# 电感

## 维基百科，自由的百科全书

${\displaystyle {\mathcal {E))=-L{\mathrm {d} i \over \mathrm {d} t))$

## 概述

### 自感

${\displaystyle {\mathcal {E))=-N((\mathrm {d} \Phi } \over \mathrm {d} t}=-N((\mathrm {d} \Phi } \over \mathrm {d} i}\ {\mathrm {d} i \over \mathrm {d} t))$

${\displaystyle L=N{\frac {\mathrm {d} \Phi }{\mathrm {d} i))}$

${\displaystyle {\mathcal {E))=-L{\mathrm {d} i \over \mathrm {d} t))$

${\displaystyle v=L((\mathrm {d} i} \over \mathrm {d} t))$

### 互感

${\displaystyle \Phi _{2}=M_{21}i_{1))$

• ${\displaystyle M_{21}={\frac {\mu _{0)){4\pi ))\oint _{\mathbb {C} _{1))\oint _{\mathbb {C} _{2)){\frac {\mathrm {d} {\boldsymbol {\ell ))_{1}\cdot \mathrm {d} {\boldsymbol {\ell ))_{2)){|\mathbf {X} _{2}-\mathbf {X} _{1}|))}$

#### 推导

${\displaystyle \Phi _{2}(t)=\int _{\mathbb {S} _{2))\mathbf {B} _{1}(\mathbf {X} _{2},t)\cdot \mathrm {d} \mathbf {a} _{2))$

${\displaystyle \mathbf {B} _{1}(\mathbf {X} _{2},t)=\nabla _{2}\times \mathbf {A} _{1}(\mathbf {X} _{2},t)}$

${\displaystyle \Phi _{2}(t)=\int _{\mathbb {S} _{2))[\nabla _{2}\times \mathbf {A} _{1}(\mathbf {X} _{2},t)]\cdot \mathrm {d} \mathbf {a} _{2}=\oint _{\mathbb {C} _{2))\mathbf {A} _{1}(\mathbf {X} _{2},t)\cdot \mathrm {d} {\boldsymbol {\ell ))_{2))$

${\displaystyle \mathbf {A} _{1}(\mathbf {X} _{2},t)\ {\stackrel {def}{=))\ {\frac {\mu _{0}i_{1)){4\pi ))\oint _{\mathbb {C} _{1)){\frac {\mathrm {d} {\boldsymbol {\ell ))_{1)){|\mathbf {X} _{2}-\mathbf {X} _{1}|))}$

${\displaystyle \Phi _{2}(t)={\frac {\mu _{0}i_{1)){4\pi ))\oint _{\mathbb {C} _{1))\oint _{\mathbb {C} _{2)){\frac {\mathrm {d} {\boldsymbol {\ell ))_{1}\cdot \mathrm {d} {\boldsymbol {\ell ))_{2)){|\mathbf {X} _{2}-\mathbf {X} _{1}|))}$

${\displaystyle M_{21}={\frac {\mathrm {d} \Phi _{2)){\mathrm {d} i_{1))}={\frac {\mu _{0)){4\pi ))\oint _{\mathbb {C} _{1))\oint _{\mathbb {C} _{2)){\frac {\mathrm {d} {\boldsymbol {\ell ))_{1}\cdot \mathrm {d} {\boldsymbol {\ell ))_{2)){|\mathbf {X} _{2}-\mathbf {X} _{1}|))}$

${\displaystyle \Phi _{1}(t)={\frac {\mu _{0}i_{1)){4\pi ))\oint _{\mathbb {C} _{1))\oint _{\mathbb {C} '_{1)){\frac {\mathrm {d} {\boldsymbol {\ell ))_{1}\cdot \mathrm {d} {\boldsymbol {\ell ))'_{1)){|\mathbf {X} _{1}-\mathbf {X} '_{1}|))}$

${\displaystyle L={\frac {\mathrm {d} \Phi }{\mathrm {d} i))={\frac {\mu _{0)){4\pi ))\oint _{\mathbb {C} }\oint _{\mathbb {C} '}{\frac {\mathrm {d} {\boldsymbol {\ell ))\cdot \mathrm {d} {\boldsymbol {\ell ))'}{|\mathbf {X} -\mathbf {X} '|))}$

${\displaystyle \mathbf {X} _{1}=\mathbf {X} '_{1))$时，这积分可能会发散，需要特别加以处理。另外，若假设闭合回路为无穷细小，则在闭合回路附近，磁场会变得无穷大，磁通量也会变得无穷大，所以，必须给予闭合回路有限尺寸，设定其截面半径${\displaystyle r_{0))$超小于径长${\displaystyle \ell _{0))$

