RLC natural response

YOU WILL BE ABLE TO

Derive the character of the series RLC's source-free response.
Compute the neper frequency α and resonant frequency ω₀.
Classify a circuit as overdamped, critically damped, or underdamped.
Compute the damped ringing frequency ω_d.

CA-5's circuits had one energy-storage element and could only do one thing: relax exponentially toward their final value. Put BOTH storage elements in one circuit and something qualitatively new appears — energy can slosh back and forth between the capacitor's electric field and the inductor's magnetic field, and the circuit can OSCILLATE. A series RLC loop, disconnected from all sources with some initial stored energy, obeys a second-order differential equation (KVL around the loop, with each element law substituted in):

\frac{d^2 i}{dt^2} + \frac{R}{L}\frac{di}{dt} + \frac{1}{LC}\, i = 0

series RLC, source-free

Trying $i = e^{st}$ (the standard move for linear ODEs) turns calculus into algebra — the characteristic equation — and its two roots are set by two frequencies that between them tell you everything about the circuit's personality:

s^2 + 2\alpha s + \omega_0^2 = 0

\alpha = \frac{R}{2L}

\omega_0 = \frac{1}{\sqrt{LC}}

characteristic equation · damping α · resonant ω₀

α (the damping) measures how fast the resistor burns the stored energy; ω₀ measures how fast L and C want to exchange it. Their contest has three outcomes. α > ω₀: OVERDAMPED — the resistor wins, and the response sags to zero without ever crossing it, like a door with too-stiff a closer. α = ω₀: CRITICALLY DAMPED — the boundary case, the fastest possible return to zero with no overshoot; this is what engineers tune shock absorbers and meter needles toward. α < ω₀: UNDERDAMPED — the exchange wins, and the response RINGS: an oscillation at the damped frequency ω_d, dying inside an exponential envelope $e^{-\alpha t}$ .

\omega_d = \sqrt{\omega_0^2 - \alpha^2}

damped ringing frequency

The three personalities of the same circuit: only R changes between these curves.

WORKED EXAMPLE

Classifying a circuit

Series RLC: L = 1 H, C = 0.25 F. ω₀ = 1/√(LC) = 1/√0.25 = 2 rad/s.
Critical damping needs α = ω₀, i.e. R/2L = 2 → R = 4 Ω.
With R = 10 Ω: α = 5 > 2 → overdamped. With R = 2 Ω: α = 1 < 2 → underdamped, ringing at ω_d = √(4 − 1) = √3 ≈ 1.73 rad/s.
Same parts bin, three behaviours — the resistor is the personality dial.

R = 4 Ω is critical; more is overdamped, less rings

WORKED EXAMPLE

Why ringing matters on real boards

Every trace has a little inductance and every pair of conductors a little capacitance — so every fast digital edge is a step hitting a tiny, badly-damped RLC.
The result is visible on any oscilloscope: edges that overshoot and ring at ω_d before settling.
Taming it means adding damping exactly as this lesson computes — a small series "termination" resistor sized toward critical. ME-5's clocks and PS-3's board layouts both inherit this arithmetic.

Ringing isn't an exam curiosity — it's what unterminated fast edges do.

PRACTICE

A series RLC has R = 6 Ω, L = 1 H, C = 1/25 F. Compute α and ω₀ and classify the response.

For practice problem 1, find the ringing frequency ω_d.

A series RLC uses L = 1 mH and C = 1 µF. Find ω₀ and the resistance that gives critical damping.

Conceptual pair: (a) physically, what is being exchanged during underdamped ringing, and what role does R play? (b) why is critical damping the engineer's usual target for anything that must settle fast?