Low Pass Filter

Circuit:

Analysis:

The left part of the circuit is simply the voltage adder, this time I added a sine wave of 1 Hz with a sine wave of 100 Hz. The goal is to filter away the 100 Hz signal.

The low pass filter I used is a simple RC filter. Denote the voltage of the opamp output to be $V_s$

$1\frac{dV_{out}}{dt} = I = \frac{V_s - V_{out}}{1}$

Set $V_{s} = e^{i \omega t}$ and $V_{out} = H(\omega) e^{i\omega t}$, we get:

$H(\omega)i\omega e^{i\omega t} = e^{i \omega t} - H(\omega) e^{i\omega t}$
$H(\omega)i\omega = 1 - H(\omega)$

$H(\omega) = \frac{1}{1 + i\omega}$

Note that $\omega$ is in the denominator, meaning for large $\omega$, the frequency response has a very small gain, so it is effectively removed. Also note that we have some phase shift because the frequency response is complex.

Simulation:

LTSpice:

The model can be downloaded here.