Second-Order RLC Response and Damping
One damping ratio sets the shape
The gold curve is the response to a step input, and the faint green curve is the critically damped response, drawn for comparison. Change the damping ratio ζ and the response either rings and overshoots or crawls up slowly; between the two lies a single point where it settles fastest with no overshoot. Find that critical point, ζ = 1, where the gold curve lies on top of the green one.
Two elements trade energy
An inductor stores energy in its current, a capacitor in its voltage. With both in one circuit, one fills as the other empties, and energy shuttles back and forth. Without resistance the circuit would ring forever, but resistance drains a little energy from each swing and calms it down. Where the 1st-order RC circuit approached its target monotonically, the 2nd-order RLC circuit can shoot past the target and swing back.
The damping ratio splits three ways
With weak resistance (ζ<1), energy shuttles back and forth many times, the response oscillates, and it overshoots the target. With strong resistance (ζ>1) there is no ringing at all, but the response crawls up 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 circuit the damping ratio is ζ = (R/2)√(C/L). Raising the resistance raises ζ and suppresses the oscillation, while raising the inductance lowers ζ and makes the circuit ring more easily. The pace of the swinging itself is set by the natural frequency ω_n = 1/√(LC), and the actual ringing frequency ω_d = ω_n√(1−ζ²) slows as damping grows. The plot's horizontal axis is time with ω_n set to 1, so any combination of R, L, and C with the same ζ gives the same curve.
Back to the first screen
With a small ζ the response overshot the target and rang several times. As ζ grew, the ringing died down, and at ζ=1 the response settled fastest with no overshoot, landing on the green reference curve. Raised further, into overdamping, it lost the ringing but became sluggish. Whatever the values of resistance, inductance, and capacitance, the single damping ratio ζ they combine into held the entire shape of the response.