The Bell System Technical Journal 



Vol. XXX July, igsi No. 3 



Copyright, 1951, American Telephone and Telegraph Company 



Reduction of Skin Effect Losses by the Use of 

 Laminated Conductors 



By A. M. CLOGSTON 



It has recently been discovered that it is possible to reduce skin effect losses 

 in transmission lines by properly laminating the conductors and adjusting the 

 velocity of transmission of the waves. The theory for such laminated transmission 

 lines is presented in the case of planar systems for both infinitesimally thin 

 laminae and laminae of finite thickness. A transmission line completely filled 

 with laminated material is discussed. An analysis is given of the modes of trans- 

 mission in a laminated line, and of the problem of terminating such a line. 



I. Introduction 



It has long been recognized that an electromagnetic wave propagating 

 in the vicinity of an electrical conductor can penetrate only a limited distance 

 into the interior of the material. This phenomenon is known as "skin effect'^ 

 and is usually measured by a so-called "skin depth" 6. If 3; is measured from 

 the surface of a conductor into its depth, the amplitude of the electro- 

 magnetic wave and the accompanying current density decreases as f^ , 

 provided the conductor is several times 5 in thickness, so that for 3; = 5 

 the ampUtude has fallen to 1/e = 0.367 times its value at the surface. The 

 skin depth 5 is given by 



= J— (I-l) 



where g is the conductivity of the material, ^i is its permeabiHty and co is 

 lir times the frequency/ under consideration. Throughout this paper ration- 

 alized MKS units are used. 



From one point of view, skin effect serves a most useful purpose; for in- 

 stance, in shielding electrical equipment or reducing crosstalk between com- 

 munication circuits. On the other hand, the effect severely limits the high 

 frequency performance of many types of electrical apparatus, including in 

 particular the various kinds of transmission Hues. 



Surprisingly enough, it has been discovered that it is possible, within 

 limits, to increase the distance to which an electromagnetic wave penetrates 



491 



