Question 7.16: Find the step response vo(t) for t > 0 in the op amp cir......
Find the step response v_{o}(t) for t > 0 in the op amp circuit of Fig. 7.59. Let v_{i}= 2u(t) V, R_{1} = 20 kΩ, R_{f}= 50 kΩ, R_{2} = R_{3} = 10 kΩ, C = 2µF.
Notice that the capacitor in Example 7.14 is located in the input loop, while the capacitor in Example 7.15 is located in the feedback loop. In this example, the capacitor is located in the output of the op amp. Again, we can solve this problem directly using nodal analysis. However, using the Thevenin equivalent circuit may simplify the problem.
We temporarily remove the capacitor and find the Thevenin equivalent at its terminals. To V_{Th} obtain consider the circuit in Fig. 7.60(a). Since the circuit is an inverting amplifier,
V_{ab} = – \frac{R_{f}}{R_{1}} v_{i}
By voltage division,
V_{Th} = \frac{R_{3}}{R_{2} + R_{3}} V_{ab} = – \frac{R_{3}}{R_{2} + R_{3}} \frac{R_{f}}{R_{1}} V_{i}
To obtain R_{Th} consider the circuit in Fig. 7.60(b), where R_{o} is the output resistance of the op amp. Since we are assuming an ideal op amp, R_{o} = 0 and
R_{Th} R_{2} \parallel R_{3} = \frac{R_{2}R_{3}}{R_{2} + R_{3}}
Substituting the given numerical values
V_{Th} = – \frac{R_{3}}{R_{2} + R_{3} } \frac{R_{f}}{R_{1}} v_{i} = – \frac{10}{20} \frac{50}{20} 2u(t) = -2.5 u(t)
R_{Th} = \frac{R_{2}R_{3}}{R_{2} + R_{3}} = 5 kΩ
The Thevenin equivalent circuit is shown in Fig. 7.61, which is similar
to Fig. 7.40. Hence, the solution is similar to that in Eq. (7.48); that is,
v(t) = V_{s} ( 1- e^{-t/\tau} ) u(t) (7.48)
v_{o}(t) = -2.5 ( 1- e^{-t/\tau} ) u(t)
where \tau = R_{Th} C = 5 × 10^{3} × 2 × 10^{-6} = 0.01 . Thus, the step response for t > 0 is
v_{o}(t) = 2.5(e^{-100t – 1}) u(t) V
