What is the damping ratio of an RLC circuit?
ζ = (R/2)·√(C/L). It decides everything about the response. Below 1 the circuit is underdamped and the voltage overshoots and rings; at exactly 1 it is critically damped and rises as fast as it possibly can without overshooting; above 1 it is overdamped and creeps up without ringing at all. The resistance that makes ζ exactly 1 is R = 2√(L/C).
Why does an RLC circuit ring?
Because the energy has two places to live and keeps moving between them. The capacitor stores it in an electric field, the inductor in a magnetic one, and neither will hold it: the capacitor discharges into the inductor, the inductor's collapsing field recharges the capacitor the other way, and round it goes. The resistor takes a bite each cycle, which is why the swings shrink. With no resistance at all it would ring forever.
What is the natural frequency of an RLC circuit?
ω₀ = 1/√(LC) radians per second, or f₀ = 1/(2π√(LC)) in hertz. That is the frequency it would ring at with no resistance. With damping present it actually rings slightly slower, at ω_d = ω₀√(1 − ζ²) — a difference that is negligible for light damping and grows quickly as the circuit approaches critical.
What is critical damping and why does it matter?
It is the exact boundary between ringing and not ringing, and it gives the fastest possible approach to the final value with no overshoot at all. That combination is what you want almost anywhere a signal must settle quickly and cleanly — a logic input, a control loop, a measuring instrument. Adding resistance beyond critical does not make things safer; it only makes them slower.
How much does an underdamped circuit overshoot?
The first peak reaches the final value multiplied by 1 + e^(−πζ/√(1−ζ²)). At ζ = 0.5 that is about 16% over; at ζ = 0.1 it is around 73%; with no resistance at all the capacitor charges to exactly twice the source voltage. That last case surprises people, and it is the reason a lightly damped supply rail can destroy a part rated only slightly above the nominal voltage.
What is the Q factor?
Q = 1/(2ζ), a measure of how lightly damped the circuit is. High Q means long ringing and a sharp response to one particular frequency — exactly what a radio tuner or an oscillator needs. Low Q means the energy is dissipated quickly. Q also equals the number of radians of oscillation over which the stored energy falls to 1/e of its starting value.
How long does an RLC transient take to settle?
For an underdamped circuit, about 4/(ζω₀) to come within 2% of the final value — the ringing envelope decays as e^(−αt) with α = R/2L. For an overdamped circuit the slower of the two exponential roots governs, because the faster one has long since vanished and the tail is entirely the slow root.
Why is an RLC circuit the same as a mass on a spring?
Because it is literally the same differential equation. LC·v″ + RC·v′ + v = V has exactly the form of mx″ + cx′ + kx = F. The inductor plays the mass (it resists changes in current the way mass resists changes in velocity), the capacitor plays the spring, and the resistor plays the damper. Every result carries across, which is why the words damping ratio, natural frequency and overshoot are shared between the two fields.
Does the current flow backwards during ringing?
Yes — that is what ringing is. The loop current is C·dv/dt, so it reverses every time the capacitor voltage stops rising and starts falling. In an underdamped circuit that happens several times before it settles. In the animation here the charge carriers reverse at exactly those instants, because their speed is the calculated current rather than a fixed drift.
What does this model leave out?
Ideal components, a perfect step from the source, and no initial charge on the capacitor. Real inductors carry series resistance and self-capacitance, real capacitors carry a little inductance, and both shift the ringing frequency and add damping you did not design in. A real switching edge also has a finite rise time, which excites the ring less than an ideal step does.