RECENT ADVANCES IN SIMULATION OF MAGNETRONS AND CROSSED-FIELD AMPLIFIERS. Abstract

Transcription

1 RECENT ADVANCES IN SIMULATION OF MAGNETRONS AND CROSSED-FIELD AMPLIFIERS George E. Dombrowski 69 Birchwood Heights Road Storrs, Connecticut (860) Abstract Various improvements and enhancements of the author's earlier code (IEEE Transactions on Electron Devices, ED-35, 1988) are presented. The details of the anode vane geometry are evaluated by application of Hockney's method using Buneman's cyclic reduction. The results of this enhancement show a significant improvement over the previously used solid anode approximation. The orbit calculation has been reformulated to allow for relativistic mass increase. It has simplicity that makes it equally adapted to ordinary voltage levels. The previous code requires large storage and computation time for systems with large numbers of rf nodes. Study of the response functions shows that values are important for only a few of these: those for sources close to the test site. Reformulation reduces the data storage and computation time by a considerable fraction. This makes it feasible to handle both the 40-vane and the 104-vane Northeastern University amplifiers. I. INTRODUCTION This paper describes three improvements in the author's computer program [1] for the simulation of magnetrons and amplifiers. Some remove limitations on accuracy. Others improve the efficiency of the computer code - a matter of importance for execution on small, 'personal' computers. II. STATIC FIELD OF THE ANODE VANES Earlier simulation calculations [2] disclose the substantial interaction between the stream and the space between vanes, yet they ignore the details of the electric field at the vane tips. They use the logarithmic potential of a solid anode. This may be justified for electrons in synchronous circular orbits, as in small-signal conditions. It fails, however, under the large-signal conditions in the spokes and especially at the terminus of the orbits. Furthermore, vanes of diverse shape have been used in both magnetron and amplifier. The anodes have often been rods rather than vanes. Vane shaping has been shown to improve device performance, likely related to secondary-electron loading [2] and multipactor 236

2 between anodes [3]. Clearly this aspect must be considered in simulation. Fig. 1. Vane geometry in real and transformed space. Buneman's Cyclic Reduction The cyclic reduction calculation can be used to solve the Laplace or Poisson equation on a binary mesh (having numbers of cells in each direction equal to a power of two). The region must have regular boundaries, as ABCDEFA in Fig. 1. The radial bounds are equipotentials; the azimuthal boundary conditions are Neumann contours ( v/ θ=0), as apply at AB and EF in view of the mirror symmetry there. It would appear that this method fails to accommodate the vane contour BB'D. Hockney [4], however, devised a method to do so. The Method of Hockney Hockney's method may be described by realizing that the mesh charges may be that of free charge - the electrons - and also the surface charges on the vanes. It becomes a task to determine the surface charge distribution that corresponds to the applied anode potential. The method of determining this charge proceeds as follows. Cyclic reduction is first used to solve the Laplace equation in the regular region with the anode voltage V^, at the outer wall, the cathode potential being zero. linear matrix The potentials on the M vane contour points form a [V] = V j, j=l,m (1) and are in general different from the anode potentials, defining a deficiency [Vx] = Vb -[V] (2) The problem becomes one of determining surface charges to eliminate these differences. 237

3 In the next phase a unit charge is placed at one of the vane mesh points (the i-th). Cyclic reduction is executed, resulting in vane mesh point potentials having the nature of reciprocal capacitances that can be designated s ij. This is repeated for all source points i. The complete array [s ij ] defines a reciprocal-capacitance matrix, [S]. Its inverse is the capacitance matrix [C] which expresses the coupling between points on the vane contour. It is required that an array [Q]=[q i ] must be such that [Q] [S] = [Vx] (3) eliminating the potential deficiency noted above. Eq. 3 has the solution [C] [Q] [S] = [Q] = [C] [Vx] (4) The final step is to assign [Q] to the vane contour and execute the cyclic reduction once more. The resulting potentials are the exact solution of Laplace's equation with the vanes at V b. The computational cost of this procedure is the execution of cyclic reduction M+2 times, the inversion of the M-square matrix [S], and the matrix multiplication to compute [Q]. Fortunately, it needs to be done only once as a preliminary calculation. During the simulation the static field is computed by interpolation, taking account of the location of the electron with respect to the vanes. This technique provides the static, space-charge-free electric field. It continues to rely on the solid anode model for space-charge field evaluation. The error in so doing is considered small, since the spacecharge fields are small in comparison to the vane-tip and rf fields. The Poisson Equation As part of the simulation process, the space-charge can be assigned to the cyclic reduction mesh and the discrepancy matrix [Vx] obtained. The surface charge [Q] is then computed from Eq. 4, the inverted matrix [C] having been saved. A second cyclic reduction with space- and surface charges in place yields the space-charge potential from which the field is evaluated. The computation burden is thus a matrix multiplication and the second cyclic reduction. This part of the procedure is applicable to the simulation of magnetrons in the pi mode. It is not practical for the amplifier, for which the previous technique can reasonably be used. This technique for magnetron simulation has only recently been implemented. Simulation Results Simulations were made for the type 4J50 magnetron, with the solid anode and with the detailed vane fields. The results are compared with each other and with measurements made by J. F. Hull [6]. Table 1 shows these results for magnetic field B = 5500 Gauss and with (pulsed) anode voltage V b =20.9 kv. This is the nominal operating point. The loaded Q is

