quad + \mathop \int \limits_{\Omega } \sigma N_{i} \left( {\frac{{\partial A_{z} }}{\partial t} + \frac{\partial }{\partial z}\left( {\frac{\partial V}{\partial t}} \right)} \right){\text{d}}v\,\varvec{k } \\ \end{aligned} $$

(2.58)

Three components of the Galerkin residual of (2.32) can be obtained by synthesizing (2.49), (2.57) and (2.58):

$$ \begin{aligned} R_{x} & = - \mathop \int \limits_{\Omega } \left( {\frac{1}{{\mu_{z} }}\frac{{\partial N_{i} }}{\partial y}\left( {\frac{{\partial A_{y} }}{\partial x} - \frac{{\partial A_{x} }}{\partial y}} \right) - \frac{1}{{\mu_{y} }}\frac{{\partial N_{i} }}{\partial z}\left( {\frac{{\partial A_{x} }}{\partial z} - \frac{{\partial A_{z} }}{\partial x}} \right)} \right){\text{d}}v \\ & \quad + \mathop \int \limits_{\Omega } \frac{1}{{\mu_{c} }}\frac{{\partial N_{i} }}{\partial x}\left( {\frac{{\partial A_{x} }}{\partial x} + \frac{{\partial A_{y} }}{\partial y} + \frac{{\partial A_{z} }}{\partial z}} \right){\text{d}}v \\ & \quad + \mathop \int \limits_{\Omega } \sigma N_{i} \left( {\frac{{\partial A_{x} }}{\partial t} + \frac{\partial }{\partial x}\left( {\frac{\partial V}{\partial t}} \right)} \right){\text{d}}v \\ \end{aligned} $$

(2.59)

$$ \begin{aligned} R_{y} & = - \mathop \int \limits_{\Omega } \left( {\frac{1}{{\mu_{x} }}\frac{{\partial N_{i} }}{\partial z}\left( {\frac{{\partial A_{z} }}{\partial y} - \frac{{\partial A_{y} }}{\partial z}} \right) - \frac{1}{{\mu_{z} }}\frac{{\partial N_{i} }}{\partial x}\left( {\frac{{\partial A_{y} }}{\partial x} - \frac{{\partial A_{x} }}{\partial y}} \right)} \right){\text{d}}v \\ & \quad + \mathop \int \limits_{\Omega } \frac{1}{{\mu_{c} }}\frac{{\partial N_{i} }}{\partial y}\left( {\frac{{\partial A_{x} }}{\partial x} + \frac{{\partial A_{y} }}{\partial y} + \frac{{\partial A_{z} }}{\partial z}} \right){\text{d}}v \\ & \quad + \mathop \int \limits_{\Omega } \sigma N_{i} \left( {\frac{{\partial A_{y} }}{\partial t} + \frac{\partial }{\partial y}\left( {\frac{\partial V}{\partial t}} \right)} \right){\text{d}}v \\ \end{aligned} $$

(2.60)

$$ \begin{aligned} R_{z} & = - \mathop \int \limits_{\Omega } \left( {\frac{1}{{\mu_{y} }}\frac{{\partial N_{i} }}{\partial x}\left( {\frac{{\partial A_{x} }}{\partial z} - \frac{{\partial A_{z} }}{\partial x}} \right) - \frac{1}{{\mu_{x} }}\frac{{\partial N_{i} }}{\partial y}\left( {\frac{{\partial A_{z} }}{\partial y} - \frac{{\partial A_{y} }}{\partial z}} \right)} \right){\text{d}}v \\ & \quad + \mathop \int \limits_{\Omega } \frac{1}{{\mu_{c} }}\frac{{\partial N_{i} }}{\partial z}\left( {\frac{{\partial A_{x} }}{\partial x} + \frac{{\partial A_{y} }}{\partial y} + \frac{{\partial A_{z} }}{\partial z}} \right){\text{d}}v \\ & \quad + \mathop \int \limits_{\Omega } \sigma N_{i} \left( {\frac{{\partial A_{z} }}{\partial t} + \frac{\partial }{\partial z}\left( {\frac{\partial V}{\partial t}} \right)} \right){\text{d}}v \\ \end{aligned} $$

