510 BELL SYSTEM TECHNICAL JOURNAL 



1 71 All Cases: 



-B = cos M -. — -7- = sin M . , . , (119) 



\ cosh A J \ sinh A J 



The latter anti-cosine formula is particularly useful when B is in the 

 neighborhood of (2« + l)7r/2, and both formulae of (119) when con- 

 sidered together determine the sign of B. 



The above formula give the solution of (114) which has a positive 

 value for A (as in the propagation constant of a passive network). 

 The other solution, since cosh ( — T) = cosh T, would have values for 

 both A and B which are the negative of those in the first solution (as 

 may be possible in an active network). 



It has been found that, when x and y are given to five or six decimals, 

 it is possible to derive A and B to about this same degree of accuracy 

 from these formulas and the Smithsonian Mathematical Tables of 

 Hyperbolic Functions. The formulae may be used to advantage in 

 accurately obtaining the propagation constant of a loaded line where 

 X and y are calculated from the known circuit constants. (See foot- 

 note 2.) 



Appendix IV 



Propagation Characteristics and Formulae for Various Lattice 



Type Networks 



Networks of the lattice type only are specifically considered here 

 since they have more general propagation characteristics than ladder 

 or bridged-T types. However, transformations of any lattice type 

 design obtained can be made to equivalent networks of these other 

 types, if physical, by means of the simple relations given in Table II 

 and the corresponding Section 2.5. 



The network drawings show only half of the elements so as to avoid 

 confusion; it is to be understood that the broken lines indicate the 

 other series and lattice branches, respectively identical. The double 

 subscript notation adopted for the elements is to be interpreted as 

 follows: the first subscript on any element denotes the general position 

 of the element in the network, 1 for the series branch and 2 for the 

 lattice branch ; the second subscript denotes the serial number of the 

 element in either branch. Elements in the two branches which have 

 the same serial numbers for their second subscripts correspond to each 

 other according to the inverse network relations. 



This group of networks, while not exhaustive, includes the simpler 

 and perhaps most useful structures, but it could readily be extended. 

 The propagation characteristics shown for each structure and derived 



