No book goes into how we jump from those simple low pass high pass band pass band stop filters like the single pole filters onto these butterworth, chebyshev e.t.c types,
No book? Of course, there a many books which in detail show how we arrive at the various approximations to come as close as possible to an ideal 2nd order lowpass function.
Some examples:
* Active network design (Claude Lindquist)
* Active and passive analog filters (Huelsman)
* Continuous-Time active Filter design (Deliyannis)
* Passive and Active Filters (W.K.Chen)
* Active RC Filter Design (Herpy)
* Modern Filter Design (Ghausi)
........
.........
- - - Updated - - -
I suspect the filters were named after the guys that did some original research long long ago, and then published the results.
Recently, I have seen a contribution from a very "smart" person writing "There was a man named Chebyshev who invited a very versatile filter long time ago".
This author did not know that P.L. Chebyshev (Tschebyscheff) was a russian mathematician who lived in the 19th century - long before the term "filter" even exists.
The problem is and was the following:
To approximate the ideal (brickwall) low pass filter function which can be realized with lumped components we need a broken rational function with posiitive coefficients only.
The magnitude of this function should be as close as possible to a horizontal line in the passband of the filter - and should cause a magnitude decrease (amplitude roll-off) beyond a certain selectable frequency (corner frequency, end of passband).
Thus, it is a mathematical problem which has several alternative solutions - each with some advantages and disadvantages. Hence, a trade-off is necessary (as always in electronics):
*
Butterworth polynoms (named after his inventor) have a maximal flat amplitude in the passband but not a very "sharp" (it`s a bad expression) transition to the stop band.
* The mathematician P.L.
Chebyshev has desribed some other polynoms (without knowing anything about filters) causing a ripple within a certain region starting with zero (defining our passband) - but with a
better transition to small values (defining our stop band).
* In some applications, it is primarily the phase (resp. the group delay) that matters for lowpass functions - and not the amplitude as in the above methods.
That means, we need a polynom (rational function) having a nearly constant group delay (linear phase) within the passband. Based on the mathematical background of the Butterworth function it is possible to find
a function with a maximum flat group delay. W.E. Thomson has used the well known Bessel functions (Mathematician F. Bessel lived in the 18th-19th century) for this purpose. The corresponding filters are called
Thomson-Bessel filter.
* A similar story applies to the elliptical functions used in
Cauer filters (W. Cauer dies in 1945; he primarily was engaged in system theory and passive lumped RLC filters)