EE 211 - Step Responses of Series and Parallel RLC Circuits

For a series RLC circuit, the differential equation is


\begin{displaymath}L C \frac{d^2 v_{\scriptscriptstyle C}}{dt^2} + R_T C \frac{dv_{\scriptscriptstyle C}}{dt} + v_{\scriptscriptstyle C} = v_T(t).\end{displaymath}

If vT(t) is a constant or zero ( vT(t) = VT), and the system has initial conditions $v_{\scriptscriptstyle C}(0) = V_o$, and $i_{\scriptscriptstyle L}(0) = I_o$, then the solution to this equation is

     
(RT C)2 - 4LC > 0 (RT C)2 - 4LC = 0 (RT C)2 - 4LC < 0
Overdamped Critically Damped Underdamped
     
$ v_{\scriptscriptstyle C} = K_1 e^{-\alpha_1 t} + K_2 e^{-\alpha_2 t} + V_T$ $ v_{\scriptscriptstyle C} = K_1 e^{-\alpha t} + K_2 t e^{-\alpha t} + V_T$ $ v_{\scriptscriptstyle C} = e^{-\alpha t} (K_1 \cos \beta t + K_2 \sin \beta t) + V_T$
     
$\alpha_1 = \frac{R_T C - \sqrt{(R_T C)^2 - 4LC}}{2LC} $ $\alpha = \frac{R_T}{2L} $ $\alpha = \frac{R_T}{2L} $

$\alpha_2 = \frac{R_T C + \sqrt{(R_T C)^2 - 4LC}}{2LC} $

   $\beta = \frac{\sqrt{4LC - (R_T C)^2}}{2LC} $

$K_1 = \frac{\alpha_2 (V_o - V_T) + I_o/C}{\alpha_2 - \alpha_1}$

 K1 = Vo - VT K1 = Vo - VT

$K_2 = \frac{-\alpha_1 (V_o - V_T) - I_o/C}{\alpha_2 - \alpha_1}$

 $K_2 = \alpha (V_o - V_T) + I_o/C$ $K_2 = \frac{\alpha (V_o - V_T) + I_o/C}{\beta}$

For a parallel RLC circuit, the differential equation is


\begin{displaymath}L C \frac{d^2 i_{\scriptscriptstyle L}}{dt^2} + G_N L \frac{di_{\scriptscriptstyle L}}{dt} + i_{\scriptscriptstyle L} = i_N(t).\end{displaymath}

If iN(t) is a constant or zero ( iN(t) = IN), and the system has initial conditions $v_{\scriptscriptstyle C}(0) = V_o$, and $i_{\scriptscriptstyle L}(0) = I_o$, then the solution to this equation is

     
(GN L)2 - 4LC > 0 (GN L)2 - 4LC = 0 (GN L)2 - 4LC < 0
Overdamped Critically Damped Underdamped
     
$ i_{\scriptscriptstyle L} = K_1 e^{-\alpha_1 t} + K_2 e^{-\alpha_2 t} + I_N$ $ i_{\scriptscriptstyle L} = K_1 e^{-\alpha t} + K_2 t e^{-\alpha t} + I_N$ $ i_{\scriptscriptstyle L} = e^{-\alpha t} (K_1 \cos \beta t + K_2 \sin \beta t) + I_N$
     
$\alpha_1 = \frac{G_N L - \sqrt{(G_N L)^2 - 4LC}}{2LC} $ $\alpha = \frac{G_N}{2C} $ $\alpha = \frac{G_N}{2C} $

$\alpha_2 = \frac{G_N L + \sqrt{(G_N L)^2 - 4LC}}{2LC} $

   $\beta = \frac{\sqrt{4LC - (G_N L)^2}}{2LC} $

$K_1 = \frac{\alpha_2 (I_o - I_N) + V_o/L}{\alpha_2 - \alpha_1}$

 K1 = Io - IN K1 = Io - IN

$K_2 = \frac{-\alpha_1 (I_o - I_N) - V_o/L}{\alpha_2 - \alpha_1}$

 $K_2 = \alpha (I_o - I_N) + V_o/L$ $K_2 = \frac{\alpha (I_o - I_N) + V_o/L}{\beta}$


Bill Rison -- December 2, 1998