What is the difference between underdamped, critically damped, and overdamped RLC circuits?

Underdamped RLC circuit:

An underdamped RLC circuit is one where the current or voltage in the circuit oscillates before reaching its final steady-state value. In this case, the energy stored in the inductor and capacitor causes the current or voltage to overshoot or undershoot the final value before settling down. The damping ratio (ζ) in the circuit is less than 1.

Critically damped RLC circuit:

A critically damped RLC circuit is one where the current or voltage in the circuit returns to its steady-state value without any oscillations or overshoot. The damping ratio (ζ) in the circuit is exactly equal to 1. This means that the circuit reaches its final value as quickly as possible without any oscillations.

Overdamped RLC circuit:

An overdamped RLC circuit is one where the current or voltage in the circuit approaches its steady-state value without any oscillations but takes a longer time to do so compared to the critically damped case. The damping ratio (ζ) in the circuit is greater than 1.

The damping ratio (ζ) determines the type of response the RLC circuit will exhibit. It is calculated as the ratio of the actual damping coefficient to the critical damping coefficient:

=

ζ=

C

c

C

where C is the damping coefficient and C_c is the critical damping coefficient. The critical damping coefficient is the value of the damping coefficient at which the circuit transitions from being underdamped to critically damped.

In summary:

Underdamped RLC circuit (ζ < 1): Oscillatory behavior, with some overshoot or undershoot before settling.

Critically damped RLC circuit (ζ = 1): No oscillations, quickest return to steady-state value.

Overdamped RLC circuit (ζ > 1): No oscillations, slower return to steady-state value.

The type of damping in the RLC circuit affects its transient response, which is how the circuit behaves during the transition from one steady-state condition to another after a sudden change in input or initial conditions.