串联和并联阻抗计算

示例 2

求下图所示并联RLC电路的复数阻抗，已知：

频率 \( f = 1.5 \; kHz \) , \( C = 15 \; \mu F \) , \( L = 20 \; mH \) 和 \( R = 50 \; \Omega \)

问题 2 的解决方案

设

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

应用并联电路的阻抗规则

\( \dfrac{1}{Z_{AB}} = \dfrac{1}{Z_R} + \dfrac{1}{Z_C} + \dfrac{1}{Z_L} \)

\( = \dfrac{1}{R} + \dfrac{1}{\dfrac{1}{j \omega C}} + \dfrac{1}{j \omega L} \)

设

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

将上述结果重写为

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

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

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

上述复数的模 \( r \) 为

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

其相位角 \( \alpha \) 为

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

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

我们现在使用复数的指数形式来表示

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

我们现在将等效阻抗 \( Z_{AB} \) 写成指数形式

\( Z_{AB} = \dfrac{1}{r} e^{-j \alpha} \)

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

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

我们现在使用已知数值

\( f = 1.5 \; kHz \) , \( C = 15 \; \mu F \) , \( L = 20 \; mH \) 和 \( R = 50 \; \Omega \)

\( X_L = \omega L = 2 \pi f L = 2 \pi 1.5 \times 10^3 \times 20 10^{-3 } = 188.50 \)

\( X_C = \dfrac{1}{\omega C} = \dfrac{1}{ 2\pi f C} = \dfrac{1}{ 2\pi 1.5 \times 10^3 \times 15 10^{-6}} = 7.07\)

模：\( \dfrac{1}{\sqrt { \left(\dfrac{1}{R}\right)^2 + \left(\dfrac{1}{{X_C}} - \dfrac{1}{ X_L} \right)^2}} \)

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

\( = 7.27 \)

相位角：\( \arctan \left(\dfrac{R}{X_L}-\dfrac{R}{X_C} \right) \)

\( = \arctan \left(\dfrac{50}{188.50}-\dfrac{50}{7.07} \right) \)

\( = - 81.64^{\circ} \)

可以使用并联RLC电路阻抗计算器进行更多练习。