(2.61)

To deduce R₀, the vector formulation is

$$ \begin{aligned} \nabla \cdot \left( {N_{i} \sigma \left( { - \frac{{\partial \varvec{A}}}{\partial t} - \nabla \left( {\frac{\partial V}{\partial t}} \right)} \right)} \right) & = \nabla N_{i} \cdot \sigma \left( { - \frac{{\partial \varvec{A}}}{\partial t} - \nabla \left( {\frac{\partial V}{\partial t}} \right)} \right) \\ & \quad + N_{i} \nabla \cdot \left( {\sigma \left( { - \frac{{\partial \varvec{A}}}{\partial t} - \nabla \left( {\frac{\partial V}{\partial t}} \right)} \right)} \right) \end{aligned} $$

(2.62)

(2.62) is sorted out to:

$$ \begin{aligned} N_{i} \nabla \cdot \left( {\sigma \left( { - \frac{{\partial \varvec{A}}}{\partial t} - \nabla \left( {\frac{\partial V}{\partial t}} \right)} \right)} \right) & = \nabla \cdot \left( {N_{i} \sigma \left( { - \frac{{\partial \varvec{A}}}{\partial t} - \nabla \left( {\frac{\partial V}{\partial t}} \right) } \right)} \right) \\ & \quad - \nabla N_{i} \cdot \sigma \left( { - \frac{{\partial \varvec{A}}}{\partial t} - \nabla \left( {\frac{\partial V}{\partial t}} \right) } \right) \end{aligned} $$

(2.63)

Substituting (2.63) into (2.38)

$$ \begin{aligned} R_{0} & = \mathop \int \limits_{\Omega } \nabla \cdot \left( {N_{i} \sigma \left( { - \frac{{\partial \varvec{A}}}{\partial t} - \nabla \left( {\frac{\partial V}{\partial t}} \right)} \right)} \right){\text{d}}v - \mathop \int \limits_{\Omega } \nabla N_{i} \cdot \sigma \left( { - \frac{{\partial \varvec{A}}}{\partial t} - \nabla \left( {\frac{\partial V}{\partial t}} \right)} \right) {\text{d}}v \\ & = \mathop {\oint }\limits_{s} N_{i} \sigma \left( { - \frac{{\partial \varvec{A}}}{\partial t} - \nabla \left( {\frac{\partial V}{\partial t}} \right)} \right) \cdot \varvec{n}{\text{d}}s - \mathop \int \limits_{\Omega } \nabla N_{i} \cdot \sigma \left( { - \frac{{\partial \varvec{A}}}{\partial t} - \nabla \left( {\frac{\partial V}{\partial t}} \right)} \right) {\text{d}}v \\ \end{aligned} $$

(2.64)

According to the conductor surface condition of $J_{n} = 0$, the surface integral term of (2.64) is zero, i.e.,

$$ R_{0} = - \mathop \int \limits_{\Omega } \nabla N_{i} \cdot \sigma \left( { - \frac{{\partial \varvec{A}}}{\partial t} - \nabla \left( {\frac{\partial V}{\partial t}} \right) } \right) {\text{d}}v $$

(2.65)

Upon a simple vector operation, (2.65) can be rewritten as

$$ \begin{aligned} R_{0} & = \mathop \int \limits_{\Omega } \sigma \left( {\frac{{\partial N_{i} }}{\partial x}\left( {\frac{{\partial A_{x} }}{\partial t} + \frac{\partial }{\partial x}\left( {\frac{\partial V}{\partial t}} \right)} \right)} \right){\text{d}}v + \mathop \int \limits_{\Omega } \sigma \left( {\frac{{\partial N_{i} }}{\partial y}\left( {\frac{{\partial A_{y} }}{\partial t} + \frac{\partial }{\partial y}\left( {\frac{\partial V}{\partial t}} \right)} \right)} \right){\text{d}}v \\ & \quad + \mathop \int \limits_{\Omega } \sigma \left( {\frac{{\partial N_{i} }}{\partial z}\left( {\frac{{\partial A_{z} }}{\partial t} + \frac{\partial }{\partial z}\left( {\frac{\partial V}{\partial t}} \right)} \right)} \right){\text{d}}v \\ \end{aligned} $$

