分类：电磁学代写

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

Posted on 2023年6月29日2023年6月29日 by statistics-lab

如果你也在怎样代写电磁学Electromagnetism 这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。电磁学Electromagnetism是物理学的一个分支，涉及到对电磁力的研究，这是一种发生在带电粒子之间的物理作用。电磁力是由电场和磁场组成的电磁场所承载的，它是诸如光这样的电磁辐射的原因。它与强相互作用、弱相互作用和引力一起，是自然界的四种基本相互作用（通常称为力）之一。在高能量下，弱力和电磁力被统一为单一的电弱力。

电磁学Electromagnetism是以电磁力来定义的，有时也称为洛伦兹力，它包括电和磁，是同一现象的不同表现形式。电磁力在决定日常生活中遇到的大多数物体的内部属性方面起着重要作用。原子核和其轨道电子之间的电磁吸引力将原子固定在一起。电磁力负责原子之间形成分子的化学键，以及分子间的力量。电磁力支配着所有的化学过程，这些过程是由相邻原子的电子之间的相互作用产生的。电磁学在现代技术中应用非常广泛，电磁理论是电力工程和电子学包括数字技术的基础。

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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

孤立的薄导电板中的涡流损耗与其厚度的平方成正比。${ }^{10}$因此，如果使用层压铁芯代替实心铁芯，则可以减少这种损失。人们注意到，除非在每个层压的两个表面上涂上厚厚的绝缘涂层，否则层压铁芯的优势就会失效。${ }^3$这是因为如果层片彼此靠近放置，层间电容占主导地位，所产生的涡流损耗往往与厚度成线性比例，而不是厚度的平方。

图5.11显示了由$n$ -绝缘层片组成的矩形芯，每个层片的宽度$W$和总厚度$T$。设层压每侧的绝缘厚度为$T_1 / 2$，其铁厚度为$T_2$。进一步，让矩形芯的角位于$(-W / 2,0),(W / 2$, $0),(-W / 2, n T)$和$(W / 2, n T)$。在此图中，绝缘区域用Region- $0^{\prime}, 1^{\prime}, 2^{\prime}, 3^{\prime}, \ldots, m^{\prime}, \ldots, n^{\prime}$表示。含铁区域用区域- $1,2, \ldots, m, \ldots, n$表示。

励磁线圈绕在长矩形铁芯上，并携带交流电$i$，其中
$$
i=I e^{j \omega t}
$$
用表面电流密度$K_o$来模拟:
$$
K_o=I \cdot N
$$

其中$N$为单位长度线圈的匝数。载流线圈将在铁芯中产生时变磁场$H_z$，在铁芯的导电区产生具有$J_x$和$J_{y^{\prime}}$分量的涡流密度，在铁芯的绝缘区产生位移电流。忽略线圈外的磁场。对于具有均匀分布电流片的长矩形铁芯，磁场沿轴向完全是轴向的，与$z$ -坐标无关。假设磁导率$\mu$，对于铁区域，介电常数$\varepsilon$，对于绝缘区域和电导率$\left(\sigma, \sigma^{\prime}\right)$，对于两种类型的区域，都是恒定的。因此，从麦克斯韦方程组的谐波场，在无电荷区域
$$
\frac{\partial^2 H_z}{\partial x^2}+\frac{\partial^2 H_z}{\partial y^2}=-\gamma^2 H_z
$$
对于铁区，在哪里
$$
\gamma=\sqrt{(-j \omega \mu) \cdot\left(\sigma+j \omega \varepsilon_o\right)}
$$
和
$$
\frac{\partial^2 H_z}{\partial x^2}+\frac{\partial^2 H_z}{\partial y^2}=-\left(\gamma^{\prime}\right)^2 H_z
$$
对于绝缘区域，其中
$$
\gamma^{\prime}=\sqrt{\left(-j \omega \mu_o\right) \cdot\left(\sigma^{\prime}+j \omega \varepsilon\right)}
$$

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

考虑一种各向异性均质介质，其特征为电导率$[\sigma]$、渗透率$[\mu]$和介电常数$[\epsilon]$
$$
\begin{gathered}
{[\sigma]=\left(\sigma_x, \sigma_y, \sigma_z\right)} \
{[\mu]=\left(\mu_x, \mu_y, \mu_z\right)} \
{[\epsilon]=\left(\epsilon_x, \epsilon_y, \epsilon_z\right)}
\end{gathered}
$$
而复合电导率的分量定义为
$$
\begin{aligned}
& \bar{\sigma}_x \stackrel{\text { def }}{=} \sigma_x+j \omega \epsilon_x \
& \bar{\sigma}_y \stackrel{\text { def }}{=} \sigma_y+j \omega \epsilon_y \
& \bar{\sigma}_z \stackrel{\text { def }}{=} \sigma_z+j \omega \epsilon_z
\end{aligned}
$$
设一个二维电磁场，它不依赖于$x$ -坐标，随$y$ -坐标和时间-t周期性变化。这种变化由因子$e^{j(\omega t-(y)}$给出，其中时间周期为$2 \pi / \omega$，波长为$2 \pi / \ell$，即两个极距。为了确定$z$ -坐标下的场变化，我们使用麦克斯韦方程:
$$
\nabla \times H=J+\frac{\partial D}{\partial t}
$$
因此，
$$
\begin{aligned}
& \frac{\partial H_z}{\partial y}-\frac{\partial H_y}{\partial z}=\bar{\sigma}_x E_x \
& \frac{\partial H_x}{\partial z}-\frac{\partial H_z}{\partial x}=\bar{\sigma}_y E_y \
& \frac{\partial H_y}{\partial x}-\frac{\partial H_x}{\partial y}=\bar{\sigma}_z E_z
\end{aligned}
$$

物理代写|电磁学代写electromagnetism代考请认准statistics-lab™

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金融工程代写

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非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

术语广义线性模型（GLM）通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归，以及方差分析和方差分析（仅含固定效应）。

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回归分析代写

多元回归分析渐进（Multiple Regression Analysis Asymptotics）属于计量经济学领域，主要是一种数学上的统计分析方法，可以分析复杂情况下各影响因素的数学关系，在自然科学、社会和经济学等多个领域内应用广泛。

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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，工具箱允许您学习和应用专业技术。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。

