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This article re-examines the Brillouin flow solutions in crossed-field diodes, with applications to magnetrons, magnetically insulated line oscillators (MILOs), and magnetically insulated transmission lines (MITLs). The Brillouin flow solutions are constructed for various geometries, including planar magnetrons, MILOs, and MITLs, cylindrical magnetrons with electrons flowing in the azimuthal direction, cylindrical MITLs and MILOs with electrons flowing in the axial direction, and radial MITLs and MILOs with electrons flowing in the radial direction. A common theme of this analysis is that two main external parameters are used to characterize the Brillouin flow: the anode–cathode voltage ($V_{a}$) and the total magnetic flux within the crossed-field diodes ($A_{a}$). These two parameters are equivalent to the gap voltage and a specification of the degree of magnetic insulation, which is approximately equal to the ratio of the magnetic field to the Hull cutoff (HC) magnetic field. The magnetic flux may be provided externally by a magnet (as in a magnetron) or by the wall currents without an external magnet (as in a MILO or MITL), or by some combination of the two, as in the intermediate case of a magnetron–MILO hybrid. Once these two parameters are specified, the electron flow speed at the top of the Brillouin hub is uniquely determined. This immediately yields the Buneman–Hartree (BH) condition according to the Brillouin flow model, whether it be a planar magnetron or a cylindrical MILO. In so doing, we have obtained, for the first time using the Brillouin flow model, the BH condition for a cylindrical MILO, and we show that the same condition is obtained from the single-particle orbit model. We also found that, in general, the electron current within the Brillouin hub contributes only to a very small fraction of the magnetic flux $A_{a}$ , regardless of the gap voltage $V_{a}$ , thereby correcting an erroneous notion that the electron flow within the crossed-field gap could be responsible for the magnetic insulation. Another counter-intuitive finding is that, for a given degree of magnetic insulation, the Brillouin hub height decreases as the gap voltage $V_{a}$ increases. These conclusions, and other results, were based on the simple, explicit analytic expressions that we have obtained for the Brillouin flow profiles, including the velocity, electron density, as well as the self-magnetic field and the self-electric field profiles due to the Brillouin hub electrons. From these analytical expressions, we deduce useful scaling laws that are applicable to the prevailing cases where the magnetic field exceeds 1.5 times the HC magnetic field, and they are valid in both relativistic and non-relativistic regimes. Thus, these scaling laws show the contrast between magnetrons and MILOs, and a ready assessment of the viability of building a moderate-current MILO, a low-voltage MILO, and a magnetron–MILO hybrid which might combine the advantages of a magnetron and a MILO. The Brillouin flow profiles in a radial MITL are explicitly calculated. Additional issues are addressed. [ABSTRACT FROM AUTHOR]

The recirculating planar crossed-field amplifier (RPCFA) was designed, constructed, and tested at the University of Michigan. The RPCFA was driven by a number of RF sources ranging in frequency from 2.40 to 3.05 GHz and powers of 1 to 800 kW. Pulsed voltage was delivered to the cathode by the Michigan electron long beam accelerator with ceramic insulator (MELBA-C) which was configured to supply pulses of −300 kV, 1–10 kA, with 0.3–1.0- $\mu \text{s}$ pulse lengths. The RPCFA demonstrated zero-drive stability and a bandwidth of 15%. Amplification of microwave signals, at the design frequency of 3 GHz, below 150 kW, was observed with a mean gain of 7.87 dB and high variability, $\sigma = 2.74$ dB. Filtering this data set to only include shots with identical voltage and current profiles yielded a gain of 6.6 ± 1.6 dB. The mean gain increased to 8.71 dB and the variability decreased to $\sigma =0.63$ dB when the injected microwave power increased beyond 150 kW. Peak output powers of nearly 6 MW were achieved with RF breakdown limiting the maximum output power of the device. [ABSTRACT FROM AUTHOR]

