i.oAiH'.D i.iMs .1X1) coMrr.xs'.ii i.w; xijhokks 421 



innK'darici's Z = K k and Z'=K' k and the relative admittances 

 U'=// h and \\" = ll' h depend on only two ratios, -namely, the 

 relative frequency r = f f^, and the appro()riale rcl.it i\e termination 

 ff or ff', — as expressed by the equations' 



^ ^ 2 ^ 1 \/i^r''+i(l-2a)r ^ (4j 



W ■v/r^+«{2ff-l)r l-4ff(l -<r)r' 



" v/l-r«+j(l-2a')r 



In particular, for a = 0.5 amd a' = 0.5, respectively, 



Z'.i = \/W'.i = y/\^-. (7) 



Equations (4) and (5) are not restricted to values of a and a' less 

 than unity. On the contrary they are valid for any (real) values of 

 these quantities — though values much exceeding unity are of infre- 

 quent occurrence in practice. 



Miscellaneous Properties and Relations 



Some of the most useful and interesting simple facts deducible from 

 equations (4) and (5) are noted in the ne.\l five paragraphs: 



In agreement with the general conclusion already reached from 

 energ\- considerations, equations (4) and (5) show that each of the 

 relative impedances and relative admittances is pure imaginary in 

 the attenuating band (r>l). In the transmitting band (0<r<l), 

 each is seen to be comple.x for all values of the relative terminations 

 (a and a'), except that each degenerates to a real \alue when the rela- 

 tive termination becomes O.o. 



Throughout the transmitting band (0<r<l), a certain conjugate 

 property is possessed by each of the quantities Z, W, Z' . W — namely, 

 each changes merely to its conjugate when c is changed to \—a, as 

 is readily seen from (4) and (5); that is, 



Z„=Z,-a. Wc='Wx-c, Z'„='Z\.a. W',= W\-,, (8) 



the bar over a symbol denoting the conjugate of the same symbol 

 without the bar. Thus, complementary characteristic impedances 

 are mutually conjugate throughout the transmitting band. 



At all values of r, 



\V,+ \\\-a = 2Wj„ Z\-VZ\-, = 2Z' .,; (9) 



•The equations were written in this sequence because, in practice, section-termi- 

 nation occurs much more frequently than load-termination. 



