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BELL SYSTEM TECHNICAL JOURNAL 



and 



sinh r = sinh 2aiL 



1 + y ^^ tanh «iL 



1 + ^^7 I coth oiiL 



£0 and r are sometimes called the equivalent line parameters. If we 

 have n sections of the type discussed above, we can write n equations 

 of the kind given by (20). If we eliminate all the terms except for the 

 first and last sections, it can be shown that 



Vn = Vi cosh nV — ~-^ sinh nV, 



^0 



pn = p\ cosh nT — ViZo 



(22) 



S, 



ih 7iT. 



We see then that F represents the propagation constant of one sec- 

 tion and Zo its specific characteristic impedance. They have the 

 physical interpretation, that Zo represents the specific impedance look- 

 ing into an infinite sequence of these sections, while T represents the 

 ratio of excess pressure or volume velocity between one section and the 

 next, when we are dealing with an infinite number of sections, or with 

 a finite number, terminated in the characteristic impedance of the filter. 



It is customary in electric filter design to determine the character- 

 istics of a dissipationless filter, and to regard dissipation as causing a 

 slight change in the filter characteristic, which usually occurs most 

 prominently in the pass bands. If we neglect dissipation, equation 

 (21) becomes 



2coL \ , i ^iPoypS2 . ( 2oiL 



cosh r = 



cos 



C 



. i ylPoypS^ 

 H ^^ r. — sm 



2Zs6i 



VPoTP 



/ , i ^lPoypS2 ^ 





V 





~C 



(23) 



The propagation constant V is in general a complex number A + iB. 

 The real part represents a diminution of the volume velocity or the 

 pressure, while the imaginary part represents a phase change, as can be 

 seen from the fact that the ratio of pressure or volume velocity is 



pi 



-r = g-(A+iB) = g-A (cos B - i sin B). 



Now cosh r = cosh {A + iB) = cosh A cos B -{- i sinh A sin B. 

 Hence we see from equation (23), if Zg is an imaginary quantity, the 

 expression for cosh V is always real, and hence either sinh A or sin B 



