State-Variable Circuit Model

For circuits, we can derive a state-variable model by:

1. Find storage elements (inductances $L$ and capacitances $C$)
2. Consider energy stored as $\frac{1}{2}C{v}^2$ and $\frac{1}{2}L{v}^2$

Consider the series RLC circuit shown below.

In this circuit the energy is stored in the capacitor and in the inductor.

• The energy stored in the capacitor is $C{v}_1^2/2$
• The energy stored in the inductor is $L{i}^2/2$.

Thus we choose $v_1$ and $i$ to be our state variables.

KVL gives us:

$$\begin{array}{rl}{v}_s& =Ri+L\frac{di}{dt}+v_1\\ L\frac{di}{dt}& =v_s-R_i-v_1\\ \frac{di}{dt}& =-\frac{R}{L}i-\frac{1}{L}{v}_1+\frac{1}{L}{v}_s\end{array}$$

Since $v_1$ is the voltage across a capacitor, we have

$$\begin{array}{rl}{v}_1& =\frac{1}{C}\int idt\\ \frac{d{v}_1}{dt}& =\frac{1}{C}i\end{array}$$

Thus, the state-variable model in matrix form is:

$$\left[\begin{array}{c}\frac{di}{dt}\\ \frac{d{v}_c}{dt}\end{array}\right]=\left[\begin{array}{cc}-\frac{R}{L}& -\frac{1}{L}\\ \frac{1}{C}& 0\end{array}\right]\left[\begin{array}{c}i\\ v_c\end{array}\right]+\left[\begin{array}{c}\frac{1}{L}\\ 0\end{array}\right]\left[\begin{array}{c}{v}_s\end{array}\right]$$