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parallel_rlc_circuit

# Parallel RLC Circuit Time Domain Analysis

## Nodal Equation

$i_C+i_L+i_R=0$ $C\frac{dv_{OUT}}{dt} + \frac{v_{OUT}}{R} + \frac{1}{L}\int_{-\infty}^t{v_{OUT}dt}=0$ Taking a derivative and dividing by C gives $\frac{d^2v_{OUT}}{dt^2} + \frac{1}{RC}\frac{dv_{OUT}}{dt} + \frac{1}{LC}v_{OUT}=0$

## Solution using method of homogeneous and particular solutions

First, notice that the nodal equation is a second order homogeneous equation. Therefore, we will not need to find a particular solution.

As usual with problems like these, guess that the solution is $e^{st}$.

Plugging in $v_{OUT}=e^{st}$ into the nodal equation yields:

$e^{st}(s^2+\frac{1}{RC}s+\frac{1}{LC})=0$

To make the variables more recognizable, let's put the equation into the standard form.

$e^{st}(s^2+2\alpha s+w_o^2)=0$

where

$\alpha=\frac{1}{2RC}$ $w_o=\sqrt{\frac{1}{LC}}$

Using the quadratic formula, one can solve for the two values of s that satisfy the equation.

$s_1=-\alpha+\sqrt{\alpha^2-w_o^2}$ $s_2=-\alpha-\sqrt{\alpha^2-w_o^2}$

To further simplify the above equations for s, one can define $w_d$ as:

$w_d=\sqrt{w_o^2-\alpha^2}$

Using the new definition yields:

$s_1=-\alpha+jw_d$ $s_2=-\alpha-jw_d$

There are two properties of linear homogeneous differential equations that are useful. A constant times the solution is still a solution, and the sum of solutions is still a solution.

Using those two properties we can write:

$v_{OUT}=A_1e^{s_1t}+A_2e^{s_2t}$

The values of $A_1$ and $A_2$ can be found using the initial conditions.

As an example, let's say that the output voltage at $t=0$ is $v_{INIT}$ and the derivative of the output voltage is 0 at $t=0$. So one can write:

$v_{INIT}=A_1+A_2$ $0=A_1s_1+A_2s_2$

Solving the above two equations for $A_1$ and $A_2$ yields:

$A_1=\frac{v_{INIT}}{1-\frac{s_1}{s_2}}$ $A_2=v_{INIT}\left(1-\frac{1}{1-\frac{s_1}{s_2}}\right)$

As an example, I used the $v_{INIT}=1$ ,$R=2.0$ ,$L=0.1$, and $C=0.5$ and the above analysis to plot the output voltage as a function of time in wxmaxima rlc_p_maxima.pdf.

to make sure the analysis was correct, I compared it to a SPICE simulation.