MAGNETRON AS GENERATOR OF CENTIMETER WAVES 211 



uniform geometry, the voltage is maximum at the input end and tlie current 

 is maximum at the shorted end, each varying sinusoidally to a node at the 

 other end of the line. 



The frequency of resonance of a section of terminated uniform line is thus 

 determined by its length. If the geometry of the line is nonuniform, the 

 frequency of resonance may be determined by the solution of Maxwell's 

 equations with the appropriate boundary conditions. In general, this 

 procedure is involved and tedious, however. One may get a reasonable 

 idea of the values of coo and Fo by assuming the half of the resonator near 

 the open end to be capacitive only, the half near the closed end inductive 

 only, and calculating C and L by appHcation of elementary formulas to an 

 equivalent parallel plate capacitance and a single turn sheet inductance of 

 height h and proper cross sectional area. In the case of geometry like that 

 of Fig. 21 (c), the division of the resonator on this basis is obvious. 



A line of physical length T, less than X/4, may be made to resonate at the 

 frequency c/\ by connecting across its input end a lumped capacitive sus- 

 ceptance, of magnitude, wC, equal to that of the inductive susceptance of 



the line, I'u cot — — [see equations (18) and (26)]. In like manner, reso- 



A 



nators like those of Fig. 21 (b) and (c), whose physical length is less than 

 X/4, may be considered to be made up of an inductive section of uniform line 

 across which additional capacitance has been inserted near the open end, 

 bringing the frequency of resonance to c/X. In Fig. 21, the three resonators 

 of different physical lengths all resonate at the same frequency. 



In addition to the resonant frequency of a resonator of distributed con- 

 stants, one may define its Qs and characteristic admittance and Hnk these 

 to the rate of change of susceptance and energy storage capacity at reso- 

 nance as was done for the circuit of lumped constants. Resonators of 

 different geometry but of the same resonant frequency dififer in character- 

 istic admittance and loss conductance and hence in the Qs and the amount 

 of energy which can be stored with unit voltage impressed across the in- 

 put end. Of the resonator types shown in Fig. 21, the slot type has the 

 largest admittance, the vane type, the smallest admittance, with the hole 

 and slot type intermediate. 



6. Resonator Systems 



6.1 Two Coupled Resonators: The resonator system of tlie magnetron 

 oscillator consists of a number of individual resonators of distributed param- 

 eters, machined into the anode block. As the simplest case of a system of 

 coupled resonators, consider that having two resonators which are coupled 



exist between oppositely traveling waves on a lossless line for constructive interference. 

 These considerations are similar to those employed later in the discussion of the mode? 

 of oscillation of the magnetron resonator system as a whole. 



