Delivered scenarios: charging V(t) = Vₛ(1 − e^(−t/RC)), discharging V(t) = V₀ e^(−t/RC), current I = (Vₛ − V)/R (or V/R), and the RC timer/smoothing circuits built on them.

C·dV/dt = (g·Vₛ − V)/R − V/R_L [A]; g = 1 charging, 0 discharging through R.

τ = C·R·R_L/(R + R_L) [s]; V∞ = g·Vₛ·R_L/(R + R_L) [V]; V(t) = V∞ + (V₀ − V∞)·e^(−t/τ).

Timer delay to a level V_th: t = τ·ln[(V∞ − V₀)/(V∞ − V_th)].

Relaxation oscillator (comparator thresholds V_lo, V_hi): T = τ·ln[(V∞ − V_lo)/(V∞ − V_hi)] + τ·ln(V_hi/V_lo); it oscillates only if V∞ > V_hi.

Honest simplifications. Ideal capacitor (no ESR or leakage), ideal resistors, ideal switch and ideal comparator with exactly known thresholds. Square-wave mode is a switch that alternates the source between Vₛ and 0 V through R. The oscillator and the switching modes are an extension of the delivered charge/discharge scenarios; they follow the same equation. Preset values and captions are those of the original simulations.

RC Timer, Flasher and Relaxation Oscillator: Capacitor Charge, Discharge, Load and Switching Case-Study Lab

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