938 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 



The admissible value of | A; |, if the fractional change in resistance is not 

 to exceed a specified value AR/Ro at a given top frequency /„» , is 



k\ = 



3\/55l /aR ^ 3V5 /aR . . 



2T\ y Ro 27rM JmTl y Ro ' ^^ ^ 



which is inversely proportional both to fm and to the square of the total 

 thickness of conducting material in the stack. If we express Ti in mils, 

 fm in Mc-sec~ , and assume the conducting layers to be copper, we get 



22.71 /AE 



(/mJMcl-' l)mi)s f Ri 



/■ 



(245) 



lo 



The variation with frequency of the surface impedance of a matched 

 stack of finite layers at moderate frequencies (say / ^ /2) is given by 

 equation (192) of Section V; but no simple formula has yet been de- 

 rived for the surface impedance of a mismatched stack of finite layers 

 in this frequency range. The derivation of such a formula would appear 

 to involve nothing more than some rather formidable algebra, the diffi- 

 culties centering around the fact that in the general case we can make 

 no a priori assumptions as to the relative magnitudes of k and (ti/8i) . 

 It is reasonable to suppose, however, that if both dielectric mismatch 

 and finite lamina thickness contribute appreciably to AR/Ro , the per- 

 missible values of I /c I and ^i individually will be less, if we are to achieve 

 a given flatness of the attenuation versus frequency curve, then the 

 permissible value of either if the other factor were unimportant. 



To exhibit the effect of dielectric mismatch from a slightly different 

 point of view, we may plot the surface resistance of an infinitely deep 

 plane stack of moderately thin layers (a finite stack several effective 

 skin depths thick would show essentially the same behavior) at a fixed 

 frequency, as a function of the mismatch parameter k. The surface re- 

 sistance is just Re Ki , which may be obtained from (217) if k and 

 ti/Si are assumed small compared to unity. Fig. 9 shows the dimension- 

 less quantity 



Re gidiK, = ;^ [Vl(ti/8^y + 9^:^ + Wi/^iYf, (246) 



for the three values h/Si = 0, /i/6i = 0.1, and ti/8i = 0.2. For an elec- 

 trically thick solid conductor we have simply 



RegAK, = 1; (247) 



hence to get any benefit from the laminated stack we must have 

 Re gidiKi smaller than unity. Actually, if we meet Clogston's condition by 



