TWO-TERMINAL BALANCING NETWORKS 279 



if the return loss is equal to or greater than some specified number of db. 

 This somewhat simplifies our problem and permits a double infinity of 

 solutions. A method has been given by Brune^ for designing such networks, 

 in which it is pointed out that there is no unique solution to the problem of 

 designing a finite two-terminal network and also states that any network 

 which satisfies the impedance function may be considered a satisfactory 

 solution to the problem. It is thought that the method which is given 

 below will provide a solution which makes maximum use of the number of 

 elements employed. That is, it will provide a given return loss with the 

 minimum number of parts. 



The required degree of approximation and the frequency range to be 

 covered determine the number of elements required in this solution. In one 

 simple case which will be discussed below in the first example, the approxi- 

 mation between the impedance of a transmission line and a network designed 

 to simulate it is the approximation between the curvature of the impedance 

 function and the arc of a circle. 



Generating Function 



The method discussed here differs from that outlined by Brune in that 

 use is made of known generating functions which are added together in series 

 to approximate the total function, similar to the manner in which sine 

 functions may be added to approximate other functions. This series type 

 network can readily be converted to the ladder type by well known net- 

 work equivalence theorems and the solution will then have the Stieltjes 

 fraction form pointed out by Fry^ and Cauer.^ 



The generating function used here is an impedance consisting of a resist- 

 ance in parallel with a pure reactance or a special case of this. This func- 

 tion plus a real corresponds to a bilinear transformation, the properties 

 of which have frequently been discussed elsewhere. This particular con- 

 figuration, for instance, has been pointed out both by Brune and by Guille- 

 min^ at M.I.T. and a discussion of the bilinear transformation has been 

 given by C. W. Carter* of the Bell Telephone Laboratories. The series 

 addition of such generating functions is similar to the form given in Foster's 

 reactance theorem except that there only pure reactances are dealt with. 

 The solution can also be worked out with admittances, but will not be dis- 

 cussed here since the average engineer is more accustomed to dealing with 

 impedances. 



In many problems, particularly those involving dissipative transmission 



1 Jour. Math. & Physics, Vol X, 1930-1931. 



2 Bull. Am. Math. Soc, 35, 1929. 



3 GuiUemin— Vol. IT. 

 *B.S.T.J., July 1925. 



