Magnetron Theory

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Magnetron theorySpilios Riyopoulosa)Science Application International Corporation, McLean, Virginia 22102

Received 28 August 1995; accepted 17 November 1995 A guiding center uid theory is applied to model steady-state, single mode, high-power magnetron operation. A hub of uniform, prescribed density, feeds the current spokes. The spoke charge follows from the continuity equation and the incompressibility of the guiding center ow. Included are the spoke self-elds DC and AC , obtained by an expansion around the unperturbed zero-spoke charge ow in powers of /V 1 , , and V 1 being the effective charge density and AC amplitude. The spoke current is obtained as a nonlinear function of the detuning from the synchronous Buneman Hartree, BH voltage V s ; the spoke charge is included in the self-consistent denition of V s . It is shown that there is a DC voltage region of width V V s V 1 , where the spoke width is constant and the spoke current is simply proportional to the AC voltage. The magnetron characteristic curves are at in that range, and are approximated by a linear expansion around V s . The derived formulas differ from earlier results J. F. Hull, in Cross Field Microwave Devices, edited by E. Okress Academic, New York, 1961 , pp. 496 527 in a there is no current cutoff at synchronism; the tube operates well below as well above the BH voltage; b the characteristics are single valued within the synchronous voltage range; c the hub top is not treated as virtual cathode; and d the hub density is not equal to the Brillouin density; comparisons with tube measurements show the best agreement for hub density near half the Brillouin density. It is also shown that at low space charge and low power the gain curve is symmetric relative to the voltage frequency detuning. While symmetry is broken at high-power/high space charge magnetron operation, the BH voltage remains between the current cutoff voltages. 1996 American Institute of Physics. S1070-664X 96 00303-4


Although magnetrons are the earliest developed sources of high-power coherent radiation, they have remained the least well understood. The theoretical description is complicated by the various time scales involved in the wave particle interaction, the high space charge, and the fundamentally two-dimensional nature of the instability: the growth rate is tied to the eld gradients both DC and AC transverse to the wave propagation direction. Among other differences, magnetrons convert potential energy to radiation, opposed to the rest of the unbound electron devices traveling wave tubes, cyclotron masers, free electron lasers , which convert kinetic energy. Furthermore, it has been recently observed experimentally, and shown theoretically, that, in the low gain, low space charge regime, the gain curve is symmetric relative to frequency detuning from synchronism; most other devices exhibit antisymmetric gain versus detuning. The rst theoretical efforts in parametrizing high-power magnetron operation resulted in the scaling laws,1 introduced by Slater 1943 and documented by Collins 1948 . They comprise a set of equations that scale the operation parameters for a desired tube design based on the operation parameters for an existing design. In effect, one deals with similarity transformations among members of the same tube family. One cannot predict the parameters for a novel design, if the latter does not extrapolate from the existing tubes. The next systematic effort on theoretical modeling wasa

undertaken by Hull.2 The magnetron characteristic curves obtained in Refs. 23, relating the DC anode current I and the RF power P to the applied DC voltage V at given RF frequency, are of the form I AV V V s 2 , P BV V V s2

1 1 CV 1/2 V V s . 2

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The current I is positive for current reaching the anode. The constants A, B, and C depend only on the anode circuit parameters, vane geometry, and RF frequency, while V s is the synchronous BunemanHartree, BH voltage, inducing E0 B0 velocity at the top of the hub that matches the RF phase velocity. The least satisfying feature of Eqs. 1 2 is that both anode current and RF power thus RF gain go to zero when the resonance condition V V s is met, contrary to experimental results and particle simulations of crossed eld devices. In addition, the operation voltage V is a doublevalued function of the current I according to 1 . As depicted in Fig. 1 there exist two, nearly symmetric operation voltages around V s at the same current I. To exclude this possibility, one must either reject operation below V s on theoretical grounds, or postulate that the second branch V V s is unstable and thus not observed during actual operation. Yet magnetrons and the related crossed-eld ampliers CFAs are known to exhibit stable operation with single valued V I characteristics well below V s , which cannot be accounted for by 1 2 . Current cutoff is also observed at voltages above the BH voltage; Eq. 1 provides no current cutoff on the high-voltage side. 1996 American Institute of Physics 1137

