What is Transient response?

The changes in voltage and current seen while a circuit containing capacitors or coils moves from one steady state to the next after a switch operates or a source changes suddenly. Initial values are set using the fact that a capacitor's voltage and a coil's current cannot change suddenly; RC and RL circuits settle exponentially with their time constant, and RLC circuits settle with or without oscillation depending on the damping ratio. It is solved with differential equations or the Laplace transform.

Japanese name: 過渡現象

Transient response appears in 1 practice question across 1 topic, including Circuit Analysis: AC, Linear Circuits and Transients.

Field: Circuit Analysis: AC, Linear Circuits and Transients

Transients in first-order circuits

In RC and RL circuits the response takes the form x(t) = x(∞) + (x(0+) − x(∞))e^(−t/τ), with the time constant τ = RC for an RC circuit and τ = L/R for an RL circuit. Knowing three things, the initial value x(0+), the final value x(∞) and the time constant, is enough to draw the waveform without solving a differential equation. About 63.2% of the change is complete at t = τ and about 99.3% at 5τ.

Finding initial and final values

A capacitor's voltage is continuous because its energy (1/2)CV² cannot change suddenly, so an uncharged capacitor can be treated as a short just after the switch closes. A coil's current is also continuous, so a coil with zero current can be treated as an open circuit just after switching. In the DC steady state after a long time it is the reverse: the capacitor acts as an open circuit and the coil as a short.

Second-order circuits and solution methods

The response of an RLC circuit is determined by the roots of the characteristic equation Ls² + Rs + 1/C = 0: distinct real roots give overdamping, a double root critical damping and complex roots a damped oscillation. With the Laplace transform, initial values are built into the s-domain circuit as sources, and the time waveform is found with algebra and partial fractions. If only the final value is needed, the final-value theorem lim(s→0) sF(s) gives it at once.

Try it: questions about Transient response

In the RC circuit in the figure, a DC voltage E is applied to an uncharged capacitor by closing the switch at t = 0. With the time constant τ = RC, which of the following is closest to the capacitor voltage vC at t = 5τ, as a percentage of the final value E?

AAbout 95%
BAbout 99.3%
C100%

Show answer and explanation

Answer: B. About 99.3%

Method 1 (substitute into the formula): the capacitor voltage is vC(t) = E(1 − e^(−t/τ)), so at t = 5τ it is 1 − e^(−5) ≈ 1 − 0.0067 = 0.993, about 99.3%. Method 2 (rule of thumb per time constant): each τ multiplies the remaining difference from the final value by e^(−1) ≈ 0.37, so the voltage reaches 63% at 1τ, 86% at 2τ, 95% at 3τ, 98% at 4τ and 99.3% at 5τ. In theory it never reaches 100%, so in practice the transient response is considered finished after about 5τ.

FAQ

Why does a high voltage appear when a switch in a circuit with a coil is opened suddenly?

Because the coil current cannot drop to 0 instantly, it tries to keep flowing at the moment the switch opens, and if it has nowhere to go, v = L di/dt produces a very large voltage. A diode is placed in parallel with a relay coil to give this current a path and limit the voltage.

When can a transient be considered finished?

In a first-order circuit the difference from the final value falls below 1% (about 0.7%) after five time constants, so in practice the transient is usually considered over after 5τ. If higher accuracy is needed, wait ln(1/ε) time constants for an allowed error ε.

Transient response in the explanations

In theory it never reaches 100%, so in practice the transient response is considered finished after about 5τ.
In the RC circuit in the figure, a DC voltage E is applied …

Questions that mention Transient response (1)

In the RC circuit in the figure, a DC voltage E is applied to an uncharged capacitor by closing the switch at t = 0. With the time constant τ = RC, which of the following is closest to the capacitor voltage vC at t = 5τ, as a percentage of the final value E?

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