Electrical System Modeling

RLC Circuit

• $g{v}_1$ is a voltage-controlled current source

We have:

• Input and output of $v_i(t)$ and $v_o(t)$
• Intermediate variables: Capacitor voltages $v_1$ and $v_2$, and inductor current $i(t)$

Then, we write down $n$ independent differential equations using Kirchhoff’s current law (KCL) and Kirchhoff’s voltage law (KVL), applied to nodes and loops involving inductors and capacitors.

KCL:

• Node 1 (between $R_1$ and $L$):

$$\frac{v_i(t)-v_1(t)}{R_1}=C_1\frac{d{v}_1(t)}{dt}+i(t)$$

• Node 2 (above $C_2$):

$$i(t)=C_2\frac{d{v}_2(t)}{dt}-g{v}_1-\frac{v_2(t)}{R_2}$$

KVL:

• Loop 1 (with $L,C_1,C_2$):

$$v_1-L\frac{di(t)}{dt}-v_2=0$$

The output can be written just as:

$$v_o(t)=v_2(t)$$

Re-writing everything:

$$\begin{array}{rl}\frac{d{v}_1(t)}{dt}& =\frac{1}{R_1{C}_1}{v}_i(t)-\frac{1}{R_1{C}_1}{v}_1(t)-\frac{1}{C_1}i(t)\\ \frac{d{v}_2(t)}{dt}& =\frac{1}{C_2}i(t)-\frac{g}{C_2}{v}_1-\frac{v_2(t)}{R_2{C}_2}\\ \frac{di(t)}{dt}& =\frac{1}{L}{v}_1-\frac{1}{L}{v}_2\end{array}$$

In matrix form:

$$\left[\begin{array}{c}\dot{v_1}(t)\\ \dot{v_2}(t)\\ \dot{i}(t)\end{array}\right]=\left[\begin{array}{ccc}-\frac{1}{R_1{C}_1}& 0& -\frac{1}{C_1}\\ -\frac{g}{C_2}& -\frac{1}{R_2{C}_2}& \frac{1}{C_2}\\ \frac{1}{L}& -\frac{1}{L}& 0\end{array}\right]\left[\begin{array}{c}{v}_1(t)\\ v_2(t)\\ i(t)\end{array}\right]+\left[\begin{array}{c}\frac{1}{R_1{C}_1}\\ 0\\ 0\end{array}\right]v_i(t)$$

and

$$v_o(t)=\left[\begin{array}{ccc}0& 1& 0\end{array}\right]\left[\begin{array}{c}{v}_1(t)\\ v_2(t)\\ i(t)\end{array}\right]$$