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## 物理代写|电磁学代写Electromagnetism代考|Idealised Model

A model machine, based on a set of simplifying assumptions, is shown in Figure $7.5 \mathrm{a}$. As the curvature of air-gap surfaces is ignored, no special functions are needed for the field expressions. The stator and rotor cores, in their developed forms, are represented by two infinitely long regions with rectangular cross-sections. The current-carrying three-phase stator winding is simulated by a current sheet, as given in Appendix 6. This current sheet is placed on the smooth stator surface, as indicated in Figure 7.5b. The separation between the stator and rotor cores is the effective air gap of the machine. The stator and rotor cores are enclosed on one side by the developed surface of the shaft and on the remaining three sides by the inner surface of the stator frame, which includes end covers. It is assumed that the stator frame, stator core and the rotor shaft are highly permeable so that the tangential components of magnetic field intensity on these surfaces are negligible.

The stator current sheet varies periodically in the peripheral direction. A sinusoidal variation is assumed; thus, effects of winding harmonics are neglected. The magnetic saturation is neglected and the rotor iron permeability is taken as a positive real number. In the machine thus idealised, rotor core is the only conducting region. In Figure 7.5a, the rotor core is shown as region 1. The remaining five regions in this figure are air regions. Because of the symmetry between these regions, field distributions in regions 5 and 6 need not be considered. The primary source for the magnetic field in all regions is the known stator current sheet.

In view of Figure $7.5 \mathrm{a}, X$ is taken parallel to the axial, $Y$ to the peripheral and $Z$ to the radial direction in the rectangular Cartesian system of space coordinates. The smooth rotor surface at the air gap is taken as the surface $z=0 ; z=-g$ and $z=D_R$ represent the stator air-gap surface and the shaft surface, respectively. The middle of the axial length of the machine is taken as the surface $x=0$ and $x= \pm L_R / 2$ and $x= \pm L_S / 2$ represent the rotor and the stator end surfaces, respectively. The two end covers are presumed to be located at $x= \pm L_O / 2$.

## 物理代写|电磁学代写Electromagnetism代考|Field Distributions

The stator current sheet simulating the stator winding, with balanced threephase currents, is the primary source for the magnetic field in all regions, that is, from region 1 to region 6. Therefore, the distribution of magnetic field in each region is characterised by the exponential factor: $\exp \cdot j(\omega t-\ell y)$, where $\ell=\pi / \tau$, $\tau$ being the pole pitch. Field expressions in this section are written in complex form without the exponential factor. Complete expressions for all field quantities can be obtained by inserting the exponential factor and then selecting the real part.

The rotor region is the only conducting region. All the remaining regions are air regions. The eddy current density is governed by the following field equations:
$$\nabla \cdot J_1=0$$
and
$$\nabla^2 J_1=\frac{j}{d^2} J_1$$
where
$$d^2=\frac{1}{s \cdot \omega \cdot \mu \cdot \sigma}$$
The magnetic field intensity $\boldsymbol{H}_1$ in the solid rotor in terms of eddy current density $J_1$ can be written as
$$\boldsymbol{H}_1=j d^2 \nabla \times \boldsymbol{J}_1$$
The axial component of the rotor eddy current density vanishes at the rotor end surfaces and is an even function of $x$. Thus, in a reference frame fixed to the rotor, the axial component of the rotor eddy current density at the rotor air-gap surface can be given by the following half-range Fourier series:
$$\left.J_{x 1}\right|{z=0}=\sum{p-\text { odd }}^{\infty} a_p \cdot \cos \left(\frac{p \pi}{L_R} \cdot x\right),$$
where $a_p$ indicates a set of arbitrary constants.

