FILTER-TYPE CIRCUITS 205 



We will go further and assume that near resonances these values of Bt 

 and Bl behave like the admittances of shunt resonant circuits, as indicated 

 in Fig. 4.15. Certainly we are right by our definition in saying that 5r === 

 at cor , and Bl = at wi, . We will assume near these frequencies a linear 

 variation of Bt and Bl with frequency, which is very nearly true for shunt 

 resonant circuits near resonance* 



Bt = 2Cr(co - cor) (4.46) 



Bl = 2Cx.(co - coO (4.47) 



Here Ct can mean twice the peak stored electric energy per section length 

 for unit peak voltage between the top of the guide and the top of the ridge R 

 when the structure resonates in the transverse mode, and Cl can mean twice 

 the stored energy per section length L for unit peak voltage across the top 



Fig. 4.16 — Longitudinal and transverse susceptances which give zero radians phase 

 shift at the lower cutoff (w = wt) and ir radians phase shift at the upper cutoff (w = cot). 



of the slot when the structure resonates in the longitudinal mode. 



In terms of Bt and Bl , expression (4.36) for the phase angle d becomes 



We see immediately that for real values of 6 (cos 6 < 1), Bt and Bl must 

 have opposite signs, making the denominator greater than the numerator. 



Figure 4.16 shows one possible case, in which cor < ool • In this case the 

 pass band {6 real) starts at the lower cutoff frequency co = cor at which Bt 

 is zero, cos ^ = 1 (from (4.48)) and ^ = 0, and extends up to the upper 

 cutoff frequency co = wz, at which Bl = 0, cos 6 = —\ and 6 = w. 



* In case the filter has a large fractional bandwidth, it may be worth while to use the 

 accurate lumped-circuit forms 



Bt = corCrCw/wr — wy/w) (4.46a) 



Bl = ulCl(.Wul - wl/«) (4.46b) 



