3.3 Transient Response (RL, RC, RLC)
Key Takeaways
- An RC circuit time constant is tau = R*C; an RL circuit time constant is tau = L/R, both in seconds.
- First-order responses follow x(t) = x(final) + [x(0+) - x(final)] * e^(-t/tau); about 63% of the change occurs in one tau and 99% in five tau.
- Capacitor voltage and inductor current cannot change instantaneously, so they set the initial conditions at t = 0+.
- A second-order RLC response is overdamped, critically damped, or underdamped depending on whether the damping ratio zeta is >1, =1, or <1.
- At steady state (DC), a capacitor behaves as an open circuit and an inductor behaves as a short circuit.
Continuity rules set the initial conditions
Two physical constraints drive every transient problem:
- Capacitor voltage cannot jump (it would require infinite current): v_C(0+) = v_C(0-).
- Inductor current cannot jump (it would require infinite voltage): i_L(0+) = i_L(0-).
To find conditions, analyze the circuit just before switching (t = 0-) at its old steady state, carry the continuous quantity across the switch, then analyze the new steady state (t = infinity). At DC steady state a capacitor is an open circuit and an inductor is a short circuit.
First-order time constants
A single-energy-storage circuit is first order. Its time constant tau (seconds) is:
- RC circuit: tau = R*C, where R is the Thevenin resistance seen by the capacitor.
- RL circuit: tau = L/R, where R is the Thevenin resistance seen by the inductor.
The universal first-order formula describes any voltage or current:
x(t) = x(infinity) + [x(0+) - x(infinity)] * e^(-t/tau).
| Elapsed time | Fraction of final change reached |
|---|---|
| 1 tau | 63.2% |
| 2 tau | 86.5% |
| 3 tau | 95.0% |
| 4 tau | 98.2% |
| 5 tau | 99.3% |
Engineers treat the transient as effectively complete after about five time constants.
Natural versus step (forced) response
The natural response is how a circuit decays with no source, driven only by stored energy (e.g., a charged capacitor discharging through R: v(t) = V_0 * e^(-t/RC)). The step (forced) response is the reaction to a sudden DC source applied at t = 0 (e.g., charging: v(t) = V_s * (1 - e^(-t/RC))). The complete response is the sum of natural (transient) and forced (steady-state) parts, which the universal formula above already combines.
Second-order RLC response
A circuit with both L and C is second order. For a series RLC the characteristic behavior depends on the damping ratio zeta, comparing the neper frequency alpha to the undamped natural frequency omega_0:
- Series RLC: alpha = R/(2L), omega_0 = 1/sqrt(LC).
- Parallel RLC: alpha = 1/(2RC), omega_0 = 1/sqrt(LC).
- zeta = alpha / omega_0.
| Condition | Damping | Behavior |
|---|---|---|
| zeta > 1 (alpha > omega_0) | Overdamped | Two real roots, slow non-oscillatory decay |
| zeta = 1 (alpha = omega_0) | Critically damped | Fastest decay without overshoot |
| zeta < 1 (alpha < omega_0) | Underdamped | Complex roots, decaying oscillation at omega_d = sqrt(omega_0^2 - alpha^2) |
Underdamped responses ring; critically damped responses settle fastest without oscillation, which is often the design target.
A 100 microfarad capacitor discharges through a 10 kilohm resistor. Approximately how long until the voltage falls to about 37% of its initial value?
In a series RLC circuit the neper frequency alpha equals the undamped natural frequency omega_0. The circuit is: