PROPORTIONING OF CIRCUITS FOR ATTENUATION 257 



Effect of Frequency on Optimum Ratio 

 It has been seen that, at the higher frequencies where the approxi- 

 mate transmission formulas may be employed, the optimum diameter 

 ratio is substantially independent of frequency. In so far as the 

 practical application of individually shielded circuits is concerned, it is 

 in these higher frequencies that interest primarily centers. Even when 

 it is desired to transmit a wide band extending from high frequencies 

 down to comparatively low ones, it is advantageous to proportion the 

 circuit so as to minimize the attenuation at the highest transmitted 

 frequency, since the attenuation at all lower frequencies will be less than 

 the value thus obtained. 



It may, however, be worth while to consider briefly the question 

 of optimum proportioning when low frequencies only are involved. 

 The appropriate transmission formulas to be used instead of the 

 approximate high-frequency expressions are known, ^ and the optimum 

 diameter ratio in any specific case may be derived from these. It will 

 be evident that, since skin efTect is present to a lesser degree at the 

 low frequencies, the diameter and thickness of the outer conductor 

 and the thickness of the inner conductor will, as the frequency is 

 decreased, have an increasing influence on the optimum proportioning. 

 Without attempting to derive precise values for the difTerent condi- 

 tions, it may be noted that the optimum diameter ratio for low fre- 

 quencies is invariably less than that for high frequencies, the high- 

 frequency value being approached asymptotically as a limit. The 

 reason for this will be readily apparent. Let the inner diameter and 

 thickness of the outer conductor be assumed fixed. At high frequencies 

 the resistance of the inner conductor varies inversely with the first 

 power of its diameter. At lower frequencies, however, this resistance 

 varies inversely with some power of the diameter greater than unity, 

 and finally, at zero frequency, assuming a solid wire, with the square 

 of the diameter. Hence it is, that, in varying the size of the inner 

 conductor in order to obtain a balance between the change of resistance 

 and change of capacity, it is advantageous to make the inner conductor 

 somewhat larger, or, in other words, to make the diameter ratio smaller, 

 at low frequencies than at high frequencies. 



Thin Walls 

 What is the result if the walls of the two coaxial conductors are made 

 very thin? Under this condition the conductor resistance, and hence 

 the attenuation, will remain substantially constant over a wide range 

 of frequencies. This constancy is realized, however, at the expense of 

 an increase in the attenuation as compared with that for thicker 

 conductor walls. 



