918 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 



to €o and then solving for eo , we get 



Moeo = M(mi + 2m2) + (mi + S^x^^i2)% . (144) 



From (142) the value of 6 is 



3mi + (mi + 8M1M2) 

 and the corresponding attenuation constant is proportional to 



(co/mo)' (co/mo}' 3mi + (mi + 8/X1/X2)'' 



%i 9i Ml + (mi + 8miM2)^ 



(146) 



It will be observed that so far we have determined only the optimum 

 value of the product moco , and so we are still free to alter the ratio of 

 Mo to Co while holding the product of these two quantities constant. For 

 given values of m and m2 , we obtain the lowest attenuation constant by 

 making eo as small as possible and mo as large as possible, subject of course 

 to the practical restriction that €0 cannot be lower than the dielectric 

 constant of free space. However if we permit m2 and mo to be simul- 

 taneously increased, as by magnetic loading of both the insulating layers 

 and the main dielectric, we find from (146) that on paper it is possible 

 to decrease the attenuation constant without any definite limit. This 

 observation is in accord wdth the fact that the attenuation constant of 

 an ordinary coaxial cable may be decreased indefinitely, with a corre- 

 sponding decrease in the velocity of propagation along the cable, if we 

 are willing to assume an unlimited amount of lossless magnetic loading. 



If Ml = M2 , (144) and (145) take the form 



Moeo = 3m262 , d = 2/3, (147) 



from which we have the result given by Clogston: If the conducting 

 and insulating layers are infinitesimally thin and have equal permea- 

 bilities, then minimum attenuation is achieved when the thickness of the 

 conducting layers is twice the thickness of the insulating layers. In this 

 case, from (146) and (147) the attenuation is proportional to 



(^o/mo)' ^ 3(eo/Mo)' Q^gs 



When Mo = M2 , corresponding to no magnetic loading, we must take 

 €0 = 3€2 , and (148) reduces to 



1* Reference 1, pp. 513-514. 



