## 物理代写|电磁学代写electromagnetism代考|Eddy Currents in Laminated Rectangular Cores

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## 物理代写|电磁学代写electromagnetism代考|Eddy Currents in Laminated Rectangular Cores

Eddy current loss in an isolated thin-conducting plate is proportional to the square of its thickness. ${ }^{10}$ This loss can thus be reduced if laminated cores are used instead of solid iron cores. It has been noticed that the advantage of laminating iron cores is defeated unless a thick insulation coating is given on the two surfaces of each lamination. ${ }^3$ This is because if laminations are placed close to one another, the interlaminar capacitance predominates, the resulting eddy current loss tends to become linearly proportional to its thickness and not to the thickness squared.

Figure 5.11 shows a rectangular core consisting of $n$-insulated laminations, each of width $W$ and overall thickness $T$. Let the insulation thickness on each side of a lamination be $T_1 / 2$ and its iron thickness be $T_2$. Further, let the corners of the rectangular core be located at $(-W / 2,0),(W / 2$, $0),(-W / 2, n T)$ and $(W / 2, n T)$. In this figure, insulation regions are indicated as Region- $0^{\prime}, 1^{\prime}, 2^{\prime}, 3^{\prime}, \ldots, m^{\prime}, \ldots, n^{\prime}$. The iron regions are indicated as Region- $1,2, \ldots, m, \ldots, n$.

The exciting coil is wound around the long rectangular core and carries an alternating current $i$, where
$$i=I e^{j \omega t}$$
It is simulated by a surface current density $K_o$ :
$$K_o=I \cdot N$$

where $N$ is the number of turns per unit length of the coil. The currentcarrying coil will produce time-varying magnetic field, $H_z$, in the core and eddy current density with components $J_x$ and $J_{y^{\prime}}$ in the conducting regions and displacement currents in the insulation regions of the core. The magnetic field outside the coil is neglected. For the long rectangular core with a uniformly distributed current sheet, the magnetic field is entirely axial and independent of $z$-coordinate, along the axial direction. It is assumed that the permeability $\mu$, for the iron regions, permittivity $\varepsilon$, for the insulation regions and conductivity $\left(\sigma, \sigma^{\prime}\right)$, for both types of regions, are constant. Thus, from Maxwell’s equations for harmonic fields, in charge-free regions
$$\frac{\partial^2 H_z}{\partial x^2}+\frac{\partial^2 H_z}{\partial y^2}=-\gamma^2 H_z$$
for iron regions, where
$$\gamma=\sqrt{(-j \omega \mu) \cdot\left(\sigma+j \omega \varepsilon_o\right)}$$
and
$$\frac{\partial^2 H_z}{\partial x^2}+\frac{\partial^2 H_z}{\partial y^2}=-\left(\gamma^{\prime}\right)^2 H_z$$
for insulation regions, where
$$\gamma^{\prime}=\sqrt{\left(-j \omega \mu_o\right) \cdot\left(\sigma^{\prime}+j \omega \varepsilon\right)}$$

## 物理代写|电磁学代写electromagnetism代考|Two-Dimensional Fields in Anisotropic Media

Consider an anisotropic homogeneous medium characterised by conductivity $[\sigma]$, permeability $[\mu]$ and permittivity $[\epsilon]$, such that
$$\begin{gathered} {[\sigma]=\left(\sigma_x, \sigma_y, \sigma_z\right)} \ {[\mu]=\left(\mu_x, \mu_y, \mu_z\right)} \ {[\epsilon]=\left(\epsilon_x, \epsilon_y, \epsilon_z\right)} \end{gathered}$$
while the components of complex conductivity are defined as
\begin{aligned} & \bar{\sigma}_x \stackrel{\text { def }}{=} \sigma_x+j \omega \epsilon_x \ & \bar{\sigma}_y \stackrel{\text { def }}{=} \sigma_y+j \omega \epsilon_y \ & \bar{\sigma}_z \stackrel{\text { def }}{=} \sigma_z+j \omega \epsilon_z \end{aligned}
Let there be a two-dimensional electromagnetic field that is independent of $x$-coordinate, varies periodically with $y$-coordinate as well as with time-t. This variation is given by the factor $e^{j(\omega t-(y)}$, where the time period is $2 \pi / \omega$ and the wave length is $2 \pi / \ell$, that is, two pole-pitches. To determine field variation with $z$-coordinate, we proceed with the Maxwell equation:
$$\nabla \times H=J+\frac{\partial D}{\partial t}$$
Thus,
\begin{aligned} & \frac{\partial H_z}{\partial y}-\frac{\partial H_y}{\partial z}=\bar{\sigma}_x E_x \ & \frac{\partial H_x}{\partial z}-\frac{\partial H_z}{\partial x}=\bar{\sigma}_y E_y \ & \frac{\partial H_y}{\partial x}-\frac{\partial H_x}{\partial y}=\bar{\sigma}_z E_z \end{aligned}

# 电磁学代考

## 物理代写|电磁学代写electromagnetism代考|Eddy Currents in Laminated Rectangular Cores

$$i=I e^{j \omega t}$$

$$K_o=I \cdot N$$

$$\frac{\partial^2 H_z}{\partial x^2}+\frac{\partial^2 H_z}{\partial y^2}=-\gamma^2 H_z$$

$$\gamma=\sqrt{(-j \omega \mu) \cdot\left(\sigma+j \omega \varepsilon_o\right)}$$

$$\frac{\partial^2 H_z}{\partial x^2}+\frac{\partial^2 H_z}{\partial y^2}=-\left(\gamma^{\prime}\right)^2 H_z$$

$$\gamma^{\prime}=\sqrt{\left(-j \omega \mu_o\right) \cdot\left(\sigma^{\prime}+j \omega \varepsilon\right)}$$

