446 BELL SYSTEM TECHNICAL JOURNAL 



2.3. Inverse Networks of Constant Impedance Product 

 We have already seen that the fundamental advantage of using 

 constant resistance networks for distortion correction lies in the fact 

 that when they are placed ahead of the receiving resistance, R, they 

 present this same impedance to the circuit proper and hence do not 

 alter the transfer exponent at that point. They can be designed to 

 have, in addition to the impedance R, a propagation constant which 

 complements this exponent and produces a resultant transfer exponent 

 at the receiving resistance which is approximately ideal. 



The possibility of physically realizing recurrent networks having a 

 constant resistance iterative impedance at all frequencies rests, as 

 will be seen, upon that of obtaining pairs of two-terminal networks the 

 product of whose impedances is constant, independent of frequency. 

 Such pairs ^ I have defined as inverse networks of impedance product R-, 

 or more concisely, inverse networks. 



In the paper just referred to it was pointed out that one elemental 

 pair of such inverse networks is composed of two resistances i?i and 

 i?2, and another is composed of an inductance L and a capacity C 

 bearing the impedance product relations at all frequencies 



i?ii?2 = Lie = R'. (9) 



The same paper gave a simple proof of the following theorem relating 

 to series and parallel combinations of networks. // 2/ and Zo' are 

 any pair of inverse networks and if Zi" and Z2" are any other pair, such 

 that Z1Z2 = Zi'zi" = R'^, then z/ and Zi" in series and 22' and 22" in 

 parallel are a pair; similarly z/ and z-l' in parallel and Zi' and 22" in 

 series are another pair. 



Without much difficulty a theorem relating to simple networks 

 having the form of a general Wheatstone bridge can also be obtained, 

 as follows : The inverse network corresponding to any given two-terminal 

 bridge network of five distinct branches is also a bridge network, and may 

 be derived by replacing the netivork in each branch of the given network by 

 its inverse network and then interchanging the networks in either opposite 

 pair of branches. By successive applications of these relations, be- 

 ginning with the elemental pairs, very complicated inverse net- 

 works can be built up. Only reactance networks were considered 

 in the paper referred to above. Ordinarily the series and parallel 



' An extensive use of inverse networks of pure reactance types was made in the 

 paper, "Theory and Design of Uniform and Composite Electric Wave- Filters," 

 O. J. Zobcl, B. S. T. J., January, 1923. Also in U. S. Patents No. 1,509,184, Sep- 

 tember 23, 1924; Nos. 1,557,229 and 1,557,230, October 13, 1925; and iNo. 1,644,004, 

 October 4, 1927. 



