Network Analysis
Transient analysis explains how voltages and currents change with time immediately after switching, source changes, or redistribution of stored energy in capacitors and inductors.
Transient analysis is the study of how voltages and currents in a circuit change with time immediately after a sudden disturbance such as switching ON or OFF, a sudden change in source, or initial energy stored in circuit elements.
It focuses on the interval between two steady states.
When a circuit changes condition, it does not instantly reach the final value. It passes through a time-varying state first. This temporary time-varying period is called the transient state.
Key idea: a circuit cannot change instantaneously if it contains energy storage elements.
A capacitor stores energy in an electric field.
W = (1 / 2) C V^2
An inductor stores energy in a magnetic field.
W = (1 / 2) L I^2
Total = Natural + Forced
V = vR + vC
V = iR + vC
i = C dvC / dt
V = RC dvC / dt + vC
dvC / dt + (1 / RC)vC = V / RC
vC(t) = V(1 - e^(-t / RC))
i(t) = (V / R)e^(-t / RC)
tau = RC
The time constant is the time required for capacitor voltage to reach about 63 percent of its final value during charging.
vC(t) = V0 e^(-t / RC)
During discharging, capacitor voltage decays exponentially to zero.
V = iR + L di / dt
di / dt + (R / L)i = V / L
i(t) = (V / R)(1 - e^(-tR / L))
vL(t) = V e^(-tR / L)
tau = L / R
| Element | At t = 0 | At t = infinity |
|---|---|---|
| Capacitor | Short circuit | Open circuit |
| Inductor | Open circuit | Short circuit |
x(t) = xf + (x0 - xf)e^(-t / tau)
Transient behavior is the adjustment of stored energy after a circuit condition changes.
tau = RC = 1 ms
v = 10(1 - e^(-1)) = 6.3 V approximately
This matches the 63 percent rule.
Transient analysis explains how circuits evolve over time due to energy storage elements, governed by differential equations and exponential responses.
