# 阻抗

## 複阻抗

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

$R= Z_m \cos\theta$
$X= Z_m \sin\theta$

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

## 歐姆定律

$v = i Z = i Z_m e^{j\theta}$

## 複值電壓與電流

$v(t) = V_m e^{j(\omega t + \phi_V)}$
$i(t) = I_m e^{j(\omega t + \phi_I)}$

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

\begin{align} 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{align}

$V_m= I_m Z_m$
$\ \phi_V = \phi_I + \theta$

$V = V_m e^{j\phi_V}$
$I = I_m e^{j\phi_I}$

$v(t) = V e^{j\omega t}$
$i(t) = I e^{j\omega t}$

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

### 複數運算的正確性

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

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

## 電路元件的阻抗

$Z_R = R$

$Z_C = \frac{1}{j\omega C}$
$Z_L = j\omega L$

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

$Z_C = \frac{e^{-j\pi/2}}{\omega C}$
$Z_L = \omega Le^{j\pi/2}$

### 電阻器

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

$v_{R}(t) = V_0 \cos(\omega t)= V_0 e^{j\omega t},\qquad V_0>0$ ，

$i_{R}(t) =\frac{V_0}{R}e^{j\omega t}$

$Z_{R} =R$

### 電容器

$i_C(t) = C \frac{\operatorname{d}v_C(t)}{\operatorname{d}t}$

$v_C(t) = V_0 \sin(\omega t)=V_0 e^{j(\omega t-\pi/2)},\qquad V_0>0$

$i_C(t) =\omega V_0 C \cos(\omega t)=\omega V_0 C e^{j\omega t}$

$\frac{v_C(t)}{i_C(t)}=\frac{V_0\sin(\omega t)}{\omega V_0 C\cos( \omega t)}= \frac{\sin(\omega t)}{\omega C \sin \left( \omega t + \frac{\pi}{2}\right)}$ 。

$v_C(t)=V_0 e^{j(\omega t-\pi/2)},\qquad V_0>0$
$i_C(t)=\omega V_0 C e^{j\omega t}$
$Z_C = \frac{e^{-j \pi/2}}{\omega C}$

$Z_C=\frac{1}{j \omega C}$

### 電感器

$v_L(t) = L \frac{\operatorname{d}i_L(t)}{\operatorname{d}t}$

$i_{L}(t) = I_0 \cos(\omega t)$

$v_L(t) = - \omega L I_0 \sin(\omega t)=\omega L I_0 \cos(\omega t + \pi/2)$

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

$i_L(t)=I_0 e^{j\omega t},\qquad I_0>0$
$v_L(t)=\omega L I_0 e^{j(\omega t+\pi/2)}$
$Z_L =\omega L e^{j \pi/2}$

$Z_L= j \omega L$

## 廣義 s-平面阻抗

$j\omega$ 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 $j\omega$ 改換為複角頻率 $s$ ，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

$Z_m= \sqrt{Z Z^*} = \sqrt{R^2 + X^2}$
$\theta = \arctan{\left(\frac{X}{R}\right)}$

$X =Z_m\sin\theta$

### 容抗

$X_C = - 1/\omega C$

### 感抗

$X_L = \omega L$

$\mathcal{E} = -{{\operatorname{d}\Phi_B} \over \operatorname{d}t}$

$\mathcal{E} = -N{\operatorname{d}\Phi_B \over \operatorname{d}t}$

## 阻抗組合

### 串聯電路

$Z_{eq}\ \stackrel{def}{=}\ Z_1 + Z_2 + \cdots + Z_n$

$Z_{eq} = R_{eq} + jX_{eq} = (R_1 + R_2 + \cdots + R_n) + j(X_1 + X_2 + \cdots + X_n)$

### 並聯電路

$\frac{1}{Z_{eq}}\ \stackrel{def}{=}\ \frac{1}{Z_1} + \frac{1}{Z_2} + \cdots + \frac{1}{Z_n}$

$Z_{eq} =\frac{Z_1 Z_2}{Z_1 + Z_2}$

$Z_{eq} = R_{eq} + j X_{eq}$

$R_{eq} = \frac{(X_1 R_2 + X_2 R_1) (X_1 + X_2) + (R_1 R_2 - X_1 X_2) (R_1 + R_2)}{(R_1 + R_2)^2 + (X_1 + X_2)^2}$
$X_{eq} = \frac{(X_1 R_2 + X_2 R_1) (R_1 + R_2) - (R_1 R_2 - X_1 X_2) (X_1 + X_2)}{(R_1 + R_2)^2 + (X_1 + X_2)^2}$

## 測量

### 電橋法

$Z_x = Z_2 Z_3/Z_1$

$Z_x =| Z_x|\angle \theta_x=|Z_2 Z_3/Z_1|\angle (\theta_2+\theta_3-\theta_1)$

### 諧振法

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

$X_{C}+X_L=0$

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

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

$X_{C'}+X_X+X_L=0$

$X_X = \frac{1}{\omega C'}-\omega L= \frac{1}{\omega C'}-\frac{1}{\omega C} =\frac{C-C'}{\omega CC'}$

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

$R_X=\frac{1}{\omega C'Q'}-\frac{1}{\omega CQ}$

$Z_X=R_X+jX_X=\left(\frac{1}{\omega C'Q'}-\frac{1}{\omega CQ}\right)+ j\left( \frac{1}{\omega C'}-\frac{1}{\omega C}\right)$

## 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 0821834657
4. ^ Katz, Eugenii, 對於電磁學有巨大貢獻的著名科學家：亞瑟·肯乃利
5. ^ 5.0 5.1 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 4th, USA: Agilent Technologies, pp.22ff, 2009
8. ^ 8.0 8.1 Bakshi, V. U.; Bakshi, U. A., Electronic Measurements, Technical Publications, pp. 68ff, 110ff, 2007, ISBN 9788189411756