A GENERAL THEORY OF ELECTRIC WAVE FILTERS 213 



The second half of the paper is devoted to an interpretation of known 

 filter theory in terms of these results and to an attempt to extend this 

 theory until it affords a definite technique for the construction of any 

 filter which the preceding analysis has shown to be physically ad- 

 missible. The method followed depends upon the fact that when a 

 number of filter structures with matched image impedances are con- 

 nected in tandem, the roots, ri • • • ;'„, of the resulting filter will be the 

 aggregate of the roots of all the individual units of which it is composed. 

 This allows us to represent the general filter as a composite structure 

 in which each constituent represents one or at most a few of the total 

 number of roots. The resulting networks are very similar to the 

 familiar Zobel type composite filter, especially when it is noticed that 

 the various required roots can be obtained from simple prototype 

 structures by transformations analogous to the w-derivation, and 

 that the preceding classification into roots of single and double mul- 

 tiplicity corresponds in the composite filter to a classification of the 

 constituent structures into half and full-sections. 



In spite of these relations, the usual composite filter theory must be 

 extended in several ways if the general filter is to be adequately 

 represented. The first extension is demanded by the fact that in the 

 general filter we must be able to assign one image impedance char- 

 acteristic of each of the constituent sections in any form compatible 

 with the preceding general equation. The required image impedances 

 are not obtainable from ordinary ladder structures. When the con- 

 stituent involved is a double root, or full-section, structure however, 

 the required impedance can readily be realized by resorting to the 

 lattice form. With half-section structures the procedure is more 

 complicated. It is necessary to make use of a combination of Dr. 

 Zobel's multiple m-derivation and a new transformation, described as 

 an /^-derivation, which alters the impedances of ladder type half- 

 sections without affecting their transfer constants. 



Similar extensions are also needed to provide the requisite variety 

 of transfer constants. Roots falling within certain ranges can be 

 provided by ordinary ladder structures, w-derived, in the usual way, 

 with a real value of m less than one. In order to complete the list, 

 however, it is also necessary to consider single sections derived with 

 real values of m greater than one, which may be realized as lattices or 

 as ladder structures with mutual inductance, and pairs of sections 

 derived with conjugate complex values of m. By including all of these 

 types of sections we can construct physical networks giving any filter 

 characteristics falling within the general limitations discovered in the 

 first part of the paper. 



