# 阻抗

## 词源

1927年，孔祥鹅《商榷电机工程译名问题》一文建议采用“电阻抗”作为该术语中文译名，以便顾及与“电抗”（electrical reactance）、“电阻”（electrical resistance）二词的学理关联[5]。这一译名得以沿用至今。

## 复阻抗

1. 直角形式：${\displaystyle R+jX}$
2. 极形式：${\displaystyle Z_{m}\angle \theta }$
3. 指数形式：${\displaystyle Z_{m}e^{j\theta ))$

${\displaystyle R=Z_{m}\cos \theta }$
${\displaystyle X=Z_{m}\sin \theta }$

${\displaystyle Z_{m}={\sqrt {R^{2}+X^{2))))$
${\displaystyle \theta =\arctan(X/R)}$

## 欧姆定律

${\displaystyle v=iZ=iZ_{m}e^{j\theta ))$

## 复值电压与电流

${\displaystyle v(t)=V_{m}e^{j(\omega t+\phi _{V})))$
${\displaystyle i(t)=I_{m}e^{j(\omega t+\phi _{I})))$

${\displaystyle Z\ {\stackrel {def}{=))\ {\frac {v(t)}{i(t)))}$

{\displaystyle {\begin{aligned}V_{m}e^{j(\omega t+\phi _{V})}&=I_{m}e^{j(\omega t+\phi _{I})}Z_{m}e^{j\theta }\\&=I_{m}Z_{m}e^{j(\omega t+\phi _{I}+\theta )}\\\end{aligned))}

${\displaystyle V_{m}=I_{m}Z_{m))$
${\displaystyle \ \phi _{V}=\phi _{I}+\theta }$

${\displaystyle V=V_{m}e^{j\phi _{V))}$
${\displaystyle I=I_{m}e^{j\phi _{I))}$

${\displaystyle v(t)=Ve^{j\omega t))$
${\displaystyle i(t)=Ie^{j\omega t))$

${\displaystyle Z\ {\stackrel {def}{=))\ {\frac {V}{I))}$

### 复数运算的正确性

${\displaystyle \cos(\omega t+\phi )={\frac {1}{2)){\Big [}e^{j(\omega t+\phi )}+e^{-j(\omega t+\phi )}{\Big ]))$

${\displaystyle \cos(\omega t+\phi )=\mathrm {Re} {\Big \{}e^{j(\omega t+\phi )}{\Big \))}$

## 电路元件的阻抗

${\displaystyle Z_{R}=R}$

${\displaystyle Z_{C}={\frac {1}{j\omega C))}$
${\displaystyle Z_{L}=j\omega L}$

${\displaystyle j=e^{j\pi /2))$
${\displaystyle -j=e^{-j\pi /2))$

${\displaystyle Z_{C}={\frac {e^{-j\pi /2)){\omega C))}$
${\displaystyle Z_{L}=\omega Le^{j\pi /2))$

### 电阻器

${\displaystyle v_{R}(t)=i_{R}(t)R}$

${\displaystyle v_{R}(t)=V_{0}\cos(\omega t)=V_{0}e^{j\omega t},\qquad V_{0}>0}$ ，

${\displaystyle i_{R}(t)={\frac {V_{0)){R))e^{j\omega t))$

${\displaystyle Z_{R}=R}$

### 电容器

${\displaystyle i_{C}(t)=C{\frac {\operatorname {d} v_{C}(t)}{\operatorname {d} t))}$

${\displaystyle v_{C}(t)=V_{0}\sin(\omega t)=\operatorname {Re} \{V_{0}e^{j(\omega t-\pi /2)}\}=\operatorname {Re} \{V_{C}e^{j\omega t}\},\quad where\quad V_{0}>0,\quad V_{C}=V_{0}e^{j(-\pi /2)))$

${\displaystyle i_{C}(t)=\omega V_{0}C\cos(\omega t)=\operatorname {Re} \{\omega V_{0}Ce^{j\omega t}\}=\operatorname {Re} \{I_{C}e^{j\omega t}\))$

