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# 物理代写|电磁学代写Electromagnetism代考|Arbitrary Orientation of Tooth and Slot

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## 物理代写|电磁学代写Electromagnetism代考|Arbitrary Orientation of Tooth and Slot

Consider the identical double-slotting shown in Figure 4.3. The toothcentres of the two equipotential surfaces are separated by a distance $\delta$, where $0 \leq \delta \leq \lambda / 2$. For $\delta=0$, the orientation will be tooth-opposite-tooth and for $\delta=\lambda / 2$ it will result in tooth-opposite-slot orientation. Let the rotor- and stator-iron slotted surfaces be at a magnetic potential of $-1 / 2$ and $+1 / 2 \mathrm{~m}$.k.s.

units respectively. The distribution of magneto-static potential in the stator slot 1 can be given as
\begin{aligned} \mathcal{V}= & \frac{1}{2}-\sum_{m=1}^{\infty} p_m \cdot \sin \left{\frac{m \pi}{s}\left(y-\frac{t}{2}+\frac{\delta}{2}\right)\right} \cdot e^{+(m \pi / s)(z+g / 2)} \ & \operatorname{over}(t / 2-\delta / 2) \leq y \leq(s+t / 2-\delta / 2) \end{aligned}
And in the rotor slot 2
\begin{aligned} \mathcal{V}2= & -\frac{1}{2}-\sum{m=1}^{\infty} p_m \cdot \sin \left{\frac{m \pi}{s}\left(y+\frac{t}{2}-\frac{\delta}{2}\right)\right} \cdot e^{-(m \pi / s)(z-g / 2)} \ & \text { over }(-s-t / 2+\delta / 2) \leq y \leq(-t / 2+\delta / 2) \end{aligned}
Note that the same set of arbitrary constants $p_m$ is involved in the two expressions.

The distributions of scalar magnetic potential on the two air-gap surfaces are periodic in the $y$ direction with a period equal to the slot-pitch $\lambda$. For $\delta=0$, these distributions are even functions of $y$ that correspond to the tooth-opposite-tooth orientation. Further, as $\delta=\lambda / 2$ corresponds to the tooth-opposite-slot orientation, the potential distributions on the airgap surfaces will be again even functions of $y$ provided that the origin is shifted to coincide with a tooth axis, that is, if $y$ is replaced by $(y \pm \delta / 2)$. These distributions can, therefore, be given by the following Fourier series expansions:
$$\left.\mathcal{V}o\right|{z=-g / 2}=q_o+\sum_{n=1}^{\infty} q_n \cdot \cos \left{\frac{n 2 \pi}{\lambda}(y+\delta / 2)\right}$$
and
$$\left.\mathcal{V}o\right|{z=g / 2}=-q_o-\sum_{n=1}^{\infty} q_n \cdot \cos \left{\frac{2 \pi n}{\lambda}(y-\delta / 2)\right}$$
where $q_n$ indicates a set of arbitrary constants. Therefore, the distribution of the scalar magnetic potential in the air-gap region is
\begin{aligned} \mathcal{V}0= & -q_0 \cdot \frac{2 z}{g}-\sum{n=1}^{\infty} q_n \cdot\left[\cos \left{\frac{n 2 \pi}{\lambda}(y-\delta / 2)\right} \cdot \frac{\sinh {(n 2 \pi / \lambda)(z+g / 2)}}{\sinh ((2 \pi n / \lambda) \cdot g)}\right. \ & \left.+\cos \left{\frac{n 2 \pi}{\lambda}(y+\delta / 2)\right} \cdot \frac{\sinh {(n 2 \pi / \lambda)(z-g / 2)}}{\sinh ((2 \pi n / \lambda) \cdot g)}\right] \end{aligned}
This expression for the air-gap potential $\mathcal{V}_0$ satisfies the requirements stated above.

## 物理代写|电磁学代写Electromagnetism代考|Evaluation of Arbitrary Constants

The arbitrary constants $p_m, q_o$ and $q_n$ involved in Equations 4.33,4.34 and 4.36 can be evaluated by using the following boundary conditions:
\begin{aligned} \begin{aligned} \left.\mathcal{V}\right|{z=-g / 2}= & \left.\mathcal{V}\right|{z=-g / 2} \quad \text { over }(t / 2-\delta / 2) \leq y \leq(s+t / 2-\delta / 2) \ = & 1 / 2 \operatorname{over}(-\delta / 2) \leq y \leq(t / 2-\delta / 2) \ & \text { and }(s+t / 2-\delta / 2) \leq y \leq(\lambda-\delta / 2) \ \left.\frac{\partial \mathcal{V}0}{\partial z}\right|{z=-g / 2}= & \left.\frac{\partial \mathcal{V}1}{\partial z}\right|{z=-g / 2} \quad \text { over }(t / 2-\delta / 2) \leq y \leq(s+t / 2-\delta / 2) \end{aligned} \end{aligned}
Thus, in view of Equations $4.37,4.38,4.35 \mathrm{a}$ and 4.33 , we get
\begin{aligned} & q_o+\sum_{n=1}^{\infty} q_n \cdot \cos \left{\frac{n 2 \pi}{\lambda}(y+\delta / 2)\right}=\frac{1}{2}-\sum_{m=1}^{\infty} p_m \cdot \sin \left{\frac{m \pi}{s}\left(y-\frac{t}{2}+\delta / 2\right)\right} \ & \quad \text { over }(t / 2-\delta / 2) \leq y \leq(s+t / 2-\delta / 2) \ & \text { and } q_o+\sum_{n=1}^{\infty} q_n \cdot \cos \left{\frac{2 \pi n}{\lambda}(y+\delta / 2)\right}=\frac{1}{2} \end{aligned}
over $(-\delta / 2) \leq y \leq(t / 2-\delta / 2)$ and $(s+t / 2-\delta / 2) \leq y \leq(\lambda-\delta / 2)$
Therefore, the Fourier coefficient $q_0$ is found as
\begin{aligned} & q_o=\frac{1}{\lambda} \cdot\left[\frac{1}{2} \int_{(-\delta / 2)}^{(\lambda-\delta / 2)} d y-\sum_{m=1}^{\infty} p_m \cdot \int_{(t / 2-\delta / 2)}^{(s+t / 2-\delta / 2)} \sin \left{\frac{m \pi}{s}\left(y-\frac{t}{2}+\frac{\delta}{2}\right)\right} d y\right] \ & \text { or, } q_o=\frac{1}{2}-\frac{s}{\lambda} \cdot \sum_{M=1}^{\infty} p_M \cdot \frac{[1-\cos (M \pi)]}{M \pi} \end{aligned}

