Answer to Question #199850 in Electrical Engineering for LUCKY

Question

Design an op-amp based integrator with a time constant of 

1 second, if the capacitor is 1µF. If a constant voltage 

of -1V is applied at t=0 seconds, how long will it take to an output of +10V (assume initial capacitor voltage 

= -10V). Sketch the input and output waveforms.

Answer 1

$A_v=\frac{V_0}{V_{in}}(jw)=\frac{-1}{|jwCR|}=\frac{-1}{wCR}=\frac{-V}{V}$

Confirming the capacitance ability

$\frac{-1}{10^4*C*100*10^3}=-1 \implies C=1nF$

For V_in(t)= 2*sin 10000t

$V_0(t)=\frac{-1}{RC}\int_0^t V_{in}(\tau) d\tau=-10000\int_0^t sin(1000\tau) d\tau=2[cos 1000t-1]$