${\displaystyle {\frac {v_{C}(t)}{i_{C}(t)))={\frac {V_{0}\sin(\omega t)}{\omega V_{0}C\cos(\omega t)))={\frac {\sin(\omega t)}{\omega C\sin \left(\omega t+{\frac {\pi }{2))\right)))}$ 。

${\displaystyle V_{C}=V_{0}e^{j(-\pi /2)},\qquad V_{0}>0}$
${\displaystyle I_{C}=\omega V_{0}Ce^{j0))$
${\displaystyle Z_{C}={\frac {e^{-j\pi /2)){\omega C))}$

${\displaystyle Z_{C}={\frac {1}{j\omega C))}$

### 电感器

${\displaystyle v_{L}(t)=L{\frac {\operatorname {d} i_{L}(t)}{\operatorname {d} t))}$

${\displaystyle i_{L}(t)=I_{0}\cos(\omega t)}$

${\displaystyle v_{L}(t)=-\omega LI_{0}\sin(\omega t)=\omega LI_{0}\cos(\omega t+\pi /2)}$

${\displaystyle {\frac {v_{L}(t)}{i_{L}(t)))={\frac {\omega L\cos(\omega t+\pi /2)}{\cos(\omega t)))}$

${\displaystyle i_{L}(t)=I_{0}e^{j\omega t},\qquad I_{0}>0}$
${\displaystyle v_{L}(t)=\omega LI_{0}e^{j(\omega t+\pi /2)))$
${\displaystyle Z_{L}=\omega Le^{j\pi /2))$

${\displaystyle Z_{L}=j\omega L}$

## 广义 s-平面阻抗

${\displaystyle j\omega }$ 定义阻抗的方法只能应用于以稳定态交流信号为输入的电路。假若将阻抗概念加以延伸，将 ${\displaystyle j\omega }$ 改换为复角频率 ${\displaystyle s}$ ，就可以应用于以任意交流信号为输入的电路。表示于时域的信号，经过拉普拉斯变换后，会改为表示于频域的信号，改成以复角频率表示。采用这更广义的标记，基本电路元件的阻抗为

## 电抗

${\displaystyle Z_{m}={\sqrt {ZZ^{*))}={\sqrt {R^{2}+X^{2))))$
${\displaystyle \theta =\arctan {\left({\frac {X}{R))\right)))$

${\displaystyle X=Z_{m}\sin \theta }$

### 容抗

${\displaystyle X_{C}=-j/\omega C}$

### 感抗

${\displaystyle X_{L}=\omega Lj}$

${\displaystyle {\mathcal {E))=-((\operatorname {d} \Phi _{B)) \over \operatorname {d} t))$

${\displaystyle {\mathcal {E))=-N{\operatorname {d} \Phi _{B} \over \operatorname {d} t))$

## 阻抗组合

### 串联电路

${\displaystyle Z_{eq}\ {\stackrel {def}{=))\ Z_{1}+Z_{2}+\cdots +Z_{n))$

${\displaystyle Z_{eq}=R_{eq}+jX_{eq}=(R_{1}+R_{2}+\cdots +R_{n})+j(X_{1}+X_{2}+\cdots +X_{n})}$

### 并联电路

${\displaystyle {\frac {1}{Z_{eq))}\ {\stackrel {def}{=))\ {\frac {1}{Z_{1))}+{\frac {1}{Z_{2))}+\cdots +{\frac {1}{Z_{n))))$

${\displaystyle Z_{eq}={\frac {Z_{1}Z_{2)){Z_{1}+Z_{2))))$

${\displaystyle Z_{eq}=R_{eq}+jX_{eq))$

${\displaystyle R_{eq}={\frac {(X_{1}R_{2}+X_{2}R_{1})(X_{1}+X_{2})+(R_{1}R_{2}-X_{1}X_{2})(R_{1}+R_{2})}{(R_{1}+R_{2})^{2}+(X_{1}+X_{2})^{2))))$
${\displaystyle X_{eq}={\frac {(X_{1}R_{2}+X_{2}R_{1})(R_{1}+R_{2})-(R_{1}R_{2}-X_{1}X_{2})(X_{1}+X_{2})}{(R_{1}+R_{2})^{2}+(X_{1}+X_{2})^{2))))$

