PROPORTIONING OF CIRCUITS FOR ATTENUATION 



253 



Figure 2 shows the vakies of the ratio p which satisfy this relation 

 plotted as a function of the conductivity ratio n. 



It is noteworthy that the optimum ratio of radii or diameters is 

 independent of (a) the diameter and thickness of outer conductor, (b) 

 the inner diameter of the inner conductor, and (r) the frequency, 

 provided the frequency is high enough for the approximate formulas 

 to hold. It follows from (a) that, assuming a fixed thickness of outer 

 conductor, moderately small in comparison with its diameter, relation 

 (7) makes it possible to find the minimum size of outer conductor with 

 which a given value of high-frequency attenuation may be realized. 

 It follows from (h) that the inner conductor may be either hollow or 

 solid, provided that the approximate resistance formulas are valid. 



5 



'la 

 '£| 



Si 



(E U 



U LD 



o z 



Q z 



z - 



O u. 



U O 



3 



0.2 0.3 0.4 0.5 I 2 3 4 5 10 20 30 40 50 



RATIO OF CONDUCTIVITIES (INNER TO OUTER CONDUCTOR, "^V-)^ ) 



Fig. 2 — Variation of optimum diameter ratio of coaxial circuit with conductivity 



ratio. 



A case of special interest arises when the two conductors have the 

 same conductivity, that is, when n equals unity. For this condition 

 the solution of (7) is * 



P = 



3.59. 



(8) 



A practical example of the case of different conductivities is a coaxial 

 structure in which the inner conductor is of copper and the outer 

 conductor of lead. For a lead outer conductor containing about 1 

 per cent of antimony, the ratio of conductivities of inner and outer 



* The existence of an optimum relation of this kind was first noted by C. S. Frank- 

 lin, who gave the value as 3.7. (See Reference 3.) Subsequently the precise value 

 was derived independently of Franklin. (See Reference 10.) 



