REDUCTION OF SKIN EFFECT LOSSES 511 



neglected compared to the first term. We have then, finally, for the char- 

 acteristic equation 



d tanh 5 + 7 = (IV-16) 



h 



For ^ much smaller than h, approximate solutions of equation (IV-16) 

 may be written 



el= -{ ' (iv-17) 



h 

 and 



'-=''"'[' +i(i)] -= 1.2.3, ••• (IV-18) 



The fundamental solution (IV-17) obviously agrees with our assumption 

 that (^s) is not large. Similarly, the higher order solutions (IV-18) are con- 

 sistent with that assumption if w is not taken too large. 



We may now return to equation (IV-6) and obtain approximate expres- 

 sions for k. 



*o = V«w[l + r;^,2-^] (IV-19) 



k. = V.w(l + -^[i+\{"7j]) «= 1.2,3 ... (IV.20) 



We see that k^ is indeed approximately equal to co^juoe. The imaginary parts 

 of (IV-19) and (IV-20) are negative and give us the desired attenuation for 

 the fundamental and higher modes of the line in nepers per meter. 



/(*o) = -JtAa/'^ (IV-21) 



2sn (J y fjLQ 



nU = -^[2 + ^ (»')'! ^4/" » =1,2,3,... (IV-22) 



2sh \_ s J 0- y Mo 



where we have assumed 8<^s <^h and ei = e. 



Let us first comment on the fact that there exist several modes of trans- 

 mission in this Une. The fundamental mode with propagation constant ^o 

 corresponds to the ordinary mode of transmission that would exist between 

 a pair of parallel plates such as shown in Fig. 8. The higher modes are 

 waves that are confined almost entirely to the laminations and are not 

 encountered in an ordinary transmission line. These modes will be more 

 fully discussed in Section VI. 



