440 BELL SYSTEM TECILMCIL JOURXAL 



For the discussion of these it should be recalled that D = },u}\'\L'C 

 and r=D/Di = f/fi=f/fc; also that Di tan £>i = \, whence Di is approx- 

 imately equal to v X when X is small. 



Equations (29) and (30) are in such form as to exhibit the manner 

 in which Z j, and Z'.s approach their simple limiting values for X = 0, 

 represented by equations (6) and (7) respectively. For when X 

 approaches 0, D cot D and D tan D approach 1 and D- respectively; 

 and for values of X even larger than the largest (about 0.12) occurring 

 in practice, D cot D and D tan D respectively are at least roughly 

 equal to 1 and to D'^ throughout even more than the first transmitting 

 band. 



The expression for Z j, reduces immediately to 1/ V 1— r^ when X is 

 zero. When X is not zero, Z,& is less than 1/V 1 — r- for all values of 

 r in the first transmitting band (0<r<l); when r increases from to 1. 

 Z.5 increases from 1 to oo. 



The expression for Z' 6 reduces immediately to V 1 — r- when X is 

 zero. Even when X is several tenths, Z'.6 is very closely equal to 

 V 1 — r^ for all values of r in the first transmitting band; when r in- 

 creases from to 1 , Z' 5 decreases from 1 to 0. 



Effects of Distributed Inductance; the "Sinuihitive Loaded Line" 



The abo\e-described relations are exemplified in l-"ig. it, which 

 gives graphs of Z ^ and Z' i, over the first iransniilling band and part 

 of the succeeding attenuating band, as functions of r, with X as para- 

 meter equal to 0.12 and to 0. It is seen that the curves of Z.s for the 

 two values of X do not difTer much in the transmitting band (0<r< 1); 

 and that the curves of Z'.s for the two values of X are indistinguish- 

 able — on the scale there used. 



In order to indicate more precisely to what extent the forms of Zj 

 and Z' 5 are affected by the presence of distributed inductance, as 

 specified by \ = LL', Fig. 10 has been prepared. This gives a graph 

 of the ratio of the values of Z •, for X = 0.12 and X = 0; and likewise 

 of Z'.6. That is, formulated in functional notation, it gives graphs 

 of Zi{r, X)/Z.6 {r, 0) and Z'.t (''. M, Z'.s ('", 0). From these it is seen 

 that, in the transmitting band, the mid-section ratio (first ratio) and 

 the mid-load ratio (second ratio) do not difTer from unity by more 

 than four per cent, and one-tenth of one per cent., respectively. 

 These observations — particularly the second— suggest that, at least 

 over the whole of the first transmitting band, the impedance of a non- 

 dissipative periodically loaded line with small distributed inductance 



