RC Charging

The RC Circuit: Charging

An RC circuit (resistor + capacitor in series) is the simplest circuit with memory — its behavior depends on history, not just the current input.

    Vs ---[R]---+---[C]--- GND
                |
               V(t)

When a voltage Vs is applied, the capacitor charges exponentially:

V(t) = Vs · (1 − e^(−t / τ))

Where τ = R · C is the time constant (seconds).

The Time Constant

At t = τ: V = Vs · (1 − 1/e) ≈ 0.632 · Vs (63.2% of final voltage)

Physical Intuition

Initially, the capacitor is empty — all voltage drops across R and current flows freely. As the cap charges, its voltage rises, reducing the driving voltage (Vs − V_cap), which slows the current. The charging slows exponentially.

General Form (arbitrary initial voltage V₀)

V(t) = Vs + (V₀ − Vs) · e^(−t / τ)

This covers both charging (V₀ < Vs) and partial charging (V₀ > 0).

Applications

• Timing circuits (555 timer)
• Debouncing switches
• Power supply filtering
• Analog delays and integrators

Your Task

Implement double rc_charge(double vs, double v0, double r, double c, double t) that returns the capacitor voltage at time t.

Use #include <math.h> for exp.