State model of parallel RLC with current source

Circuit. Current source i(t), resistor R, inductor L and capacitor C all in parallel across one pair of nodes. Choose the capacitor voltage v (= v_C) and inductor current i_L as the state variables (one for each energy-storing element), with i(t) as the input.

Equations

• Inductor: L di_L/dt = v
• KCL at the top node: i(t) = v/R + i_L + C dv/dt

So

dv/dt   = −(1/RC) v − (1/C) i_L + (1/C) i(t)
di_L/dt = (1/L) v

State model

| v'  |   | −1/(RC)  −1/C | | v   |   | 1/C |
|     | = |               | |     | + |     | i(t)
| iL' |   |   1/L      0  | | iL  |   |  0  |

x = [v, i_L]ᵀ, ẋ = A x + B u with

A = [[−1/RC, −1/C], [1/L, 0]], B = [[1/C], [0]], u = i(t)

Output equations

• Voltage across the network: y₁ = v = [1 0] x
• Resistor current: i_R = v/R = [1/R 0] x
• Capacitor current: i_C = i(t) − v/R − i_L = [−1/R −1] x + [1] u

Check: the characteristic equation det(sI − A) = s² + s/(RC) + 1/(LC) = 0 is the standard parallel RLC equation, giving damping α = 1/(2RC) and ω₀ = 1/√(LC).