Analyzing Complex RC Circuits Using Thevenin’s Theorem

When a circuit features resistors in both series and parallel arrangements around a reactive component, the standard time‑constant formulas (τ=RC for RC, τ=L/R for RL) no longer apply directly. The solution lies in reducing the network to a simple equivalent that preserves the voltage across the reactive element.

Consider the following example:

Step 1 – Identify the Load

In this network the capacitor (C1 = 100 µF) is the load. We temporarily remove it to determine the Thevenin equivalent seen from its terminals.

Step 2 – Compute the Thevenin Voltage (V_th)

With the switch closed and no load present, the voltage at the capacitor terminals (node 2–node 3) equals the voltage across resistor R2. Calculations give:

V_th = 1.8182 V

Because a fully charged capacitor behaves as an open circuit (zero current), this is also the final steady‑state voltage the capacitor will reach.

Step 3 – Compute the Thevenin Resistance (R_th)

All independent sources are turned off (voltage sources shorted, current sources opened). The resulting resistance seen from the capacitor terminals is:

R_th = 454.545 Ω

Step 4 – Calculate the Time Constant

With C1 = 100 µF, the circuit’s time constant is:

τ = R_th · C1 = 454.545 Ω × 100 µF = 0.0454545 s (45.45 ms)

Step 5 – Determine the Transient Response

Assuming the capacitor starts uncharged (V₀=0 V), the voltage at any time t is:

V(t) = V_th · (1 – e^–t/τ)

At t = 60 ms:

V(60 ms) = 1.8182 V × (1 – e^{–0.06/0.0454545}) ≈ 1.3325 V

Verification with SPICE Simulation

The following netlist was used to confirm the analytical result. The transient analysis samples every 5 ms over 370 ms.

* Original circuit
v1 1 0 dc 20
r1 1 2 2k
r2 2 3 500
r3 3 0 3k
c1 2 3 100u ic=0

* Thevenin equivalent
v2 4 0 dc 1.818182
r4 4 5 454.545
c2 5 0 100u ic=0

.tran 0.005 0.37 uic
.print tran v(2,3) v(5,0)
.end

The output shows identical capacitor voltages in both networks at every sampled instant, confirming the equivalence of the Thevenin reduction.

Review

For any RC or RL network that is not a simple series configuration, treat the reactive element as the load.
Reduce the remaining network to its Thevenin equivalent (one voltage source + series resistor).
Apply the standard time‑constant formula to find the transient response.