the_integrator

The Integrator

The integrator is the basic building block for all of our analog computers.

Analysis

The current through the resistor is:

$i=\frac{v_{IN}}{R}$

which is the same current that flows into the capacitor.

$i=\frac{v_{IN}}{R}=C\frac{dv_C}{dt}$

Notice that $v_C=-v_{OUT}$.

$i=\frac{v_{IN}}{R}=-C\frac{dv_{OUT}}{dt}$

Now we just have to get the expression into the final form.

$\frac{-v_{IN}}{RC}dt=dv_{OUT}$

$\frac{-1}{RC}\int_{t_0}^t{v_{IN}dt}=\int_{v_0}^v{dv_{OUT}}$

$\frac{-1}{RC}\int_{t_0}^t{v_{IN}dt}+v_{0}=v_{OUT}$