The harmonic recirculating planar magnetron (HRPM) is a frequency-agile, multispectral HPM source. The HRPM implements an $L$ -band oscillator (LBO) and an $S$ -band oscillator (SBO) near 1 and 2 GHz, respectively. The novel coaxial-all-cavity-extraction (CACE) method was implemented to extract power from the SBO. The two oscillators demonstrated harmonic frequency locking, where the SBO frequency locked to the LBO second harmonic frequency. The two oscillators are concluded to operate as a damped, driven, harmonic oscillator system. In the locked state, the LBO acts as the driving oscillator, the SBO acts as the driven oscillator, and the coupling mechanism between the two oscillators is the harmonic content in the electron spokes as they propagate directly from the LBO to the adjacent SBO. The two primary variables studied in this system are the LBO harmonic frequency and the SBO quality factor, $Q$. In isolated SBO experiments, the dominant operating state of the magnetron was the $5\pi $ /6 mode. However, when operated in tandem with the LBO, the SBO was forced to operate in the $\pi $ -mode. Output powers in HRPM experiments generated SBO powers of 9.5 ± 1.4 MW at high $Q$ , 19 ± 6 MW at moderate $Q$ , and 28 ± 9 MW at low $Q$. Output powers in the isolated SBO configuration were not significantly different from the HRPM. By experimentally manipulating the hub drift direction and altering the evolution of harmonic content received by the SBO, the locked state was significantly diminished, suggesting that the beam spokes play a crucial role. [ABSTRACT FROM AUTHOR]

A phase-locking model has been developed from circuit theory to qualitatively explain the various regimes observed in magnetron injection-locking experiments. The experiments utilize two continuous-wave oven magnetrons: one functions as an oscillator and the other as a driver. The model includes both magnetron-specific electronic conductance and frequency-pulling parameter. Both time and frequency domain solutions are developed from the model, allowing investigations into the growth and saturation as well as the frequency response of the output signal. This simplified model recovers qualitatively many of the phase-locking frequency characteristics observed in the experiments. [ABSTRACT FROM AUTHOR]

This paper presents experimental results that demonstrate the first known instance of harmonic frequency locking in a magnetron. The prototype crossed-field high-power microwave source, termed the multifrequency recirculating planar magnetron, consisted of two planar cavity arrays having design frequencies near 1 and 2 GHz, respectively, coupled by smooth-bore electron beam recirculation bends. The magnetron was driven by the MELBA-C modified Marx generator using −300-kV pulses supplying up to 3 kA for approximately 300 ns. Using a 0.17-T axial magnetic field, the harmonic frequency-locked state simultaneously produced microwave pulses of 32± 3 MW at 0.984± 0.001 GHz and 13±2 MW at 1.970±0.002 GHz. In addition, changes in the relative phase difference between the magnetron slow wave structures (SWSs) were correlated with changes in the axial magnetic field magnitude. This correlation, along with the results from experiments that tested each individual planar cavity array in isolation, suggested the locking was facilitated through the second harmonic content of the 1-GHz modulated electron beam recirculating from the 1-to 2-GHz SWS. Analysis of shots conducted near 0.17 T showed a ±17° shot-to-shot variation in the average relative phase difference between the SWSs, and the average variation in this phase difference during a locked shot was ±8°. [ABSTRACT FROM AUTHOR]

The recirculating planar crossed-field amplifier (RPCFA) was designed and simulated using the finite-element frequency-domain code ANSYS HFSS and the particle-in-cell (PIC) code MAGIC. The RPCFA is a high-power microwave device adapted from the recirculating planar magnetron, developed at the University of Michigan, Ann Arbor, MI, USA. Electromagnetic (EM) PIC simulations of a planar, meander line, and slow wave structure demonstrated 13.5-dB amplification of a 1.3-MW, 3-GHz signal to approximately 29 MW. The RPCFA is designed to be driven by pulsed power from the Michigan electron long beam accelerator-ceramic insulator, which is currently configured to deliver pulses at −300 kV, 1–10 kA, with 0.3– $1~\mu \text{s}$ pulse lengths. The RF input-drive signal will be provided by an MG5193 magnetron which delivers 5- $\mu \text{s}$ pulses up to 2.6 MW at 3 GHz. EM PIC simulations also demonstrated zero-drive stability of the design and were used to evaluate changes in performance resulting from variations of several experimental parameters. Variation of the drive frequency suggested that the RPCFA is expected to have a 3-dB amplification bandwidth of 300 MHz or 10%. [ABSTRACT FROM AUTHOR]

