FILTER-TYPE CIRCUITS 195 



alteration we can have below the boundary is one in which fields in alternate 

 spaces follow a +, — , +, — pattern. Thus, the rapid variations of field above 

 the boundary predicted by (4.14) for values of ^^h which make ^t greater 

 than TT cannot be matched below the boundary. The frequency at which 

 l3i = T constitutes the cutoff frequency of the structure regarded as a filter. 

 There is another pass band in the region x < ^oh < Sir/l, in which the ratio 

 oi Eto H below the boundary has the same sign as the ratio oi Eto H above 

 the boundary. 



A more elaborate matching of fields would show that our expression is 

 considerably in error near cutoff. This matter will not be pursued here; the 

 behavior of filters near cutoff will be considered in connection with lumped 

 circuit representations. 



We can obtain the complex power flow P by integrating the Poynting 

 vector over a plane normal to the z direction in the region y > 0. Let us 

 consider the power flow over a depth W normal to the plane of the paper. 

 Then 



P = 1 f f {E,H* - EyHt) dx dy (4.15) 



I Jo Jo 



Using (4.1) and (4.3), we obtain 



2 Jo W€ 



4 coe7 



(4.16) 



We will express this in terms of E the magnitude of the z component of 

 the field at y = 0, which, according to (4.5), is 



E=^Ho (4.17) 



We will also note that 



coe = coVjUe/ V w/e 



= {<^/c)/VljTe = ^o/V/V^ 

 and that 



VaiA = 377 ohms (4.19) 



By using (4.17)-(4.18) in connection with (4.16), we obtain 



£-//3'P = (4//^oTr)(T//3)' V/Ve (4.20) 



We notice that this impedance is very small for low frequencies, at which 



(4.18) 