## 物理代写|电磁学代写electromagnetism代考|Two-Dimensional Fields in Anisotropic Media

$$\begin{gathered} {[\sigma]=\left(\sigma_x, \sigma_y, \sigma_z\right)} \ {[\mu]=\left(\mu_x, \mu_y, \mu_z\right)} \ {[\epsilon]=\left(\epsilon_x, \epsilon_y, \epsilon_z\right)} \end{gathered}$$

\begin{aligned} & \bar{\sigma}_x \stackrel{\text { def }}{=} \sigma_x+j \omega \epsilon_x \ & \bar{\sigma}_y \stackrel{\text { def }}{=} \sigma_y+j \omega \epsilon_y \ & \bar{\sigma}_z \stackrel{\text { def }}{=} \sigma_z+j \omega \epsilon_z \end{aligned}

$$\nabla \times H=J+\frac{\partial D}{\partial t}$$

\begin{aligned} & \frac{\partial H_z}{\partial y}-\frac{\partial H_y}{\partial z}=\bar{\sigma}_x E_x \ & \frac{\partial H_x}{\partial z}-\frac{\partial H_z}{\partial x}=\bar{\sigma}_y E_y \ & \frac{\partial H_y}{\partial x}-\frac{\partial H_x}{\partial y}=\bar{\sigma}_z E_z \end{aligned}

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 物理代写|电磁学代写electromagnetism代考|Eddy Currents in Cores with Regular Polygonal Cross-Sections

statistics-lab™ 为您的留学生涯保驾护航 在代写电磁学electromagnetism方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写电磁学electromagnetism代写方面经验极为丰富，各种代写电磁学electromagnetism相关的作业也就用不着说。

## 物理代写|电磁学代写electromagnetism代考|Eddy Currents in Cores with Regular Polygonal Cross-Sections

Distributions of magnetic fields in solid cores with rectangular and circular cross-sections due to alternating current excitation have been analytically determined. ${ }^{2,3}$ For cores with uncommon cross-sections, field distributions are usually evaluated using numerical methods., ${ }^{4,8}$ Analytical solutions are available $e^{5-7}$ for field distributions in cores with cross-sections in the shape of isosceles right-angled triangles. A quasi-analytical method for the determination of the approximate distribution of magnetic field intensity in cores with regular polygonal cross-sections is presented in this section as an alternative to the existing numerical methods. Although only three types of core sections, namely, cores with triangular, hexagonal and octagonal cross-sections, as shown in Figures 5.6 through 5.8 are considered, the method can be readily extended for other regular polygonal sections.

Consider a long conducting core carrying a surface current sheet with density $K$ simulating a uniformly distributed current-carrying winding wound around the core. The winding current is at power frequency. The magnetic field outside the core will be zero if the displacement currents are neglected. Inside the core, the magnetic field will be axial, that is in the $z$-direction such that just under the current sheet
$$\left.H_z\right|_{\text {core surface }}=K$$
where $|K|$ indicates the root mean square (rms) value of the surface current density on the conductor surface flowing in the anticlockwise direction, and $\mathrm{H}_z$ indicates the magnetic field in the axial direction, both in phasor form.
The eddy current equation for the magnetic field is
$$\nabla^2 H_z=\eta^2 H_z$$
where
$$\eta^2=j \omega_0 \cdot \mu \sigma$$
$\omega_0=$ frequency of the sinusoidally time-varying field
$\mu=$ permeability of the core
$\sigma=$ conductivity of the core
This is a two-dimensional problem as fields vary along $x$ – and $y$-directions only. Thus,
$$\frac{\partial H_z}{\partial x^2}+\frac{\partial H_z}{\partial y^2}=\eta^2 H_z$$
The solutions of this equation for solid cores with triangular, hexagonal and octagonal cross-sections are discussed in the following three subsections.

## 物理代写|电磁学代写electromagnetism代考|Cores with Triangular Cross-Sections

Consider a long solid-conducting core with a triangular cross-section shown in Figure 5.6. Let the length of each side of the triangle be $L$. A rectangle constructed using the base of this equilateral triangle is shown by dotted lines. Let the torch function be defined by the finite Fourier series:
$$\left.H_z^{\prime}\right|{y=L / \sqrt{3}}=\sum{m-\text { odd }}^{(2 M-1)} T_m \cdot \cos \left(\frac{m \pi}{L} \cdot x\right)$$
where $T_m$ indicates a set of Fourier coefficients.

On setting
$$\left.H_z^{\prime}\right|{x= \pm L / 2}=\left.H_z^{\prime}\right|{y=-L /(2 \sqrt{3})}=0$$
The solution of eddy current equation for the rectangular region can be given as
$$H_z^{\prime}=\sum_{m-\alpha d d d}^{(2 M-1)} T_m \cdot \cos \left(\frac{m \pi}{L} \cdot x^{\prime}\right) \cdot \frac{\sinh \left[\alpha_m \cdot\left{y^{\prime}+L /(2 \sqrt{ } 3)\right}\right]}{\sinh \left(\alpha_m \cdot L \sqrt{3} / 2\right)}$$
where
$$\begin{gathered} \alpha_m=\sqrt{\left(\frac{m \pi}{L}\right)^2+\eta^2} \ x^{\prime}=x \ y^{\prime}=y \end{gathered}$$
Next, we construct two more similar rectangles, each containing one or the other of the two remaining sides of the equilateral triangle. Let the field distributions in these regions be
$$H_z^{\prime \prime}=\sum_{m-\text {-odd }}^{(2 M-1)} T_m \cdot \cos \left(\frac{m \pi}{L} \cdot x^{\prime \prime}\right) \cdot \frac{\sinh \left[\alpha_m \cdot\left{y^{\prime \prime}+L /(2 \sqrt{3})\right}\right]}{\sinh \left(\alpha_m \cdot L \cdot \sqrt{3} / 2\right)}$$

$$H_z^{\prime \prime}=\sum_{m-\text { odd }}^{(2 M-1)} T_m \cdot \cos \left(\frac{m \pi}{L} \cdot x^{\prime \prime}\right) \cdot \frac{\sin \left[\alpha_m \cdot\left{y^{\prime \prime}+L /(2 \sqrt{3})\right}\right]}{\sinh \left(\alpha_m \cdot L \cdot \sqrt{3} / 2\right)}$$
where
\begin{aligned} & x^{\prime \prime}=y \cdot \frac{\sqrt{3}}{2}-x \cdot \frac{1}{2} \ & y^{\prime \prime}=-y \cdot \frac{1}{2}-x \cdot \frac{\sqrt{3}}{2} \ & x^{\prime \prime}=-y \cdot \frac{\sqrt{3}}{2}-x \cdot \frac{1}{2} \ & y^{\prime \prime}=-y \cdot \frac{1}{2}+x \cdot \frac{\sqrt{3}}{2} \end{aligned}

