This paper gives a theoretical treatment of some recently developed wave filter terminating sections whose application is discussed in the accompanying paper on "Impedance Correction of Wave Filters." The sections consist primarily of non-recurrent ladder networks which operate, over the transmission bands of the associated filters, as transformers whose ratio varies with frequency. The transformation ratio of the network is specified, as a function of frequency, by a power series containing a limited number of terms and the design procedure therefore depends upon the construction of power series approximations to the ratio between the resistance of the filter proper and the desired resistance. A separate network is added to secure control of the reactance component. An increased number of terms in the power series, and therefore an improved approximation to the desired transformation ratio, can be obtained by increasing the number of branches in the network. The method thus leads to a series of sections of progressively increasing complexity and with progressively improving impedance characteristics. By an inversion of the analysis a second series of sections can also be obtained. The paper is chiefly devoted to a discussion of these two series of filter sections, but other possible applications of the method are also described briefly.T H E analysis of transmission circuits with which telephone engineers are familiar is an outgrowth of the general physical theory of the propagation of wave disturbances in continuous media. Problems analogous to the analysis of a smooth transmission line are found, for example, in optical and acoustical theory and in the theory of the vibrations of a taut string. The situations of most importance from the standpoint of general physics are those in which the continuous medium extends indefinitely in at least one direction. Since, moreover, this is also the simplest case, it has been customary to base our transmission analysis upon the analogous concept of an infinite line with distributed constants. The analysis of such a structure, since it depends upon only two quantities, the characteristic impedance and the propagation constant, is of course very simple.An actual telephone transmission circuit, however, is by no means an infinite structure containing distributed constants. Many lines, for example, are loaded. Whether loaded or unloaded, they do not extend indefinitely, but are interrupted by terminal apparatus and intermediate repeaters. Each of these, moreover, contains a miscellany of apparatus, such as modulators, transformers, amplifiers, filters, equalizers, by-pass circuits, and the like, having little physical resemblance to a continuous medium. 794
