i.o.inr.i) i.i.\r.s .i.\i> co.u/'/i.v.v.r/v.vc xr:iir()h!Ks a.\7 



I'lif Rflatixr Impedamfs 



TIk- torimilas lor the impedances and adniiltanres of a non-dissipa 

 live poriixlically loaded line (Fig. 1) with any amount of distributed 

 inductance L will next be set down, and discussed somewhat, with 

 piirticular regard to the transmitting and the attenuating bands of 

 the loaded line. 



As before, it is convenient to deal with the relative impedances 

 Z, Z' and the relative admittances 11', W defined by equations (2). 

 Special attention is given to the particular \alues Zb, Z' >, W i. W t, 

 corresponding to mid-point terminations. 



It is found that Z,Z', W, W can be expressed in terms of three 

 independent quantities — namely, the relative frequency r = f/Jc, the 

 inductance ratio \ = L, L', and the relative termination a or <t'. For 

 most applications the quantity r = f'fc is more significant than any 

 other quantity proportional to the frequency /, and on that score it 

 would be desirable to employ it explicitly in the formulas for the 

 impedances and admittances. However, the formulas are rendered 

 considerably more compact b>- employing the quantity D defined by 

 equation (16). Whenever desired, D can be expressed in terms of r, 

 X, and p by means of (16); and thence in terms of r and X by means 

 of (22). 



Because of their special importance the formulas for the mid-point 

 relative impedances and relative admittances will be set down first. 

 From Appendix D these formulas are found to be 



_ J_ _ i J^ i \+D cot D 

 '~Wi ~ \X+1\X-I> tan D' 



(25) 



1 |(X +£>cot£>)(X-£>tanZ?) 



'■'=w:r\ Mx+T) "• ^^^^ 



=v 



X^+2X£)cot2£)-D» 



X(X4-1) 



(26.1) 



From these formulas it can be verified that Z.j and Z' s are pure 

 imaginary throughout every attenuating band, and it can be seen 

 that they are pure real throughout every transmitting band. 



A study of equations (25) and (26) brings out also the following 

 facts regarding the variation of Zs and Z's in the transmitting and 

 the attenuating bands, with increasing frequency: 



In the first transmitting band, Zs ranges from 1 to », but in all 

 of the other odd transmitting bands it ranges from Jc to <^, through 

 finite intervening values; in the even transmitting bands it ranges 