# 电磁学代考

## 物理代写|电磁学代写electromagnetism代考|Eddy Currents in Cores with Regular Polygonal Cross-Sections

$$\left.H_z\right|_{\text {core surface }}=K$$

$$\nabla^2 H_z=\eta^2 H_z$$

$$\eta^2=j \omega_0 \cdot \mu \sigma$$
$\omega_0=$正弦时变场的频率
$\mu=$岩心渗透率
$\sigma=$芯的电导率

$$\frac{\partial H_z}{\partial x^2}+\frac{\partial H_z}{\partial y^2}=\eta^2 H_z$$

## 物理代写|电磁学代写electromagnetism代考|Cores with Triangular Cross-Sections

$$\left.H_z^{\prime}\right|{y=L / \sqrt{3}}=\sum{m-\text { odd }}^{(2 M-1)} T_m \cdot \cos \left(\frac{m \pi}{L} \cdot x\right)$$

$$\left.H_z^{\prime}\right|{x= \pm L / 2}=\left.H_z^{\prime}\right|{y=-L /(2 \sqrt{3})}=0$$

$$H_z^{\prime}=\sum_{m-\alpha d d d}^{(2 M-1)} T_m \cdot \cos \left(\frac{m \pi}{L} \cdot x^{\prime}\right) \cdot \frac{\sinh \left[\alpha_m \cdot\left{y^{\prime}+L /(2 \sqrt{ } 3)\right}\right]}{\sinh \left(\alpha_m \cdot L \sqrt{3} / 2\right)}$$

$$\begin{gathered} \alpha_m=\sqrt{\left(\frac{m \pi}{L}\right)^2+\eta^2} \ x^{\prime}=x \ y^{\prime}=y \end{gathered}$$

$$H_z^{\prime \prime}=\sum_{m-\text {-odd }}^{(2 M-1)} T_m \cdot \cos \left(\frac{m \pi}{L} \cdot x^{\prime \prime}\right) \cdot \frac{\sinh \left[\alpha_m \cdot\left{y^{\prime \prime}+L /(2 \sqrt{3})\right}\right]}{\sinh \left(\alpha_m \cdot L \cdot \sqrt{3} / 2\right)}$$

$$H_z^{\prime \prime}=\sum_{m-\text { odd }}^{(2 M-1)} T_m \cdot \cos \left(\frac{m \pi}{L} \cdot x^{\prime \prime}\right) \cdot \frac{\sin \left[\alpha_m \cdot\left{y^{\prime \prime}+L /(2 \sqrt{3})\right}\right]}{\sinh \left(\alpha_m \cdot L \cdot \sqrt{3} / 2\right)}$$

\begin{aligned} & x^{\prime \prime}=y \cdot \frac{\sqrt{3}}{2}-x \cdot \frac{1}{2} \ & y^{\prime \prime}=-y \cdot \frac{1}{2}-x \cdot \frac{\sqrt{3}}{2} \ & x^{\prime \prime}=-y \cdot \frac{\sqrt{3}}{2}-x \cdot \frac{1}{2} \ & y^{\prime \prime}=-y \cdot \frac{1}{2}+x \cdot \frac{\sqrt{3}}{2} \end{aligned}

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 物理代写|电磁学代写electromagnetism代考|Boundary Conditions

statistics-lab™ 为您的留学生涯保驾护航 在代写电磁学electromagnetism方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写电磁学electromagnetism代写方面经验极为丰富，各种代写电磁学electromagnetism相关的作业也就用不着说。

## 物理代写|电磁学代写electromagnetism代考|Boundary Conditions

In view of the assumption that the relative permeability for iron is large (i.e. $\mu_r \gg 1$ ), two sets of boundary conditions are specified. The first of these is used for the selection of field expressions, whereas the second can be used to evaluate the arbitrary constants.
Selection of Field Expressions
For selecting the field expressions, the following boundary conditions are assumed. These boundary conditions are to be identically satisfied by the selected expressions.
\begin{aligned} & \left.H_{4 y}\right|{x=d}=0 \ & \left.H{4 z}\right|{x=d}=0 \ & \left.H{3 y}\right|{z=-g}=0 \ & \left.H{3 x}\right|{z=0}=0 \ & \left.H{3 x}\right|_{z=-g}=0 \end{aligned}

The various arbitrary constants used to describe magnetic fields in different regions can be evaluated by using the following boundary conditions:
$$\begin{gathered} \left.H_{4 x}\right|{z=0}=\left.H{1 x}\right|{z=0}-K{o y} \quad \text { over } 0 \leq x \leq d \ \left.H_{4 x}\right|{z=-8}=\left.H{2 x}\right|{z=-g} \quad \text { over } 0 \leq x \leq d \ \left.H{4 x}\right|{x=0}=\left.H{3 x}\right|{x=0} \quad \text { over }-g \leq z \leq 0 \ \left.H{4 y}\right|{z=0}=\left.H{1 y}\right|{z=0}+K{o x} \quad \text { over } 0 \leq x \leq d \ \left.H_{4 y}\right|{z=-8}=\left.H{2 y}\right|{z=-g} \quad \text { over } 0 \leq x \leq d \ \left.H{4 y}\right|{x=0}=\left.H{3 y}\right|_{x=0} \quad \text { over }-g \leq z \leq 0 \end{gathered}$$

$$\begin{gathered} \left.H_{4 z}\right|{z=0}=\left.H{1 z}\right|{z=0} \quad \text { over } 0 \leq x \leq d \ \left.H{4 z}\right|{z=-g}=\left.H{2 z}\right|{z=-g} \quad \text { over } 0 \leq x \leq d \ \left.H{4 z}\right|{x=0}=\left.H{3 z}\right|{x=0} \quad \text { over }-g \leq z \leq 0 \ \left.H{3 y}\right|{z=0}=K_x \quad \text { over } 0 \leq x \leq d \end{gathered}$$ From Equations $4.117 \mathrm{j}, 4.114 \mathrm{a}, 4.109 \mathrm{c}$ and $4.108 \mathrm{a}$ $$a_m=k_m=\left(j \frac{m \pi}{\lambda}\right) \cdot \sum{n-\text { odd }}^{\infty} \ell_{m-n} \quad \text { for } m=1,2,3, \ldots$$

