180 BELL SYSTEM TECHNICAL JOURNAL 



By elementary methods it may be shown that if the real part of 

 this expression is constant for all real frequencies then Ai — Bi, 

 • • • An = Bn, and the imaginary part is zero. All other possibilities 

 involve special relations between the B's, which correspond to a F(X) 

 with poles on the imaginary axis. This has been excluded by the 

 condition of no purely reactive shunt across the terminals. The 

 study of networks of constant admittance may then be restricted to 

 the study of the conditions for constant conductance. 



We will consider, then, the problem of designing two passive net- 

 works of linear elements such that, when connected in parallel, they 

 will have constant conductance. The value of the constant con- 

 ductance may be taken as unity without loss of generality. 



The conductance of a finite network may be written as a ratio of 

 two polynomials in frequency. Its value must always be positive for 

 real frequencies, and for the case under consideration it may never 

 exceed unity, since otherwise the conductance of the second network 

 to make up the constant resistance pair would be required to be 

 negative. The expression for the conductance of the first network 

 may be written in the form 



<^i = 1 + F(x) ' ^^^ 



where X may be i(co/coo) as above, or it may be taken as any imaginary 

 function of frequency which may be realized by the impedance of a 

 reactive network, and F{\) is the ratio of two polynomials in even 

 powers of X. By subtracting Gi from unity the required expression 

 for d may be obtained: 



G2 = ^ . - (2) 



1 + 



Fix) 



An investigation of general networks of an arbitrary number of 

 resistance and reactance elements fulfilling the relations (1) and (2) 

 would take the present investigation too far from its main objective. 

 If the networks are to have the general properties of wave filters with 

 a minimum of loss in a band, they may be restricted to reactive net- 

 works having a single resistance. Furthermore, both resistances may 

 be taken as unity, for, in cases where this is not necessary, a trans- 

 formation to some other value may be made after the design is com- 

 pleted on the unit resistance basis. We assume, too, that when 

 X = 0, F{\) = and Gi = 1, G2 = 0. This implies the proper choice 

 of the expression for X. 



