• BME 210
  • Handout 5: Forced and Natural Response

    Steps for analyzing driven RL circuits

    1. With all independent sources zeroed out, simplify the circuit to determine \(R_\textrm{eq}\), \(L_\textrm{eq}\), and the time constant \(\tau= L_\textrm{eq}/R_\textrm{eq}\).
    2. 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.
    3. 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 \(f(\infty)\).
    4. Write the total response as the sum of the forced and natural responses: \(f(t)=f(\infty) + Ae^{-t/\tau}\).
    5. 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.
    6. \(f(0^+) = f(\infty) + A\) 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

    1. With all independent sources zeroed out, simplify the circuit to determine \(R_\textrm{eq}\), \(C_\textrm{eq}\), and the time constant \(\tau = R_\textrm{eq}C_\textrm{eq}\).
    2. 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.
    3. 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 \(f(\infty)\).
    4. Write the total response as the sum of the forced and natural responses: \(f(t)=f(\infty) + Ae^{-t/\tau}\).
    5. 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.
    6. \(f(0^+) = f(\infty) + A\) 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