(2.66)

4.2.2 Non-Eddy Current Region

In non-eddy current region ω, conductivity σ = 0; the third term in (2.59), (2.60) and (2.61) will disappear, and μ = μ₀, based on the above deduction, the corresponding Galerkin residuals of (2.34) are as

$$ \begin{aligned} R_{x}^{\prime} & = - \mathop \int \limits_{\omega } \left( {\frac{1}{{\mu_{0} }}\frac{{\partial N_{i} }}{\partial y}\left( {\frac{{\partial A_{y} }}{\partial x} - \frac{{\partial A_{x} }}{\partial y}} \right) - \frac{1}{{\mu_{0} }}\frac{{\partial N_{i} }}{\partial z}\left( {\frac{{\partial A_{x} }}{\partial z} - \frac{{\partial A_{z} }}{\partial x}} \right)} \right){\text{d}}v \\ & \quad + \mathop \int \limits_{\omega } \frac{1}{{\mu_{0} }}\frac{{\partial N_{i} }}{\partial x}\left( {\frac{{\partial A_{x} }}{\partial x} + \frac{{\partial A_{y} }}{\partial y} + \frac{{\partial A_{z} }}{\partial z}} \right){\text{d}}v \\ \end{aligned} $$

(2.67)

$$ \begin{aligned} R_{y}^{\prime} & = - \mathop \int \limits_{\omega } \left( {\frac{1}{{\mu_{0} }}\frac{{\partial N_{i} }}{\partial z}\left( {\frac{{\partial A_{z} }}{\partial y} - \frac{{\partial A_{y} }}{\partial z}} \right) - \frac{1}{{\mu_{0} }}\frac{{\partial N_{i} }}{\partial x}\left( {\frac{{\partial A_{y} }}{\partial x} - \frac{{\partial A_{x} }}{\partial y}} \right)} \right){\text{d}}v \\ & \quad + \mathop \int \limits_{\omega } \frac{1}{{\mu_{0} }}\frac{{\partial N_{i} }}{\partial y}\left( {\frac{{\partial A_{x} }}{\partial x} + \frac{{\partial A_{y} }}{\partial y} + \frac{{\partial A_{z} }}{\partial z}} \right){\text{d}}v \\ \end{aligned} $$

(2.68)

$$ \begin{aligned} R_{z}^{\prime} & = - \mathop \int \limits_{\omega } \left( {\frac{1}{{\mu_{0} }}\frac{{\partial N_{i} }}{\partial x}\left( {\frac{{\partial A_{x} }}{\partial z} - \frac{{\partial A_{z} }}{\partial x}} \right) - \frac{1}{{\mu_{0} }}\frac{{\partial N_{i} }}{\partial y}\left( {\frac{{\partial A_{z} }}{\partial y} - \frac{{\partial A_{y} }}{\partial z}} \right)} \right){\text{d}}v \\ & \quad + \mathop \int \limits_{\omega } \frac{1}{{\mu_{0} }}\frac{{\partial N_{i} }}{\partial z}\left( {\frac{{\partial A_{x} }}{\partial x} + \frac{{\partial A_{y} }}{\partial y} + \frac{{\partial A_{z} }}{\partial z}} \right){\text{d}}v \\ \end{aligned} $$

(2.69)

On the treatment of complicated material properties in the formulation and numerical implementation based on the potential sets, and the measurement and prediction of the corresponding material properties, can be found in the related chapters of this book or other related literatures.