RLC Damping

Solved the RLC circuit's differential equation for the damped current, then built the circuit and pulled damping coefficients off an oscilloscope to verify ζ = R/2L.

Testing LC oscillators in my radio builds, the carrier amplitude decayed instead of holding. Applying Kirchhoff’s voltage law to the RLC loop gives a second-order homogeneous ODE, and solving it yields a damped sinusoid whose decay envelope is governed by a damping coefficient ζ = R/2L. For the measured L and C, theory predicts ζ = (495 ± 5 H⁻¹) · R.

Bench setup: DC power supply, oscilloscope showing a damped ring-down, and a soldered RLC circuit on a cutting mat — Charge the capacitor, close the loop, catch the ring-down on the scope.

Ten series resistances up to the edge of critical damping, peak voltages read off the oscilloscope, and the natural log of the peaks plotted against time so each run’s ζ falls out as a line’s slope. Fitting ζ against total resistance gives a slope of 505 ± 22 H⁻¹ — a 2.04% deviation from theory.

Damping coefficient against resistance: experimental points with error bars against the theoretical line — Experimental ζ against R (orange fit) on the theoretical line (blue).

The full derivation, raw peak tables, and uncertainty analysis are in the paper:

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