GENERAL CIRCUIT CONSIDERATIONS 235 



We see that, for given values of v and Q, decreasing the group velocity, 

 which increases E-/0^P, also increases the attenuation per wavelength. 



5.5a Attenuation of Circuits 



For various structures, Q can be evaluated in terms of surface resistivity, 

 R, the intrinsic resistance of space, v yu/^ ~ >^^^ ohms, and varous other 

 parameters. For instance, Schelkunoff- gives for the Q of a pill-box resona- 

 tor 



1 -1- a/h 



Here a is the radius of the resonator and h is the height. If we express the 

 radius in terms of the resonant wavelength Xo {a — 1.2Xo/7r), we obtain 



(1 + h/a)n 



Here n is the number of resonators per wavelength (assuming the walls 

 separating the resonators to be of negligible thickness); thus 



n = h/\ = (VXo)(c/^) (5.41) 



From (5.40) and (5.38) we obtain for a series of pill-box resonators 



db/ wavelength = ^M{R/\^^e){c/vg){\ + h/a)n (5.42) 



In Appendix III an estimate of the Q of an array of fine half-wave paral- 

 lel wires is made by assuming conduction in one direction with a surface 

 resistance R. On this basis, Q is found to be 



Q = (VMR){v/c) (5.43) 



and hence 



db/wavelength = 27.3{R/\/M{c/v,) (5.44) 



For non-magnetic materials, surface resistance varies as the square root 

 of the resistivity times the frequency. The table below gives R for copper 

 and db/wavelength for pill-box resonators for h/a « 1 (5.42) and for wires 

 (5.44) for several frequencies 



f, mc R, Ohms (db/wavelength)/ (c/vg) 



Pill-box Resonators Wires 



i.i X 10-^w 10.3 X 10-^ 



6.0 X 10-% 18.1 X 10-^ 



10.4 X 10-% 32.6 X 10-^ 



In Section 3.3b a circuit made up of resonators, with a group velocity 

 .031 times the phase velocity, was discussed. Suppose such a circuit were 

 2 Electromagnetic Waves, S. A. Schelkunoff, Van Nostrand, 1943. Page 269. 



