State Space Model of Circuits

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

Tips for modelling circuit elements using state variables

The goal here is to create an ODE that has derivatives of the output variable(s) with no derivatives of the input variable(s)

For C and L, we go with the expression that works with the output (for example, if V is out output we would use the v expression for C and L)

The representation of C and L come from Impedance

We use KVL and KCL to form these expressions and models

We can also take our time derivatives to the LaPlace domain to solve.

The Impedance Method

Basically using ohms law and equivalent impedance to create transfer functions.

1. Find $Z_{eq}$
2. Define $\frac{V_s\left(s\right)}{I\left(s\right)}=Z_{eq}$
3. Define output with respect to a known term
4. Perform substitution and solve for the required transfer function

Force to Voltage Analogy

For a mass, spring damper system:
$f=m\frac{dv}{dt}+cv+k{\int }_{}^{}vdt$
For an LRC circuit:
$V_s=L\frac{di}{dt}+Ri+\frac{1}{C}{\int }_{}^{}idt$
You can view the similarities by comparing constants and independent variables

We can also create transfer functions of ${\frac{V_O}{V_1}}_{}$ in the same manner as we do in mechanical systems and signals.