# Handout 5: Forced and Natural Response

## Steps for analyzing driven RL circuits

- With all independent sources zeroed out, simplify the circuit to determine \(R_\textrm{eq}\), \(L_\textrm{eq}\), and the time constant \(\tau= L_\textrm{eq}/R_\textrm{eq}\).
- Viewing \(L_\textrm{eq}\) as a short circuit, use DC analysis methods to find \(i_\textrm{L}(0^-)\), the inductor current just prior to the discontinuity.
- Again viewing \(L_\textrm{eq}\) as a short circuit, use DC analysis methods to find the forced response. This is the value approached by \(f(t)\) as \(t\rightarrow\infty\), or \(f(\infty)\).
- Write the total response as the sum of the forced and natural responses: \(f(t)=f(\infty) + Ae^{-t/\tau}\).
- Find \(f(0^+)\) by using the condition that \(i_\textrm{L}(0^+)=i_\textrm{L}(0^-)\). If desired, \(L_\textrm{eq}\) may be replaced by a current source \(i_\textrm{L}(0^+)\) [an open circuit if \(i_\textrm{L}(0^+)\) = 0] for this calculation. With the exception of inductor currents (and capacitor voltages), other currents and voltages in the circuit may change abruptly.
- \(f(0^+) = f(\infty) + A\) and \(f(t) = f(\infty) + [f(0^+) - f(\infty)]e^{-t/\tau}\), or total response = final value + (initial value – final value) \(e^{-t/\tau}\).

## Steps for analyzing driven RC circuits

- With all independent sources zeroed out, simplify the circuit to determine \(R_\textrm{eq}\), \(C_\textrm{eq}\), and the time constant \(\tau = R_\textrm{eq}C_\textrm{eq}\).
- Viewing \(C_\textrm{eq}\) as an open circuit, use DC analysis methods to find \(v_\textrm{C}(0^-)\), the capacitor voltage just prior to the discontinuity.
- Again viewing \(C_\textrm{eq}\) as an open circuit, use DC analysis methods to find the forced response. This is the value approached by \(f(t)\) as \(t\rightarrow\infty\), or \(f(\infty)\).
- Write the total response as the sum of the forced and natural responses: \(f(t)=f(\infty) + Ae^{-t/\tau}\).
- Find \(f(0^+)\) by using the condition that \(v_\textrm{C}(0^+)=v_\textrm{C}(0^-)\). If desired, \(C_\textrm{eq}\) may be replaced by a voltage source \(v_\textrm{C}(0^+)\) [a short circuit if \(v_\textrm{C}(0^+)\) = 0] for this calculation. With the exception of capacitor voltages (and inductor currents), other currents and voltages in the circuit may change abruptly.
- \(f(0^+) = f(\infty) + A\) and \(f(t) = f(\infty) + [f(0^+) - f(\infty)]e^{-t/\tau}\), or total response = final value + (initial value – final value) \(e^{-t/\tau}\).

Last updated:

January 6, 2018