LAMIXATKD TKAXSMISSION LINES. II 11// 



does not exceed a specified value." We suppose tliat the tliickuess /i 

 ot" tlie conductors is fixed, but that the proportion ot" conductinj^' ma- 

 teiial in the cable may be adjusted by changing the thickness of the 

 insulators. Let am be the value of the attenuation constant which is not 

 t o be exceeded in the operating frequency range, and let Jm be the f re- 

 (luency at which this maximum attenuation is reached. What should 

 l)e the fraction 6 of conducting material in the cable in order to maxi- 

 mize Jm ? It is tacitly assumed that «,„ is at least slightly greater than 

 the minimum "flat" attenuation constant which is possible with a cable 

 of the given diameter, since obviously we do not wish to work ontii-ely 

 ill the very-low-frequency range. 



In the frequency range of interest the attenuation constant of the 79th 

 motle is given bj' equation (484), which may be written 



2 rfifi 2 2 2 J.2 



2\/m/€ g Gx/m/c g 



where Xp i^^ ti root of (475) and indepcnident of 0. Sohing (495) for the 

 frequency fm at which a. is equal to am , and su])stituting for e, jl, and 

 g from (268), we obtain 



J m — 



V3 r2[^Mi + (1 - 0)m2]^[(1 - d)/e2fgiam xl"" 



iriJLigiti 



(4%) 



A routine calculation shows tliat fm is a maximum, considered as a 

 function of 6, when 6 satisfies 



amgieid^i -f 2(1 - 0)m-->] _ .^ 2 (,,.^. 



[0M1 + (1 - ^Wd - ^)-4 ■^'''' ^ ^^ 



Equation (497) is easih' reduced to a quartic equation in 6, which may 

 be soh-ed without difficulty when the other parameters are gi\'en. The 

 maximum value of /„ is then obtained by substituting 6 back into (49G). 

 We shall now investigate in more detail the case in which 



Ml = M2 , (498) 



that is, the permeabilities of the conducting and insulating layers arc 

 equal. In this case the low-frequency attenuation constant ao , which is 

 just tlio first term on t)ie right side of eciuation (495), is given by 



«„ = ^"^ .^ ^- , (499) 



20\/l - d VfJ^-i/e-i gx ' 



2" A similar problem was first investigated in an uniJiihlished memnr.indiim 

 l)y H. S. Black. 



