LAMINATED TRANSMISSION LINES. II 



no: 



© = (1 + /)/i\/coMi(7i'2 = (1 + 0/i\/7rM,(7,/, 



(452) 



whicli is proportional to the s(|iiaro root of IVcmjikmicx-. For small (-) 

 O(|uatioii (451) may \)c \vritt(Mi 



Q 



2(ch 



1) + 4 shr P^ 

 4n 



4 sill 



2 pir 

 4n 



(-) 



_1_ 

 12 



o + 



csch 

 



1 2 . 2 px 



1 - - sin ^ 



3 4n 



(453) 



1 14 . 2 PTT 



1 - Y= Sin ^ 

 15 4/1 



0' + 



on expanding th(> right side in powers of by Dwight 657.2 and 657.8. 

 If we replace sin pw/^n by p7r/4/i and neglect the square of this quantity 

 in comparison with unitj^, (453) becomes 



4n2 e' 12 



(454) 



Iiitrodncing this expression for q into equation (443) and substituting 

 for from (452), we get for the propagation constant, 



1 



n V4wWifi'i^? 12" 



(455) 



As the frequency approaches zero in a Clogston 2 line of finite thick- 

 ness, the term in l/w dominates the other terms in square brackets in 

 eciuation (455). Thus at veiy low frequencies the attenuation and phase 

 constants of the pth mode are given approximately by 



_ pird /we _ pt 

 " ~ 27\ y 2^ ~ "a 



/3 



pird 



we 

 97i 



(456) 

 (457) 



Trif 

 9 



■zg a y g 



where 2Ti (= 2nti) is the total thickness of conducting material in the 

 stack of thickness a. To this approximation the attenuation and phase 

 constants are equal, and are proportional to the square root of frequency 

 We note that at very low frequencies, 



7^ 2ip~T^d^(j:e 1 dp-ir-e/fii 



2 



0-1 



^Tlg 



tcofjLigi 



2-2 



ag 



(458) 



which will be very small compared to unity for lines of all reasonable 

 dimensions, in agreement with our assumption (ii) above. 



