证明 (1) 对于图 2-12(a)所示的无源网络,根据复数阻抗的方法可得
\[
\frac{U_2(s)}{U_1(s)} = \frac{R_2}{\dfrac{R_1 \cdot \dfrac{1}{C_1 s}}{R_1 + \dfrac{1}{C_1 s}} + R_2} = \frac{R_2}{\dfrac{R_1}{R_1 C_1 s + 1} + R_2} = \frac{1 + R_1 C_1 s}{\dfrac{R_1 + R_2}{R_2} + R_1 C_1 s}
\]
\[
= \frac{R_2}{R_1 + R_2} \cdot \frac{1 + R_1 C_1 s}{1 + \dfrac{R_2}{R_1 + R_2} R_1 C_1 s}
\]
(2) 对于图 2-12(b)所示的无源网络,根据复数阻抗的方法可得
\[
\frac{U_2(s)}{U_1(s)} = \frac{R_2 + \dfrac{1}{C_2 s}}{\dfrac{\left(R_3 + \dfrac{1}{C_1 s}\right) \times R_1}{\left(R_3 + \dfrac{1}{C_1 s}\right) + R_1} + \left(R_2 + \dfrac{1}{C_2 s}\right)} = \frac{R_2 + \dfrac{1}{C_2 s}}{\dfrac{R_1 + R_1 R_3 C_1 s}{1 + R_3 C_1 s + R_1 C_1 s} + R_2 + \dfrac{1}{C_2 s}}
\]
\[
= \frac{R_2(R_1 + R_3)C_1 C_2 s^2 + (R_1 C_1 + R_2 C_2 + R_3 C_1)s + 1}{(R_1 R_2 + R_2 R_3 + R_1 R_3)C_1 C_2 s^2 + (R_1 C_1 + R_2 C_2 + R_1 C_2 + R_3 C_1)s + 1}
\]
2-17 试求图 2-13 所示网络的传递函数 \(U_2(s)/U_1(s)\)。
解 对于图 2-13 所示的无源网络,根据复数阻抗的方法可得
\[
\frac{U_2(s)}{U_1(s)} = \frac{\dfrac{Z}{ZCs+1}}{\left\{\left[\left(\dfrac{Z}{ZCs+1}+R\right)\cdot\dfrac{1}{Cs}\right]\Big/\left[\left(\dfrac{Z}{ZCs+1}+R\right)+\dfrac{1}{Cs}\right]\right\}+R}
\]
\[
= \frac{Z}{ZCs+1} \cdot \frac{ZRC^2s^2 + (2Z+R)Cs + 1}{ZR^2C^2s^2 + (3ZRC+R^2C)s + (Z+2R)}
\]
\[
= \frac{1}{ZCs+1} \cdot \frac{ZRC^2s^2 + (2Z+R)Cs + 1}{R^2C^2s^2 + \left(3RC + \dfrac{R^2C}{Z}\right)s + \left(1 + \dfrac{2R}{Z}\right)}
\]
图 2-13 无源网络电路图
图 2-14 无源网络电路图
2-18 试证明图 2-14 所示网络的传递函数为
\[
\frac{U_2(s)}{U_1(s)} = \frac{\dfrac{L_1}{R_1}s + 1}{\dfrac{L_1}{R_1}R_2 C_1 s^2 + \left(\dfrac{L_1}{R_1} + R_2 C_1\right)s + \dfrac{R_1+R_2}{R_1}}
\]
证明 对于图 2-14 所示的无源网络,根据复数阻抗的方法可得
· 29 ·