462 BELL SYSTEM TECHNICAL JOURNAL 



zn = (c sinh (Ao/2) + cosh {Ao/2)yzn 



+ 2 sinh Uo/2)(c sinh {Ao/2) + cosh {Ao/2))R, 

 and (30) 



c + tanh Uo/2) 



c = 



c tanh iAo/2) + 1 



In the above process we would generally be increasing the number 

 of network parameters without changing the number or magnitude of 

 the phase coefficients. 



2.83. Phase Constant Comparisons of Certain Pairs of Lattice Type 



Netivorks 



It has already been stated that there are usually two physical net- 

 works of the same structural lattice form which have identical attenua- 

 tion constants but different phase constants. They are derivable as 

 two physical solutions from the same attenuation coefficients. In the 

 case of a limited class of these networks, an interesting relation exists 

 between the phase constants of such a pair which may be stated as 

 follows. 



Theorem. — The two lattice type networks of every pair having the same 

 attenuation characteristic in each of which the series impedance (zn) 

 consists of a resistance in parallel luith any pure reactance network, of 

 different proportions in each, have phase constants such that their sum or 

 difference is identical with that of a non-dissipative lattice phase network 

 zvhose series impedance {zu) is a pure reactance network proportional to 

 that in the series impedance of either of the pair. 



A corollary results from this. 



One netivork of the pair is equivalent to the tandem combination of 

 the other and the related phase network. 



It should be pointed out here that results for the case in which Zu 

 is a resistance in series with a reactance network are similar, except 

 for a phase change of -k, since then the lattice impedance 221, the 

 inverse network of Su, is a resistance in parallel with a reactance 

 network. 



A procedure for proving the theorem will be sketched briefly. 

 Assume as given one network in which Zn is made up of a resistance in 

 parallel with a pure reactance network whose impedance is imy, 

 where m is a positive constant and y \s a. function of frequency. This 

 gives a form 



1 + Q.f ^ ^ 



Reversing the process, we obtain from the same coefficients P» and Q» 

 a second similarly constructed network besides the original one. The 



