LOAi^F.n i.iM.s .i\i> coMrr.xs.iTi.w: xr/nroRks 4.i.t 



imiiu'dialolv from (20.1): and/, from (2'.V), sim'c /?„ = r„ + (M— Dir 2. 

 Ill (larlicular. tho Rraph for « = 1 is a graph of Di; hut Di — and hence 

 /i ran lie evaluated much tnore precisely liy means of I'ig. 7 dcscrihecf 

 in the preceding paragra|)h. 



The boundary transition frecjuencies /„.| „ and /„„+i of the wth 

 compound band (any compound band) depend on only one para- 

 meter (besides w) — namely, the product LC. The internal transi- 

 tion frequency/, depends on two independent parameters (besides n) 



namely, the product LC and the ratio \ = L/L'. Hence, fixing LC 

 fixes all of the lH)undary frequencies of the compound bands; fixing 

 LC and X fixes all of the transition frequencies — boundary and in- 

 ternal. Fixing any one l)oundary frequency fixes LC and thereby 

 fixes all of the remaining lioundary frequencies; fixing any two transi- 

 tion freciuencies of which at least one is an internal transition frequency 

 fi.\es LC and X and therein' fixes all of the remaining transition fre- 

 quencies — boundary and internal. 



The relative widths of all the transmit ting and attenuatiiii; hands 

 tiepend on only one parameter — namely, the ratio \ = LL'. Hence, 

 fixing X fixes the relative widths of all these bands; fixing the ratio 

 of the widths of any two bands not both of which are compound 

 bands fi.xes X and thereby fixes the relative widths of all the trans- 

 mitting anrl attenuating Imnds. 



The effect of increasing X, when L'C is fixed, is to lower the critical 

 frequency fc=f\, the critical frequency approaching zero when X 

 approaches infinity. But for even the largest values of X met in 

 practice the critical frequency is not much lower than for X = 0; the 

 fractional decrease (// —fc)/fc' produced in the critical frequency by 

 increasing X from to any value X is exactly equal to 1 — /> and hence 

 for any ordinary value of X is, by (22), closely equal to X 6 (which 

 is only 0.02 for X = 0.12). It is interesting to note that the nominal 

 impedance — defined by equation (1) — is increased about three times 

 as much as the critical frequency is decreased; for the fractional 

 increase in the nominal impedance is exactly vl -f-X— 1, and hence 

 approximately X 2. 



.-\ll the transition fretiuencies are reduced by increasing X, when L'C 

 is fixed. The transition freciuencies l)ounding the compound bands, 

 and hence the widths of the compound bands, decrease in direct pro- 

 portion to an increase ofV X. But the values of the internal transi- 

 tion frefjuencies do not decrease so rapidly; for the ratio of transmitting 

 band width to attenuating band width increases with increasing X. 



