A 2-pole (second order) active bandpass filter

Show that the transfer function of the 2-stage op-amp amplifier is:

Hint: Use standard network analysis and the magic rules governing ideal op-amps. With op-amps it is often convenient to use nodal analysis (i.e. Kirchoff's current law that says that as much current flows out of a node as in.) Ideal op-amps have infinite input impedance so no current flows in to an input. In this circuit use at the + input of the first op-amp and at the - input of the second op-amp.

It can be shown that an amplifier with a transfer function of the form:

is a bandpass filter. is called the resonant frequency (in ). Q is the quality factor discribing how peaked the response is as a function of frequency. Q is related to the bandwidth by where f and are in Hertz ( ). The bandwidth is the frequency difference between half-power points (or amplitude points).

So design a second order bandpass filter with resonant frequency 10kHz and bandwidth 200Hz. Arbitrarily choose .

SOLUTION:

Feedback maintains the inverting and non-inverting inputs of op-amps at the same voltage. In the circuit diagram the two inputs of the first op-amp are both at volts.
The output of the first op-amp is also at voltage because it is in the voltage- follower configuration.
The voltage at the inverting input of the second op-amp because the non-inverting input is grounded.
Using Laplace Transform network theory and generalized impedances, (at op-amp 1 non-inverting input) gives:

Using and :

Using Laplace Transform network theory and generalized impedances, (at op-amp 2 inverting input) gives:

Now eliminate between (i) and (ii) to find .
Substitute in (i) for using (ii).

Dividing numerator and denominator by :

Bandpass filter design:
We want .
We want .

So we must have :

(Arbitrarily choosing ).

(Again arbitrarily choosing ).