电容

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${\displaystyle C={\frac {Q}{V))}$

1法拉等于1库仑伏特，即电容为1法拉的电容器，在正常操作范围内，每增加1伏特的电势差可以多储存1库仑的电荷。

${\displaystyle \mathrm {d} W={\frac {q}{C))\,\mathrm {d} q}$

${\displaystyle W_{\text{charging))=\int _{0}^{Q}{\frac {q}{C))\,\mathrm {d} q={\frac {Q^{2)){2C))={\frac {1}{2))CV^{2}=U_{\text{stored))}$

单位

${\displaystyle {\mbox{1 F))={10^{6)){\mbox{μF))={10^{9)){\mbox{nF))={10^{12)){\mbox{pF))}$ [2]

电容器

${\displaystyle \sigma =Q/A}$

${\displaystyle E=\sigma /\varepsilon =Q/\varepsilon A}$

${\displaystyle V=Ed=\sigma d/\varepsilon =Qd/\varepsilon A}$

${\displaystyle C=Q/V=\varepsilon A/d}$

电压依赖性电容器

${\displaystyle \mathrm {d} Q=C(V)\ \mathrm {d} V}$

${\displaystyle E=V/d}$

${\displaystyle P=f(E)=f(V/d)=g(V)}$

${\displaystyle D=P+\epsilon _{0}E=g(V)+\epsilon _{0}V/d}$

${\displaystyle Q=DA=(g(V)+\epsilon _{0}V/d)A}$

${\displaystyle C(V)={\frac {Q}{V))={\frac {g(V)A}{V))+{\frac {\epsilon _{0}A}{d))}$

${\displaystyle C={\frac {Q}{V))=kA+{\frac {\epsilon _{0}A}{d))}$

${\displaystyle Q=\int _{0}^{V}C(V')\ \mathrm {d} V'}$

${\displaystyle \mathrm {d} U_{\text{stored))=Q\mathrm {d} V''=\left[\int _{0}^{V''}\ C(V')\ \mathrm {d} V'\right]\mathrm {d} V''}$

${\displaystyle \int _{a}^{z}f(x)g'(x)\ \mathrm {d} x=\left[f(x)g(x)\right]_{a}^{z}-\int _{a}^{z}f'(x)g(x)\ \mathrm {d} x}$

${\displaystyle \int _{a}^{z}\int _{a}^{x}\ h(y)\ \mathrm {d} y\ \mathrm {d} x=\left[\int _{a}^{x}\ xh(y)\ \mathrm {d} y\right]_{a}^{z}-\int _{a}^{z}xh(x)\ \mathrm {d} x=\int _{a}^{z}zh(y)\ \mathrm {d} y-\int _{a}^{z}yh(y)\ \mathrm {d} y=\int _{a}^{z}\ \left(z-y\right)h(y)\ \mathrm {d} y}$

${\displaystyle U_{\text{stored))=\int _{0}^{V}\ \left[\int _{0}^{V''}\ C(V')\ \mathrm {d} V'\right]\mathrm {d} V''=\int _{0}^{V}\ \left(V-V'\right)C(V')\ \mathrm {d} V'}$

频率依赖性电容器

${\displaystyle {\boldsymbol {D))(t)={\frac {\varepsilon _{0)){\sqrt {2\pi ))}\int _{-\infty }^{t}\mathrm {d} t'\ \varepsilon _{r}(t-t'){\boldsymbol {E))(t')}$

${\displaystyle {\boldsymbol {D))(t)={\frac {\varepsilon _{0)){\sqrt {2\pi ))}\int _{-\infty }^{\infty }\mathrm {d} t'\ \varepsilon _{r}(t-t'){\boldsymbol {E))(t')}$

${\displaystyle {\boldsymbol {D))(\omega )=\varepsilon _{0}\varepsilon _{r}(\omega ){\boldsymbol {E))(\omega )}$

${\displaystyle \varepsilon _{r}(\omega )}$复函数，其虚值部分与介质的电场能量吸收有关。更详尽细节，请参阅条目电容率。由于电容与电容率成正比，电容也具有这频率行为。对于时间做傅里叶变换于高斯定律：

${\displaystyle Q(\omega )=\oint _{\mathbb {S} }\mathbf {D} (\mathbf {r} ,\omega )\cdot \mathrm {d} \mathbf {a} }$

