# 集膚效應

## 理论

$J= J_s \exp(-{x \over \delta} )$

$\delta=\sqrt{{2\rho}\over{\omega \mu}}$

ρ =导体的电阻率
ω = 交流电的角频率 = 2π ×频率
μ = 导体的绝对磁导率 = $\mu_0 \cdot \mu_r$，其中$\mu_0$真空磁导率$\mu_r$是导体的相对磁导率

$R={{\rho \over \delta}\left({L\over{\pi (D-\delta)}}\right)}\approx{{\rho \over \delta}\left({L\over{\pi D}}\right)}$

L=导线的长度
D=导线直径

$\frac{I(r)}{I} = \frac{Ber(\frac{\sqrt{2}\,a}{\delta})-Ber(\frac{\sqrt{2}\,r}{\delta}) + i \, [Bei(\frac{\sqrt{2}\,a}{\delta}) - Bei(\frac{\sqrt{2}\,r}{\delta})]}{Ber(\frac{\sqrt{2}\,a}{\delta}) + i \, Bei(\frac{\sqrt{2}\,a}{\delta})}$

### 圆柱形导体的模型

$\mathrm{rot} \, \mathbf{E} = - i \, \omega \, \mathbf{B}$

$\mathrm{rot} \, \mathbf{B} = \mu_0 \mathbf{J}$

$\mathbf{J} = \sigma \, \mathbf{E}$

σ是导体的电导率

$\mathrm{rot} \, \mathbf{J} = - i \, \omega \, \sigma \, \mathbf{B}$
$\mathrm{rot} \, \mathbf{B} = \mu \, \mathbf{J}$

$\mathbf{J} = \begin{pmatrix}0\\0\\j(r)\end{pmatrix}$

$\mathrm{rot} \, \mathrm{rot} \, \mathbf{J} = - i \, \omega \, \sigma \, \mathrm{rot} \, \mathbf{B}$

$\nabla \, \mathrm{div} \, \mathbf{J} - \Delta \mathbf{J} = - i \, \omega \, \sigma \, \mu \, \mathbf{J}$

$\Delta \mathbf{J} = i \, \omega \, \sigma \, \mu \, \mathbf{J}$

$\frac{d^2\,j}{dr^2}(r) + \frac{1}{r} \, \frac{d\,j}{dr}(r) = i \, \omega \, \sigma \, \mu \, j(r)$

$k^2 = i \, \omega \, \sigma \, \mu$，再将方程两边乘上r2就得到电流密度应该满足的方程：

$r^2 \, \frac{d^2\,j}{dr^2}(r) + r \, \frac{d\,j}{dr}(r) - r^2 \, k^2 \, j(r) = 0$

$\xi^2 \, \frac{d^2\,j}{d\xi^2}(\xi) + \xi \, \frac{d\,j}{d\xi}(\xi) + \xi^2 \, j(\xi) = 0$

$j(r) = j_0 \, J_0(i \, k \, r)$

$k = \sqrt{i} \, \sqrt{\omega \, \sigma \, \mu} = \frac{1+i}{\sqrt{2}} \, \sqrt{\omega \, \sigma \, \mu} = \frac{1+i}{\delta}$

$i \, k = \frac{-1+i}{\delta} = e^{i \, 3 \, \pi/4} \, \frac{\sqrt{2}}{\delta}$

$\begin{matrix}j(r) &=& j_0 \, J_0(e^{i \, 3 \, \pi/4} \, \frac{\sqrt{2} \, r}{\delta})\\ &=& j_0 \, (ber(\frac{\sqrt{2} \, r}{\delta}) + i \, bei(\frac{\sqrt{2} \, r}{\delta}))\end{matrix}$

$\begin{matrix}I &=& \int_0^a j(r) \, 2 \, \pi \, r \, dr\\ &=& 2 \, \pi \, j_0 \int_0^a J_0(e^{i \, 3 \, \pi/4} \, \frac{\sqrt{2} \, r}{\delta}) \, r \, dr\\ &=& \pi \, \delta^2 \, j_0 \, \int_0^{\sqrt{2} \, a / \delta}(ber(x) + i \, bei(x)) \, x \, dx\end{matrix}$

BerBei为相应的原函数

$Ber(x) = \int_0^x ber(x^\prime)\, x^\prime \, dx^\prime \qquad \mbox{ et } \qquad Bei(x) = \int_0^x bei(x^\prime) \, x^\prime \, dx^\prime$

$I = \pi \, \delta^2 \, j_0 \, \left(Ber(\frac{\sqrt{2}\, a}{\delta}) + i \, Bei(\frac{\sqrt{2}\, a}{\delta})\right)$

$\begin{matrix}I(r)&=& \int_{a-r}^a j(r^\prime) \, 2 \, \pi \, r^\prime \, dr^\prime\\ &=& \pi \, \delta^2 \, j_0 \, \left( Ber(\frac{\sqrt{2}\, a}{\delta})- Ber(\frac{\sqrt{2}\, r}{\delta}) + i \, [Bei(\frac{\sqrt{2}\, a}{\delta}) - Bei(\frac{\sqrt{2}\, r}{\delta})] \right)\end{matrix}$

$\frac{I(r)}{I} = \frac{Ber(\frac{\sqrt{2}\,a}{\delta})-Ber(\frac{\sqrt{2}\,r}{\delta}) + i \, [Bei(\frac{\sqrt{2}\,a}{\delta}) - Bei(\frac{\sqrt{2}\,r}{\delta})]}{Ber(\frac{\sqrt{2}\,a}{\delta}) + i \, Bei(\frac{\sqrt{2}\,a}{\delta})}$

$D_\mathrm{W} = {\frac{200~\mathrm{mm}}{\sqrt{f/\mathrm{Hz}}}}$

0.80
0.65
0.79
0.64

60 Hz 8.57 mm
10 kHz 0.66 mm
100 kHz 0.21 mm
1 MHz 66 µm
10 MHz 21 µm

## 相關參考

• William Hart Hayt, Engineering Electromagnetics Seventh Edition, (2006), McGraw Hill, New York ISBN 0073104639
• Paul J. Nahin, Oliver Heaviside: Sage in Solitude, (1988), IEEE Press, New York, ISBN 0879422386
• Terman, F.E. Radio Engineers' Handbook, McGraw-Hill 1943 -- for the Terman formula mentioned above