# 阻抗

## 複阻抗

1. 直角形式：${\displaystyle R+jX}$
2. 極形式：${\displaystyle Z_{m}\angle \theta }$
3. 指數形式：${\displaystyle Z_{m}e^{j\theta }}$

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

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

## 歐姆定律

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

## 複值電壓與電流

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

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

{\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}}}

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

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

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

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

### 複數運算的正確性

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

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

## 電路元件的阻抗

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

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

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

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

### 電阻器

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

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

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

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

### 電容器

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

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

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

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

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

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

### 電感器

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

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

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

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

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

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

## 廣義 s-平面阻抗

${\displaystyle j\omega }$ 定義阻抗的方法只能應用於以穩定態交流信號為輸入的電路。假若將阻抗概念加以延伸，將 ${\displaystyle j\omega }$ 改換為複角頻率 ${\displaystyle s}$ ，就可以應用於以任意交流信號為輸入的電路。表示於時域的信號，經過拉普拉斯變換後，會改為表示於頻域的信號，改成以複角頻率表示。採用這更廣義的標記，基本電路元件的阻抗為

## 電抗

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

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

### 容抗

${\displaystyle X_{C}=-1/\omega C}$

### 感抗

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

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

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

## 阻抗組合

### 串聯電路

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

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

### 並聯電路

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

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

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

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

## 測量

### 電橋法

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

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

### 諧振法

1. 調整可調電容器的電容 ${\displaystyle C}$ ，使得RLC電路進入共振狀況。用測Q計測量電容器的品質因子 ${\displaystyle Q}$
2. 如右圖所示，將阻抗為 ${\displaystyle Z_{x}}$ 的被測元件串聯於RLC電路，調整可調電容器的電容 ${\displaystyle C'}$ ，使得電路進入共振狀況。用測Q計測量電容器的品質因子 ${\displaystyle Q'}$

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

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

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

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

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

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

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

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

## 參考文獻

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, （原始內容存檔於2009-10-27）
5. Horowitz, Paul; Hill, Winfield. 1. The Art of Electronics. Cambridge University Press. 1989: 31–33. ISBN 0-521-37095-7.
6. ^ Alexander, Charles; Sadiku, Matthew, Fundamentals of Electric Circuits 3, revised, McGraw-Hill: pp. 829–830, 2006, ISBN 9780073301150
7. ^ Agilent Impedance Measurement Handbook (PDF) 4th, USA: Agilent Technologies: pp.22ff, 2009, （原始內容 (PDF)存檔於2011-05-16）
8. Bakshi, V. U.; Bakshi, U. A., Electronic Measurements, Technical Publications: pp. 68ff, 110ff, 2007, ISBN 9788189411756