### 耦合系数

${\displaystyle k={\frac {M}{\sqrt {L_{1}L_{2))))}$

### 电感与磁场能量

${\displaystyle N_{k}\Phi _{k}=\sum _{n=1}^{K}L_{k,n}i_{n))$

${\displaystyle v_{k}=-{\mathcal {E))_{k}=N_{k}{\frac {\mathrm {d} \Phi _{k)){\mathrm {d} t))=\sum _{n=1}^{K}L_{k,n}{\frac {\mathrm {d} i_{n)){\mathrm {d} t))=L_{k}{\frac {\mathrm {d} i_{k)){\mathrm {d} t))+\sum _{n=1,\ n\neq k}^{K}M_{k,n}{\frac {\mathrm {d} i_{n)){\mathrm {d} t))}$

${\displaystyle k}$条闭合回路的电功率${\displaystyle p_{k))$

${\displaystyle p_{k}=i_{k}v_{k))$

${\displaystyle W_{1}=\int i_{1}v_{1}\mathrm {d} t=\int _{0}^{I_{1))i_{1}L_{1}\mathrm {d} i_{1}={\frac {1}{2))L_{1}I_{1}^{2))$

${\displaystyle W_{2}=\int i_{2}v_{2}\mathrm {d} t=\int _{0}^{I_{2))i_{2}L_{2}\mathrm {d} i_{2}+\int _{0}^{I_{2))I_{1}M_{1,2}\mathrm {d} i_{2}={\frac {1}{2))L_{2}I_{2}^{2}+M_{1,2}I_{1}I_{2))$

${\displaystyle W_{k}=\int i_{k}v_{k}\mathrm {d} t=\int _{0}^{I_{k))i_{k}L_{k}\mathrm {d} i_{k}+\sum _{n=1}^{k-1}\int _{0}^{I_{k))I_{n}M_{n,k}\mathrm {d} i_{k}={\frac {1}{2))L_{k}I_{k}^{2}+\sum _{n=1}^{k-1}M_{n,k}I_{n}I_{k))$

${\displaystyle W={\frac {1}{2))\sum _{k=1}^{K}L_{k}I_{k}^{2}+\sum _{k=1}^{K}\sum _{n=1}^{k-1}M_{n,k}I_{n}I_{k}={\frac {1}{2))\sum _{k=1}^{K}L_{k}I_{k}^{2}+{\frac {1}{2))\sum _{k=1}^{K}\sum _{n=1,n\neq k}^{K}M_{n,k}I_{n}I_{k))$

## 串联与并联电路

### 串联电路

#### 自感现象

${\displaystyle L_{eq}=L_{1}+L_{2}+\cdots +L_{n))$

${\displaystyle n}$电感器串联在一起，并在这个串联电路的两端加上电源。按照电感的定义，第${\displaystyle k}$个电感器两端的电压${\displaystyle v_{k))$等于其电感${\displaystyle L_{k))$乘以通过的电流的变率${\displaystyle {\frac {\mathrm {d} i_{k)){\mathrm {d} t))}$

${\displaystyle v_{k}=L_{k}{\frac {\mathrm {d} i_{k)){\mathrm {d} t))}$

${\displaystyle i=i_{1}=i_{2}=\cdots =i_{n))$

${\displaystyle v=v_{1}+v_{2}+\cdots +v_{n}=L_{1}{\frac {\mathrm {d} i_{1)){\mathrm {d} t))+L_{2}{\frac {\mathrm {d} i_{2)){\mathrm {d} t))+\cdots +L_{n}{\frac {\mathrm {d} i_{n)){\mathrm {d} t))=(L_{1}+L_{2}+\cdots +L_{n}){\frac {\mathrm {d} i}{\mathrm {d} t))}$

${\displaystyle L_{eq}=L_{1}+L_{2}+\cdots +L_{n))$

#### 互感现象

• 假设两个电感器分别产生的磁场或磁通量，其方向相同，则称为“串联互助”，其等效电感
${\displaystyle L_{eq}=L_{1}+L_{2}+2M}$
• 假设两个电感器分别产生的磁场或磁通量，其方向相反，则称为“串联互消”，其等效电感
${\displaystyle L_{eq}=L_{1}+L_{2}-2M}$

${\displaystyle L_{eq}=(M_{11}+M_{22}+M_{33})+(M_{12}+M_{13}+M_{23})+(M_{21}+M_{31}+M_{32})}$

${\displaystyle L_{eq}=(M_{11}+M_{22}+M_{33})+2(M_{12}+M_{13}+M_{23})}$

#### 互感公式推导

${\displaystyle -v+L_{1}{\frac {\mathrm {d} i}{\mathrm {d} t))+M{\frac {\mathrm {d} i}{\mathrm {d} t))+L_{2}{\frac {\mathrm {d} i}{\mathrm {d} t))+M{\frac {\mathrm {d} i}{\mathrm {d} t))=0}$