## 测量

### 电桥法

${\displaystyle Z_{x}=Z_{2}Z_{3}/Z_{1))$

${\displaystyle Z_{x}=|Z_{x}|\angle \theta _{x}=|Z_{2}Z_{3}/Z_{1}|\angle (\theta _{2}+\theta _{3}-\theta _{1})}$

### 谐振法

1. 调整可调电容器的电容 ${\displaystyle C}$ ，使得RLC电路进入共振状况。用测Q计测量电容器的品质因子 ${\displaystyle Q}$
2. 如右图所示，将阻抗为 ${\displaystyle Z_{x))$ 的被测元件串联于RLC电路，调整可调电容器的电容 ${\displaystyle C'}$ ，使得电路进入共振状况。用测Q计测量电容器的品质因子 ${\displaystyle Q'}$

${\displaystyle X_{C}+X_{L}=0}$

${\displaystyle {\frac {1}{\omega C))=\omega L}$

${\displaystyle Q={\frac {|X_{C}|}{R))={\frac {1}{\omega CR))={\frac {\omega L}{R))}$

${\displaystyle X_{C'}+X_{X}+X_{L}=0}$

${\displaystyle X_{X}={\frac {1}{\omega C'))-\omega L={\frac {1}{\omega C'))-{\frac {1}{\omega C))={\frac {C-C'}{\omega CC'))}$

${\displaystyle Q'={\frac {|X_{C'}|}{R_{X}+R))={\frac {1}{\omega C'(R_{X}+R)))}$

${\displaystyle R_{X}={\frac {1}{\omega C'Q'))-{\frac {1}{\omega CQ))}$

${\displaystyle Z_{X}=R_{X}+jX_{X}=\left({\frac {1}{\omega C'Q'))-{\frac {1}{\omega CQ))\right)+j\left({\frac {1}{\omega C'))-{\frac {1}{\omega C))\right)}$

## 参阅

• 阻抗匹配
• 阻抗心动描记术（impedance cardiography
• 阻抗电桥（impedance bridging
• 特性阻抗（characteristic impedance
• 负阻抗变换器impedance bridging

## 参考文献

1. ^ Alexander, Charles; Sadiku, Matthew, Fundamentals of Electric Circuits 3, revised, McGraw-Hill: pp. 387–389, 2006, ISBN 9780073301150
2. ^ Science, p. 18, 1888
3. ^ Oliver Heaviside, The Electrician, p. 212, 23rd July 1886 reprinted as Electrical Papers, p64, AMS Bookstore, ISBN 978-0-8218-3465-7
4. ^ Katz, Eugenii, 對於電磁學有巨大貢獻的著名科學家：亞瑟·肯乃利, （原始内容存档于2009-10-27）
5. ^ 存档副本. [2019-12-14]. （原始内容存档于2019-05-12）.
6. Horowitz, Paul; Hill, Winfield. 1. The Art of Electronics. Cambridge University Press. 1989: 31–33. ISBN 0-521-37095-7.
7. ^ Alexander, Charles; Sadiku, Matthew, Fundamentals of Electric Circuits 3, revised, McGraw-Hill: pp. 829–830, 2006, ISBN 9780073301150
8. ^ Agilent Impedance Measurement Handbook (PDF) 4th, USA: Agilent Technologies: pp.22ff, 2009, （原始内容 (PDF)存档于2011-05-16）
9. Bakshi, V. U.; Bakshi, U. A., Electronic Measurements, Technical Publications: pp. 68ff, 110ff, 2007, ISBN 9788189411756