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

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

Posted on 2023年6月29日2023年6月29日 by statistics-lab

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

本文用解析法确定了矩形和圆形实心在交流励磁作用下的磁场分布。${ }^{2,3}$对于不常见截面的岩心，通常使用数值方法评估场分布。， ${ }^{4,8}$对于横截面为等腰直角三角形的岩心中的场分布有解析解$e^{5-7}$。本文提出了一种准解析方法来确定正多边形截面岩心磁场强度的近似分布，作为现有数值方法的一种替代方法。虽然只考虑图5.6 ～ 5.8所示的三角形、六角形和八角形截面三种类型的岩心截面，但该方法可以很容易地推广到其他正多边形截面。

考虑一个带表面电流片的长导电铁芯，其密度为$K$，模拟铁芯周围均匀分布的载流绕组。绕组电流在工频。如果忽略位移电流，磁芯外的磁场将为零。在磁芯内部，磁场将是轴向的，即在$z$ -方向，这样就在电流片下面
$$
\left.H_z\right|_{\text {core surface }}=K
$$
其中$|K|$为导体表面沿逆时针方向流动的表面电流密度的均方根值，$\mathrm{H}_z$为轴向磁场，均为相量形式。
磁场的涡流方程为
$$
\nabla^2 H_z=\eta^2 H_z
$$
在哪里
$$
\eta^2=j \omega_0 \cdot \mu \sigma
$$
$\omega_0=$正弦时变场的频率
$\mu=$岩心渗透率
$\sigma=$芯的电导率
这是一个二维问题，因为场仅沿$x$ -和$y$ -方向变化。因此，
$$
\frac{\partial H_z}{\partial x^2}+\frac{\partial H_z}{\partial y^2}=\eta^2 H_z
$$
在接下来的三个小节中讨论了具有三角形、六边形和八边形截面的实心岩心的这个方程的解。

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

考虑一个三角形截面的长固体导电铁芯，如图5.6所示。设三角形每条边的长度为$L$。用等边三角形的底边构成的矩形用虚线表示。设火炬函数由有限傅里叶级数定义:
$$
\left.H_z^{\prime}\right|{y=L / \sqrt{3}}=\sum{m-\text { odd }}^{(2 M-1)} T_m \cdot \cos \left(\frac{m \pi}{L} \cdot x\right)
$$
其中$T_m$表示一组傅里叶系数。

设置
$$
\left.H_z^{\prime}\right|{x= \pm L / 2}=\left.H_z^{\prime}\right|{y=-L /(2 \sqrt{3})}=0
$$
矩形区域涡流方程的解为
$$
H_z^{\prime}=\sum_{m-\alpha d d d}^{(2 M-1)} T_m \cdot \cos \left(\frac{m \pi}{L} \cdot x^{\prime}\right) \cdot \frac{\sinh \left[\alpha_m \cdot\left{y^{\prime}+L /(2 \sqrt{ } 3)\right}\right]}{\sinh \left(\alpha_m \cdot L \sqrt{3} / 2\right)}
$$
在哪里
$$
\begin{gathered}
\alpha_m=\sqrt{\left(\frac{m \pi}{L}\right)^2+\eta^2} \
x^{\prime}=x \
y^{\prime}=y
\end{gathered}
$$
接下来，我们再构造两个类似的矩形，每个矩形包含等边三角形剩下的两条边中的一条或另一条。让这些区域的场分布不变
$$
H_z^{\prime \prime}=\sum_{m-\text {-odd }}^{(2 M-1)} T_m \cdot \cos \left(\frac{m \pi}{L} \cdot x^{\prime \prime}\right) \cdot \frac{\sinh \left[\alpha_m \cdot\left{y^{\prime \prime}+L /(2 \sqrt{3})\right}\right]}{\sinh \left(\alpha_m \cdot L \cdot \sqrt{3} / 2\right)}
$$

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

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

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

Posted on 2023年6月29日2023年6月29日 by statistics-lab

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

考虑到铁的相对磁导率较大(即$\mu_r \gg 1$)的假设，规定了两组边界条件。其中第一个用于选择字段表达式，而第二个可用于计算任意常数。
字段表达式的选择
为了选择字段表达式，假设以下边界条件。所选表达式必须同样满足这些边界条件。
$$
\begin{aligned}
& \left.H_{4 y}\right|{x=d}=0 \ & \left.H{4 z}\right|{x=d}=0 \ & \left.H{3 y}\right|{z=-g}=0 \ & \left.H{3 x}\right|{z=0}=0 \ & \left.H{3 x}\right|_{z=-g}=0
\end{aligned}
$$

用来描述不同区域磁场的各种任意常数可以用以下边界条件求值:
$$
\begin{gathered}
\left.H_{4 x}\right|{z=0}=\left.H{1 x}\right|{z=0}-K{o y} \quad \text { over } 0 \leq x \leq d \
\left.H_{4 x}\right|{z=-8}=\left.H{2 x}\right|{z=-g} \quad \text { over } 0 \leq x \leq d \ \left.H{4 x}\right|{x=0}=\left.H{3 x}\right|{x=0} \quad \text { over }-g \leq z \leq 0 \ \left.H{4 y}\right|{z=0}=\left.H{1 y}\right|{z=0}+K{o x} \quad \text { over } 0 \leq x \leq d \
\left.H_{4 y}\right|{z=-8}=\left.H{2 y}\right|{z=-g} \quad \text { over } 0 \leq x \leq d \ \left.H{4 y}\right|{x=0}=\left.H{3 y}\right|_{x=0} \quad \text { over }-g \leq z \leq 0
\end{gathered}
$$

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

在受时变电磁场作用的导电区产生涡流。第七章讨论了瞬变电磁场引起的涡流。本节专门讨论由稳态正弦时变电磁场引起的涡流感应。对于工频激励，位移电流通常被忽略。因此，磁场强度$\boldsymbol{H}$满足下式:
$$
\begin{gathered}
\nabla^2 H=\eta^2 H \
\nabla \cdot H=\mathbf{0} \
\eta^2=-j \omega_0 \cdot \mu \sigma
\end{gathered}
$$

式中$\omega_o$为正弦时变场的频率，$\mu$为磁导率，$\sigma$为材料的电导率。

一旦解出式5.1a和$\mathrm{b}$，涡流密度可以很容易地从
$$
J=\nabla \times H
$$
通过以下边值问题讨论了磁场强度$H$的方程5.1a和$\mathrm{b}$的解。