## 物理代写|电磁学代写electromagnetism代考|Eddy Current Machines (Solid Rotor Induction Machines)

Eddy currents are induced in conducting regions subjected to time-varying electromagnetic fields. Eddy currents due to transient electromagnetic fields are discussed in Chapter 7. This section is devoted to the induction of eddy currents due to steady-state sinusoidally time-varying electromagnetic fields. For power frequency excitation, the displacement currents are usually neglected. Therefore, the magnetic field intensity, $\boldsymbol{H}$, satisfies the following equations:
$$\begin{gathered} \nabla^2 H=\eta^2 H \ \nabla \cdot H=\mathbf{0} \ \eta^2=-j \omega_0 \cdot \mu \sigma \end{gathered}$$

where $\omega_o$ is the frequency of the sinusoidally time-varying field, and $\mu$ is the permeability and $\sigma$ is the conductivity of the material.

Once Equations 5.1a and $\mathrm{b}$ are solved, the eddy current density can be readily found from
$$J=\nabla \times H$$
The solution of Equations 5.1a and $\mathrm{b}$ for the magnetic field intensity $H$ is discussed through the following boundary-value problems.

Figure 5.1 shows a simplified two-dimensional model of a polyphase solid rotor induction machine with its armature winding simulated by a surface current sheet on a smooth highly permeable stator surface at $z=-g$. Let the surface current density in the reference frame fixed on the rotor at $z=0$ be given as
$$K_x=K_v \cdot e^{j\left(t y-\omega_0 \cdot t\right)}$$
Where
$$\begin{gathered} \omega_0=s \cdot \omega \ s=\operatorname{slip} \stackrel{\operatorname{def}}{=} 1-\frac{\text { rotor speed }}{\text { synchronous speed }}=1-\frac{v}{(\omega / \ell)} \ \omega=\text { supply frequency } \ \ell=\frac{\pi}{\text { pole pitch }}=\frac{\pi}{\tau} \end{gathered}$$
and $\left|k_o\right|$ indicates the amplitude of the surface current density, with currents flowing in the $x$ (or axial) direction. This simplified treatment neglects the curvature of air-gap surfaces. The analysis that takes cognizance of curvature is available in the literature.

# 电磁学代考

## 物理代写|电磁学代写electromagnetism代考|Boundary Conditions

\begin{aligned} & \left.H_{4 y}\right|{x=d}=0 \ & \left.H{4 z}\right|{x=d}=0 \ & \left.H{3 y}\right|{z=-g}=0 \ & \left.H{3 x}\right|{z=0}=0 \ & \left.H{3 x}\right|_{z=-g}=0 \end{aligned}

$$\begin{gathered} \left.H_{4 x}\right|{z=0}=\left.H{1 x}\right|{z=0}-K{o y} \quad \text { over } 0 \leq x \leq d \ \left.H_{4 x}\right|{z=-8}=\left.H{2 x}\right|{z=-g} \quad \text { over } 0 \leq x \leq d \ \left.H{4 x}\right|{x=0}=\left.H{3 x}\right|{x=0} \quad \text { over }-g \leq z \leq 0 \ \left.H{4 y}\right|{z=0}=\left.H{1 y}\right|{z=0}+K{o x} \quad \text { over } 0 \leq x \leq d \ \left.H_{4 y}\right|{z=-8}=\left.H{2 y}\right|{z=-g} \quad \text { over } 0 \leq x \leq d \ \left.H{4 y}\right|{x=0}=\left.H{3 y}\right|_{x=0} \quad \text { over }-g \leq z \leq 0 \end{gathered}$$

$$\begin{gathered} \left.H_{4 z}\right|{z=0}=\left.H{1 z}\right|{z=0} \quad \text { over } 0 \leq x \leq d \ \left.H{4 z}\right|{z=-g}=\left.H{2 z}\right|{z=-g} \quad \text { over } 0 \leq x \leq d \ \left.H{4 z}\right|{x=0}=\left.H{3 z}\right|{x=0} \quad \text { over }-g \leq z \leq 0 \ \left.H{3 y}\right|{z=0}=K_x \quad \text { over } 0 \leq x \leq d \end{gathered}$$ 从方程$4.117 \mathrm{j}, 4.114 \mathrm{a}, 4.109 \mathrm{c}$和 $4.108 \mathrm{a}$ $$a_m=k_m=\left(j \frac{m \pi}{\lambda}\right) \cdot \sum{n-\text { odd }}^{\infty} \ell_{m-n} \quad \text { for } m=1,2,3, \ldots$$

## 物理代写|电磁学代写electromagnetism代考|Eddy Current Machines (Solid Rotor Induction Machines)

$$\begin{gathered} \nabla^2 H=\eta^2 H \ \nabla \cdot H=\mathbf{0} \ \eta^2=-j \omega_0 \cdot \mu \sigma \end{gathered}$$

$$J=\nabla \times H$$

$$K_x=K_v \cdot e^{j\left(t y-\omega_0 \cdot t\right)}$$

$$\begin{gathered} \omega_0=s \cdot \omega \ s=\operatorname{slip} \stackrel{\operatorname{def}}{=} 1-\frac{\text { rotor speed }}{\text { synchronous speed }}=1-\frac{v}{(\omega / \ell)} \ \omega=\text { supply frequency } \ \ell=\frac{\pi}{\text { pole pitch }}=\frac{\pi}{\tau} \end{gathered}$$
$\left|k_o\right|$表示表面电流密度的幅值，电流沿$x$(或轴向)方向流动。这种简化处理忽略了气隙表面的曲率。在文献中有考虑曲率的分析。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 物理代写|电磁学代写electromagnetism代考|Fringing Flux for Tooth-Opposite-Tooth Orientation with Small Air Gap

statistics-lab™ 为您的留学生涯保驾护航 在代写电磁学electromagnetism方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写电磁学electromagnetism代写方面经验极为丰富，各种代写电磁学electromagnetism相关的作业也就用不着说。

