Mad Science Lab Archives/BiQuadMath follows so you are warnedsome knowledge of La Place transform and s plane is neededClassic Tow Thomas Bi Quad Filter

This filter is a classic filter design. The following illustrates how a physics equation specifying a 2 pole filter can be converted into an actual circuit. Why is this important.....1. it can be solved by mortals , 2 the results can be tuned to produce a wide range of filters with individual control of gain and resonance as well as frequency, and 3. by analyzing this solvable circuit one can gain insight into other more complex filter types. and lastly by combining different second pole filters other order filters can be easily created The following draws heavily from Design of Analog Filters by Rolf Schaumann and Mac E. Valkenburg a great book on filters also..Active Filter Cookbook by Dan Lancaster is a great reference on filters 
Assuming we trust the physics people, which I generally do, a second order filter can be described mathematically by the following equation where we assume the input V1 is changed by some process that is frequency dependent, has some Q dependence, a gain H, and is magically transformed into V2, which if everything is working, is a filtered version of the original in the frequency domain. Basically, the output has some of its frequency content changed. 
Above: to make a simpler equation the frequency is scaled. This is a standard technique in filter design. 
Above: Rewrite the equation to get it into a form where we can identify circuit elements in s plane remembering that in the La Place transform 1/s is like a capacitor further develop equation below 
Above: identifying V sub B and V sub L and how to get them 
Above:Basic definitions of voltages needed. 
Above: Summing node realizes the weighted sum of three voltages V1 VL and VB. Here the equation is shown in blocks that will then be replaced by op amps to perform the block functions. 
Above: by rewriting the equation to isolate certain terms it can be shown that the same equation can produce other filter types. See below for proof that these are different filter types. 
Above: since we want to use op amps we use only inverting amplifier configurations. Now have the three op amps defined to create the Low Pass filter 
Above: to prove these equations logically define filter types we look at the extremes. s = 0 corresponds to a very low frequency, s = infinity a very high frequency 
Above: circuit designed in Orcad Layout. Now to prove this thing works with simulations in PSpice 
Above: graph showing a frequency sweep at input and the corresponding output. A second order filter is defined as 40 db per decade (here 3 kHz to 30 KHz) 
Above: Band Pass output, graph showing a frequency sweep at input and the corresponding output. A first order filter is defined as 20 db per decade (here 30 kHz to 300 KHz) 
Above: Two Biquad low pass filters cascaded to create a four pole filter and the resulting graph showing the results below 
http://en.wikipedia.org/wiki/Biquad_filter 
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