Including a slow-wave structure (SWS) on the anode in the conventional, planar, and inverted magnetron, we systematically study the linear stability of Brillouin flow, which is the prevalent flow in crossed-field devices. The analytic treatment is fully relativistic and fully electromagnetic, and it incorporates the equilibrium density profile, flow profile, and electric field and magnetic field profiles in the linear stability analysis. Using parameters similar to the University of Michigan's recirculating planar magnetron, the numerical data show that the resonant interaction of the vacuum circuit mode and the corresponding smooth-bore diocotron-like mode is the dominant cause for instability. This resonant interaction is far more important than the intrinsic negative (positive) mass property of electrons in the inverted (conventional) magnetron geometry. It is absent in either the smooth-bore magnetron or under the electrostatic assumption, one or both of which was almost always adopted in prior analytical formulation. This resonant interaction severely restricts the wavenumber for instability to the narrow range in which the cold tube frequency of the SWS is within a few percent of the corresponding smooth bore diocotron-like mode in the Brillouin flow. [ABSTRACT FROM AUTHOR]

The multi-frequency recirculating planar magnetron (MFRPM) is the first magnetron capable of simultaneous generation of significantly different output frequencies (1 and 2 GHz) in a single operating pulse. Design and simulation of a prototype MFRPM were followed by hardware fabrication and experimental verification using the Michigan Electron Long Beam Accelerator with a Ceramic insulator at -300 kV, 1-5 kA, and 0.14-0.23 T axial magnetic field. Preliminary results demonstrated simultaneous generation of microwave pulses near 1GHz and 2GHz at powers up to 44MW and 21MW, respectively, with peak total efficiencies up to 9%. [ABSTRACT FROM AUTHOR]

The Brillouin flow is the prevalent flow in crossed-field devices. We systematically study its stability in the conventional, planar, and inverted magnetron geometry. To investigate the intrinsic negative mass effect in Brillouin flow, we consider electrostatic modes in a nonrelativistic, smooth bore magnetron. We found that the Brillouin flow in the inverted magnetron is more unstable than that in a planar magnetron, which in turn is more unstable than that in the conventional magnetron. Thus, oscillations in the inverted magnetron may startup faster than the conventional magnetron. This result is consistent with simulations, and with the negative mass property in the inverted magnetron configuration. Inclusion of relativistic effects and electromagnetic effects does not qualitatively change these conclusions. [ABSTRACT FROM AUTHOR]

Calibrated microwave power and phase measurements are presented for the first recirculating planar magnetron prototype consisting of two coupled six-cavity 1-GHz planar cavity arrays. The results are presented for a solid cathode and two mode-control cathodes (MCCs) with aluminum or velvet electron emitters. The measurements were conducted using a prototype coaxial microwave power extraction scheme. The experimental operating parameters included: pulsed cathode voltages between −250 and −300 kV, voltage pulselengths of 200–600 \mu \texts , axial magnetic fields of 0.1–0.32 T, and entrance currents of 1–10 kA. The results showed improved oscillator frequency locking for the MCCs and increases in power and efficiency using the velvet electron emitter. [ABSTRACT FROM PUBLISHER]

Microwave oscillation has been measured for the first time in a 12-cavity axial-magnetic-field recirculating planar magnetron, designed to operate in \pi mode at 1 GHz. The device operates with a -300-kV pulsed cathode voltage and a 0.2-T axial magnetic field, and oscillates at transverse currents exceeding 1 kA when driven by an electron beam pulselength between 0.5 and 1 \mu\s. Microwave pulses were measured at frequencies between 0.97–1 GHz and achieved several hundred nanoseconds in length. Mode competition was observed between the \pi and 5 \pi/6 modes. [ABSTRACT FROM PUBLISHER]

