506 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1951 



With the results obtained in Sections II and III, we can compare the 

 curves of attenuation as a function of frequency for conventional and lam- 

 inated lines. Let us consider a transmission line such as that shown in Fig. 1, 

 where we may imagine the center conductor to be either laminated as shown 

 or made of solid metal. Let us suppose, as in the introduction, that most of 

 the power loss is in the center conductor and that the distance between the 

 stack and outer conductor is large compared to the depth of the stack. 

 Clearly, the attenuation of the line will be proportional to the power per 

 unit area flowing into the center conductor for a given power flow in the 

 line. If Ex is the transverse magnetic field and Ex is the longitudinal electric 

 field, the power flow into the center conductor per unit area will be given by 



^ I H, I . I £x I cos $ = i Real part {H,-Et) (III-45) 



where ^ is the phase angle between E^ and Ex and (*) indicates the con- 

 jugate of a complex quantity. If C is the circumference of the center con- 

 ductor, and Z is the characteristic impedance of the line, the attenuation of 

 the line will be given by 



7 = ^^^ L^ 1 2 Real part {E.-Et) (III-46) 



First, let us suppose that the inner conductor is solid and that the fre- 

 quency is very low. In that case, the uniform current density in the metal 

 will be Ez/D, and therefore Ex will be Ez/aD. Hence, for this case the 

 attenuation will be 



^'(^ ^"^") = 2-ctp ("^-^'^ 



In a similar manner, the attenuation of the line when the center con- 

 ductor is laminated will be for very low frequencies 



7 J/ small) = ^-^1^ (III-48) 



Next, let us consider the solid conductor again but for frequencies where 

 6 « D. Then we have from equations (II-9) and (11-15) 



Ex= - ^-t-* Ex (III-49) 



(TO 



Hence, 



-" = 2cL ("^-^°^ 



