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the_integrator

The Integrator

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

The coolest thing about this circuit is that it does calculus!

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.

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

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

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

the_integrator.txt · Last modified: 2015/05/25 20:06 by admin