图5.1为多相固体转子感应电机的简化二维模型，电枢绕组采用光滑的高导磁定子表面($z=-g$)上的表面电流片进行模拟。令在$z=0$处固定在转子上的参照系中的表面电流密度为
$$
K_x=K_v \cdot e^{j\left(t y-\omega_0 \cdot t\right)}
$$
在哪里
$$
\begin{gathered}
\omega_0=s \cdot \omega \
s=\operatorname{slip} \stackrel{\operatorname{def}}{=} 1-\frac{\text { rotor speed }}{\text { synchronous speed }}=1-\frac{v}{(\omega / \ell)} \
\omega=\text { supply frequency } \
\ell=\frac{\pi}{\text { pole pitch }}=\frac{\pi}{\tau}
\end{gathered}
$$
$\left|k_o\right|$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。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

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

Posted on 2023年6月9日2023年6月9日 by statistics-lab

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

由图4.9可以看出，$z$ -平面的实轴和虚轴分别用$x$和$y$轴表示，$w$ -平面的实轴和虚轴分别用$u$和$v$轴表示。将$z$ -平面(图4.9 a)转换为$w$ -平面(图$4.9 \mathrm{~b}$)，将$z$ -平面的实轴映射到$w$ -平面的正实轴上，将$z$ -平面的齿廓映射到w-平面的负实轴上。这要求将z平面上的实轴放置在$w$ -平面上实轴的正部分上。进一步，矫直$z$ -平面上的位形和$z$ -平面上的齿廓要映射到$w$ -平面实轴的负部分。在$z$ -平面上方便点$w=0, w=-\infty$和$w=\infty$如图4.9 a所示。$z=j \omega, j g / 2$和$\infty$在$w$ -平面上的结果点如图4.9b所示。在将结构拉出成一条直线时，两个角要被拉直。这些角度在图$4.9 \mathrm{a}$中显示为$\alpha$和$\beta$。

从$z$ -plane到$w$ -plane的Schwarz-Christoffel变换通过以下关系进行:
$$
\frac{d z}{d w}=A(w-a)^{(\alpha-\pi / \pi)} \cdot(w-b)^{(\beta-\pi / \pi)}
$$
其中$a$和$b$为两个内角$\alpha$和$\beta$在$w$ -平面上的位置。因此，
$$
\begin{gathered}
a=-1, \quad \alpha=3 \pi / 2 \
b=0, \quad \beta=0
\end{gathered}
$$
将这些值插入公式4.85，我们得到
$$
d z=A \cdot \frac{(w+1)^{1 / 2}}{w} \cdot d w
$$
关于积分，我们有
$$
z=A \cdot\left[2(w+1)^{1 / 2}+\log \left{\frac{(w+1)^{1 / 2}-1}{(w+1)^{1 / 2}+1}\right}\right]+C
$$
其中$C$为积分常数。对于$z$ -平面的原点，如图$4.9 \mathrm{a}$所示，$C$的值为零。设置$w=-1$时，得到:
$$
\left.z\right|{w=-1}=A \cdot \log (-1)=A \cdot j \pi $$ 而从图$4.9 \mathrm{a}$ $$ \left.z\right|{w=-1}=j(g / 2)
$$
因此，
$$
A=\frac{g}{2 \pi}
$$
给出变换关系为
$$
z=\frac{g}{2 \pi} \cdot\left[2(w+1)^{1 / 2}+\log \left{\frac{(w+1)^{1 / 2}-1}{(w+1)^{1 / 2}+1}\right}\right]
$$
在$w$ -平面上，实轴(或$u$)的正负部分处于不同的等电位值;也就是说，正的$u$为零，负的$u$为$-1 / 2$。因此，这个平面上的通量线是半圆。

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

接下来，考虑如图4.10所示的$\chi(=\varphi+j \psi)$ -平面。在该平面中，等势面平行于$\psi=0$(或1)平面，磁通线平行于$\psi$轴。电势值随距离$\psi=0$平面的距离呈线性变化。因此，这个平面称为正则场平面。该平面上的方便值$w(-\infty,-1,0$和$\infty)$如图4.10所示。

将$\chi$ -平面变换为$w$ -平面只需要矫直一个内角，即$\alpha=0$。因此，
$$
\frac{d \chi}{d w}=A(w-a)^{(\alpha-\pi / \pi)}
$$
其中$a=0$和$\alpha=0$。

因此，
$$
d \chi=\frac{A}{w} d w
$$
积分得到
$$
\chi=A \log (w)
$$
或者
$$
w=e^{(\chi / A)}
$$
由于，如图$4.10 w=-1$ at $\chi=j$所示，因此
$$
A=1 / \pi
$$
因此，
$$
w=e^{\pi \cdot \chi}
$$
代入式4.92，得到
$$
z=\frac{g}{2 \pi} \cdot\left[2\left(e^{\pi \cdot \chi}+1\right)^{1 / 2}+\log \left{\frac{\left(e^{\pi \cdot \chi}+1\right)^{1 / 2}-1}{\left(e^{\pi \cdot \chi}+1\right)^{1 / 2}+1}\right}\right]
$$
由于$z=x+j y$和$\chi=\varphi+j \psi$，通过为$\varphi$选择一个常数值可以得到一条通量线的图，对于不同的$\psi$值，由该方程求出其在$z$ -平面上坐标的连接点。对于不同的$\varphi$值可重复此过程，每个值产生不同的通量线。同样地，等势线的图可以通过互换$\varphi$和$\psi$的作用得到。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

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

Posted on 2023年6月9日2023年6月9日 by statistics-lab

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

气隙场沿周边方向周期性变化，周期为一个齿距 $\lambda$．在没有开槽的情况下，气隙场不沿外围方向变化。假设气隙表面的电势 $z=-g / 2$ as $+1 / 2$，在气隙表面 $z=+g / 2$ as $-1 / 2$时，气隙内的电位分布为
$$
\mathcal{V}0=-\frac{z}{g} $$ 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
$$
P_\lambda=\varphi_\lambda=\mu_o(\lambda / g)
$$
在式4.36中， $\delta$ 为两个开槽等势面齿心之间的距离，如图4.3所示。牙对牙的方向对应于 $\delta$ 等于零，而对于齿对槽方向，值为 $\delta$ 一半是齿距吗 $\lambda$．在前一种情况下，间隙渗透， $P_1$，为最大值。导电性值随 $\delta$ 从零开始增加。它达到一个最小值 $P_2$，牙对槽方向。根据式4.36，隙磁导率 $P$ 结束 $\lambda$ 对于双开槽气隙表面，发现如下:
$$
P=-\mu_o \int_0^\lambda \frac{\partial \mathcal{V}_o}{\partial z} \cdot d y=\mu_o 2 q_o(\lambda / g)
$$