4 Table 1. 4J50 Magnetron: B= 5500 Gauss. The solid-anode simulation shows results - anode current and rf power - that are surprisingly greater than the vane-field simulation. These in turn are much closer to the measured result of Hull. Results for higher anode voltage (not shown here) are even more dramatically different. The inference is clearly that the vane fields serve a very useful role. Further comparison with the simulation and measurement is made in Table 2, in which the slope of the Gauss line is determined. Table 2. 4J50 Magnetron: Gauss line slope data. The slope data are in good agreement here, providing further confidence in the simulation process. III. RELATIVISTIC ORBIT ALGORITHM Simulation of the relativistic magnetron [7,8] has called for suitable modification of the orbit calculation. As in the past, rf magnetic fields are assumed to have negligible effects. Also as in the past, the trajectory variables are expressed as power series in 'cyclotron time', T=ω c t. As may be needed, the orbit during a simulation interval may be subdivided into several smaller intervals in order that T«l for each. This ensures the convergence of the series. For each charge in the stream, a radially aligned Cartesian coordinate system is assigned as shown in Fig

5 Fig. 2.Coordinates for trajectory calculation These energy relations serve to monitor the orbit calculation. In particular the X and Y coordinates are accepted as computed above, and the velocity components are modified to preserve energy balance. This algorithm performs well for simulation of the MIT A6 relativistic magnetron, for which 7 may at times be as high as 3. It is used also for conventional magnetrons; in the 4J50 the relativistic mass increase may be as much as 2 percent. 240

6 IV. AMPLIFIER: RESPONSE FUNCTIONS u An essential part of the simulation of the cfa is the determination of the behavior of the rf network. This system consists of a number N of nodes corresponding to the anode vanes. At any discrete time t i. The state of the network is defined by the set of rf voltage and their time integrals, vdt V i, i =1,N (16) u i, i =1,N (17) The state of the rf network at the end of a time interval At is determined by the state at its beginning and also depends on the extent of electron stream excitation, as manifested by the electronic currents I, entering the vanes during the interval I i, I =1,N (18) There are thus two variables (v and u) to define the state, and three 'stimuli' or sources of network excitation: the initial v s and u s, and the electron current. For the amplifier, rf input signals must be added. Simulation calculations rely on a prior computation of network response functions, which are the state variables v and u resulting from independent unit stimuli. Thus, RV denotes a voltage response, i.e., an rf voltage produced by a stimulus; RU denotes a value of the time-integral u produced by a stimulus. Indices denote the source node and the node where the response appears. Lastly, subscripts are used to denote the type of stimulus producing the response. For example, The complexity of these response functions leads to limitations, especially with small computers. First, the one-time calculation of the functions can be tedious. Secondly, their use in the simulation process requires considerable memory storage. Finally, for each stored function there must be at least one multiplication and one addition; this can result in slow calculation. Each of these factors becomes more serious because the time required increases as the square of the number of vanes, N. Further study of the nature of the response functions discloses what should already be apparent from the nature of the rf network as a ladder. Table 3 lists, for a 40-vane network, a typical response function, viz., RV v (12,i). 241

7 Table 3. Response of a 40-vane network These and similar data show that for short time steps the slow-wave system responses are negligible outside the range of 5 vanes on either side of the stimuli. They make no contribution to the signals, and simply waste computer time. The computer code has accordingly been revised to eliminate them. Table 4 presents further insight into the nature of the response functions. Table 4. Response functions of a 40-vane network These data show that the response functions are principally identical functions depending on i-j. Thus the functions RV v (i,j), i=l,n, j=1,n having N 2 values, can be replaced by a simpler function RV v ( i-j ), i-j = 0,5 with 6 values. Exceptions to this rule are apparent at the ends of the rf network, where the response is influenced by connections to source and/or load. V. Conclusions Simulation of magnetrons has been improved by accurate calculation of the anode vane fields and also by accounting for relativistic electron dynamics. The effectiveness of the simulation code for amplifiers has been enhanced by use of the response functions of uniform rf networks. 242

8 VI. Acknowledgment The author is pleased to recognize the debt he owes to the University of Michigan - especially to Professor William G. Dow - for his education in physical electronics. VI. References 1. G. E. Dombrowski, "Simulation of Magnetrons and Crossed-Field Amplifiers,"IEEE Trans. Electron Devices, vol. ED-35, p. 2060, G. E. Dombrowski, "Computer Simulation of Primary and Secondary Anode Loading in Magnetrons," IEEE Trans. Electron Devices, vol. ED- 38, p. 2234, J. R. M. Vaughan, "Observations of Multipactor in Magnetrons,"IEEE Trans. Electron Devices, vol. ED-15, p. 883, Buneman, "A Compact Non-iterative Poisson Solver," SUIPR Report No. 294, Stanford University Institute for Plasma Research, R. W. Hockney, "The Potential Calculation and Some Applications," in Methods in Computational Physics, vol. 9, p. 162, (1970); Academic Press, New York 6. J. F. Hull, "Crossed Field Electron Interaction in Space Charge Limited Beams", D.E.E. Dissertation at Polytechnic Institute of Brooklyn, T. E. Ruden, "Relativistic Magnteron Interaction,"IEEE Conference on Plasma Science, Conf. Rec., (1982) 8. T. E. Ruden and G. E. Dombrowski, "Simulation of High-Power Relativistic Magnetron Interaction," IEEE Conference on Plasma Science, Conf. Rec., (1991) 237