## 物理代写|电磁学代写Electromagnetism代考|Field Distributions

$$\nabla \cdot J_1=0$$

$$\nabla^2 J_1=\frac{j}{d^2} J_1$$

$$d^2=\frac{1}{s \cdot \omega \cdot \mu \cdot \sigma}$$

## 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。

Posted on Categories:Electromagnetism, 物理代写, 电磁学

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

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•70分以下全额退款

## 物理代写|电磁学代写Electromagnetism代考|Idealised Model

A model machine, based on a set of simplifying assumptions, is shown in Figure $7.5 \mathrm{a}$. As the curvature of air-gap surfaces is ignored, no special functions are needed for the field expressions. The stator and rotor cores, in their developed forms, are represented by two infinitely long regions with rectangular cross-sections. The current-carrying three-phase stator winding is simulated by a current sheet, as given in Appendix 6. This current sheet is placed on the smooth stator surface, as indicated in Figure 7.5b. The separation between the stator and rotor cores is the effective air gap of the machine. The stator and rotor cores are enclosed on one side by the developed surface of the shaft and on the remaining three sides by the inner surface of the stator frame, which includes end covers. It is assumed that the stator frame, stator core and the rotor shaft are highly permeable so that the tangential components of magnetic field intensity on these surfaces are negligible.

The stator current sheet varies periodically in the peripheral direction. A sinusoidal variation is assumed; thus, effects of winding harmonics are neglected. The magnetic saturation is neglected and the rotor iron permeability is taken as a positive real number. In the machine thus idealised, rotor core is the only conducting region. In Figure 7.5a, the rotor core is shown as region 1. The remaining five regions in this figure are air regions. Because of the symmetry between these regions, field distributions in regions 5 and 6 need not be considered. The primary source for the magnetic field in all regions is the known stator current sheet.

In view of Figure $7.5 \mathrm{a}, X$ is taken parallel to the axial, $Y$ to the peripheral and $Z$ to the radial direction in the rectangular Cartesian system of space coordinates. The smooth rotor surface at the air gap is taken as the surface $z=0 ; z=-g$ and $z=D_R$ represent the stator air-gap surface and the shaft surface, respectively. The middle of the axial length of the machine is taken as the surface $x=0$ and $x= \pm L_R / 2$ and $x= \pm L_S / 2$ represent the rotor and the stator end surfaces, respectively. The two end covers are presumed to be located at $x= \pm L_O / 2$.

## 物理代写|电磁学代写Electromagnetism代考|Field Distributions

The stator current sheet simulating the stator winding, with balanced threephase currents, is the primary source for the magnetic field in all regions, that is, from region 1 to region 6. Therefore, the distribution of magnetic field in each region is characterised by the exponential factor: $\exp \cdot j(\omega t-\ell y)$, where $\ell=\pi / \tau$, $\tau$ being the pole pitch. Field expressions in this section are written in complex form without the exponential factor. Complete expressions for all field quantities can be obtained by inserting the exponential factor and then selecting the real part.

The rotor region is the only conducting region. All the remaining regions are air regions. The eddy current density is governed by the following field equations:
$$\nabla \cdot J_1=0$$
and
$$\nabla^2 J_1=\frac{j}{d^2} J_1$$
where
$$d^2=\frac{1}{s \cdot \omega \cdot \mu \cdot \sigma}$$
The magnetic field intensity $\boldsymbol{H}_1$ in the solid rotor in terms of eddy current density $J_1$ can be written as
$$\boldsymbol{H}_1=j d^2 \nabla \times \boldsymbol{J}_1$$
The axial component of the rotor eddy current density vanishes at the rotor end surfaces and is an even function of $x$. Thus, in a reference frame fixed to the rotor, the axial component of the rotor eddy current density at the rotor air-gap surface can be given by the following half-range Fourier series:
$$\left.J_{x 1}\right|{z=0}=\sum{p-\text { odd }}^{\infty} a_p \cdot \cos \left(\frac{p \pi}{L_R} \cdot x\right),$$
where $a_p$ indicates a set of arbitrary constants.

