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 10*kHz*
and bandwidth 200*Hz*. 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 ).

Tue May 16 09:27:54 EST 2000