# 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/inductorβs perspective, not from like the sourceβs perspective.

R is the resistance $R_{th}$ as seen from the capacitor/inductorβs perspective. Review Theveninβs Theorem

**General solution for capacitors (RC Circuits)**
$v_{c}(t)=v_{c}(β)+[v_{c}(t_{0})βv_{c}(β)]e_{Οtβtβ}tβ₯t_{0}$

where

- $v_{c}(β)$ = the final value or DC steady-state value when $tββ$
- $v_{c}(t_{0})$ = the initial value
- $Ο=RC$ = Time Constant

$i(t)=Cdtdv(t)β$ using the above formula for $v(t)$, DONβT use the follow below
**General solution for inductors (RL Circuits)**
$i(t)=i(β)+[i(t_{0})βi(β)]e_{Οtβtβ}tβ₯t_{0}$
where

- $v_{c}(β)$ = the final value or DC steady-state value when $tββ$
- $v_{c}(t_{0})$ = the initial value
- $Ο=RLβ$ = Time Constant, where $R$ is the thevenin resistance as seen by the inductor

$v(t)=Ldtdi(t)β$ using the above formula for $v(t)$, DONβT use the formula of capacitor for voltage.

### General Guidelines for solving these problems

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

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

Remember, an inductor acts like a short circuit in DC constant current. β 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.

β This got me quite confused.