Abstract: A comparative study is given on electron bunching in transit-time tubes with uniform and stepped RF electric-field profiles. It is shown that the stepped ...
P1-7: Efficiency Enhancement in Transit-Time Microwave Tubes by RF - Field Stepping Joaquim J. Barroso Associated Plasma Laboratory, National Institute for Space Research - INPE 12227-010 São José dos Campos, SP, Brazil p p / mc is the normalized momentum, g the where ~ normalized field amplitude and f(z) specifies the electricfield axial distribution. For the TM010 mode the field is uniform, f(z) =1, and to describe the axial distribution of the S mode we use the analytical function profile f ( z ) tanh[100( z d / 2)] as illustrated in Fig. 1.
Abstract: A comparative study is given on electron bunching in transit-time tubes with uniform and stepped RF electric-field profiles. It is shown that the stepped profile provides a two-fold enhancement in the conversion efficiency in comparison with that obtained with a longitudinally uniform electric-field distribution. This improvement arises from a phase synchronization process due to the action of RF electric fields of reversed signs along the interaction space.
profile amplitude
1
Keywords: electron bunching; microwave generation; conversion efficiency; transit-time microwave tubes Introduction Previous conceptual design studies [1-2] have demonstrated that in the monotron (the simplest of the microwave tubes) the conversion efficiency is essentially limited to 20%. This efficiency is achieved in circular cavities operating in the TM010 mode for which the axial electric-field distribution is uniform along the interaction space. Nevertheless, this limitation is overcome by using a cavity bisected by a diaphragm or a conducting foil [3-4] so as to support a Smode standing wave in the partitioned cavity thereby yielding a two-fold increase in the conversion efficiency, thus providing a 40% efficiency level [5]. These previous results were obtained through optimization schemes on the basis of particle-in cell (PIC) simulations without much emphasis on the physical mechanisms underpinning the efficiency enhancement process. By examining the kinematical ballistic bunch in both interaction systems, the present paper shows that the efficiency enhancement arises from an extra group of bunched electrons that is formed in the second region where the electric field reverses sign.
-1 0.0
0.5
1.0
1.5
2.0
2.5
3.0
normalized axial distance zZ/c
Figure 1. Stepped field profile represented by the ~ ~ function f ( ~z ) tanh[100( ~z d / 2) with d 2.90 . Setting g = 0.045 (to be justified later) for the uniform [f(z)=1] and stepped profiles the force and trajectories equations are numerically solved subject to the initial p (I0 ) p0 where the initial conditions ~z (I0 ) 0 , ~ momentum p0 is related to the injection energy through p ( 0)
[511 W0 (keV )]2 5112 / 511 .
(2)
For 33 electrons at W0=10 keV with I0 uniformly distributed over 2S, the exit energy-entrance phase plots are displayed in Fig. 2 for both profiles. We see that under the action of the stepped field profile, seven electrons (#15-#21 indicated by blue squares) out of a total of 33 remain as fast particles. Also we note that electrons #6-#9 (red circles) reach energies above the 10 keV input energy while interacting with uniform-profile electric field. By contrast, this group of electrons is brought to a lower energy level (~2.5 keV) when interacting with the stepped-profile field. Also we note that electrons #6-#9 (red circles) reach energies above the 10 keV input energy while interacting with uniform-profile electric field. In this way, the stepped profile with g=0.045 converts a group of 13 fast dispersed electrons into a bunch of 13 slow particles.
Efficiency and Electron Energy The conversion efficiency is calculated by numerically integrating the relativistic equation of motion for an ensemble of electrons injected at given energy, W0, streaming along a cavity of length d, and averaging out the work done on the beam electrons by the RF fields during an oscillation period [6]. Assuming one-dimensional motion and that the space-force forces are sufficiently weak such that the cavity normal modes are not modified by the presence of the beam, the normalized force equation is [6] ~ ~ d~ p / d t g f ( z ) cos( t I0 )
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0
113
Table 1. Injection energy, field amplitude, interaction length and the resulting conversion efficiency
20 stepped profile uniform
exit energy [keV]
#15 #6
#21
#9
Field amplitude g (normalized)
Interaction length (normalized)
10
0.045
2.90
43.50
20
0.068
3.96
44.50
50
0.112
5.74
45.00
Injection energy (keV)
15
Conversion efficiency (%)
10
5
0 0.0
0.5
1.0
1.5
2.0
Conclusion In the absence of space-charge forces a kinematical formulation has been developed to examine the ballistic bunching in a one-dimensional RF fields for monotrons with uniform and stepped electric-field profiles. Describing the injection of electrons at a prescribed initial energy into the interaction region in terms of their entrance phases relative to a cycle of the driving oscillation, this formalism has given a good insight into energy exchange of an electron stream with a standing-wave electric field. For the input energy of 10 keV a maximum conversion efficiency of 43.50 % is achieved at the optimum values of ~ g = 0.045 and interaction length, d 2.90 , by noting that that the optimum length is uniquely determined from the beam’s input energy. Alternatively, input energies of 20 keV and 50 keV would yield optimum interaction lengths of 3.96 and 5.74, corresponding to field amplitudes 0.068 and 0.112, respectively, and with conversion efficiencies of about 44.8 %
entrance phase/S
Figure 2. Exit energy as function of entrance phase for 33 electrons after the interaction with uniform (red circles) and stepped (blue circles) electric field profiles. It is apparent that two groups of bunched electrons (with entrance phases in the ranges 0.00.5S and 1.5S-2.0S ) For the uniform profile, the efficiency (Fig. 3) attains the maximum of 18.20 % at g=0.053 and drops to zero at g=0.074, a value far below the corresponding threshold amplitude gth!0.194 for electron reflection to occur [6]. By contrast, in addition to a higher efficiency of 43.50% at g=0.045, for the stepped profile the efficiency falls to zero at g=0.069, very close to the threshold value of 0.067. Other conversion efficiency results are given in Tab. 1 50
efficiency [%]
40
Acknowledgment This work is supported by the National Council for Scientific and Technological Development (CNPq), Brazil.
stepped profile uniform
30
20
References 1 J. J. Müller and E. Rostas, Helvet. Phys. Acta,, vol. 13, no. 3, pp. 435-450, 1940.
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2. Y. K. Yulpatov, Radiophys. Quantum Electron., vol. 13, pp. 1374-1378, 1970.
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3. B. M. Marder, M. C. Clark, L. D. Bacon, J. M. Hoffman, R. W. Lemke, and P. D. Coleman, IEEE Trans. Plasma Sci. , vol., 20, pp. 312-331, June 1992.
0.08
field amplitude qE/(mZc)
4. R. W. Lemke, J. Appl. Phys., vol. 72, pp. 4422-4428, Nov. 1992.
Figure 3. Efficiency dependence on the electric-field amplitude for uniform (dashed line) and stepped (solid line) profiles with optimum interactions lengths of ~ ~ d 1.40 and d 2.90 , respectively, at initial energy of 10 keV.
5. J. J. Barroso, IEEE Trans. Plasma Sci., vol. 32, no. 3, pp. 1205-1211, June 2004 6. J. J. Barroso, IEEE Trans. Electron Devices, vol. ED56, no. 9, pp. 2150-2154, Sept. 2009.
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