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

在齿对齿的情况下，气隙表面 $z=0$ 是一个零电位曲面， $\mathcal{V}=0$．宽槽和深槽的方向如图4.5所示。在这个表面上，势分布是一个奇函数 $z$．此外，势分布是的偶函数 $y$．因此，气隙内的电位分布可以表示为
$$
\mathcal{V}0=-\frac{2}{\pi} \int_0^{\infty} F(u) \cdot \cos (u \cdot y) \cdot \frac{\sinh {u \cdot z}}{\sinh (u \cdot g / 2)} \cdot d u-\frac{z}{g} $$ 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
$$
F(u)=\int_0^{\infty}\left(\left.\mathcal{V}o\right|{z=-g / 2}-1 / 2\right) \cdot \cos (u \cdot y) \cdot d u
$$
槽1区域内的电位分布可以表示为
$$
\mathcal{V}_1=\frac{2}{\pi} \int_0^{\infty} f(w) \cdot \sin {w \cdot(y-t / 2)} \cdot e^{w(z+g / 2)} \cdot d w-\frac{1}{\pi} \cdot \tan ^{-1}\left(\frac{z+g / 2}{y-t / 2}\right)
$$
结束 $t / 2 \leq y<\infty$ 和 $z \ngtr-g / 2$．
式4.49中第二项表示铁表面上的电位值 $y=t / 2$，为 $z<-g / 2$．第一项允许在开槽处电位分布的一般变化而不干扰铁表面上的电位。这个术语可以被认为是一个纠正术语。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

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

Posted on 2023年6月9日2023年6月9日 by statistics-lab

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

考虑矢量磁势的泊松方程。设这个方程的解，对于体积为$v$的区域，满足边界条件近似为$A$。我们定义绝对误差$\alpha$为
$$
\alpha \stackrel{d e f}{=} A-A_o
$$
式中$A_o$为该区域泊松方程的解，该解精确地满足其边界面上的边界条件$s$。向量误差$\alpha$是$v$中任意点的空间坐标的函数。边界面上点的误差定义为$\alpha^s$，
$$
\left.\alpha^s \stackrel{d e f}{=}\left(A-A_o\right)\right|_s
$$
自从
$$
\nabla^2 A=-\mu J
$$
和
$$
\nabla^2 A_o=-\mu J
$$
因此，
$$
\nabla^2 \alpha=0
$$
此外，我们有
$$
\nabla \cdot \alpha=\nabla \cdot A-\nabla \cdot A_o=0-0=0
$$
设置
$$
\beta \stackrel{d e f}{=} \nabla \times \alpha
$$
让我们考虑一下同一性
$$
\begin{aligned}
\nabla \cdot(\alpha \times \beta) & \equiv \beta \cdot(\nabla \times \alpha)-\alpha \cdot(\nabla \times \beta) \
& =|\beta|^2-\alpha \cdot[\nabla \times \nabla \times \alpha] \
& \equiv|\beta|^2-\alpha \cdot\left[\nabla(\nabla \cdot \alpha)-\nabla^2 \alpha\right]
\end{aligned}
$$
因为散度和拉普拉斯向量误差$\alpha$都是零，
$$
\nabla \cdot(\alpha \times \beta)=|\beta|^2
$$

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

考虑以因子$e^{-j \omega t}$为特征的调和场的相量形式的麦克斯韦的两个旋度方程
$$
\begin{gathered}
\nabla \times E=j \omega \mu H \
\nabla \times H=J-j \omega \varepsilon E
\end{gathered}
$$
令这些方程中涉及的场向量仅近似地满足规定的边界条件。完全满足给定边界条件的场矢量麦克斯韦方程为
$$
\begin{gathered}
\nabla \times \boldsymbol{E}_o=j \omega \mu \boldsymbol{H}_o \
\nabla \times \boldsymbol{H}_o=\boldsymbol{J}_o-j \omega \varepsilon \boldsymbol{E}_o
\end{gathered}
$$
因此，
$$
\begin{gathered}
\nabla \times e=j \omega \mu h \
\nabla \times h=j-j \omega \varepsilon e
\end{gathered}
$$
在哪里
$$
\begin{gathered}
\boldsymbol{e}=\boldsymbol{E}-\boldsymbol{E}_o \
\boldsymbol{h}=\boldsymbol{H}-\boldsymbol{H}_o \
j=\boldsymbol{J}-\boldsymbol{J}_o
\end{gathered}
$$
和
$$
j=\sigma e
$$
现在，既然
$$
-\nabla \cdot\left(e \times h^\right) \equiv-h^ \cdot(\nabla \times e)+e \cdot\left(\nabla \times h^\right) $$因此，根据式3.138c和3.139c，我们得到$$ -\nabla \cdot\left(\mathbf{e} \times h^\right)=\sigma e^2+j \omega\left(\varepsilon e^2-\mu h^2\right)
$$
在哪里
$$
\begin{aligned}
& e^2 \stackrel{d e f}{=} \boldsymbol{e} \cdot \boldsymbol{e}^* \
& h^2 \stackrel{d e f}{=} \boldsymbol{h} \cdot \boldsymbol{h}^*
\end{aligned}
$$

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

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

Posted on 2023年5月22日2023年5月22日 by statistics-lab

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

考虑图3.2所示的矩形区域$(-W / 2 \leq x \leq W / 2,0 \leq y \leq h)$。该区域分段均匀，介电常数为$(0<y<g)$为$\varepsilon_1$, $(g<y<h)$为$\varepsilon_2$。指定的边界条件为
$$
\begin{gathered}
\left.V\right|{x= \pm W / 2}=0 \ \left.V\right|{y=0}=0
\end{gathered}
$$
在顶面，即$y=h$处没有指定边界条件。然而，没有另一个边界条件，而是给出
$$
\left.V\right|{y=k}=\sum{m-o d d}^{\infty} a_m \cos \left(\frac{m \pi}{W} \cdot x\right)+\sum_{n=1}^{\infty} b_n \sin \left(\frac{n 2 \pi}{W} \cdot x\right)
$$
在哪里$g<k<h$。
让我们把这个矩形区域分成两个子区域，这样子区域1延伸到$(-W / 2 \leq x \leq W / 2,0 \leq y \leq k)$上，子区域2延伸到$(-W / 2 \leq x \leq W / 2, k \leq y \leq h)$上。对于子区域1，确定了所有四个边界边的潜在分布;因此，如果它是一个均匀区域，就应该有可能得到势分布的唯一解。由于子区域1是分段均匀的，$y$$=g$作为两个均匀带的边界，即1a延伸到$0<y<g$上，$1 \mathrm{~b}$延伸到$g<y<k$上，如果已知沿边界$y=g$的电位分布，则可以很容易地获得这些区域的电位分布。让我们假设一个伪火炬函数
$$
\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)
$$
其中$c_m$和$d_n$表示两组未知常数。

