38 BELL SYSTEM TECHNICAL JOURNAL 



The general propagation constant formula is 



cosh {A+i B)= cosh A cos B-\-i sinh A sin B, 



-i+ifi-i+iQ+S* 



_ r 2 (ri + 2r 2 )+x 2 (x 1 + 2:x; 2 ) . (r^i — r^) 

 2(r 2 2 +x 2 2 ) + * 2(> 2 2 +.r 2 2 ) ; 



(14) 



f\, r 2 and aci, x 2 being the resistance and reactance components of the 

 two impedances, Z\ and z 2 . 



In the transmitting bands of non-dissipative wave-filters where 

 ri = r 2 = : o, the formula becomes 



cos £ = l + i-. 



Differentiating this equation and introducing the above facts, we 

 obtain for the rate of change of the phase constant with frequency 



dB (l-cos5) 



df Xi sin B 



(s 2 sin 2 B+t 2 cos 2 B). (15) 



This rate obviously has the same sign as the denominator which we 

 shall show is positive. For, the non-dissipative wave-filter being 

 considered as the limiting case of one having small positive dissipa- 

 tion, we may temporarily return to the general propagation constant 

 relations (14) in which r x and r 2 are assumed to be positive infinitesi- 

 mals. Then in the limit when ri = r 2 = 0, since also X\ and x% are of 

 opposite signs in any transmitting band, it follows that X\ and 

 sin B are of the same sign and that their product is positive. 