{\displaystyle {\begin{aligned}I(\omega )&=j\omega Q(\omega )=j\omega \oint _{\mathbb {S} }\mathbf {D} (\mathbf {r} ,\omega )\cdot \mathrm {d} \mathbf {a} \\&=\left[G(\omega )+j\omega C(\omega )\right]V(\omega )={\frac {V(\omega )}{Z(\omega )))\\\end{aligned))}

${\displaystyle \varepsilon _{r}(\omega )=\varepsilon _{r}'(\omega )-j\varepsilon _{r}''(\omega )={\frac {1}{j\omega Z(\omega )C_{0))}={\frac {C(\omega )}{C_{0))))$

电容矩阵

${\displaystyle V_{1}=P_{11}Q_{1}+P_{12}Q_{2}+P_{13}Q_{3))$
${\displaystyle V_{2}=P_{21}Q_{1}+P_{22}Q_{2}+P_{23}Q_{3))$
${\displaystyle V_{3}=P_{31}Q_{1}+P_{32}Q_{2}+P_{33}Q_{3))$

${\displaystyle Q_{1}=C_{11}V_{1}+C_{12}V_{2}+C_{13}V_{3))$
${\displaystyle Q_{2}=C_{21}V_{1}+C_{22}V_{2}+C_{23}V_{3))$
${\displaystyle Q_{3}=C_{31}V_{1}+C_{32}V_{2}+C_{33}V_{3))$

${\displaystyle V_{i}=\sum _{j=1}^{n}P_{ij}Q_{j},\qquad \qquad i=1,2,\dots ,n}$
${\displaystyle Q_{i}=\sum _{j=1}^{n}C_{ij}V_{j},\qquad \qquad i=1,2,\dots ,n}$

${\displaystyle P_{ij}\ {\stackrel {def}{=))\ {\frac {\partial V_{i)){\partial Q_{j))))$

${\displaystyle C_{ij}\ {\stackrel {def}{=))\ {\frac {\partial Q_{i)){\partial V_{j))))$

${\displaystyle P_{ij}=P_{ji))$
${\displaystyle C_{ij}=C_{ji))$

${\displaystyle C\ {\stackrel {def}{=))\ Q/\Delta V}$

${\displaystyle V_{i}=-P_{ii}Q+P_{ij}Q}$
${\displaystyle V_{j}=-P_{ji}Q+P_{jj}Q}$

${\displaystyle C=Q/(V_{j}-V_{i})=1/(P_{ii}+P_{jj}-2P_{ij})}$

自电容

${\displaystyle V=Q/4\pi \varepsilon _{0}R}$

${\displaystyle C=Q/V=4\pi \varepsilon _{0}R}$

范例

${\displaystyle C=4\pi \varepsilon _{0}R=4\pi \times 8.85\times 10^{-12}\times 0.2\approx 22[pF]}$

${\displaystyle C=4\pi \times 8.85\times 10^{-12}\times 6.378\times 10^{6}\approx 700[\mu F]}$

简单系统的电容

${\displaystyle \varepsilon }$: 介质的电容率

${\displaystyle \varepsilon }$: 介质的电容率

${\displaystyle d}$: 距离， ${\displaystyle d>a}$
${\displaystyle l}$: 电线长度

${\displaystyle w_{i))$: 导板板宽
${\displaystyle k_{i}=d/(2w_{i}+d)}$
${\displaystyle k^{2}=k_{1}k_{2))$
${\displaystyle K}$: 椭圆积分
${\displaystyle l}$: 长度

${\displaystyle \varepsilon }$: 介质的电容率

${\displaystyle =2\pi \varepsilon a\left\{1+{\frac {1}{2D))+{\frac {1}{4D^{2))}+{\frac {1}{8D^{3))}+{\frac {1}{8D^{4))}+{\frac {3}{32D^{5))}+O\left({\frac {1}{D^{6))}\right)\right\))$
${\displaystyle =2\pi \varepsilon a\left\{\ln 2+\gamma -{\frac {1}{2))\ln \left(2D-2\right)+O\left(2D-2\right)\right\))$

${\displaystyle a}$: 半径
${\displaystyle d}$: 距离，${\displaystyle d>2a}$
${\displaystyle D=d/2a>1}$
${\displaystyle \gamma }$欧拉-马歇罗尼常数

${\displaystyle d}$: 距离，${\displaystyle d>a}$
${\displaystyle D=d/a}$

${\displaystyle l}$: 电线长度
${\displaystyle \Lambda =\ln(l/a)}$

参考文献

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中华人民共和国法定计量单位. 维基文库. 1984-02-27.

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