Second-Order RLC Response and Damping
One damping ratio sets the shape
Slide the damping ratio ζ and watch the step response change. It overshoots and rings, until one point settles fastest with no overshoot at all. Find that critical point, ζ = 1.
Two elements trade energy
An inductor stores energy in its current, a capacitor in its voltage. In one circuit, as one empties the other fills, and energy shuttles back and forth. With no resistance it would slosh forever, but resistance drains a little energy from each swing and brings it to rest. Where a first-order RC eased in monotonically, a second-order RLC can overshoot and swing back.
The damping ratio splits three ways
With weak resistance (ζ<1) the energy shuttles several times: the response oscillates and overshoots the target. With strong resistance (ζ>1) no sloshing appears at all and it crawls in slowly. At exactly one point between them, ζ=1, the response settles fastest with no overshoot. This is critical damping.
ζ comes from R, L and C
In a series RLC the damping ratio is ζ = (R/2)√(C/L). Raising the resistance increases ζ and suppresses the oscillation; raising the inductance lowers ζ and makes it slosh more readily. The bare speed of the sloshing is set by the natural frequency ω_n = 1/√(LC), and the actual ringing frequency is ω_d = ω_n√(1−ζ²), which slows as damping grows.
Back to the first screen
When ζ was small the response shot past the target and rang several times; as you raised ζ the ringing shrank, and at ζ=1 it settled fastest with no overshoot. Raise it further (overdamped) and there is no oscillation but it slows down. Whatever the values of R, L and C, the single damping ratio ζ their combination makes held the entire shape of the response.