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 .
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
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 ).