LAMINATED TRANSMISSION LINES. I 925 



It will bo instructive to comi)aro the impedanee of a laminated plane 

 stack with the impedance of a solid metal ])lat(^ over the full fre(iuency 

 ran^e from zero to ver>- lii,u;li ficMiueiicies."' If the stack contains /( con- 

 ducting laycM's, each of thickness d , and the metal plate is of thickness 

 J\ = nti , the impedances of the plate and of the stack will be e(|ual at 

 zero frequency, and also at \-ery high fre(iuencies where the fii'st layer 

 of the stack is alnnidy many skin depths thick. For simplicity we assume 

 that l)()th the plate and the stack are backed by infinite-impedance 

 surfaces at all fretiuencies. 



To orient ourselves we shall (l(>fine tln-ee critical frequencies, for which 

 n^spectix'cly the thickness of the solid plate is equal to one skin depth in 

 the metal, th(^ thickness of the stack isecjual to one "effective skin depth", 

 and the thickness of a single conducting layer is eciual to \/3 skin depths 

 in the metal. These frequencies are 



/2 = VS/(jrfiigihT^ = VSnfi {T, = Ta), (178) 



fz = 3/(TrnigA) = Sn'fy {h - V35,). 



The approximate forms of the surface impedance functions of the plate 

 and the stack in the various freciuency ranges are then quite simple. 



In the range ^ / ^ /i , the surface impedance of the solid plate is 

 approximately constant and given by 



Z„(Tn) ^ l/giT, , (179) 



while in the I'ange.f ^ A we see approximately the surface impedance of 

 an infinite plate, 



Zo(to) ;^ (1 + i)/gA = (1 + iW^i^^^J/gi , (180) 



which is proportional to \/f. The surface impedance of the stack is 

 approximately constant in the range ^ / ^ /2 , where 



Zo(Tn) ^ l/r/iT^i , (181) 



while in the range /2 ^ / ^ /s it is approximately eciual to the impedance 

 A'l of an infinitely deep stack of moderately thin layers as given by the 

 first of equations (164), namely 



Zo(7o) ^ (l/VS + i)TrnAf, (182) 



'* In this connection see also ■Reference 1, Fig. 2, ]). 494. Clogston compares 

 a laminated stack with a solid i)late of the same total thickness as the stack, hence 

 a plate which contains more conducting material than the stack. 



