# 物理代写|电磁学代写electromagnetism代考|Cage or Wound Rotor Induction Machines

#### Doug I. Jones

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## 物理代写|电磁学代写electromagnetism代考|Cage or Wound Rotor Induction Machines

A simplified treatment, based on a homogeneous anisotropic region for the slotted part of the cage or wound rotor, is presented in this section. The developed view for these machines is shown in Figure 6.1. The curvature of air-gap surfaces is neglected and the two-dimensional treatment assumes unskewed slots resulting in zero variation of electromagnetic fields in the axial (or $x$ ) direction. Field analysis for machines with skewed rotor slots is presented in Section 6.5. The symbol $y$ indicates the peripheral and $z$ indicates the radial direction. The idealized machine, shown in Figure 6.1, assumes infinitely permeable stator iron and the rotor core beyond its slotted region. The axial length of the machine is considered as infinite with stator and rotor slots running parallel to the axis of the machine. The currentcarrying polyphase stator winding is simulated by a suitable current sheet on the smooth stator air-gap surface. The air-gap length is corrected using Carter’s coefficient.

Let the stator current sheet in a reference frame moving with the rotor be given as
$$K_x=K_o \cdot e^{j(s-t-t-\ell \cdot y)}$$
where
\begin{aligned} & \left|K_0\right|=\text { peak surface current density of known value } \ & s=\text { slip } \ & \omega=\text { angular frequency of the supply } \end{aligned}
$$\ell \stackrel{\operatorname{def}}{=} \frac{\pi}{\tau}$$
$\tau=$ pole pitch
First, let us consider the air-gap region extended over $0 \geq z \geq-g$, in the radial direction. There being no conduction currents, from Maxwell’s equations one gets
$$\nabla \times H=\frac{\partial D}{\partial t}=j s \cdot \omega \cdot \epsilon_o E$$
and
$$\nabla \times E=-\frac{\partial B}{\partial t}=-j s \cdot \omega \cdot \mu_o H$$
Since
$$\nabla \cdot H=0$$
$$\nabla \times \nabla \times H \equiv \nabla(\nabla \cdot H)-\nabla^2 H=-\nabla^2 H$$
Also, using Equations 6.15a and $6.15 b$
$$\nabla \times \nabla \times \boldsymbol{H}=j s \cdot \omega \cdot \epsilon_o \nabla \times \boldsymbol{E}=(s \cdot \omega)^2 \cdot \mu_o \cdot \epsilon_0 \boldsymbol{H}=\left(\frac{s \cdot \omega}{c}\right)^2 \boldsymbol{H}$$
where $c$ indicates the velocity of light in free space.

## 物理代写|电磁学代写electromagnetism代考| Rotor Parameters

Let us choose the following notations:
$d_{\mathrm{s}}=$ rotor slot depth
$w_s=$ rotor slot width
$\lambda=$ rotor slot pitch
$\gamma=$ rotor slot space factor (copper area/slot area)
$\varepsilon=$ rotor slot insulation permittivity
$\mu=$ rotor tooth iron permeability
$\sigma=$ conductivity for rotor conductors
Now, for the anisotropic homogeneous region, it may be seen that
$$\begin{gathered} \epsilon_x=\epsilon_o+\left(\epsilon-\epsilon_o\right) \cdot(1-\gamma) \cdot w_s / \lambda \ \epsilon_y=\frac{\epsilon_o \cdot w_s+\epsilon \cdot(1-\gamma) \cdot\left(\lambda-w_s\right)}{\epsilon_0 \cdot \epsilon \cdot(1-\gamma) / \lambda} \ \epsilon_z \cong \epsilon_x \end{gathered}$$

$$\begin{gathered} \sigma_x=\gamma \cdot \frac{\sigma \cdot w_s}{\lambda} \ \mu_x=\mu_o \cdot\left(w_s / \lambda\right)+\mu \cdot\left(1-w_s / \lambda\right) \ \mu_y=\frac{\mu \cdot \mu_o \cdot \lambda}{\mu_o \cdot \lambda+\left(\mu-\mu_o\right) \cdot w_s} \ \mu_z=\mu-\left(\mu-\mu_o\right) \cdot w_s / \lambda \end{gathered}$$
where $\gamma<1$ for wound rotor
$$\cong 1 \text {, for cage rotor }$$

# 电磁学代考

## 物理代写|电磁学代写electromagnetism代考|Cage or Wound Rotor Induction Machines

$$K_x=K_o \cdot e^{j(s-t-t-\ell \cdot y)}$$

\begin{aligned} & \left|K_0\right|=\text { peak surface current density of known value } \ & s=\text { slip } \ & \omega=\text { angular frequency of the supply } \end{aligned}
$$\ell \stackrel{\operatorname{def}}{=} \frac{\pi}{\tau}$$
$\tau=$极距

$$\nabla \times H=\frac{\partial D}{\partial t}=j s \cdot \omega \cdot \epsilon_o E$$

$$\nabla \times E=-\frac{\partial B}{\partial t}=-j s \cdot \omega \cdot \mu_o H$$

$$\nabla \cdot H=0$$
$$\nabla \times \nabla \times H \equiv \nabla(\nabla \cdot H)-\nabla^2 H=-\nabla^2 H$$

$$\nabla \times \nabla \times \boldsymbol{H}=j s \cdot \omega \cdot \epsilon_o \nabla \times \boldsymbol{E}=(s \cdot \omega)^2 \cdot \mu_o \cdot \epsilon_0 \boldsymbol{H}=\left(\frac{s \cdot \omega}{c}\right)^2 \boldsymbol{H}$$

## 物理代写|电磁学代写electromagnetism代考| Rotor Parameters

$d_{\mathrm{s}}=$转子槽深
$w_s=$转子槽宽
$\lambda=$转子槽距
$\gamma=$转子槽空间系数(铜面积/槽面积)
$\varepsilon=$转子槽绝缘介电常数
$\mu=$转子齿铁磁导率
$\sigma=$转子导体的导电性

$$\begin{gathered} \epsilon_x=\epsilon_o+\left(\epsilon-\epsilon_o\right) \cdot(1-\gamma) \cdot w_s / \lambda \ \epsilon_y=\frac{\epsilon_o \cdot w_s+\epsilon \cdot(1-\gamma) \cdot\left(\lambda-w_s\right)}{\epsilon_0 \cdot \epsilon \cdot(1-\gamma) / \lambda} \ \epsilon_z \cong \epsilon_x \end{gathered}$$

$$\begin{gathered} \sigma_x=\gamma \cdot \frac{\sigma \cdot w_s}{\lambda} \ \mu_x=\mu_o \cdot\left(w_s / \lambda\right)+\mu \cdot\left(1-w_s / \lambda\right) \ \mu_y=\frac{\mu \cdot \mu_o \cdot \lambda}{\mu_o \cdot \lambda+\left(\mu-\mu_o\right) \cdot w_s} \ \mu_z=\mu-\left(\mu-\mu_o\right) \cdot w_s / \lambda \end{gathered}$$

$$\cong 1 \text {, for cage rotor }$$

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