r.n.tDF.n I.IXES AND COMPENSATING NETWORKS 4W 



clearly that: one and only one transition value of D lies within each 

 segment of width r 2; sinh-T <0 when i>,_i „<D <Z)„, and sinh'r>0 

 when D^<D<D,„j^\: the zeros of X-D tan D and of X + ZJ cot Z> 

 are situated in the odd and even numbered segments, respectively ; 

 with increasing D, the transmitting bands continually decrease in 

 width and the attenuating bands continually increase in width, the 

 change taking place rapidly at first and then more and more slowly; 

 the mid-point relative impedances are pure imaginary throughout 

 every attenuating band and pure real throughout every transmitting 

 band, and. they have the ranges stated in the third and fourth para- 

 graphs following equation (26.1). The graphical methods are useful 

 also for showing the nature of the effects produced by varying the 

 parameter X. 



Discussion of the Disposition of the Bands 



The rest of this Appendix will be devoted to a discussion of the 

 most salient properties of the compound bands and their constituent 

 transmitting and attenuating bands. 



The ratio of transmitting band width to compound band wiiitti 

 continually decreases with increasing D and becomes zero when D 

 becomes infinite; that is, the transmitting bands vanish and the 

 compound bands become pure attenuating bands. These facts can 

 be seen graphically, or analytically from equation (14-A). 



The ratio of transmitting band width to compound band width 

 continually increases with increasing X; this ratio ranging from zero 

 when X is zero to unity when X is infinite. These facts can be seen 

 graphically, or from equation (14-A). When X approaches zero the 

 /-width of each compound band approaches infinity; the /-width of 

 each transmitting band approaches zero, e.xcept for the first trans- 

 mitting band, whose width approaches a value equal to f'\=f'c — 

 for equation (14-A) shows that D„{Dn — D„_i„)/\ approaches unity, 

 and hence that fn(fn—fn-i.n) approaches l/7r-L'C=/'l^ whence 

 fn—fn-in approaches zero for w=^l and approaches /'i for n = l. 



The effects of varying the parameter X will now be outlined briefly, 

 in the next two paragraphs, for the cases respectively of L'C fixed and 

 LC fixed. The conclusions reached depend partly on the equation 

 D = kuV LC = kuiy/xL/C defining D; partly on the fact already de- 

 duced that the i?-width of each compound band is an absolute con- 

 stant (ir 2); and partly on equation (14-A). 



When L'C is fixed, increasing X reduces all of the transition fre- 

 quencies. The transition frequencies bounding the compound bands, 



