\[
R'=\dfrac{R\dfrac{\theta_1}{\theta_{\max}}\cdot R\dfrac{\theta_{\max}-\theta_1}{\theta_{\max}}}{R\dfrac{\theta_1}{\theta_{\max}}+R\dfrac{\theta_{\max}-\theta_1}{\theta_{\max}}}+\dfrac{R\dfrac{\theta_2}{\theta_{\max}}\cdot R\dfrac{\theta_{\max}-\theta_2}{\theta_{\max}}}{R\dfrac{\theta_2}{\theta_{\max}}+R\dfrac{\theta_{\max}-\theta_2}{\theta_{\max}}}
\]
\[
=\dfrac{R}{\theta_{\max}^2}\theta_1(\theta_{\max}-\theta_1)+\dfrac{R}{\theta_{\max}^2}\theta_2(\theta_{\max}-\theta_2)
\]
此时,电位器误差角与输出电压之间的传递函数为
\[
\dfrac{U_o(s)}{\Delta\Theta(s)}=K_1\dfrac{Z(s)}{R'+Z(s)}
\]
2-15 图2-11为以恒速运转的他激直流发电机,输入为激磁绕组电压\(u_i(t)\),输出为端电压\(u_o(t)\)。求传递函数\(U_o(s)/U_i(s)\)。
图2-10 电位器误差检测器原理图
图2-11 他激直流发电机原理图
解 设他激直流发电机阻抗为\(Z_i\),则
\[
\dfrac{U_i(s)}{R_f+L_f s}=\dfrac{U_o(s)}{Z_i}
\]
即传递函数为
\[
\dfrac{U_o(s)}{U_i(s)}=\dfrac{Z_i}{R_f+L_f s}=\dfrac{Z_i/R_f}{1+sL_f/R_f}
\]
2-16 试证明图2-12(a)和(b)所示网络的传递函数分别为
(1) \(\dfrac{U_2(s)}{U_1(s)}=\dfrac{R_2}{R_1+R_2}\dfrac{1+R_1C_1s}{1+\dfrac{R_2}{R_1+R_2}R_1C_1s}\);
(2) \(\dfrac{U_2(s)}{U_1(s)}=\dfrac{R_2(R_1+R_3)C_1C_2s^2+(R_1C_1+R_2C_2+R_3C_1)s+1}{(R_1R_2+R_2R_3+R_1R_3)C_1C_2s^2+(R_1C_1+R_2C_2+R_1C_2+R_3C_1)s+1}\)。
图2-12 无源网络电路图
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