# 串联和并联阻抗计算

   

## 串联阻抗的等效阻抗

A和B点之间的等效阻抗为
$Z_{AB} = Z_1 + Z_2 + ... + Z_n$

$Z_R = R$ , $Z_C = \dfrac{1}{j \omega C} = \dfrac{-j}{\omega C}$ , $Z_L = j \omega L$

$Z_{AB} = Z_R + Z_C + Z_L = R - \dfrac{j}{\omega C} + j \omega L$

$X_L = \omega L = 2 \pi f L$ 和 $X_C = \dfrac{1}{\omega C} = \dfrac{1}{2 \pi f C}$

$Z_{AB} = R + j(X_L - X_C)$ 代入数值
$X_L = \omega L = 2 \pi f L = 2 \pi 10^3 10^{-2} = 62.83 \Omega$
$X_C = \dfrac{1}{\omega C} = \dfrac{1}{2 \pi f C} = \dfrac{1}{2 \pi 10^3 10^{-5} } = 15.92 \Omega$

$Z_{AB} = 100 + j ( 62.83 - 15.92 )$

$Z_{AB} = 100 + j ( 62.83 - 15.92 ) = 100 + 46.91 j$

$Z_{AB} = \sqrt {100^2 + 46.91^2} e^{j \arctan{\dfrac{46.91}{100}}} = 110.45 e^{j 0.44}$
$Z_{AB}$ 写成相量形式
$Z_{AB} = 110.45 \angle 0.44 \; 弧度 = 110.45 \angle 25.13^{\circ}$

## 并联阻抗的等效阻抗

A和B点之间的等效阻抗为
$\dfrac{1}{Z_{AB}} = \dfrac{1}{Z_1} + \dfrac{1}{Z_2} + ... + \dfrac{1}{Z_n}$

$Z_R = R$ , $Z_C = \dfrac{1}{j \omega C}$ , $Z_L = j \omega L$

$\dfrac{1}{Z_{AB}} = \dfrac{1}{Z_R} + \dfrac{1}{Z_C} + \dfrac{1}{Z_L}$
$= \dfrac{1}{R} + \dfrac{1}{\dfrac{1}{j \omega C}} + \dfrac{1}{j \omega L}$

$X_L = \omega L$ 和 $X_C = \dfrac{1}{\omega C}$

$\dfrac{1}{Z_{AB}} = \dfrac{1}{R} + \dfrac{1}{\dfrac{X_C}{j}} + \dfrac{1}{j X_L}$
$\dfrac{1}{Z_{AB}} = \dfrac{1}{R} + \dfrac{j}{{X_C}} - j \dfrac{1}{ X_L}$
$= \dfrac{1}{R} + j (\dfrac{1}{{X_C}} - \dfrac{1}{ X_L} )$

$r = \sqrt { (\dfrac{1}{R})^2 + (\dfrac{1}{{X_C}} - \dfrac{1}{ X_L} )^2}$

$\alpha = \arctan \left(\dfrac{\dfrac{1}{{X_C}} - \dfrac{1}{ X_L}}{\dfrac{1}{R}} \right)$

$= \arctan \left(\dfrac{R}{X_C}-\dfrac{R}{X_L} \right)$

$\dfrac{1}{Z_{AB}} = r e^{j\alpha}$

$Z_{AB} = \dfrac{1}{r} e^{-j \alpha}$
$= \dfrac{1}{\sqrt { \left(\dfrac{1}{R}\right)^2 + \left(\dfrac{1}{{X_C}} - \dfrac{1}{ X_L} \right)^2}} e^{-j \arctan \left(\dfrac{R}{X_C}-\dfrac{R}{X_L} \right) }$
$= \dfrac{1}{\sqrt { \left(\dfrac{1}{R}\right)^2 + \left(\dfrac{1}{{X_C}} - \dfrac{1}{ X_L} \right)^2}} e^{j \arctan \left(\dfrac{R}{X_L}-\dfrac{R}{X_C} \right) }$

$f = 1.5 \; kHz$ , $C = 15 \; \mu F$ , $L = 20 \; mH$ 和 $R = 50 \; \Omega$
$X_L = \omega L = 2 \pi f L = 2 \pi 1.5 \times 10^3 \times 20 10^{-3 } = 188.50$
$X_C = \dfrac{1}{\omega C} = \dfrac{1}{ 2\pi f C} = \dfrac{1}{ 2\pi 1.5 \times 10^3 \times 15 10^{-6}} = 7.07$

$= \dfrac{1}{\sqrt { \left(\dfrac{1}{50}\right)^2 + \left(\dfrac{1}{{7.07}} - \dfrac{1}{ 188.50} \right)^2}}$
$= 7.27$

$= \arctan \left(\dfrac{50}{188.50}-\dfrac{50}{7.07} \right)$
$= - 81.64^{\circ}$