GENERAL CIRCUIT CONSIDERATIONS 233 



For a series of resonators dg long with infinitely thin walls E?/fi^P will be 

 less than the values given by (5.24) and (5.25) by a factor M"^'^. This is 

 plotted vs. dg in Fig. 5.10. 



5.4c Fixed Gap Spacing 



Suppose it is decided in advance to put only one gap in a length specified 

 by the transit angle dt . How wide should the gap be made, and how much 

 will F?/^^P be reduced below the value for very thin resonators and infi- 

 nitely thin walls? 



Let us assume that all the stored energy is energy stored between parallel 

 planes separated by the gap thickness, expressed in radians as 6 or in dis- 

 tance as L 



9t = i3e 



dg = /3L 



Here ^ is the gap spacing and L is the spacing between resonators. 



From Section 4.4 of Chapter IV we see that if V is the gap voltage, the 

 field strength E is given by 



E = MV/L 



The stored energy per unit length, W, will be 



W = W^VyiL (5.30) 



Here Pf^o is a constant depending on the cross-section of the resonators. 

 Thus, for unit field strength, the stored energy will be 



W = WoL/m^ 



(5.31) 

 W = Wo(d^/dg)(dg/2y/smHdg/2) 



We see that Wo is merely the value of W when dt = 9g and dg = 0, or, 

 for zero wall thickness and very thin resonators. Thus, the ratio W/Wo re- 

 lates the actual stored energy per unit length per unit field to this optimum 

 stored energy for resonators of the same cross section. 



For dt < 2.33, W/Wo is smallest (best) for dg = dt (zero wall thickness). 

 For larger values oi dt , the optimum value of dg is 2.33 radians and for 

 this optimum value 



(Wo/Wy = (lASO/dtY'^ (5.32) 



If 0i < 2.33, it is thus best to make dg = dt. Then {F?/l3^Py'^ is re- 

 duced by the factor [sm{d/2)/{d/2)Y'\ which is plotted in Fig. 5.10. If 

 dt > 2.33, it is best to make d = 2.33. Then {E?/0^Pyi^ is reduced from the 



