Second-Order RLC Transients
Two storage elements let energy slosh between them. Overdamped, critical, underdamped — and a car suspension is the same problem in a different costume.
Skip to the animationTwo storage elements can exchange energy, so the response can overshoot and oscillate — characterised by ω₀ = 1/√(LC), which sets the speed, and a damping ratio ζ containing the resistance, which sets the shape.
Why two elements change everything
With one storage element energy can only accumulate or drain, so the response is monotonic. With two, the inductor's field can charge the capacitor and the capacitor can rebuild the field — an exchange mechanism, which is what makes overshoot possible.
ω₀ sets the speed and ζ sets the shape, independently. Resistance appears only in ζ — it decides the character and never the natural frequency.
The three cases
| ζ > 1 overdamped | ζ = 1 critical | ζ < 1 underdamped | |
|---|---|---|---|
| Roots | Two real | Repeated | Complex conjugate |
| Overshoot | None | None | Yes, then ringing |
| Speed | Slowest | Fastest with no overshoot | Reaches target first |
| Used where | Overshoot unacceptable, speed irrelevant | A design target, never exactly achieved | Most designs — ζ ≈ 0.7 |
Critical damping is a knife edge: component tolerances guarantee landing slightly to one side. Around ζ = 0.7 gives roughly 5% overshoot with a much faster rise, which is why most designs accept some overshoot deliberately.
Ringing as a failure mode
Overshoot is an actual overvoltage, not a plotting artefact. A ringing digital edge can cross the logic threshold repeatedly, so one transition is read as several. In power switching, stray inductance rings with device capacitance and can exceed the voltage rating outright.
The fix is deliberate damping — a snubber across a switch, or source termination on a digital line — raising ζ at the cost of edge speed. Every unwanted LC in a layout has an ω₀ and a ζ whether anyone designed them or not.
Zero damping
With ζ = 0 no energy leaves and the oscillation continues indefinitely at ω₀. Real circuits always have some resistance, but it can be extraordinarily small — a quartz crystal's Q of tens of thousands corresponds to ζ near 10⁻⁵.
Q = 1/2ζ, so high Q and light damping are the same statement seen from resonance and from transients respectively.
The same equation elsewhere
The RLC equation is identical to a mass-spring-damper's, with L for mass, C for compliance and R for damping — and identical again to a control loop's step response, with the same ω₀, the same ζ and the same overshoot formula.
A car suspension, a moving-coil meter and an RLC circuit are one problem in three costumes, so intuition built in any of them transfers intact.
The numbers you will be asked for
- Natural frequency
ω₀ = 1/√(LC)
- Damping ratio, series RLC
ζ = (R/2)·√(C/L)
- Damped frequency
ω_d = ω₀√(1 − ζ²)
- Overshoot
M_p = e^(−πζ/√(1−ζ²))
- Settling time
t_s ≈ 4 / (ζω₀)
- Q and damping
Q = 1 / 2ζ
Watch it work
Check yourself
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