• 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