## 物理代写|电磁学代写electromagnetism代考|Fringing Flux for Tooth-Opposite-Tooth Orientation with Small Air Gap

As can be seen from Figure 4.9 the real and imaginary axes of $z$-plane are represented by $x$ and $y$ axes, and the real and imaginary axes of $w$-plane by $u$ and $v$ axes, respectively. For transforming $z$-plane (Figure 4.9 a) to $w$-plane (Figure $4.9 \mathrm{~b}$ ), the real axis in $z$-plane is to be mapped on the positive real axis in $w$-plane, while the tooth contour in the $z$-plane is to be mapped on the negative real axis in w-plane. This calls for placing the real axis in z-plane on the positive part of the real axis in $w$-plane. Further, straightening the configuration in $z$-plane and the tooth contour on the $z$-plane is to be mapped on the negative part of the real axis in $w$-plane. On $z$-plane the convenient points $w=0, w=-\infty$ and $w=\infty$ are shown in Figure 4.9 a. The resulting points on $w$-plane for $z=j \omega, j g / 2$ and $\infty$ are shown in Figure 4.9b. In pulling out the configuration to a straight line two angles are to be straightened. These angles are shown in Figure $4.9 \mathrm{a}$ as $\alpha$ and $\beta$.

The Schwarz-Christoffel transformation from $z$-plane to $w$-plane is performed through the relation:
$$\frac{d z}{d w}=A(w-a)^{(\alpha-\pi / \pi)} \cdot(w-b)^{(\beta-\pi / \pi)}$$
where $a$ and $b$ are the locations of the two internal angles $\alpha$ and $\beta$ in the $w$-plane. Therefore,
$$\begin{gathered} a=-1, \quad \alpha=3 \pi / 2 \ b=0, \quad \beta=0 \end{gathered}$$
Inserting these values in Equation 4.85, we get
$$d z=A \cdot \frac{(w+1)^{1 / 2}}{w} \cdot d w$$
On integrating, we have
$$z=A \cdot\left[2(w+1)^{1 / 2}+\log \left{\frac{(w+1)^{1 / 2}-1}{(w+1)^{1 / 2}+1}\right}\right]+C$$
where $C$ indicates the constant of integration. For the origin in the $z$-plane, shown in Figure $4.9 \mathrm{a}$, the value of $C$ is zero. On setting $w=-1$ this equation results:
$$\left.z\right|_{w=-1}=A \cdot \log (-1)=A \cdot j \pi$$
While from Figure $4.9 \mathrm{a}$
$$\left.z\right|_{w=-1}=j(g / 2)$$
Therefore,
$$A=\frac{g}{2 \pi}$$
Giving the transformation relation as
$$z=\frac{g}{2 \pi} \cdot\left[2(w+1)^{1 / 2}+\log \left{\frac{(w+1)^{1 / 2}-1}{(w+1)^{1 / 2}+1}\right}\right]$$
In the $w$-plane, the positive and negative parts of the real (or $u$ ) axis are at different equipotential values; that is, zero for positive $u$ and $-1 / 2$ for negative $u$. Therefore, flux lines in this plane are semicircles.

## 物理代写|电磁学代写electromagnetism代考|Transformation from χ Plane to w Plane

Next, consider the $\chi(=\varphi+j \psi)$-plane shown in Figure 4.10. In this plane, the equipotential surfaces are parallel to $\psi=0$ (or 1) plane, while flux lines are parallel to the $\psi$-axis. The value of potential varies linearly with the distance from $\psi=0$ plane. This plane is, therefore, called regular field plane. The convenient values of $w(-\infty,-1,0$ and $\infty)$ on this plane are shown in Figure 4.10 .

The transformation of $\chi$-plane into $w$-plane requires straightening only one internal angle, namely, $\alpha=0$. Therefore,
$$\frac{d \chi}{d w}=A(w-a)^{(\alpha-\pi / \pi)}$$
where $a=0$, and $\alpha=0$.

Thus,
$$d \chi=\frac{A}{w} d w$$
On integrating, we get
$$\chi=A \log (w)$$
Or
$$w=e^{(\chi / A)}$$
Since, as shown in Figure $4.10 w=-1$ at $\chi=j$, thus
$$A=1 / \pi$$
Therefore,
$$w=e^{\pi \cdot \chi}$$
Substituting in Equation 4.92, we get
$$z=\frac{g}{2 \pi} \cdot\left[2\left(e^{\pi \cdot \chi}+1\right)^{1 / 2}+\log \left{\frac{\left(e^{\pi \cdot \chi}+1\right)^{1 / 2}-1}{\left(e^{\pi \cdot \chi}+1\right)^{1 / 2}+1}\right}\right]$$
Since $z=x+j y$ and $\chi=\varphi+j \psi$, plot for a flux line can be obtained by choosing a constant value for $\varphi$, and joining points whose coordinates in the $z$-plane are found from this equation for different values of $\psi$. The procedure could be repeated for different values of $\varphi$, each value results in a different flux line. Similarly, plots for equipotential lines can be obtained by interchanging the roles of $\varphi$ and $\psi$.

