RLC transients and resonant response

Consider a single series resistor, inductor, and capacitor with constant positive component values. Choose capacitor charge q so that the series current is its time derivative. The loop equation becomes a driven second-order equation for charge, analogous to a mass on a damped spring. Initial charge and initial current are independent data. For an unforced circuit, the resistor dissipates energy at a rate proportional to current squared, so the total capacitor-plus-inductor energy cannot increase.

Dividing by inductance defines the undamped angular frequency and damping ratio. Underdamping gives oscillatory decay, critical damping gives repeated real characteristic roots, and overdamping gives two real decay rates. The damped frequency is smaller than the undamped frequency and exists as an oscillation frequency only in the underdamped regime. A resistor does not simply lower amplitude while leaving the period exactly unchanged; that description is only an approximation when damping is weak.

When the source is removed and damping is below critical, charge is an exponentially decaying combination of sine and cosine. If current initially vanishes, the sine coefficient must compensate the derivative of the exponential. Writing only a decaying cosine would generally impose a nonzero initial current. This matters when comparing a charged-capacitor release with a driven sinusoidal fit: their phases are determined by different information and should not be borrowed from one another.

The envelope of charge or voltage decays with rate alpha. Energy is quadratic and its slowly varying envelope decays twice as fast in a weakly damped oscillator. Instantaneous electric and magnetic energies exchange within each cycle, so the exact energy need not be a perfect exponential at every instant for appreciable damping. What is exact is its negative derivative from resistive dissipation. These distinctions prevent a factor-of-two error when extracting a damping rate from a ringdown trace.

For a fixed-amplitude sinusoidal voltage source, the series impedance is smallest when inductive and capacitive reactances cancel. The current amplitude therefore peaks at the undamped frequency, independently of resistance in this ideal model. Capacitor voltage and charge have an additional inverse-frequency factor relative to current. Their peak occurs at a lower frequency when damping is sufficiently weak, and disappears as an interior nonzero-frequency peak once the damping ratio reaches the relevant threshold.

A frequency sweep measures steady-state response only if each point allows enough settling or uses an appropriate dynamic measurement model. Comparing that sweep with free decay is a strong diagnostic: ringdown probes characteristic roots, while the swept response probes a specified transfer function. Real parasitic capacitance, frequency dependent resistance, and source impedance can shift all these predictions. A fitted resonance should therefore be reported together with circuit topology, observable, drive convention, and damping assumptions.