LAMINATED TRANSMISSION LINES. II 



1199 



shows the pormissiblc total variation in e for each of thoso typos of lum- 

 uuiformitv. 



Case 



Symmetric discontinuity. . 

 Unsymmetric discontinuity 



Linear 



Half-cycle cosine 



Rectangular step 



One-cycle cosine 



Three-cycle cosine 



16.5 



28.0 

 42.6 

 29.5 

 53.0 

 59.8 

 78.9 



Cmax — Cm in 



r = Vi 



0.0167 

 0.0295 

 0.0430 

 0.0298 

 0.0535 

 0.0604 

 0.0797 



= H 



0.0027 

 0.0047 

 0.0069 

 0.0048 

 0.0086 

 0.0097 

 0.0128 



Ho 



0.0007 



0.0012 

 0.0017 

 0.0012 

 0.0021 

 0.0024 

 0.0032 



It would be easy to construct a similar table for any other values of the 

 attenuation ratio r, and for any specified degradation due to nonuniformity. 

 It is, howe\'er, already ob^^ous that the greater the improvement for ^^'hich 

 one strives, that is, the smaller the ratio r, the more stringent will be the 

 requirement on (emax — emin)/€o ; in fact, the permissible value of this quan- 

 tity is proportional to r". In an}^ practical case the value of e will have to 

 be controlled against long-range variations within a fraction of a per 

 cent, and if attenuation reduction factors of the order of one-fifth or one- 

 tenth are contemplated, the variations probably cannot exceed a few hun- 

 dredths of a per cent. It also appears that a steadj^ increase or decrease in 

 the value of I across the stack will be the most serious type of nonuni- 

 formity, since the effects of very rapid fluctuations A\ill tend to average 

 out. 



Clearly the nonuniform laminated transmission lines which we have been 

 considering in this section are \'ery highly idealized, even if we disregard the 

 geometrical differences between plane and coaxial stiiictures. Any real 

 Clogston cable ^^■ill be built up of layers of finite thickness with unavoid- 

 able random fluctuations from layer to layer, superimposed on slower 

 variations in the average properties of the layers from one side of the 

 stack to the other. The thickness of an individual layer will also vary more 

 or less in both directions parallel to the layer, so that the properties of the 

 stack will be functions of the coordinates <^ and z as well as of p. A few 

 qualitative remarks are in order concerning these neglected effects. 



The effect of finite lamina thickness in a nonuniform stack can be cal- 

 culated, by the method employed in Section XI for a uniform coaxial 

 stack, if we make the plausible assumption that the macroscopic current 

 distribution remains the same as for infinitesimally thin layers. The results 



