NONLINEAR ELECTRON-WAVE INTERACTION IN CROSSED ELECTRIC AND MAGNETIC FIELDS. TECHNICAL REPORT NO. 39
A general two-dimensional large-signal theory was developed for electron- wave interaction in crossed d-c electric and magnetic fields. The six nonlinear integrodifferential equations were adapted to study thin- and thickstream forward- and backward-wave interactions and interactions with multiple streams in crossed fields. The large-signal equations were derived by the self-consistent field method with the space-charge fields evaluated by means of a three-dimensional Green's function in an equivalent rectangular tube, the walls of which coincided with the sole, the anode plane, and the two sole end hats. The error involved in the space-charge field expressions due to replacing the slow-wave structure by a smooth conducting electrode is shown to be negligible. The space-charge fields were simplified to the two-dimensional case in terms of a weighting function dependent upon the ratio of the beam width in the direction of the magnetic field to the spacing between the end hats. The electron stream at the input to the device was divided into a number of layers, each effectively representing approximately 4 to 5% of the interaction region. The large-signal equations were solved on a high-speed digital computer for the electron trajectories, the r-f phase lead, and gain characteristics by following the individual charge elements into which the stream is divided. The effects of magnetic field, space-charge forces, injection velocity parameter, and stream thickness on the interaction were studied for both Brillouin and nonslipping laminar streams. The retention of acceleration terms in the equations of motion was necessary to account for the excitation of cyclotron waves, which are important for cases where omega / omega D < 5. The improvement in phase focusing with the inclusion of space-charge forces resulted in an improvement of the interaction. Gain characteristics for nonlaminar streams are presented for varying degrees of nonlaminarity at the input. The growth factor for thick Brillouin streams is somewhat reduced from that for thin beams due to a wide velocity variation which proved a hindrance to phase focusing. The start-oscillation length of a backward-wave oscillator was determined for varying r-f power levels and found to he shortest for a power level such that there is very little electron collection on the r-f structure. In the double-stream crossed-field interaction, the increased growth parameter was due to the coulomb force between bunches in the two streams, causing the upper bunch to move closer to the circuit and thus enhancing the growth mechanism. The small-signal equations developed for the positive-sole crossed-field interaction indicated the excitation of beating velocity waves on the electron stream under such an interaction. The "complementary behavior" of positive-sole crossed-field interaction with normal (negative-sole) interaction was established on the basis of small-signal equations and some of the results were compared with those obtsined from the large-signal equations adapted to the positive-sole case. On the basis of a small-signal premodulation analysis it was found that the drift regions are fairly broadband in phase characteristics. The phase-length characteristics were obtained for maximum growing-wave excitation at the output of these regions. (auth)
- Research Organization:
- Michigan. Univ., Ann Arbor. Research Inst.
- DOE Contract Number:
- NO. DA-36-039 SC-78260
- NSA Number:
- NSA-15-024075
- OSTI ID:
- 4032767
- Report Number(s):
- AD-249933
- Resource Relation:
- Other Information: Orig. Receipt Date: 31-DEC-61
- Country of Publication:
- United States
- Language:
- English
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