In any time‑varying electrical network, the voltage does not jump instantaneously to its final value. Instead, it rises gradually—often following an exponential curve—until the circuit reaches a steady‑state condition where the voltage becomes constant.

For a simple resistor‑capacitor (RC) network, the time it takes to reach steady state is governed by the product of the resistance (R) and the capacitance (C), known as the time constant τ = RC. By selecting appropriate values for R and C, designers can tailor the transient response to meet specific performance criteria.

Step 1 – Define the Source Voltage

Identify the DC supply that powers the RC network. In our illustrative example, we choose a source voltage Vs = 100 V.

Step 2 – Choose R and C

Select realistic component values. Here we use R = 10 Ω and C = 6 µF (6 × 10⁻⁶ F). The resulting time constant is:

τ = R × C = 10 Ω × 6 µF = 0.00006 s (60 µs).

Step 3 – Compute the Steady‑State Voltage

The capacitor voltage at any instant t after the supply is applied is given by:

V(t) = Vs [1 – e^(–t/τ)]

Using this expression, we can evaluate the voltage at several key moments:

• t = 0 s (supply just switched on) τ = 0.00006 s → t/τ = 0 → e^(–0) = 1 V(0) = 100 V [1 – 1] = 0 V
• t = 5 µs t/τ = 5 µs / 60 µs ≈ 0.083 e^(–0.083) ≈ 0.920 V(5 µs) = 100 V [1 – 0.920] ≈ 8 V
• t = 1 s t/τ = 1 s / 60 µs ≈ 16 667 → e^(–16 667) ≈ 0 V(1 s) = 100 V [1 – 0] = 100 V

As time progresses beyond a few time constants (typically 5τ ≈ 0.3 ms for this example), the exponential term vanishes and the capacitor voltage settles at the supply value—here, 100 V—indicating that the circuit has reached steady state.

By adjusting R or C, you can accelerate or delay the approach to steady state. For instance, doubling the resistance to 20 Ω would double the time constant to 120 µs, making the voltage rise more slowly.

These calculations provide a reliable basis for predicting transient behavior in RC circuits, which is essential for designing stable, high‑performance electronic systems.