i.o.inr.i) UM.s ./.v/) co.\iri:\s.n i.\i. M.niouKS 455 



Disposition of the Transmitting and the Alleniiating Bands 



The propanation ronslaiU V=A-{-iB of a non-dissipative loaded 

 line (pvr periodic interval) can he cxpresseii in terms of \ = L L' arid 

 the quantity D tlelined l)y equation (1()). From Appendix I). 



cosh r = cos 2D - -^ .si n ■2D, ( 1-A) 



sinlv r = (sin= 2D){D tan D-\)(D cot D + \) r- (2-A) 



= (sin' 2D){D--\--2\D cot 2D)/\- (3-A) 



= (-sin= 2D)(l + l/\)Z'.l (3.1-A) 



Thus, for a non-dissipative loaded line, cosh V and sinh- 1' are both 

 pure real. 



When cosh I" is known. .1 and B can 1)0 e\aliiated In' means of the 

 identity 



cosh r =cosh {A +iB) =cosh A cos B + i sinh .1 sin B. (4-A) 



In particular, when cosh F is pure real — as for a non-dissipative 

 loaded line — the values of A and B must evidently be such as to 

 satisfy the pair of equations 



sinh A sin 5 = 0, (o-A) cosh A cos 5 = cosh T; (6-A) 



with, of cf)urse, the added restriction that A must be real and positive, 



and B real. Thence it is readily found that: 



When cosh- r<l, that is, sinh= r<0, 



then A=:Oand B = cos"' cosh F ; (7-A) 



When cosh^ r>l, that is, sinh- r>0, 



then A =cosh~' [cosh r| and B = qir; (8-A) 



cosh r being real, and q being an even or an odd integer according as 

 cosh r is positive or negative, respectively. 



Before continuing with the general case (X^O) it seems worth while 

 to digress long enough to apply the preceding general formulas to the 

 limiting case where \ = 0. For it, formula (1-A) reduces to 



coshr = l-2r-, (9-A) 



where r=f fc = D/Dc, and fc is given by (3). Application of (7-A) 



and (8-A) to (9-A) shows that : 



When 0<r<l, then A =0 and 5 = 2 sin 'r; (10-A) 



When r>l, than ^ =2 cosh'r and B^qir, (H-A) 



where q is an odd integer. 



