# Calculate Gain of Modified Band-Pass Filter

I would like to calculate the gain of this simple band-pass filter.

simulate this circuit – Schematic created using CircuitLab

What I'm interested in knowing is, in order to find this gain, would I have to set up a set of differential equations using Kirchoff's rules, or could I avoid this through some other trick? I'm particularly interested in the instance when the input angular frequency is equal to the characteristic frequency, $$\omega=\frac{1}{RC}$$

NOTE: I'm not asking for the answer, but rather for a hint a possible route to getting there (a trick). Also, if this question is not fit for EESE, then I apologize.

• There is no "trick" necessary. The circuit is nothing else than a primary voltage divider, where one part (the most right part) is a "secondary" voltage divider. Use the capacitive impedance Z=1/(jωC) – LvW Feb 26 '15 at 11:20

You need the complex frequency response, which you can get by using the voltage divider rule with complex impedances. You probably know that for a capacitor you have $Z=1/(j\omega C)$. This way you can express the ratio $V_{out}/V_{in}$ as a complex function of the independent variable $\omega$ (this is the complex frequency response). The gain for a given frequency is then the magnitude of the frequency response evaluated at that frequency.
If you do the math you should get for $\omega=1/RC$ a gain
$$G=\frac{A}{1+2A}$$