${\displaystyle v=(L_{1}+L_{2}+2M){\frac {\mathrm {d} i}{\mathrm {d} t))}$

${\displaystyle L_{eq}=L_{1}+L_{2}+2M}$

### 并联电路

#### 自感现象

${\displaystyle {\frac {1}{L_{eq))}={\frac {1}{L_{1))}+{\frac {1}{L_{2))}+\cdots +{\frac {1}{L_{n))))$

#### 互感现象

• 假设两个电感器分别产生的磁场或磁通量，其方向相同，则称为“并联互助”，其等效电感
${\displaystyle L_{eq}={\frac {L_{1}L_{2}-M^{2)){L_{1}+L_{2}-2M))}$
• 假设两个电感器分别产生的磁场或磁通量，其方向相反，则称为“并联互消”，其等效电感
${\displaystyle L_{eq}={\frac {L_{1}L_{2}-M^{2)){L_{1}+L_{2}+2M))}$

#### 互感公式推导

${\displaystyle -v+L_{1}{\frac {\mathrm {d} i_{1)){\mathrm {d} t))+M{\frac {\mathrm {d} i_{2)){\mathrm {d} t))=0}$
${\displaystyle -v+L_{2}{\frac {\mathrm {d} i_{2)){\mathrm {d} t))+M{\frac {\mathrm {d} i_{1)){\mathrm {d} t))=0}$

${\displaystyle {\frac {di_{1)){dt))={\frac {v(L_{2}-M)}{L_{1}L_{2}-M^{2))))$
${\displaystyle {\frac {di_{2)){dt))={\frac {v(L_{1}-M)}{L_{1}L_{2}-M^{2))))$

${\displaystyle {\frac {di}{dt))={\frac {di_{1)){dt))+{\frac {di_{2)){dt))={\frac {v(L_{1}+L_{2}-2M)}{L_{1}L_{2}-M^{2))))$

${\displaystyle v={\frac {L_{1}L_{2}-M^{2)){L_{1}+L_{2}-2M))\ {\frac {\mathrm {d} i}{\mathrm {d} t))}$

${\displaystyle L_{eq}={\frac {L_{1}L_{2}-M^{2)){L_{1}+L_{2}-2M))}$

## 镜像法

• 一条笔直的载流导线与导体墙之间的距离为${\displaystyle d/2}$
• 两条互相平行、载有异向电流的导线，彼此之间的距离为${\displaystyle d}$

## 非线性电感

“大信号电感”是用来计算磁通量，以方程定义为

${\displaystyle L_{s}(i)\ {\stackrel {\mathrm {def} }{=))\ {\frac {N\Phi }{i))={\frac {\Lambda }{i))}$

“小信号电感”是用来计算电压，以方程定义为

${\displaystyle L_{d}(i)\ {\stackrel {\mathrm {def} }{=))\ {\frac {\mathrm {d} (N\Phi )}{\mathrm {d} i))={\frac {\mathrm {d} \Lambda }{\mathrm {d} i))}$

${\displaystyle v(t)={\frac {\mathrm {d} \Lambda }{\mathrm {d} t))={\frac {\mathrm {d} \Lambda }{\mathrm {d} i)){\frac {\mathrm {d} i}{\mathrm {d} t))=L_{d}(i){\frac {\mathrm {d} i}{\mathrm {d} t))}$

## 简单电路的自感

${\displaystyle \quad {\frac {r^{2}N^{2)){3\ell ))\left\{-8w+4{\frac {\sqrt {1+m)){m))\left(K\left({\sqrt {\frac {m}{1+m))}\right)-\left(1-m\right)E\left({\sqrt {\frac {m}{1+m))}\right)\right)\right\))$

${\displaystyle ={\frac {r^{2}N^{2}\pi }{\ell ))\left\{1-{\frac {8w}{3\pi ))+\sum _{n=1}^{\infty }{\frac {\left(2n\right)!^{2)){n!^{4}\left(n+1\right)\left(2n-1\right)2^{2n))}\left(-1\right)^{n+1}w^{2n}\right\))$
${\displaystyle ={\begin{cases}{\frac {r^{2}N^{2}\pi }{\ell ))\left(1-{\frac {8w}{3\pi ))+{\frac {w^{2)){2))-{\frac {w^{4)){4))+{\frac {5w^{6)){16))-{\frac {35w^{8)){64))+...\right)\ ,&w\ll 1\\rN^{2}\left\{\left(1+{\frac {1}{32w^{2))}+O({\frac {1}{w^{4))})\right)\ln \left(8w\right)-{\frac {1}{2))+{\frac {1}{128w^{2))}+O({\frac {1}{w^{4))})\right\}\ ,&w\gg 1\end{cases))}$

