并联RLC二阶电路

零状态RLC并联电路的自然响应

• 拓扑约束
  • KVL: $u_C=u_L=-u_R$
  • KCL: $i_C+i_L=i_R$
• 元件约束
  • $i_C=C\frac{d{u}_C}{dt}$
  • $u_L=L\frac{d{i}_L}{dt}$
  • $R=\frac{u_R}{i_R}$

列写方程：

$$\begin{array}{r}{i}_C+i_L=i_R\\ C\frac{d{u}_C}{dt}+i_L& =\frac{u_R}{R}\\ C\frac{d(u_L)}{dt}+i_L& =\frac{(-u_L)}{R}\\ C\frac{d(L\frac{d{i}_L}{dt})}{dt}+i_L& =\frac{-(L\frac{d{i}_L}{dt})}{R}\\ LC\frac{d^2{i}_L}{d^{2}t}+i_L& =-\frac{L}{R}\frac{d{i}_L}{dt}\\ LC\frac{d^2{i}_L}{d^{2}t}+\frac{L}{R}\frac{d{i}_L}{dt}+i_L& =0\\ \frac{d^2{i}_L}{d^{2}t}+\frac{1}{RC}\frac{d{i}_L}{dt}+\frac{1}{LC}{i}_L& =0\end{array}$$

定义：$2\alpha =\frac{1}{RC}$，${\omega }_0^2=\frac{1}{LC}$，得：

$$\frac{d^2{u}_C}{d^{2}t}+2\alpha \frac{d{u}_C}{dt}+{\omega }_0^2{u}_C=0$$

特征方程为

$${\lambda }^2+2\alpha \lambda +{\omega }_0^2=0$$

......剩余推导过程和结论和之前的串联RLC完全相同。

串联RLC和并联RLC的区别

相同

• 串联RLC和并联RLC的震荡项都是 ${\omega }_0^2=\frac{1}{LC}$

不同

• 串联RLC的阻尼项为 $2\alpha =\frac{R}{L}$
• 并联RLC的阻尼项为 $2\alpha =\frac{1}{RC}$