Using a hybrid approach, 3-D simulations of magnetic priming of a relativistic magnetron have been performed. The primed magnetic field values were calculated using a magnetostatics code (Magnum) and then imported into a particle-in-cell code (Magic PIC) and run for the case of a six-vane relativistic magnetron. The magnetically perturbative structures chosen for implementation in these simulations were sets of three highpermeability wires of various lengths, which would be placed within the cathode, the anode, or, in the combined case, both the cathode and anode. In the best-performing cathode-wire case (three 4-cm wires), magnetic priming was found to reduce the start-oscillation time of the magnetron to 50% that of the unprimed case. When wires were embedded in both the cathode and the anode, the best-performing case (4-cm cathode wires and 4-cm anode wires) was found to start oscillating at 30% of the start-oscillation time of the unprimed case. The cases of magnetically primed magnetrons were found to exhibit slightly reduced equilibrium power levels, compared with the unprimed case. [ABSTRACT FROM AUTHOR]

The Buneman–Hartree condition is re-examined in a cylindrical, smooth-bore, relativistic magnetron using both the conventional, single particle model, and the Brillouin flow model. These two models yield the same result for the Buneman–Hartree condition only in the limit of a planar magnetron. When b/a=1.3, where a is the cathode radius and b (>a) is the anode radius, the difference in the two models becomes significant. When b/a=4 the difference is acute, the Buneman–Hartree magnetic field at a given voltage in the Brillouin flow model exceeds four times that in the single particle model. Such a difference is always present, whether the voltage is relativistic or not. These results are quantified for b/a>1 using Davidson’s model, conveniently cast in terms of the normalized gap voltage and normalized magnetic flux imposed on the cylindrical magnetron. A comparison with the University of Michigan/L-3 relativistic magnetron experiment is given. [ABSTRACT FROM AUTHOR]

Experiments have been performed on the UM/L-3 (6-vane, L-band) relativistic magnetron to test a new microwave window configuration designed to limit vacuum side breakdown. In the baseline case, acrylic microwave windows were mounted between three of the waveguide coupling cavities in the anode block vacuum housing and the output waveguides. Each of the six 3 cm deep coupling cavities is separated from its corresponding anode cavity by a 1.75 cm wide aperture. In the baseline case, vacuum side window breakdown was observed to initiate at single waveguide output powers close to 20 MW. In the new window configuration, three Air Force Research Laboratory-designed, vacuum-rated directional coupler waveguide segments were mounted between the coupling cavities and the microwave windows. The inclusion of the vacuum side power couplers moved the microwave windows an additional 30 cm away from the anode apertures. Additionally, the Lucite microwave windows were replaced with polycarbonate windows and the microwave window mounts were redesigned to better maintain waveguide continuity in the region around the microwave windows. No vacuum side window breakdown was observed in the new window configuration at single waveguide output powers of 120+MW (a factor of 3 increase in measured microwave pulse duration and factor of 3 increase in measured peak power over the baseline case). Simulations were performed to investigate likely causes for the window breakdown in the original configuration. Results from these simulations have shown that in the original configuration, at typical operating voltage and magnetic field ranges, electrons emitted from the anode block microwave apertures strike the windows with a mean kinetic energy of 33 keV with a standard deviation of 14 keV. Calculations performed using electron impact angle and energy data predict a first generation secondary electron yield of 65% of the primary electron population. The effects of the primary aperture electron impacts, combined with multiplication of the secondary populations, were determined to be the likely causes of the poor microwave window performance in the original configuration. [ABSTRACT FROM AUTHOR]

The condition for mutual, or peer-to-peer, locking of two magnetrons is derived. This condition reduces to Adler’s classical phase-locking condition in the limit where one magnetron becomes the “master” and the other becomes the “slave.” The formulation is extended to the peer-to-peer locking of N magnetrons, under the assumption that the electromagnetic coupling among the N magnetrons is modeled by an N-port network. [ABSTRACT FROM AUTHOR]

The injection locking of a magnetron is theoretically analyzed when either the free running oscillator or the drive signal has a frequency chirp. It is found that complete phase locking of the signal cannot be achieved in either case. However, as long as the locking condition of Adler is well-satisfied instantaneously, a high degree of locking occurs during a major duration of the frequency chirps. The expected output phase variation is computed in terms of the noise in the free-running magnetron oscillator for the case of constant drive frequency. [ABSTRACT FROM AUTHOR]

