886 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 



we discuss losses due to imperfect dielectrics and lossy magnetic ma- 

 terials. 



Part II will be largely devoted to transmission lines of the so-called 

 "Clogston 2" type, in which the entire propagation space is filled with 

 the laminated medium, though to a lesser extent we shall also consider 

 transmission lines having an arbitrary fraction of their total volume 

 filled with laminations and the I'est with dielectric. We shall first derive 

 expressions for the propagation constant and the fields of the lowest 

 Clogston 2 mode assuming infinitesimally thin laminae, so that the 

 attenuation constant is essentially independent of freciuency, and then 

 go on to investigate the transition of the lowest Clogston 1 mode into 

 the lowest Clogston 2 mode as the space occupied by the main dielectric 

 is gradually filled with laminations. We shall also discuss the higher 

 modes which can exist in Clogston 1 and Clogston 2 lines with infinitesi- 

 mally thin laminae. Next the effect of finite lamdna thickness on the 

 variation of attenuation with frecjuency in a Clogston 2 will be investi- 

 gated, and then the important cjuestion of the influence of nonuni- 

 formity of the laminated medium on the transmission properties of the 

 line. We shall conclude with a short section on dielectric and magnetic 

 losses. 



Insofar as possible, plane and coaxial lines will be treated together 

 throughout the paper. Since however Bessel functions are not so easy 

 to manipulate as hyperbolic functions, there will be a few cases where 

 explicit formulas are not yet available for the cylindrical geometry. In 

 these cases the formulas derived for the parallel-plane geometry usually 

 provide reasonably good approximations, or if greater accuracy is desired 

 specific examples may be worked out numerically from the fundamental 

 equations in cylindrical coordinates. 



The purpose of the present paper is to set up a general mathematical 

 framework for the analysis of laminated transmission lines, and to treat 

 the major theoretical questions which arise in connection with these 

 lines. In view of the length of the mathematical analysis, we have not 

 devoted much space to numerical examples, although a large number of 

 specific formulas are given which may be used to calculate the theoretical 

 performance of almost any Clogston-type line that happens to be of 

 interest. A considerable part of our work is directed toward evaluating 

 the effects of deviations from the ideal Clogston structure. Both theoreti- 

 cal and experimental results suggest that the limitations on the ultimate 

 applications of the Qogston cable ai-e likely to be imposed by practical 

 problems of manufacture. These limitations, however, depend upon 

 engineering (questions which we shall not consider here. 