# 电磁学代考

## 物理代写|电磁学代写electromagnetism代考|Fringing Flux for Tooth-Opposite-Tooth Orientation with Small Air Gap

$$\frac{d z}{d w}=A(w-a)^{(\alpha-\pi / \pi)} \cdot(w-b)^{(\beta-\pi / \pi)}$$

$$\begin{gathered} a=-1, \quad \alpha=3 \pi / 2 \ b=0, \quad \beta=0 \end{gathered}$$

$$d z=A \cdot \frac{(w+1)^{1 / 2}}{w} \cdot d w$$

$$z=A \cdot\left[2(w+1)^{1 / 2}+\log \left{\frac{(w+1)^{1 / 2}-1}{(w+1)^{1 / 2}+1}\right}\right]+C$$

$$\left.z\right|{w=-1}=A \cdot \log (-1)=A \cdot j \pi$$ 而从图$4.9 \mathrm{a}$ $$\left.z\right|{w=-1}=j(g / 2)$$

$$A=\frac{g}{2 \pi}$$

$$z=\frac{g}{2 \pi} \cdot\left[2(w+1)^{1 / 2}+\log \left{\frac{(w+1)^{1 / 2}-1}{(w+1)^{1 / 2}+1}\right}\right]$$

## 物理代写|电磁学代写electromagnetism代考|Transformation from χ Plane to w Plane

$$\frac{d \chi}{d w}=A(w-a)^{(\alpha-\pi / \pi)}$$

$$d \chi=\frac{A}{w} d w$$

$$\chi=A \log (w)$$

$$w=e^{(\chi / A)}$$

$$A=1 / \pi$$

$$w=e^{\pi \cdot \chi}$$

$$z=\frac{g}{2 \pi} \cdot\left[2\left(e^{\pi \cdot \chi}+1\right)^{1 / 2}+\log \left{\frac{\left(e^{\pi \cdot \chi}+1\right)^{1 / 2}-1}{\left(e^{\pi \cdot \chi}+1\right)^{1 / 2}+1}\right}\right]$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 物理代写|电磁学代写electromagnetism代考|Air-Gap Permeance

statistics-lab™ 为您的留学生涯保驾护航 在代写电磁学electromagnetism方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写电磁学electromagnetism代写方面经验极为丰富，各种代写电磁学electromagnetism相关的作业也就用不着说。

## 物理代写|电磁学代写electromagnetism代考|Air-Gap Permeance

The air-gap field varies periodically along the peripheral direction with a period of one tooth-pitch $\lambda$. In the absence of slotting, the air-gap field does not vary along the peripheral direction. Assuming that the potential at the air-gap surface $z=-g / 2$ as $+1 / 2$, and at the air-gap surface $z=+g / 2$ as $-1 / 2$, the potential distribution in the air gap can be given as
$$\mathcal{V}0=-\frac{z}{g}$$ Thus, net flux over a tooth-pitch $\lambda$ can be given as $$\varphi\lambda=-\mu_o \int_0^\lambda \frac{\partial \mathcal{V}o}{\partial z} \cdot d y=\mu_o(\lambda / g)$$ Since the potential difference between the two smooth air-gap surfaces is unity, the gap permeance $P\lambda$, over $\lambda$ is numerically equal to the flux over $\lambda$. Thus, in view of Equation $4.48 \mathrm{a}$, we have
$$P_\lambda=\varphi_\lambda=\mu_o(\lambda / g)$$
In Equation 4.36, $\delta$ is the distance between tooth-centres of two slotted equipotential surfaces as shown in Figure 4.3. The tooth-opposite-tooth orientation corresponds to $\delta$ equal to zero, while for the tooth-opposite-slot orientation the value for $\delta$ is one-half the tooth-pitch $\lambda$. In the former case, the gap permeance, $P_1$, is maximum. The value of permeance decreases as $\delta$ increases from zero. It reaches to a minimum value $P_2$, at the tooth-opposite-slot orientation. In view of Equation 4.36, the gap permeance $P$ over $\lambda$ for the double slotted air-gap surfaces is found as follows:
$$P=-\mu_o \int_0^\lambda \frac{\partial \mathcal{V}_o}{\partial z} \cdot d y=\mu_o 2 q_o(\lambda / g)$$

## 物理代写|电磁学代写electromagnetism代考|Tooth-Opposite-Tooth Orientation

In the case of tooth-opposite-tooth, the air-gap surface $z=0$ is a zero potential surface, $\mathcal{V}=0$. This orientation for wide and deep slots is shown in Figure 4.5. Across this surface, the potential distribution is an odd function of $z$. Further, the potential distribution is an even function of $y$. Therefore, the potential distribution in the air gap can be expressed as
$$\mathcal{V}0=-\frac{2}{\pi} \int_0^{\infty} F(u) \cdot \cos (u \cdot y) \cdot \frac{\sinh {u \cdot z}}{\sinh (u \cdot g / 2)} \cdot d u-\frac{z}{g}$$ over $-\infty \leq y<\infty$. The first term on the right-hand side is the Fourier integral representation that accounts for variation of the potential along the $y$ direction at $z= \pm g / 2$. The function $F(u)$ is the Fourier cosine transform of $\left(\left.\mathcal{V}_o\right|{z=-g / 2}-1 / 2\right)$, thus
$$F(u)=\int_0^{\infty}\left(\left.\mathcal{V}o\right|{z=-g / 2}-1 / 2\right) \cdot \cos (u \cdot y) \cdot d u$$
The potential distribution in the region for slot 1 can be given as
$$\mathcal{V}_1=\frac{2}{\pi} \int_0^{\infty} f(w) \cdot \sin {w \cdot(y-t / 2)} \cdot e^{w(z+g / 2)} \cdot d w-\frac{1}{\pi} \cdot \tan ^{-1}\left(\frac{z+g / 2}{y-t / 2}\right)$$
over $t / 2 \leq y<\infty$ and $z \ngtr-g / 2$.
In Equation 4.49 the second term ensures the potential value on the iron surface at $y=t / 2$, for $z<-g / 2$. The first term permits a general variation in the potential distribution at the slot-opening without disturbing the potential on the iron surface. This term may be considered as a corrective term.

