CONSTANT RESISTANCE NETWORKS 



179 



Zi and Z2 are built up of resistive and reactive elements in the usual 

 way. 



The lattice type will not be considered here. The first step in 

 extending the other two is shown in Fig. 2, where the networks shown 



Fig. 2 — The first step in extending the fundamental forms of the constant 

 resistance networks. 



have a constant resistance if Z1Z2 = E}. The networks have now 

 taken on the form of two half-section filters in parallel or series, 

 provided that Zi and Z2 are purely reactive. This suggests the 

 possibility of an extension to more complicated configurations having 

 the general properties of wave filters with constant resistance. Since 

 the shunt and series types are analytically the same, only the former 

 will be considered in detail. 



Use will be made of the following theorem : 



Any finite network of linear elements having a constant conductance at 

 all frequencies, and no purely reactive shunt across its terminals, has 

 zero susceptance. 



The admittance may be written ^ 



F(X) = 



^0 + ^lX + 



+ A^n^' 



B, + B,\+ ■■■ + 5„X» ' 



where X — i{co/coo) and m is equal to or one greater or one less than n. 

 ojo is a constant which fixes the frequency scale. If the real part of 

 Y is to be a constant other than zero, Ao cannot be zero and m must 

 be equal to or greater than n. If there is no purely reactive shunt 

 across the terminals, -So cannot be zero and m cannot be greater than n. 

 The expression for the admittance may then be written 



F(X) =G 



1 + ^iX + 



+ ^.X" 



1 + ^iX + 



+ B„\- 



^ See "Synthesis of a Finite Two Terminal Network Whose Driving Point Im- 

 pedance is a Prescribed Function of Frequency," Otto Brune, M. I. T. Journal of 

 Mathematics and Physics, vol. 10, 1931. 