## 物理代写|电磁学代写Electromagnetism代考|Field Distributions

$$\nabla \cdot J_1=0$$

$$\nabla^2 J_1=\frac{j}{d^2} J_1$$

$$d^2=\frac{1}{s \cdot \omega \cdot \mu \cdot \sigma}$$

## 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。

Posted on Categories:Electromagnetism, 物理代写, 电磁学

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

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## 物理代写|电磁学代写Electromagnetism代考|Physical Description

Consider a solid-rotor induction machine with three-phase stator winding housed in open rectangular stator slots. The three-phase current flowing in the three-phase winding causes the magnetic field in the air gap as well as in the rotor iron to vary periodically in the peripheral direction. The discrete nature of the winding introduces winding harmonics. These harmonic fields will be present even if the current-carrying conductors are replaced with current strips on smooth stator air-gap surface. Because of the slotted stator air-gap surface, the air-gap permeance varies periodically in the peripheral direction. Owing to this combined effect, the magnetic field, in the air gap as well as in the solid rotor, contains nontriplen odd space harmonics. Each harmonic field rotates at different speeds; some harmonic fields rotate in the direction that the fundamental field rotates in and others in the opposite direction. The former is termed as positive direction and the latter as negative direction. Each harmonic field associated with a specific eddy current density in the rotor iron contributes to the torque developed in the machine. At stand still, torques produced by some of these harmonic fields, called positive, are in the direction that the fundamental field rotates in and the others, called negative, act in the opposite direction. The net torque is the vector sum of all these torques. Consider the machine working as a motor with its rotor rotating at near synchronous speed of the fundamental field. Fields due to 5 th, 11th and other negative revolving harmonics cause braking torque. Certain harmonic fields, viz., 7th, 13th and other positive revolving harmonic fields rotate at $1 / 7$ th, $1 / 13$ th and such fraction of the synchronous speeds of the fundamental field. With reference to these harmonic fields, the machine is rotating at super-synchronous speeds. Therefore, with reference to these harmonic fields, the machine is operating as a synchronous generator; each inducing negative torque. Thus, the effects of each harmonic field are to produce $I^2 R$ loss in the rotor iron and to reduce the net torque developed by the machine.

## 物理代写|电磁学代写Electromagnetism代考|Slip/Torque Characteristics

Figure 7.1 shows the slip/torque characteristics ${ }^1$ of a typical solid-rotor induction machine evaluated with and without considering the harmonic fields. This figure also shows the result of an approximate treatment that accounts for winding harmonics; by simulating current-carrying conductors in stator slots with current filaments on smooth stator air-gap surface. This approximate treatment, therefore, ignores the slotting effect that causes the air-gap permeance to vary periodically in the peripheral direction. In a wound-rotor machine, the design of rotor winding ensures that these unwanted features are reduced.

For the analysis presented here, the machine is idealised by assuming infinite permeability for the shaft and the stator iron, constant rotor iron permeability, negligible curvature for the air-gap surfaces and negligible variation of fields in the axial direction. The idealised machine is illustrated in Figure 7.2. The treatment for the magnetic field in stator slots is a modified form of Roth’s ${ }^{1,2}$ analysis. An alternative treatment based on Rogowski’s ${ }^{3,4}$ method of analysis is also available. ${ }^5$

In a reference frame shown in Figure 7.2, let $z=0$ be the rotor air-gap surface, and $z=-g$ the stator air-gap surface. In the coordinate system chosen, the axial direction is along the $X$-axis, the peripheral direction is along the $Y$-axis and the $Z$-axis indicates the radial direction. In solid-rotor induction machines, the rotor air-gap surface is smooth, but the stator air-gap surface is slotted. Three-phase winding is housed in the stator slots.

## 物理代写|电磁学代写Electromagnetism代考|Slip/Torque Characteristics

$$\mathcal{F}_0=\mathcal{F}_1-\mathcal{F}_2$$

$$\nabla \cdot \mathcal{F}_0=0$$

$$\nabla \times \mathcal{F}_0=0$$

$$\mathcal{F}_0 \stackrel{\text { def }}{=} \nabla \varphi$$

$$\nabla^2 \varphi=0$$

$$\nabla \cdot(\varphi \nabla \varphi) \equiv \varphi \nabla^2 \varphi+(\nabla \varphi)^2=(\nabla \varphi)^2=\left(\mathcal{F}_0\right)^2$$

## 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。