REDUCTION OF SKIN EFFECT LOSSES 493 



It will be shown in the following sections that the electromagnetic wave and 

 the accompanying currents will penetrate most deeply into the center con- 

 ductor if the wave travels through the line with a velocity 



V = -^ (1-3) 



One way to make the wave assume this velocity is to let the dielectric con- 

 stant ci have the value 



„ = . = ,(. + H) 



(1-4) 



If the depth of the stack of laminations D is small compared to the dis- 

 tance between the stack and the outer conductor, and if the wave travels 

 with the velocity given in equation (1-3), it will be shown that the wave 

 decreases with distance into the center conductor as e~^'8y, where 6„, is 

 given by 



5„, = V3 (1 + t/W)i8/W)8; W « d (1-5) 



1 . . 



Here 5 = / ^ is the skin depth appropriate to the material of the con- 



ducting laminae and the frequency / under consideration. Let us now also 

 associate with the center conductor an average longitudinal conductivity 

 given by 



^ (1-6) 



W + / 



We will suppose for the present case that most of the attenuation of the 

 transmission line results from the currents flowing in the inner conductor. 

 It is easy to see that the attenuation of the line for very low frequencies will 

 be A/dD where ^4 is a constant depending on the impedance of the line. 

 As the frequency increases, 8w decreases, and when 6«, becomes several 

 times smaller than D it will be shown that the attenuation becomes A/adu,. 

 At still higher frequencies 8 will similarly become several times smaller than 

 W, and the attenuation then becomes A /ad. From these considerations, a 

 qualitative picture of the attenuation of the laminated line can be sketched 

 as in Fig. 2. 



For comparison, we have also sketched in Fig. 2 the attenuation that 

 would be obtained if the laminations in Fig. 1 were replaced with solid metal. 

 At low frequencies, the attenuation of this line would clearly be A/aD. 

 When the frequency becomes high enough for 5 to be several times smaller 

 than D the attenuation will be shown to become A/a8. 



It will be observed how the attenuation of the unlaminated line remains 