${\displaystyle N}$：卷绕匝数
${\displaystyle r}$：半径
${\displaystyle \ell }$：长度
${\displaystyle w=r/\ell }$
${\displaystyle m=4w^{2))$
${\displaystyle E,K}$椭圆积分

（高频率）
${\displaystyle {\frac {\mu _{0}\ell }{2\pi ))\,\ln {\frac {\;r_{\text{o))}{\;r_{i))))$ ${\displaystyle r_{\text{o))}$：外半径
${\displaystyle r_{i))$：内半径
${\displaystyle \ell }$：长度

${\displaystyle a}$：导线半径

${\displaystyle {\frac {\mu _{0)){\pi ))\left(b\ln {\frac {2b}{a))+d\ln {\frac {2d}{a))-\left(b+d\right)\left(2-{\frac {Y}{2))\right)+2{\sqrt {b^{2}+d^{2))}\right.}$

${\displaystyle \left.-b\cdot \operatorname {arsinh} {\frac {b}{d))-d\cdot \operatorname {arsinh} {\frac {d}{b))+O\left(a\right)\right)}$

${\displaystyle a}$：导线半径
${\displaystyle b}$：边长
${\displaystyle d}$：边宽
${\displaystyle b,d\gg a}$

${\displaystyle {\frac {\mu _{0}\ell }{\pi ))\left(\ln {\frac {d}{a))+Y/2\right)}$ ${\displaystyle a}$：导线半径
${\displaystyle d}$：距离
${\displaystyle d\geq 2a}$
${\displaystyle \ell }$：长度

（高频率）
${\displaystyle {\frac {\mu _{0}\ell }{\pi ))\operatorname {arcosh} \left({\frac {d}{2a))\right)={\frac {\mu _{0}\ell }{\pi ))\ln \left({\frac {d}{2a))+{\sqrt ((\frac {d^{2)){4a^{2))}-1))\right)}$ ${\displaystyle a}$：导线半径
${\displaystyle d}$：距离
${\displaystyle d\geq 2a}$
${\displaystyle \ell }$：长度

${\displaystyle {\frac {\mu _{0}\ell }{2\pi ))\left(\ln {\frac {2d}{a))+Y/2\right)}$ ${\displaystyle a}$：导线半径
${\displaystyle d}$：距离
${\displaystyle d\geq a}$
${\displaystyle \ell }$：长度

（高频率）
${\displaystyle {\frac {\mu _{0}\ell }{2\pi ))\operatorname {arcosh} \left({\frac {d}{a))\right)={\frac {\mu _{0}\ell }{2\pi ))\ln \left({\frac {d}{a))+{\sqrt ((\frac {d^{2)){a^{2))}-1))\right)}$ ${\displaystyle a}$：导线半径
${\displaystyle d}$：距离
${\displaystyle d\geq a}$
${\displaystyle \ell }$：长度

• ${\displaystyle Y=1/2}$：电流均匀地分布于整个导体截面。
• ${\displaystyle Y=0}$：集肤效应，电流均匀地分布于导体表面。
• 对于高频率案例，假若导体彼此移向对方，另外会有屏蔽电流流动于导体表面，含有参数${\displaystyle Y}$的表达式不成立。

## 参考资料

1. ^ Heaviside, O. Electrician. Feb. 12, 1886, p. 271.见该文集的再版页面存档备份，存于互联网档案馆
2. ^ Glenn Elert. The Physics Hypertextbook: Inductance. 1998–2008 [2010-04-08]. （原始内容存档于2009-06-02）.
3. ^ Michael W. Davidson. Molecular Expressions: Electricity and Magnetism Introduction: Inductance. 1995–2008 [2010-04-08]. （原始内容存档于2016-03-03）.
4. ^ Bansal, Rajeev, Fundamentals of engineering electromagnetics illustrated, CRC Press: pp. 154, 2006, ISBN 9780849373602
5. ^ Alexander, Charles; Sadiku, Matthew, Fundamentals of Electric Circuits 3, revised, McGraw-Hill: pp. 564–565, 2006, ISBN 9780073301150
6. ^ Ghosh, Smarajit, Fundamentals of Electrical and Electronics Engineering, PHI Learning Pvt. Ltd.: pp. 113–117, 2004, ISBN 9788120323162
7. ^ Lorenz, L. Über die Fortpflanzung der Elektrizität. Annalen der Physik. 1879, VII: 161–193.（这表达式给出面电流流动于圆柱体表面的电感）.
8. ^ Elliott, R. S. Electromagnetics. New York: IEEE Press. 1993.对于均匀电流分布，答案里不应该有常数 -3/2。
• Frederick W. Grover. Inductance Calculations. Dover Publications, New York. 1952.
• Griffiths, David J. Introduction to Electrodynamics (3rd ed.). Prentice Hall. 1998. ISBN 0-13-805326-X.