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

一般来说，向量磁势的拉普拉斯方程和泊松方程有无穷个解$A$。这里回顾的唯一性定理${ }^2$描述了这些方程的唯一解所满足的边界条件。

设$A_1$和$A_2$为方程2.40的任意两个解，给定以封闭表面$s$为界的体积$v$中的磁势分布。差电位$A_o$是
$$
A_o=A_1-A_2
$$
可以注意到，差分势$A_o$满足拉普拉斯方程
$$
\nabla^2 A_o=0
$$
鉴于式2.38
$$
\nabla \cdot A_1=\nabla \cdot A_2=0
$$
因此，
$$
\nabla \cdot A_o=0
$$
考虑同一性 ${ }^3$
$$
\iiint_v[(\nabla \times \boldsymbol{P}) \cdot(\nabla \times \mathbf{Q})-\boldsymbol{P} \cdot(\nabla \times \nabla \times \mathbf{Q})] d v \equiv \oiint_s[\boldsymbol{P} \times(\nabla \times \mathbf{Q})] \cdot d s
$$

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

物理代写|电磁学代写electromagnetism代考|Periodic Fields, Field Equations in Phasor Form

Posted on 2023年5月22日2023年5月22日 by statistics-lab

物理代写|电磁学代写electromagnetism代考|Periodic Fields, Field Equations in Phasor Form

Maxwell’s equations in point (or differential) form for harmonic fields can be readily obtained from Equations 2.9 and 2.10. We define field vectors as complex phasors instead of instantaneous sinusoidal quantities. The moduli of these complex phasors give root mean square values. If these complex phasors are multiplied with $\sqrt{2}$ times the exponential factor $\exp (j \omega t)$, the real part gives the instantaneous expressions for harmonic field of frequency $\omega$. The time derivative of field is simply the phasor field multiplied by $j \omega$. Thus, Maxwell’s equations for harmonic fields are
$$
\begin{gathered}
\nabla \cdot \tilde{\boldsymbol{B}}=0 \
\nabla \cdot \tilde{\boldsymbol{D}}=\tilde{\rho} \
\nabla \times \tilde{\boldsymbol{E}}=-j \omega \cdot \tilde{B} \
\nabla \times \tilde{\boldsymbol{H}}=\tilde{J}+j \omega \cdot \tilde{D}
\end{gathered}
$$
In the above equations, phasor quantities are distinguished from instantaneous quantities by placing a cap on each symbol. In subsequent treatment, however, no such distinction is made.

Maxwell’s equations for harmonic fields in homogeneous source-free regions are given as
$$
\nabla \cdot H=0
$$

$$
\begin{gathered}
\nabla \cdot E=0 \
\nabla \times E=-j \omega \mu \cdot H \
\nabla \times H=\sigma E+j \omega \varepsilon \cdot E
\end{gathered}
$$
Using Equations 2.50 and 2.51a
$$
\nabla \times \nabla \times E=-j \omega \mu \cdot \nabla \times H=-j \omega \mu \cdot(\sigma+j \omega \varepsilon) E
$$
Also,
$$
\nabla \times \nabla \times E \equiv \nabla(\nabla \cdot E)-\nabla^2 E
$$

物理代写|电磁学代写electromagnetism代考|Retarded Potentials

Earlier, Maxwell’s equations for time-varying fields were given by Equations 2.9 a through $2.9 \mathrm{~d}$ and the current density by Equation 2.10. Also $B$ was given by Equation 2.35. These are reproduced below:
$$
\begin{gathered}
\nabla \cdot \boldsymbol{B}=0 \
\nabla \cdot \boldsymbol{D}=\rho \
\nabla \times \boldsymbol{E}=-\frac{\partial \boldsymbol{B}}{\partial t} \
\nabla \times \boldsymbol{H}=\boldsymbol{J}+\frac{\partial \boldsymbol{D}}{\partial t} \
\boldsymbol{J}=\boldsymbol{J}_o+\sigma \boldsymbol{B} \stackrel{\text { def }}{=} \nabla \times A
\end{gathered}
$$
From Equations $2.9 \mathrm{c}$ and 2.35, we get
$$
\nabla \times E=-\frac{\partial}{\partial t}(\nabla \times A)
$$
For stationary medium, space and time coordinates being independent of each other, we may rewrite this equation as
$$
\nabla \times E=-\nabla \times\left(\frac{\partial A}{\partial t}\right)
$$
Therefore,
$$
\nabla \times\left(\boldsymbol{E}+\frac{\partial \boldsymbol{A}}{\partial t}\right)=0
$$
In view of Equation 2.54c, we may define
$$
E+\frac{\partial A}{\partial t} \stackrel{\operatorname{def}}{=}-\nabla V
$$
or
$$
\boldsymbol{E}=-\nabla V-\frac{\partial \boldsymbol{A}}{\partial t}
$$

电磁学代考

物理代写|电磁学代写electromagnetism代考|Periodic Fields, Field Equations in Phasor Form

谐波场的点(或微分)形式麦克斯韦方程组可由式2.9和式2.10得到。我们将场矢量定义为复相量，而不是瞬时正弦量。这些复相量的模得到均方根值。如果将这些复相量与$\sqrt{2}$乘以指数因子$\exp (j \omega t)$相乘，则实部给出频率$\omega$的谐波场的瞬时表达式。场的时间导数就是相量场乘以$j \omega$。因此，调和场的麦克斯韦方程组为
$$
\begin{gathered}
\nabla \cdot \tilde{\boldsymbol{B}}=0 \
\nabla \cdot \tilde{\boldsymbol{D}}=\tilde{\rho} \
\nabla \times \tilde{\boldsymbol{E}}=-j \omega \cdot \tilde{B} \
\nabla \times \tilde{\boldsymbol{H}}=\tilde{J}+j \omega \cdot \tilde{D}
\end{gathered}
$$
在上述方程中，相量与瞬时量的区别是在每个符号上加一个帽。然而，在随后的治疗中，没有作出这样的区分。

