A Method of Impedance Correction 



By H. W. BODE 



This paper gives a theoretical treatment of some recently developed wave 

 filter terminating sections whose application is discussed in the accompany- 

 ing 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 pro- 

 gressively increasing complexity and with progressively improving imped- 

 ance 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. 



THE analysis of transmission circuits with which telephone engin- 

 eers are famiUar 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 trans- 

 mission 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 ex- 

 tend indefinitely, but are interrupted by terminal apparatus and inter- 

 mediate repeaters. Each of these, moreover, contains a miscellany of 

 apparatus, such as modulators, transformers, amplifiers, filters, equal- 

 izers, by-pass circuits, and the like, having little physical resemblance 

 to a continuous medium. 



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