FILTER-TYPE CIRCUITS 



197 



p - ^ ^ n si"h y{d - y) j^, 



±Lz — —J — Ho r — -. — e 



coe cosh yd 



/3o = /3- - T 

 At y = we have from (4.23) and (4.12) 



E, = -j - Hoe~'^' tanh yd 



E, = -i ^ ^oe"'^' tan /3o/^ 

 coe 



Hence, we must have 



yh tanh {{d/h)yh) = /5o/? tan /Jq/^ 



(4.23) 

 (4.24) 



(4.25) 

 (4.12) 



(4.26) 



Fig. 4.5 — The transverse mode of the circuit of Fig. 4.3 exists in this circuit also. 



Here we have added parameter, (d/h). For any value of d/h, we can obtain 

 yh vs f^oh; and we can obtain (Sh in terms of yh by means of 4.24 



0h = ({yhY + (l3ohYy" 



We see that for small values of jSah (low frequencies) 



7- = (I'/d) 0l 



1^ ^ 1^0 



h + d 



(4.27) 



(4.28) 

 (4.29) 



If we examine Fig. 4.5, to which this applies, we find (4.28) easy to explain. 

 At low frequencies, the magnetic field is essentially constant from y = d 

 to y = —h, and hence the inductance is proportional to the height h + d. 

 The electric field will, however, extend only from y = to y == ^; hence 

 the capacitance is proportional to \/d. The phase constant is proportional 

 to \/LC, and hence (4.29). At higher frequencies the electric and magnetic 

 fields vary with y and (4.29) does not hold. 



We see that (4.26) predicts infinite values of y for j3h = -kJI. As in the 

 previous cases, cutoff occurs at ,3^ = tt. 



