# 集膚效應

## 理论

${\displaystyle J=J_{s}\exp(-{x \over \delta })}$

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

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 = ${\displaystyle \mu _{0}\cdot \mu _{r}}$，其中${\displaystyle \mu _{0}}$真空磁导率${\displaystyle \mu _{r}}$是导体的相对磁导率

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

L=导线的长度
D=导线直径

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

### 圆柱形导体的模型

${\displaystyle \nabla \times \,\mathbf {E} =-i\,\omega \,\mathbf {B} }$

${\displaystyle \nabla \times \,\mathbf {B} =\mu _{0}\mathbf {J} }$

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

σ是导体的电导率

${\displaystyle \nabla \times \,\mathbf {J} =-i\,\omega \,\sigma \,\mathbf {B} }$
${\displaystyle \nabla \times \,\mathbf {B} =\mu \,\mathbf {J} }$

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

${\displaystyle \nabla \times \,(\nabla \times \,\mathbf {J} )=-i\,\omega \,\sigma \,(\nabla \times \,\mathbf {B} )}$

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

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

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

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

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

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

${\displaystyle j(r)=j_{0}\,J_{0}(i\,k\,r)}$

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

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

${\displaystyle {\begin{matrix}j(r)&=&j_{0}\,J_{0}(e^{i\,3\,\pi /4}\,{\frac {{\sqrt {2}}\,r}{\delta }})\\&=&j_{0}\,(ber({\frac {{\sqrt {2}}\,r}{\delta }})+i\,bei({\frac {{\sqrt {2}}\,r}{\delta }}))\end{matrix}}}$

${\displaystyle {\begin{matrix}I&=&\int _{0}^{a}j(r)\,2\,\pi \,r\,dr\\&=&2\,\pi \,j_{0}\int _{0}^{a}J_{0}(e^{i\,3\,\pi /4}\,{\frac {{\sqrt {2}}\,r}{\delta }})\,r\,dr\\&=&\pi \,\delta ^{2}\,j_{0}\,\int _{0}^{{\sqrt {2}}\,a/\delta }(ber(x)+i\,bei(x))\,x\,dx\end{matrix}}}$

BerBei为相应的原函数

${\displaystyle Ber(x)=\int _{0}^{x}ber(x^{\prime })\,x^{\prime }\,dx^{\prime }\qquad {\mbox{ et }}\qquad Bei(x)=\int _{0}^{x}bei(x^{\prime })\,x^{\prime }\,dx^{\prime }}$

${\displaystyle I=\pi \,\delta ^{2}\,j_{0}\,\left(Ber({\frac {{\sqrt {2}}\,a}{\delta }})+i\,Bei({\frac {{\sqrt {2}}\,a}{\delta }})\right)}$

${\displaystyle {\begin{matrix}I(r)&=&\int _{a-r}^{a}j(r^{\prime })\,2\,\pi \,r^{\prime }\,dr^{\prime }\\&=&\pi \,\delta ^{2}\,j_{0}\,\left(Ber({\frac {{\sqrt {2}}\,a}{\delta }})-Ber({\frac {{\sqrt {2}}\,r}{\delta }})+i\,[Bei({\frac {{\sqrt {2}}\,a}{\delta }})-Bei({\frac {{\sqrt {2}}\,r}{\delta }})]\right)\end{matrix}}}$

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

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

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
1. ^ 林漢年. 《電磁相容分析與設計 : 從PI與SI根因探討》. 滄海圖書. 2021: 第B–3頁. ISBN 9789865647735.