928 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 



which can be reduced to 



^0(70) = 



^ (3?i - 1) q2 _ (on'' - 1) ^4 , 



ngi^i L 6 180 



.2,2 





1 



gVT^iL^ ' h\ ' 9^1 



(192) 



the last expression being valid if the number of double layers is not too 

 small {n ^ 5, say). To this approximation the fractional changes in the 

 resistance and reactance of the stack are 



(193) 



^ =T\A ^ T\t\.Mg\f 

 Ro 981 9 



AX T:ti 



^ = -TT = TAttm^U (194) 



where 



i?o = l/giTx (195) 



is the dc resistance. From the exact calculations described above it 

 appears that (193) and (194) are valid up to the neighborhood of the 

 critical frequency 



h = V3/(TMihTi), (196) 



at which frequency the approximate formulas yield 



AR/Ro = 1/3, AX/Ro = V3. (197) 



For/ > /2 , however, these approximations rapidly break down. 



We may now answer the question: What must be the thickness ti of 

 the individual conducting layers in a plane stack which contains a given 

 total thickness Ti of conducting material, if at a specified top frequency 

 fm the resistance of the stack is not to have increased by more than a 

 specified small fraction of its dc value? We find that the permissible 

 value of ^1 is 



TTHigiliJm y lio 



and we note that this value of ^i is inversely proportional both to fm and 

 to Ti . If we measure h and Ti in mils and/„ in Mc -sec^^ then on putting 



