Parallel RLC-circuit

Consider a street lamp with an RLC surge protector to protect its components from sudden voltage spikes.

This setup can be approximated by the simplest parallel RLC circuit with an input DC source generating a step response.

When the switch is flipped on, applying Kirchhoff's current law at the top node yields a second-order differential equation.

The complete solution to this equation is a combination of transient and steady-state responses.

The transient response diminishes over time and resembles the source-free series RLC solutions in various damping scenarios.

If the damping factor exceeds the resonant frequency, the response is overdamped.

When the damping factor equals the resonant frequency, the response is critically damped.

If the damping factor is less than the resonant frequency, the response is underdamped.

The steady-state response corresponds to the final inductor current, matching the source current.

The constants involved can be deduced from the initial conditions of the circuit.

Eliminating the input source current results in solely a transient response.