Path: utzoo!utgpu!jarvis.csri.toronto.edu!mailrus!tut.cis.ohio-state.edu!ucbvax!hplabs!hpfcso!hpfcdj!myers
From: myers@hpfcdj.HP.COM (Bob Myers)
Newsgroups: sci.electronics
Subject: Re: State Variable Filters
Message-ID: <17660018@hpfcdj.HP.COM>
Date: 7 Sep 89 17:59:50 GMT
References: <2376@radio.oakhill.UUCP>
Organization: Hewlett Packard -- Fort Collins, CO
Lines: 37

>For years I have seen a 3 op-amp filter structure known as a
>'state variable' or 'biquad' filter.  The filter provides simultaneous
>highpass, lowpass, and bandpass outputs.  I understand how the filter
>works but WHY is it called a 'state variable' / 'biquad' filter???
 
>Biquadratic relates to the fact that the filter has 2 integrators
>making it 2nd order...o.k. lots of filter structures can be 2nd order
>and they don't even need 2 op-amps to be 2nd order!  So why the name
>biquad?

No, a circuit can be "2nd order" without being "biquadratic", or even,
for that matter, having two integrators.  It's the order of the polynomials
in the transfer function that counts (basically, the number of poles/zeroes),
which pretty much translates into the number of reactive components (outside
of things like bypass caps and the like).  Two integrators certainly can
make a second-order system, but it's not the only one possible.

Actually, the name "biquadratic" comes from the fact that the design
realizes a biquadratic transfer function; i.e., one of the form

                              2
                F(s)  =   K (s  + (z1+z2)s + z1z2)
                          ------------------------
                              2
                             s  + (p1+p2)s + p1p2 

where z1, z2, p1, and p2 are the zeroes and poles, respectively.  The
"state-variable biquad" is one name for this three-amplifier realization of
a biquadratic transfer function, but I'm not certain how "state variable"
got hung on this particular circuit.  (There is at least one other three-amp
biquad circuit to which I have found reference: the "feedforward three-amp
biquad".)

 
Bob Myers  KC0EW   HP Graphics Tech. Div.|  Opinions expressed here are not
                   Ft. Collins, Colorado |  those of my employer or any other
myers%hpfcla@hplabs.hp.com               |  sentient life-form on this planet.