齐次无源区域谐波场的麦克斯韦方程为
$$
\nabla \cdot H=0
$$

$$
\begin{gathered}
\nabla \cdot E=0 \
\nabla \times E=-j \omega \mu \cdot H \
\nabla \times H=\sigma E+j \omega \varepsilon \cdot E
\end{gathered}
$$
使用公式2.50和2.51a
$$
\nabla \times \nabla \times E=-j \omega \mu \cdot \nabla \times H=-j \omega \mu \cdot(\sigma+j \omega \varepsilon) E
$$
还有，
$$
\nabla \times \nabla \times E \equiv \nabla(\nabla \cdot E)-\nabla^2 E
$$

物理代写|电磁学代写electromagnetism代考|Retarded Potentials

之前，时变场的麦克斯韦方程组由式2.9 a至$2.9 \mathrm{~d}$给出，电流密度由式2.10给出。$B$由式2.35给出。现将这些资料转载如下:
$$
\begin{gathered}
\nabla \cdot \boldsymbol{B}=0 \
\nabla \cdot \boldsymbol{D}=\rho \
\nabla \times \boldsymbol{E}=-\frac{\partial \boldsymbol{B}}{\partial t} \
\nabla \times \boldsymbol{H}=\boldsymbol{J}+\frac{\partial \boldsymbol{D}}{\partial t} \
\boldsymbol{J}=\boldsymbol{J}_o+\sigma \boldsymbol{B} \stackrel{\text { def }}{=} \nabla \times A
\end{gathered}
$$
由式$2.9 \mathrm{c}$和2.35，我们得到
$$
\nabla \times E=-\frac{\partial}{\partial t}(\nabla \times A)
$$
对于静止介质，空间坐标和时间坐标相互独立，我们可以将方程改写为
$$
\nabla \times E=-\nabla \times\left(\frac{\partial A}{\partial t}\right)
$$
因此，
$$
\nabla \times\left(\boldsymbol{E}+\frac{\partial \boldsymbol{A}}{\partial t}\right)=0
$$
根据公式2.54c，我们可以定义
$$
E+\frac{\partial A}{\partial t} \stackrel{\operatorname{def}}{=}-\nabla V
$$
或
$$
\boldsymbol{E}=-\nabla V-\frac{\partial \boldsymbol{A}}{\partial t}
$$

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

物理代写|电磁学代写electromagnetism代考|Maxwell’s Equations in Integral Form

Posted on 2023年5月22日2023年5月22日 by statistics-lab

物理代写|电磁学代写electromagnetism代考|Maxwell’s Equations in Integral Form

Electric lines of force, also called flux lines, emanate from positive electric charge and terminate on negative electric charge. Likewise, in the case of permanent magnets, magnetic flux lines outside the magnet emanate from its north pole and terminate on the south pole. Inside the magnet, the direction of flux lines are reversed, that is, these lines emanate from south pole and terminate on the north pole. Therefore, magnetic flux lines are closed curves. Clearly, the electric flux lines and the magnetic flux lines behave differently. This is because the positive and negative charges exist independently while north and south magnetic poles are not. Therefore, it is possible that only one type of electric charge may reside inside a closed surface. However, no closed surface can enclose only one type of magnetic pole. The total electric flux emanating from a closed surface is defined as the total charge inside the surface. If both positive and negative charges are present inside the surface, the net charge will be the algebraic sum of discrete charges. In the case of distributed charge, the net charge will be the integrated value of the charge density. The magnetic poles being inseparable (magnetic monopoles do not exist in nature even though we may sometimes postulate their existence purely as a mathematical construct), the total magnetic flux emanating from any closed surface is always zero. This leads to the Gauss’s law, or the integral form of Maxwell’s ${ }^1$ equations for magnetic and electric fields
$$
\begin{aligned}
& \oiint_s B \cdot d s=0 \
& \oiint_s \boldsymbol{D} \cdot d s=Q
\end{aligned}
$$
In these equations, $\boldsymbol{B}$ and $\boldsymbol{D}$, respectively, indicate magnetic and electric flux density on the closed surface $s$, and $Q$ indicates net charge inside this surface.
Faraday’s law of electromagnetic induction states that electro-motive-force (e.m.f.), in a closed contour (i.e. closed path), is equal to the rate of decrease of magnetic flux $\varphi$, linking (or passing through) the closed contour. The e.m.f. is defined as the integrated value of the electric field intensity along the closed contour. Therefore,
$$
\text { e.m.f. }=-\frac{d \varphi}{d t}
$$
or
$$
\oint_c E \cdot d l=-\frac{d}{d t} \iint_s B \cdot d s
$$
In these equations, the electric field intensity $E$ is on the closed contour $c$ and the magnetic flux density $\boldsymbol{B}$ is on the surface s, while the contour $c$ is the edge of the open surface s. Equation 2.4 is identified as Maxwell’s third equation in integral form.

物理代写|电磁学代写electromagnetism代考|Maxwell’s Equations in Point Form

Maxwell’s equations in point (or differential) form can be readily obtained by applying the divergence theorem to Equations 2.1 and 2.2, and using Stokes theorem to Equations 2.4 and 2.8. The resulting equations are
$$
\begin{gathered}
\nabla \cdot B=0 \
\nabla \cdot D=\rho \
\nabla \times E=-\frac{\partial B}{\partial t} \
\nabla \times H=J+\frac{\partial D}{\partial t}
\end{gathered}
$$
In Equation 2.9b, the volume charge density $\rho$, at a given point, is a source for electric field. In Equation 2.9d, the current density $J$ has two components
$$
J=J_o+\sigma E
$$
where $J_o$ indicates current source density, while the second term $(\sigma E)$ indicates induced current density in a conductor with conductivity $\sigma$.