## 物理代写|电磁学代写Electromagnetism代考|Arbitrary Orientation of Tooth and Slot

\begin{aligned} \mathcal{V}= & \frac{1}{2}-\sum_{m=1}^{\infty} p_m \cdot \sin \left{\frac{m \pi}{s}\left(y-\frac{t}{2}+\frac{\delta}{2}\right)\right} \cdot e^{+(m \pi / s)(z+g / 2)} \ & \operatorname{over}(t / 2-\delta / 2) \leq y \leq(s+t / 2-\delta / 2) \end{aligned}

\begin{aligned} \mathcal{V}2= & -\frac{1}{2}-\sum{m=1}^{\infty} p_m \cdot \sin \left{\frac{m \pi}{s}\left(y+\frac{t}{2}-\frac{\delta}{2}\right)\right} \cdot e^{-(m \pi / s)(z-g / 2)} \ & \text { over }(-s-t / 2+\delta / 2) \leq y \leq(-t / 2+\delta / 2) \end{aligned}

$$\left.\mathcal{V}o\right|{z=-g / 2}=q_o+\sum_{n=1}^{\infty} q_n \cdot \cos \left{\frac{n 2 \pi}{\lambda}(y+\delta / 2)\right}$$

$$\left.\mathcal{V}o\right|{z=g / 2}=-q_o-\sum_{n=1}^{\infty} q_n \cdot \cos \left{\frac{2 \pi n}{\lambda}(y-\delta / 2)\right}$$

\begin{aligned} \mathcal{V}0= & -q_0 \cdot \frac{2 z}{g}-\sum{n=1}^{\infty} q_n \cdot\left[\cos \left{\frac{n 2 \pi}{\lambda}(y-\delta / 2)\right} \cdot \frac{\sinh {(n 2 \pi / \lambda)(z+g / 2)}}{\sinh ((2 \pi n / \lambda) \cdot g)}\right. \ & \left.+\cos \left{\frac{n 2 \pi}{\lambda}(y+\delta / 2)\right} \cdot \frac{\sinh {(n 2 \pi / \lambda)(z-g / 2)}}{\sinh ((2 \pi n / \lambda) \cdot g)}\right] \end{aligned}

## 物理代写|电磁学代写Electromagnetism代考|Evaluation of Arbitrary Constants

\begin{aligned} \begin{aligned} \left.\mathcal{V}\right|{z=-g / 2}= & \left.\mathcal{V}\right|{z=-g / 2} \quad \text { over }(t / 2-\delta / 2) \leq y \leq(s+t / 2-\delta / 2) \ = & 1 / 2 \operatorname{over}(-\delta / 2) \leq y \leq(t / 2-\delta / 2) \ & \text { and }(s+t / 2-\delta / 2) \leq y \leq(\lambda-\delta / 2) \ \left.\frac{\partial \mathcal{V}0}{\partial z}\right|{z=-g / 2}= & \left.\frac{\partial \mathcal{V}1}{\partial z}\right|{z=-g / 2} \quad \text { over }(t / 2-\delta / 2) \leq y \leq(s+t / 2-\delta / 2) \end{aligned} \end{aligned}

\begin{aligned} & q_o+\sum_{n=1}^{\infty} q_n \cdot \cos \left{\frac{n 2 \pi}{\lambda}(y+\delta / 2)\right}=\frac{1}{2}-\sum_{m=1}^{\infty} p_m \cdot \sin \left{\frac{m \pi}{s}\left(y-\frac{t}{2}+\delta / 2\right)\right} \ & \quad \text { over }(t / 2-\delta / 2) \leq y \leq(s+t / 2-\delta / 2) \ & \text { and } q_o+\sum_{n=1}^{\infty} q_n \cdot \cos \left{\frac{2 \pi n}{\lambda}(y+\delta / 2)\right}=\frac{1}{2} \end{aligned}

\begin{aligned} & q_o=\frac{1}{\lambda} \cdot\left[\frac{1}{2} \int_{(-\delta / 2)}^{(\lambda-\delta / 2)} d y-\sum_{m=1}^{\infty} p_m \cdot \int_{(t / 2-\delta / 2)}^{(s+t / 2-\delta / 2)} \sin \left{\frac{m \pi}{s}\left(y-\frac{t}{2}+\frac{\delta}{2}\right)\right} d y\right] \ & \text { or, } q_o=\frac{1}{2}-\frac{s}{\lambda} \cdot \sum_{M=1}^{\infty} p_M \cdot \frac{[1-\cos (M \pi)]}{M \pi} \end{aligned}

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