# 电磁学代考

## 物理代写|电磁学代写electromagnetism代考|Air-Gap Permeance

$$\mathcal{V}0=-\frac{z}{g}$$ 因此，净通量在一个齿距上 $\lambda$ 可以表示为 $$\varphi\lambda=-\mu_o \int_0^\lambda \frac{\partial \mathcal{V}o}{\partial z} \cdot d y=\mu_o(\lambda / g)$$ 由于两个光滑气隙表面之间的电位差为一，故气隙渗透率为 $P\lambda$，完毕 $\lambda$ 在数值上等于通量除以 $\lambda$． 因此，鉴于式 $4.48 \mathrm{a}$，我们有
$$P_\lambda=\varphi_\lambda=\mu_o(\lambda / g)$$

$$P=-\mu_o \int_0^\lambda \frac{\partial \mathcal{V}_o}{\partial z} \cdot d y=\mu_o 2 q_o(\lambda / g)$$

## 物理代写|电磁学代写electromagnetism代考|Tooth-Opposite-Tooth Orientation

$$\mathcal{V}0=-\frac{2}{\pi} \int_0^{\infty} F(u) \cdot \cos (u \cdot y) \cdot \frac{\sinh {u \cdot z}}{\sinh (u \cdot g / 2)} \cdot d u-\frac{z}{g}$$ 结束 $-\infty \leq y<\infty$． 右边的第一项是傅里叶积分表示它解释了沿 $y$ 方向: $z= \pm g / 2$． 函数 $F(u)$ 的傅里叶余弦变换是什么 $\left(\left.\mathcal{V}_o\right|{z=-g / 2}-1 / 2\right)$，因此
$$F(u)=\int_0^{\infty}\left(\left.\mathcal{V}o\right|{z=-g / 2}-1 / 2\right) \cdot \cos (u \cdot y) \cdot d u$$

$$\mathcal{V}_1=\frac{2}{\pi} \int_0^{\infty} f(w) \cdot \sin {w \cdot(y-t / 2)} \cdot e^{w(z+g / 2)} \cdot d w-\frac{1}{\pi} \cdot \tan ^{-1}\left(\frac{z+g / 2}{y-t / 2}\right)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 物理代写|电磁学代写electromagnetism代考|Approximation Theorem for Vector Magnetic Potential

statistics-lab™ 为您的留学生涯保驾护航 在代写电磁学electromagnetism方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写电磁学electromagnetism代写方面经验极为丰富，各种代写电磁学electromagnetism相关的作业也就用不着说。

## 物理代写|电磁学代写electromagnetism代考|Approximation Theorem for Vector Magnetic Potential

Consider Poisson’s equation for the vector magnetic potential. Let a solution of this equation, for a region of volume $v$, that satisfies boundary conditions approximately be $A$. We define absolute error $\alpha$ as
$$\alpha \stackrel{d e f}{=} A-A_o$$
where $A_o$ is the solution of Poisson’s equation for this region that accurately satisfies the boundary conditions on its bounding surface $s$. The vector error $\alpha$ is a function of space coordinates of any point in $v$. This error for a point on the bounding surface is defined as $\alpha^s$,
$$\left.\alpha^s \stackrel{d e f}{=}\left(A-A_o\right)\right|_s$$
Since
$$\nabla^2 A=-\mu J$$
and
$$\nabla^2 A_o=-\mu J$$
Thus,
$$\nabla^2 \alpha=0$$
Further, we have
$$\nabla \cdot \alpha=\nabla \cdot A-\nabla \cdot A_o=0-0=0$$
Setting
$$\beta \stackrel{d e f}{=} \nabla \times \alpha$$
Let us consider the identity
\begin{aligned} \nabla \cdot(\alpha \times \beta) & \equiv \beta \cdot(\nabla \times \alpha)-\alpha \cdot(\nabla \times \beta) \ & =|\beta|^2-\alpha \cdot[\nabla \times \nabla \times \alpha] \ & \equiv|\beta|^2-\alpha \cdot\left[\nabla(\nabla \cdot \alpha)-\nabla^2 \alpha\right] \end{aligned}
Since the divergence as well as the Laplacian of the vector error $\alpha$ is zero,
$$\nabla \cdot(\alpha \times \beta)=|\beta|^2$$

## 物理代写|电磁学代写electromagnetism代考|Approximation Theorem for Maxwell’s Equations

Consider Maxwell’s two curl equations in phasor form for harmonic fields characterised by the factor $e^{-j \omega t}$
$$\begin{gathered} \nabla \times E=j \omega \mu H \ \nabla \times H=J-j \omega \varepsilon E \end{gathered}$$
Let the field vectors involved in these equations only approximately satisfy the prescribed boundary conditions. Maxwell’s equations for field vectors exactly satisfying the given boundary conditions are
$$\begin{gathered} \nabla \times \boldsymbol{E}_o=j \omega \mu \boldsymbol{H}_o \ \nabla \times \boldsymbol{H}_o=\boldsymbol{J}_o-j \omega \varepsilon \boldsymbol{E}_o \end{gathered}$$
Therefore,
$$\begin{gathered} \nabla \times e=j \omega \mu h \ \nabla \times h=j-j \omega \varepsilon e \end{gathered}$$
where
$$\begin{gathered} \boldsymbol{e}=\boldsymbol{E}-\boldsymbol{E}_o \ \boldsymbol{h}=\boldsymbol{H}-\boldsymbol{H}_o \ j=\boldsymbol{J}-\boldsymbol{J}_o \end{gathered}$$
and
$$j=\sigma e$$
Now, since
$$-\nabla \cdot\left(e \times h^\right) \equiv-h^ \cdot(\nabla \times e)+e \cdot\left(\nabla \times h^\right)$$ Therefore, in view of Equations 3.138c and 3.139c, we get $$-\nabla \cdot\left(\mathbf{e} \times h^\right)=\sigma e^2+j \omega\left(\varepsilon e^2-\mu h^2\right)$$
where
\begin{aligned} & e^2 \stackrel{d e f}{=} \boldsymbol{e} \cdot \boldsymbol{e}^* \ & h^2 \stackrel{d e f}{=} \boldsymbol{h} \cdot \boldsymbol{h}^* \end{aligned}