电磁学代考

物理代写|电磁学代写electromagnetism代考|Maxwell’s Equations in Integral Form

电力线，也称为通量线，由正电荷发出，以负电荷终止。同样，在永久磁铁的情况下，磁铁外部的磁通量线从其北极发出，终止于南极。在磁体内部，磁通线的方向是相反的，即磁通线从南极发出，终止于北极。因此，磁通量线是封闭曲线。显然，电通量线和磁通量线的表现是不同的。这是因为正电荷和负电荷是独立存在的，而南北磁极则不是。因此，有可能在一个封闭的表面内只存在一种电荷。然而，没有一个封闭的表面可以只包含一种类型的磁极。从封闭表面发出的总电通量定义为表面内的总电荷。如果表面内同时存在正电荷和负电荷，则净电荷将是离散电荷的代数和。在分布电荷的情况下，净电荷将是电荷密度的积分值。磁极是不可分割的(磁单极子在自然界中并不存在，尽管我们有时可以把它们纯粹作为一种数学结构来假设)，从任何封闭表面发出的总磁通量总是零。这就引出了高斯定律，或者麦克斯韦的${ }^1$磁场和电场方程的积分形式
$$
\begin{aligned}
& \oiint_s B \cdot d s=0 \
& \oiint_s \boldsymbol{D} \cdot d s=Q
\end{aligned}
$$
式中$\boldsymbol{B}$和$\boldsymbol{D}$分别表示封闭表面$s$上的磁通密度和电通密度，$Q$表示封闭表面内的净电荷。
法拉第电磁感应定律指出，在封闭轮廓(即封闭路径)中，电动势(e.m.f)等于连接(或穿过)封闭轮廓的磁通量降低率$\varphi$。电动势定义为电场强度沿闭合轮廓的积分值。因此，
$$
\text { e.m.f. }=-\frac{d \varphi}{d t}
$$
或
$$
\oint_c E \cdot d l=-\frac{d}{d t} \iint_s B \cdot d s
$$
其中，电场强度$E$在闭合轮廓$c$上，磁通密度$\boldsymbol{B}$在曲面s上，而轮廓$c$为开放表面s的边缘。将式2.4确定为麦克斯韦第三方程的积分形式。

物理代写|电磁学代写electromagnetism代考|Maxwell’s Equations in Point Form

将散度定理应用于方程2.1和2.2，将Stokes定理应用于方程2.4和2.8，可以很容易地得到点(或微分)形式的Maxwell方程。得到的方程是
$$
\begin{gathered}
\nabla \cdot B=0 \
\nabla \cdot D=\rho \
\nabla \times E=-\frac{\partial B}{\partial t} \
\nabla \times H=J+\frac{\partial D}{\partial t}
\end{gathered}
$$
在式2.9b中，给定点的体积电荷密度$\rho$是电场的来源。在式2.9d中，电流密度$J$有两个分量
$$
J=J_o+\sigma E
$$
其中$J_o$表示电流源密度，而第二项$(\sigma E)$表示导电率$\sigma$的导体中的感应电流密度。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

物理代写|电磁学代写electromagnetism代考|ELEC3104

Posted on 2023年3月14日2023年3月14日 by statistics-lab

如果你也在怎样代写电磁学electromagnetism这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

电磁学是电荷、磁矩和电磁场之间的物理互动。电磁场可以是静态的，缓慢变化的，或形成波。电磁波一般被称为光，遵守光学定律。

我们提供的电磁学electromagnetism及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(Generalized) Linear Models 广义线性模型
Statistical Machine Learning 统计机器学习
Longitudinal Data Analysis 纵向数据分析
Foundations of Data Science 数据科学基础

物理代写|电磁学代写electromagnetism代考|The pH and Equilibrium Constant

The dissociation constant of water molecules manifests the competition between the energy of binding and the entropy of charge liberation. It requires considering the interchange of energetic $(E)$ and entropic $(S)$ effects in controlling the separation to examine $\mathrm{pH}$ from a quantitative viewpoint. The competition between energy minimization (bound water molecule state) and entropy maximum (ionic dissociation of water) sets the stage for many biological reactions.
The reaction for dissociation of a water molecule is
$$
\mathrm{H}_2 \mathrm{O} \rightleftharpoons \mathrm{H}^{+}+\mathrm{OH}^{-}
$$

The problem is to find the fraction of water molecules that are in dissociated state in a sample of water. Equilibrium constant is gives as
$$
K_{\mathrm{eq}}=\left(\prod_i^N c_{i 0}^{\nu_i}\right) \exp \left(-\beta \sum_i^N \mu_{i 0} \nu_i\right)
$$
In Eq. (11.2), $\beta=1 / k_B T$ (where $k_B$ is Boltzmann’s constant and $T$ is the temperature), $\mu_{i 0}$ is the standard chemical potential, and
$$
\mu_i=\mu_{i 0}+k_B T \ln \left(\frac{c_i}{c_{i 0}}\right)
$$
where $c_{i 0}$ is the standard state concentration of the $i$ component. $\nu_i$ is the stoichiometric coefficient that equals the change in the number of particles of the $i$-th component during reaction:
$$
\mathrm{A}+\mathrm{B} \rightleftharpoons \mathrm{AB}
$$
In Eq. (11.4), $\nu_{\mathrm{A}}=\nu_{\mathrm{B}}=-1$ and $\nu_{\mathrm{AB}}=1$. On the other hand, for reaction given by Eq. (11.1), $\nu_{\mathrm{H}^{+}}=\nu_{\mathrm{OH}^{-}}=+1$ and $\nu_{\mathrm{H}2 \mathrm{O}}=-1$. The dissociation constant is defined as $$ K{\mathrm{d}}=\frac{1}{K_{\mathrm{eq}}}
$$
The law of mass action implies that
$$
K_{\mathrm{eq}}=\prod_i^N c_i^{\nu_i}
$$
which is known as the law of mass action.

