parallel_rlc_circuit

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

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.

parallel_rlc_circuit.txt · Last modified: 2015/11/25 06:20 by admin