Transient and Steady-State Analysis

Syllabus: Time and frequency domain analysis of linear circuits: RL, RC and RLC circuits, solution of network equations using Laplace transform.

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▶Interactive labSwitching transients: RC, RL and series RLC

First-order RL and RC transients

Every first-order response has the same shape: start value, final value, one time constant. Find those three and write the answer — no differential equation needed.

$x(t)=x(\infty)+\left[x(0^{+})-x(\infty)\right]e^{-t/\tau}$
the universal form
$\tau=RC\quad\text{or}\quad \tau=L/R$
$R$ is the Thevenin resistance seen by $C$ or $L$
$x(t)=x(\infty)+\left[x(t_0)-x(\infty)\right]e^{-(t-t_0)/\tau}$

Where marks are lost

Capacitor voltage and inductor current cannot jump. Everything else can.

See it

Initial and final conditions

At $t=0^{+}$ a capacitor holds its previous voltage and an inductor its previous current. After a long time, DC sources leave the capacitor open and the inductor shorted. GATE often asks for the rate of change at $0^{+}$ , not just the value: find the capacitor current or inductor voltage in the $0^{+}$

Second-order RLC

The damping ratio decides the shape. Critically damped is the boundary between overshoot and none, and is the condition GATE most often asks you to impose.

$\omega_0=\frac{1}{\sqrt{LC}}$

Network equations by Laplace transform

Transform the elements, not the differential equation: $R$ stays, $L$ becomes $sL$ , $C$ becomes $1/sC$ , and the initial conditions enter as sources

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