Experiments have been performed in testing magnetic priming at the cathode of a relativistic magnetron to study the effects on high-power microwave performance. Magnetic priming consists of N/2 azimuthal magnetic perturbations applied to an N-cavity magnetron for rapid generation of the desired number of electron spokes for the π-mode. Magnetic perturbations were imposed by utilizing three high-permeability nickel-iron wires embedded beneath the emission region of the cathode, spaced 1200 apart. Magnetic priming was demonstrated to increase the percentage of π-mode shots by 15% over the baseline case. Mean peak power for π-mode shots was found to be higher in the magnetically primed case by almost a factor of two. Increases in mean microwave pulsewidth were also observed in the magnetically primed case when compared to the unprimed case (66-ns primed versus 50-ns unprimed). Magnetron starting current for the magnetically primed π-mode exhibited a reduction to 69% of the unprimed baseline starting current. [ABSTRACT FROM AUTHOR]

Previous experiments demonstrated that imposing an azimuthally varying axial magnetic field, axially asymmetric, in dc-operating oven magnetrons causes rapid mode growth (by magnetic priming) and significant noise reduction. This configuration was previously implemented by adding five perturbing magnets on the upper existing magnet of the magnetron. Experiments reported here add five perturbing magnets on each of the two existing magnets of the magnetron, restoring the axial symmetry of the magnetic field, while maintaining the five-fold azimuthal magnetic field symmetry. Compared with the unperturbed magnetic field case, it has been observed that the noise close to the carrier is reduced by up to 20 dB, while the sidebands are not completely eliminated for medium and high currents. Magnetron start-oscillation currents are somewhat higher for this axially symmetric, azimuthally varying magnetic field as compared to the baseline unperturbed magnetic field. [ABSTRACT FROM AUTHOR]

Two-dimensional simulations have been performed on a six-vane relativistic magnetron with uniform axial magnetic fields versus azimuthally varying axial magnetic fields, defined as magnetic priming. Electron phase-space plots show rapid growth of the π-mode when the axial magnetic field has three-azimuthal perturbations: it takes 36 ns for the π-mode to dominate in the uniform magnetic field case versus only 13 us for the π-mode to dominate in the case with magnetic priming imposed. Rf electric field plots versus time show the suppression of the 2π/3-mode when magnetic priming is imposed. [ABSTRACT FROM AUTHOR]

Azimuthally varying axial magnetic fields have been utilized to perform "magnetic priming" of magnetrons for rapid startup, low noise, and mode control. An overview of the latest magnetic priming experimental and simulation results are presented in this paper. Magnetic priming experiments in dc-operating microwave oven magnetrons show sideband elimination, even with the cathode heater turned off. Simulations using three three-dimensional (3-D) improved concurrent electromagnetic particle-in-cell (ICEPIC) codes with two different computational algorithms recover the oven magnetron experimental results obtained with magnetic priming including fast mode growth, rapid spoke formation, and the tendency toward lower noise operation. A new, axially symmetric, azimuthally varying magnetic field geometry for oven magnetrons is explored and preliminary results are reported. Simulations using two-dimensional (2-D) MAGIC code for the University of Michigan/Titan relativistic magnetron show that the oscillation startup time can be dramatically decreased (almost by a factor of 3) and mode competition can be suppressed with magnetic priming. [ABSTRACT FROM PUBLISHER]

Peer-to-peer locking of two magnetrons is analyzed including the effects of a frequency chirp and of low frequency noise. It is found that complete phase locking cannot be achieved in either case. However, as long as the locking condition is well satisfied instantaneously, a high degree of locking occurs. This analysis in the time domain is adapted to locking in the spatial domain, in particular to the interpretation of some recent experiments on the spatial correlation of two ablating current-carrying wires that are placed sufficiently close to each other. [ABSTRACT FROM AUTHOR]

