LAMINATED TKAxNSMISSION^LINES. I 937 



impedance ^0(70) cannot be represented bj'' a simple power of / in the 

 range jt ^ / ^ /s . At frequencies above /» , if the hiyer thickness is 

 finite, the impedance is approximately that of a solid conductor, namely 



ZoCto) ^^ (1 + 0/{/i5i = (1 + i)V7rfIJ7i, , (240) 



whicii is proportional to \/J. 



Since in general the surface resistance depends upon the two param- 

 eters ti/8i and k, it is not possible to plot a single curve which shows the 

 variation of resistance with frequency under all possible conditions of 

 dielectric mismatch. However if we compare a matched stack of finite 

 layers with a similar mismatched stack, we see that the asymptotic 

 behavior of Zoiyo) is the same for both stacks at very low and very 

 high frequencies. A numerical study of the exact equation for Zu(yo) 

 shows that in the neighborhood of the critical frequency /2 , the resist- 

 ance of the mismatched stack is higher than the resistance of the matched 

 stack. (The critical frequency /2 as defined in (235) is a function of the 

 mismatch parameter k, but will be of the same order of magnitude for 

 a slightly mismatched stack as for a perfectly matched stack.) The 

 resistance of the mismatched stack exhibits relatively small fluctuations 

 above and below the resistance of the matched stack in the neighborhood 

 of the upper critical frequency /s , but this region is not of as much prac- 

 tical interest as the region near /2 , where the stack resistance is defi- 

 nitely increased by the effect of dielectric mismatch. 



An explicit expression for the rate at which the surface impedance of 

 a mismatched stack begins to depart from its dc value as the frequency 

 is increased has been worked out only for the ideal case of infinitesimally 

 thin layers. For a plane stack of infinitesimal layers backed by an in- 

 finite-impedance surface, equation (228) gives, at moderately low fre- 

 quencies. 



K r. . (r^sY (TrsY 







2ikTl 4k^T{ 

 381 4551 



from which the fractional changes in resistance and reactance are 

 AR 4k'n ^kW^glTlf 



(241) 



Ro 4551 45 ' 



AX ^ j2kTl ^ J2kiniigyTlf 

 Ro 35i 3 



(242) 

 (243) 