# 电磁学代考

## 物理代写|电磁学代写electromagnetism代考|Approximation Theorem for Vector Magnetic Potential

$$\alpha \stackrel{d e f}{=} A-A_o$$

$$\left.\alpha^s \stackrel{d e f}{=}\left(A-A_o\right)\right|_s$$

$$\nabla^2 A=-\mu J$$

$$\nabla^2 A_o=-\mu J$$

$$\nabla^2 \alpha=0$$

$$\nabla \cdot \alpha=\nabla \cdot A-\nabla \cdot A_o=0-0=0$$

$$\beta \stackrel{d e f}{=} \nabla \times \alpha$$

\begin{aligned} \nabla \cdot(\alpha \times \beta) & \equiv \beta \cdot(\nabla \times \alpha)-\alpha \cdot(\nabla \times \beta) \ & =|\beta|^2-\alpha \cdot[\nabla \times \nabla \times \alpha] \ & \equiv|\beta|^2-\alpha \cdot\left[\nabla(\nabla \cdot \alpha)-\nabla^2 \alpha\right] \end{aligned}

$$\nabla \cdot(\alpha \times \beta)=|\beta|^2$$

## 物理代写|电磁学代写electromagnetism代考|Approximation Theorem for Maxwell’s Equations

$$\begin{gathered} \nabla \times E=j \omega \mu H \ \nabla \times H=J-j \omega \varepsilon E \end{gathered}$$

$$\begin{gathered} \nabla \times \boldsymbol{E}_o=j \omega \mu \boldsymbol{H}_o \ \nabla \times \boldsymbol{H}_o=\boldsymbol{J}_o-j \omega \varepsilon \boldsymbol{E}_o \end{gathered}$$

$$\begin{gathered} \nabla \times e=j \omega \mu h \ \nabla \times h=j-j \omega \varepsilon e \end{gathered}$$

$$\begin{gathered} \boldsymbol{e}=\boldsymbol{E}-\boldsymbol{E}_o \ \boldsymbol{h}=\boldsymbol{H}-\boldsymbol{H}_o \ j=\boldsymbol{J}-\boldsymbol{J}_o \end{gathered}$$

$$j=\sigma e$$

$$-\nabla \cdot\left(e \times h^\right) \equiv-h^ \cdot(\nabla \times e)+e \cdot\left(\nabla \times h^\right)$$因此，根据式3.138c和3.139c，我们得到$$-\nabla \cdot\left(\mathbf{e} \times h^\right)=\sigma e^2+j \omega\left(\varepsilon e^2-\mu h^2\right)$$

\begin{aligned} & e^2 \stackrel{d e f}{=} \boldsymbol{e} \cdot \boldsymbol{e}^* \ & h^2 \stackrel{d e f}{=} \boldsymbol{h} \cdot \boldsymbol{h}^* \end{aligned}

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 物理代写|电磁学代写electromagnetism代考|Example of a Rectangular Region

statistics-lab™ 为您的留学生涯保驾护航 在代写电磁学electromagnetism方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写电磁学electromagnetism代写方面经验极为丰富，各种代写电磁学electromagnetism相关的作业也就用不着说。

## 物理代写|电磁学代写electromagnetism代考|Example of a Rectangular Region

Consider the rectangular region $(-W / 2 \leq x \leq W / 2,0 \leq y \leq h)$ shown in Figure 3.2. The region is piecewise homogeneous, with permittivity $\varepsilon_1$ for $(0<y<g)$ and $\varepsilon_2$ for $(g<y<h)$. The boundary conditions specified are
$$\begin{gathered} \left.V\right|{x= \pm W / 2}=0 \ \left.V\right|{y=0}=0 \end{gathered}$$
No boundary condition is specified on the top surface, that is, at $y=h$. However, instead of another boundary condition, it is given that
$$\left.V\right|{y=k}=\sum{m-o d d}^{\infty} a_m \cos \left(\frac{m \pi}{W} \cdot x\right)+\sum_{n=1}^{\infty} b_n \sin \left(\frac{n 2 \pi}{W} \cdot x\right)$$
where $g<k<h$.
Let us divide this rectangular region into two subregions, such that the subregion 1 extends over $(-W / 2 \leq x \leq W / 2,0 \leq y \leq k)$ and the subregion 2 extends over $(-W / 2 \leq x \leq W / 2, k \leq y \leq h)$. For the subregion 1 , the potential distributions along all the four boundary sides are specified; therefore, it should be possible to obtain a unique solution for the potential distribution had it been a homogeneous region. Since subregion 1 is piecewise homogeneous with $y$ $=g$ serving as a boundary between two homogeneous zones, viz. 1a stretching over $0<y<g$, and $1 \mathrm{~b}$ stretching over $g<y<k$ the potential distributions in these zones can be readily obtained if the potential distribution is known along the boundary $y=g$. Let us assume a pseudo-torch function
$$\left.V\right|{y=g}=\sum{m=\text { odd }}^{\infty} c_m \cos \left(\frac{m \pi}{W} \cdot x\right)+\sum_{n=1}^{\infty} d_n \sin \left(\frac{n 2 \pi}{W} \cdot x\right)$$
where $c_m$ and $d_n$ indicate two sets of unknown constants.

## 物理代写|电磁学代写electromagnetism代考|Uniqueness Theorem for Vector Magnetic Potentials

In general, there are infinite solutions of Laplace and Poisson equations for the vector magnetic potential $A$. The uniqueness theorem ${ }^2$ reviewed here describes boundary conditions to be satisfied for a unique solution of these equations.

Let $A_1$ and $A_2$ be any two solutions for Equation 2.40, given the distribution of magnetic potential in a volume $v$ bounded by the closed surface $s$. The difference potential $A_o$ is
$$A_o=A_1-A_2$$
It may be noted that the difference potential $A_o$ satisfies the Laplace equation
$$\nabla^2 A_o=0$$
In view of Equation 2.38
$$\nabla \cdot A_1=\nabla \cdot A_2=0$$
Therefore,
$$\nabla \cdot A_o=0$$
Consider the identity ${ }^3$
$$\iiint_v[(\nabla \times \boldsymbol{P}) \cdot(\nabla \times \mathbf{Q})-\boldsymbol{P} \cdot(\nabla \times \nabla \times \mathbf{Q})] d v \equiv \oiint_s[\boldsymbol{P} \times(\nabla \times \mathbf{Q})] \cdot d s$$

# 电磁学代考

## 物理代写|电磁学代写electromagnetism代考|Example of a Rectangular Region

$$\begin{gathered} \left.V\right|{x= \pm W / 2}=0 \ \left.V\right|{y=0}=0 \end{gathered}$$

$$\left.V\right|{y=k}=\sum{m-o d d}^{\infty} a_m \cos \left(\frac{m \pi}{W} \cdot x\right)+\sum_{n=1}^{\infty} b_n \sin \left(\frac{n 2 \pi}{W} \cdot x\right)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。