Transient Analysis

First-order circuits (one capacitor or inductor) respond exponentially:

$x(t) = x(\infty) + \left[x(0^+) - x(\infty)\right] e^{-t/\tau}$

with $\tau = RC$ or $\tau = L/R$. Just find the initial value, the final value and the time constant.

Second-order RLC circuits are described by the damping ratio $\zeta$ and natural frequency $\omega_0 = 1/\sqrt{LC}$. For a series RLC: $\zeta = \dfrac{R}{2}\sqrt{\dfrac{C}{L}}$.

• $\zeta > 1$ — overdamped (slow, no overshoot)
• $\zeta = 1$ — critically damped (fastest without overshoot)
• $\zeta < 1$ — underdamped (rings and overshoots)

Remember: capacitor voltage and inductor current cannot change instantly.