OTHER L-C FILTERS 

 The design equations are 



^ 2(oj2 — coi) ^ a>ift>2 



_ 1 ^ _ 2(0^2 — COi) 



^ 2(w2 — (Oi)R ^ oj^co^R 



Filters to work between unequal resistances 



We have dealt at some length with the classical high- and low-pass filters, 

 and have mentioned briefly the band-pass, band-stop, and m-derived filters. 

 All these are supposed to work between equal generator and load resistances, 

 and the reasons for making this provision are: (1) the case is theoretically 

 important because it is the condition for maximum power transfer; and (2) 

 it enables a filter of complicated transmission characteristic to be assembled 

 from simple sections, the overall performance being merely the sum of the 

 several performances (on a dB scale). 



It often happens in electronics that filters are required for parts of a circuit 

 where maximum power transfer conditions do not obtain. In valve circuits 

 we are usually more interested in voltage than in power, and maximum 

 voltage transfer occurs when the load resistance is much larger than the 

 generator resistance. Further, such filters are often of a simple kind, requiring 

 only one section. It is therefore pertinent to inquire whether filters can be 

 built when the resistances between which they are to work are different. The 

 answer is that they can, and we shall indicate the general procedure and 

 illustrate with an example. We restrict ourselves to sections having a 

 Butterworth response, i.e. whose transmission characteristic is similar to 

 that for a classical filter. 



For a Butterworth response, if the filter possesses n reactive elements, the 

 transmission characteristic is 3 dB's down at the cut-off" frequency and is 

 asymptotic to a slope of 6n dB's per octave, and 



Kout 

 Kin 



^-i72 (high-pass) or / ,„^^n^l|% (low-pass) 



l-(')T l-(?J 



In our case, « = 3, so for a low-pass filter 



Fout 



V 



m 



1 



6\l/2 



Looking back to the calculations for the transmission characteristic of the 

 classical low-pass filter, we see that the penultimate stage is of the form 



Kout 1 



m 



(1 + ylco2 + Boi'^ + Doj^fl^ 



and happily A and B turn out to be zero, and D to be {IJojq). This happens 

 because of the particular values we gave to the reactive elements of the 

 section, in terms of Mq and R. Had the filter elements or the R been different, 



93 



