next up previous
Next: About this document Up: No Title Previous: A natural oscillator

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 tex2html_wrap_inline267 at the + input of the first op-amp and tex2html_wrap_inline271 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. tex2html_wrap_inline275 is called the resonant frequency (in tex2html_wrap_inline277 ). Q is the quality factor discribing how peaked the response is as a function of frequency. Q is related to the bandwidth tex2html_wrap_inline283 by tex2html_wrap_inline285 where f and tex2html_wrap_inline283 are in Hertz ( tex2html_wrap_inline291 ). The bandwidth is the frequency difference between half-power points (or tex2html_wrap_inline293 amplitude points).

So design a second order bandpass filter with resonant frequency 10kHz and bandwidth 200Hz. Arbitrarily choose tex2html_wrap_inline299 .


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 tex2html_wrap_inline301 volts.
The output of the first op-amp is also at voltage tex2html_wrap_inline303 because it is in the voltage- follower configuration.
The voltage at the inverting input of the second op-amp tex2html_wrap_inline305 because the non-inverting input is grounded.
Using Laplace Transform network theory and generalized impedances, tex2html_wrap_inline267 (at op-amp 1 non-inverting input) gives:


Using tex2html_wrap_inline309 and tex2html_wrap_inline311 :


Using Laplace Transform network theory and generalized impedances, tex2html_wrap_inline271 (at op-amp 2 inverting input) gives:



Now eliminate tex2html_wrap_inline315 between (i) and (ii) to find tex2html_wrap_inline321 .
Substitute in (i) for tex2html_wrap_inline315 using (ii).



Dividing numerator and denominator by tex2html_wrap_inline329 :


Bandpass filter design:
We want tex2html_wrap_inline331 .
We want tex2html_wrap_inline333 .


So we must have :


(Arbitrarily choosing tex2html_wrap_inline335 ).


(Again arbitrarily choosing tex2html_wrap_inline337 ).

next up previous
Next: About this document Up: No Title Previous: A natural oscillator

Keith Jones
Tue May 16 09:27:54 EST 2000