Transient Analysis

Time-varying currents and voltages resulting from the abrupt change of the circuit (or due to switching) are called transients.

Important in many applications

• Digital Circuits
• Timing Circuit
• Power System

Usually, transient analysis requires Ordinary Differential Equations to solve.

General Solutions

These general solutions can apply for both charging or discharging circuits.

Finding R

When you calculate $R$, which is really $R_{th}$, make sure to look at it from the capacitor/inductor’s perspective, not from like the source’s perspective.

R is the resistance $R_{th}$ as seen from the capacitor/inductor’s perspective. Review Thevenin’s Theorem

General solution for capacitors (RC Circuits) $v_c(t)=v_c(\infty )+[v_c(t_0)-v_c(\infty )]e^{\frac{t-t_0}{\tau }}t\ge t_0$

where

• $v_c(\infty )$ = the final value or DC steady-state value when $t\to \infty$
• $v_c(t_0)$ = the initial value
• $\tau =RC$ = Time Constant

$i(t)=C\frac{dv(t)}{dt}$ using the above formula for $v(t)$, DON’T use the follow below General solution for inductors (RL Circuits) $i(t)=i(\infty )+[i(t_0)-i(\infty )]e^{\frac{t-t_0}{\tau }}t\ge t_0$ where

• $v_c(\infty )$ = the final value or DC steady-state value when $t\to \infty$
• $v_c(t_0)$ = the initial value
• $\tau =\frac{L}{R}$ = Time Constant, where $R$ is the thevenin resistance as seen by the inductor

$v(t)=L\frac{di(t)}{dt}$ using the above formula for $v(t)$, DON’T use the formula of capacitor for voltage.

General Guidelines for solving these problems

Capacitors Capacitors act like open-circuit at $i(\infty )$, so current becomes 0 at infinity. You solve for voltage. → ALWAYS Treat conductor like open circuit to solve for voltage

Inductors In general, for inductors, to solve $i(\infty )$, the inductor will be short circuited as the circuit stabilizes.

Remember, an inductor acts like a short circuit in DC constant current. → AWAYS Treat inductor like short circuit and solve for current

Note on R_{th}

Just solve it the way you solve thevenin, easy.

Only exception is for finding the equivalent resistance, you treat sources and the inductor/capacitor as open-circuit as that is the definition of an equivalent/thevenin resistance.

→ This got me quite confused.