Question 9.9: Objective: Determine the time constant required in an integr...
Objective: Determine the time constant required in an integrator.
Consider the integrator shown in Figure 9.30. Assume that voltage VC across the capacitor is zero at t = 0. A step input voltage of v_{I} = −1 V is applied at t = 0. Determine the time constant required such that the output reaches +10 V at t = 1 ms
From Equation (9.70), we have
v_{o} = \frac{- 1}{R_{1} C_{2}} \int_{0}^{t}{(- 1) d t´} = \left.\frac{1}{R_{1} C_{2}} t´ \right|_{0}^{t} = \frac{t}{R_{1} C_{2}}
At t = 1 ms, we want v_O = 10 V. Therefore,
10 = \frac{10^{−3}}{R_{1} C_{2}}
which means the time constant is R_{1} C_{2} = 0.1 ms.
Comment: As an example, for a time constant of 0.1 ms, we could have R_{1} = 10 k \Omega and C_{2} = 0.01 µF, which are reasonable values of resistance and capacitance.