物理代写|电磁学代写electromagnetism代考|Charge on DNA and Proteins

The charge state of a macromolecule depends on $\mathrm{pH}$ of the solution. On the other hand, the charge state of the macromolecule is important in determining both their structure and function in solution. Here, we discuss how the charge state is tuned, by considering the following reaction:
$$
\mathrm{HM} \rightleftharpoons \mathrm{H}^{+}+\mathrm{M}^{-} .
$$
Here, $M$ is the macromolecule of interest. From the equation of the law of mass action, the dissociation constant becomes:
$$
K_{\mathrm{d}}=\frac{c_{\mathrm{H}^{+}} \cdot c_{\mathrm{M}^{-}}}{c_{\mathrm{HM}}}
$$
We introduce $\mathrm{pK}$ to measure the tendency of the macromolecule to undergo the dissociation reaction as
$$
\mathrm{pK}=-\log {10} K{\mathrm{d}} .
$$”

Or,
$$
\begin{aligned}
+\log {10} K{\mathrm{d}} & =\log {10} c{\mathrm{H}^{+}}+\log {10} c{\mathrm{M}^{-}}-\log {10} c{\mathrm{HM}} \
& =-\mathrm{pH}+\log {10} c{\mathrm{M}^{-}}-\log {10} c{\mathrm{HM}} .
\end{aligned}
$$
Combining Eqs. (11.16) and (11.17), we obtain
$$
\mathrm{pH}=\mathrm{pK}+\log {10}\left(\frac{c{\mathrm{M}^{-}}}{c_{\mathrm{HM}}}\right)
$$
which is known as Hendersen-Hasselbalch equation.
If $c_{\mathrm{M}^{-}}=c_{\mathrm{HM}}$, then $\mathrm{pH}=\mathrm{pK}$. That corresponds to the $\mathrm{pH}$ at which half of $\mathrm{HM}$ molecules are dissociated. Therefore, $\mathrm{pK}$ equals $\mathrm{pH}$ at which half of macromolecules have been dissociated.
Consider a DNA, which has a $\mathrm{pK}$ such that $\mathrm{pK} \approx 1$. Then,
$$
\mathrm{pH}=1+\log {10}\left(\frac{c{\mathrm{DNA}^{-}}}{c_{\mathrm{HDNA}}}\right) \text {. }
$$
Using Eq. (11.19), at normal pH (that is, $\mathrm{pH}=7$ ), phosphates on the DNA backbones are fully dissociated. That is, two electronic charges for every base pair, or a linear charge density:
$$
\lambda=\frac{2 e}{0.34 \mathrm{~nm}}
$$

电磁学代考

物理代写|电磁学代写electromagnetism代考|The pH and Equilibrium Constant

水分子的解离常数体现了结合能与电荷释放熵之间的竞争。它需要考虑能量的交换 $(E)$ 和樀 $(S)$ 控制分离检查的效果 $\mathrm{pH}$ 从定量的角度来看。能量最小化 (结合水分子状态) 和樀最大 (水的离子解离) 之间的竞争为许多生物反应奠定了基础。水分子的解离反应是
$$
\mathrm{H}2 \mathrm{O} \rightleftharpoons \mathrm{H}^{+}+\mathrm{OH}^{-} $$ 问题是找到水样中处于离解状态的水分子的分数。平衡常数为 $$ K{\mathrm{eq}}=\left(\prod_i^N c_{i 0}^{\nu_i}\right) \exp \left(-\beta \sum_i^N \mu_{i 0} \nu_i\right)
$$
在等式中。(11.2), $\beta=1 / k_B T$ (在哪里 $k_B$ 是玻尔兹曼常数，并且 $T$ 是温度)， $\mu_{i 0}$ 是标准化学势，并
$$
\mu_i=\mu_{i 0}+k_B T \ln \left(\frac{c_i}{c_{i 0}}\right)
$$
在哪里 $c_{i 0}$ 是标准状态浓度 $i$ 成分。 $\nu_i$ 是化学计量系数，等于粒子数的变化 $i$ – 反应过程中的第一个组分:
$$
A+B \rightleftharpoons A B
$$
在等式中。(11.4), $\nu_{\mathrm{A}}=\nu_{\mathrm{B}}=-1$ 和 $\nu_{\mathrm{AB}}=1$. 另一方面，对于方程式给出的反应。(11.1), $\nu_{\mathrm{H}^{+}}=\nu_{\mathrm{OH}^{-}}=+1$ 和 $\nu_{\mathrm{H} 2 \mathrm{O}}=-1$. 解离常数定义为
$$
K \mathrm{~d}=\frac{1}{K_{\mathrm{eq}}}
$$
群众行动定律意味着
$$
K_{\mathrm{eq}}=\prod_i^N c_i^{\nu_i}
$$
这就是众所周知的质量作用定律。

物理代写|电磁学代写electromagnetism代考|Charge on DNA and Proteins

大分子的电荷态取决于 $\mathrm{pH}$ 的解决方案。另一方面，大分子的电荷状态对于确定它们在溶液中的结构和功能很重要。在这里，我们通过考虑以下反应来讨论如何调整电荷状态:
$$
\mathrm{HM} \rightleftharpoons \mathrm{H}^{+}+\mathrm{M}^{-} .
$$
这里， $M$ 是感兴趣的大分子。根据质量作用定律的方程式，解离常数变为:
$$
K_{\mathrm{d}}=\frac{c_{\mathrm{H}^{+}} \cdot c_{\mathrm{M}^{-}}}{c_{\mathrm{HM}}}
$$
我们介绍pK测量大分子发生解离反应的趋势
$$
\mathrm{pK}=-\log 10 K \mathrm{~d}
$$
或者，
$$
+\log 10 K \mathrm{~d}=\log 10 c \mathrm{H}^{+}+\log 10 c \mathrm{M}^{-}-\log 10 c \mathrm{HM} \quad=-\mathrm{pH}+\log 10 c \mathrm{M}^{-}-\log 10 c \mathrm{HM} .
$$
结合方程式。(11.16) 和 (11.17)，我们得到
$$
\mathrm{pH}=\mathrm{pK}+\log 10\left(\frac{c \mathrm{M}^{-}}{c_{\mathrm{HM}}}\right)
$$
称为 Hendersen-Hasselbalch 方程。
如果 $c_{\mathrm{M}^{-}}=c_{\mathrm{HM}}$ ，然后 $\mathrm{pH}=\mathrm{pK}$. 这对应于 $\mathrm{pH}$ 在哪一半 $\mathrm{HM}$ 分子解离。所以， $\mathrm{pK}$ 等于 $\mathrm{pH}$ 其中一半的大分子已经解离。
考虑一个 DNA，它有一个 $\mathrm{pK}$ 这样 $\mathrm{pK} \approx 1$. 然后，
$$
\mathrm{pH}=1+\log 10\left(\frac{c \mathrm{DNA}^{-}}{c_{\mathrm{HDNA}}}\right) .
$$
使用方程式。(11.19)，在正常 $\mathrm{pH}$ 值下（即， $\mathrm{pH}=7$ )， DNA 主链上的磷酸盐完全解离。也就是说，每个碱基对有两个电子电荷，或线性电荷密度:
$$
\lambda=\frac{2 e}{0.34 \mathrm{~nm}}
$$

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