An electron rotating under a uniform axial magnetic field and a radial electric field exhibits an effective mass that may be negative, positive, or infinite, in response to an azimuthal electric field. This paper reports simulation results that show instability and stability when the effective mass are negative and positive, respectively, depending on the magnitude and orientation of the radial electric field. Thus, the inverted magnetron would have a much faster startup than the conventional magnetron, an important consideration for pulsed operation. When the effective mass is infinite, the electrons hardly respond to an azimuthal ac electric field. [ABSTRACT FROM AUTHOR]

Experiments on peer-to-peer locking of 2 kW magnetrons are performed. These experiments verify the recently developed theory on the condition under which the two nonlinear oscillators may be locked to a common frequency. Dependent on the coupling, the frequency of oscillation when locking occurs does not necessarily lie between the free running frequencies of the two isolated, stand-alone magnetrons. Likewise, when the locking condition is violated, the beat frequency is not necessarily equal to the difference between these free running frequencies. [ABSTRACT FROM AUTHOR]

A relativistic magnetron priming technique using multiple cathodes is simulated with a three-dimensional, fully electromagnetic, particle-in-cell code. This technique is based on electron emission from N/2 individual cathodes in an N-cavity magnetron to prime the π mode. In the case of the six-cavity relativistic magnetron, π-mode start-oscillation times are reduced up to a factor of 4, and mode competition is suppressed. Most significantly, the highest microwave field power is observed by utilizing three cathodes compared to other recently explored priming techniques. [ABSTRACT FROM AUTHOR]

A cathode priming technique of a relativistic magnetron is analyzed via a three-dimensional particle-in-cell simulation. By imposing a threefold azimuthal variation on the emitting cathode of a six-cavity relativistic magnetron, the electrons are prebunched at birth. This leads to fast startup of the pi mode with three electron spokes. Suppression of unwanted modes during startup is observed in the simulation. [ABSTRACT FROM AUTHOR]

A three-dimensional parallel particle-in-cell code, ICEPIC, is used to simulate the geometry and the magnetic field profiles of the recent low-noise, fast startup magnetron experiments at the University of Michigan. The fast startup, the power levels, and the starting currents that have been observed in these experiments are quantitatively reproduced in the simulations. The tendency for low noise operation has also been reproduced with the use of an azimuthally varying magnetic field. [ABSTRACT FROM AUTHOR]

Preliminary experiments of the recirculating planar magnetron microwave source have demonstrated that the device oscillates but is susceptible to intense mode competition due, in part, to poor coupling of RF fields between the two planar oscillators. A novel method of improving the cross-oscillator coupling has been simulated in the periodically slotted mode control cathode (MCC). The MCC, as opposed to a solid conductor, is designed to electromagnetically couple both planar oscillators by allowing for the propagation of RF fields and electrons through resonantly tuned gaps in the cathode. Using the MCC, a 12-cavity anode block with a simulated 1 GHz and 0.26 c phase velocity (where c is the speed of light) was able to achieve in-phase oscillations between the two sides of the device in as little as 30 ns. An analytic study of the modified resonant structure predicts the MCC's ability to direct the RF fields to provide tunable mode separation in the recirculating planar magnetron. The self-consistent solution is presented for both the degenerate even (in phase) and odd (180° out of phase) modes that exist due to the twofold symmetry of the planar magnetrons. [ABSTRACT FROM AUTHOR]

It is shown that the vector potential and the scalar potential satisfy a Buneman-Hartree like (BHL) condition and a Hull-cutoff like (HCL) condition everywhere within the Brillouin flow of a cylindrical, relativistic magnetron, when the phase velocity in the Buneman-Hartree condition is replaced by the laminar, local flow velocity of the Brillouin flow. The vector potential and the scalar potentials include the Brillouin flow's self magnetic field and the self electric fields. The HCL condition reduces to the conventional Hull cutoff condition derived from single particle orbit theory when the Brillouin hub extends to the anode. However, the BHL condition reduces to the conventional Buneman-Hartree condition only in the planar magnetron limit but may be substantially different for a cylindrical magnetron, as demonstrated recently [Lau et al., Phys. Plasmas 17, 033102 (2010)]. These conclusions apply also to the inverted magnetron configuration. The effects of ions are discussed. [ABSTRACT FROM AUTHOR]