Phys. Plasmas 3 (3), March 1996



FIG. 1. The typical magnetron V I curve according to earlier results, given by Eq. 1 , against experimental operation points Xs , taken from Ref. 2, Fig. 6.44 Litton 4J52 at Q L 166 .

has been shown experimentally7,8 and theoretically9,10 that the low space charge, small signal operation exhibits symmetric gain versus voltage detuning V V s , with maximum gain at resonance. The symmetry is broken at high-power, high space charge operation, but the fact remains that magnetrons operate well below, as well as above, the Hartree voltage. The range of below-synchronism operation becomes more extended when the spoke charge effects are included in the denition of the Hartree voltage; that pushes V s to higher values, moving synchronism farther up from the low-voltage current cutoff point. Rather than a point-by-point comparison between the present approach and Refs. 23, some of the main differences that are responsible for the new results will be emphasized. First and most obvious is the assumption in Refs. 23 that the hub surface acts as a perfect conductor for the RF; the RF mode structure is then taken similar to the vacuum structure, with the hub surface treated as a virtual cathode. To see why this is misleading, consider the linear equation for the small amplitude RF mode structure in the presence of a uniform hub,11142 p 2

x A guiding center uid approach has recently been developed and applied4 6 in the study of CFAs with distributed emission thermionic and/or secondary cathodes. Simulation results,6 from the numerical implementation of the guiding center GC uid model, have showed good agreement between the GC model predictions and experimental characteristic curves for the SFD 262 tube. In the present work we apply the guiding center uid approach to address the performance of the magnetron. The GC uid equations can actually be solved analytically for a magnetron at steady-state operation, taking advantage of the uniformity and periodicity around the tube. The results lead to the following new characteristic equations to replace 1 2 : I V 1 B V Vs , Rs V2 ss

1 k2 1

ku 0 x

2 2 p 2

2 p

V 1 x2 p 2 p 2

ku 0 x x




kV 1 ku 0 x

ku 0 x


2 p






1 A I Is ,


where A, B, and are constants, R has units of resistance, and ( s ) stands for values at synchronism. These equations hold within the voltage range V V s V 1 kD/sinh(kD) around the synchronous voltage; operation in that range is characterized by the maximum achievable spoke current at a given AC voltage. Outside that range the departure from synchronism limits the current that reaches the anode. The current cutoff voltages V c , located well above and below V s , are also estimated using the GC theory. The central result from the GC uid model is that the operation range for magnetrons as well as CFAs is centered around the BH voltage V s ; nite AC power and anode current are drawn at synchronism V V s . Although zero gain at synchronism applies to other microwave devices traveling wave tubes, TWTs; free electron lasers, FELs , recently it1138 Phys. Plasmas, Vol. 3, No. 3, March 1996

where u 0 (x) cE 0 (x)/B 0 is the local drift velocity and 2 2 p 4 e n 0 (x)/m e . For a uniform hub density truncated at x d, one has dn 0 /dx n 0 (x d) and the coefcient on the ku 0 (d)] right-hand side RHS becomes singular, 1/[ , when the drift velocity at the hub top approaches synchronism. The singularity forces the local RF voltage to zero, V 1(d) 0, in order to balance the left- and right-hand side of 5 . However, when the hub top is nonsynchronous, ku 0 (d)] 0, the RF voltage proles V 1(x) exhibit a [ maximum at the hub surface, as one should expect, since that is where the surface charge perturbation peaks. It has been recently shown15 that the singularity at resonance is an artifact of the usual perturbative expansion, and is removed with the proper mathematical treatment of the resonant uid motion. Both the corrected linear RF voltage V 1 and the parallel RF eld E 1y V 1/ y exhibit a local maximum at the hub top, regardless of synchronism. In the nonlinear regime, including the effects of fully developed spokes, the local AC maximum is shifted near the spoke center; the AC voltage at the hub top remains nite and, in fact, larger than the empty cavity value. The treatment of the hub surface as a virtual cathode, rooted in the singular behavior of 5 , is removed from the present work. A second major difference fro