Crossed-field Devices Research Articles

Limiting current in a collisional crossed-field gap

Crossed-field devices are often used in pulsed power and high-power microwave applications. Previous studies derived closed-form solutions for the limiting current of a vacuum crossed-field system, corresponding to the maximum permissible current for laminar flow, below and above the Hull cutoff BH for magnetic insulation. We extend these studies by introducing collision frequency into the electron force law as a friction term to derive the limiting current in a collisional crossed-field gap. The resulting solution recovers the vacuum crossed-field case in the limit of no collisions and the collisional space-charge limited current with general initial velocity for magnetic field B→0. In the limit of infinite collisions, we obtain a crossed-field equivalent to the Mott–Gurney law for the maximum current permissible in a collisional, nonmagnetic diode. When the collision frequency ν is less than the electron cyclotron frequency Ω, increasing initial velocity makes the critical current nonmonotonic with increasing ν with the critical current higher at B=BH for ν=Ω. As for a misaligned crossed-field gap where a component of the magnetic field was introduced parallel to the electric field across the gap, magnetic insulation is eliminated and the discontinuity at B=BH for limiting current observed in a vacuum crossed-field gap vanishes. As B→∞, the limiting current approaches a constant that depends on the initial velocity and the collision frequency.

The Orbitron: A crossed-field device for co-confinement of high energy ions and electrons

To explore the confinement of high-energy ions above the space charge limit, we have developed a hybrid magnetic and electrostatic confinement device called an Orbitron. The Orbitron is a crossed-field device combining aspects of magnetic mirrors, magnetrons, and orbital ion traps. Ions are confined in orbits around a high-voltage cathode with co-rotating electrons confined by a relatively weak magnetic field. Experimental and computational investigations focus on reaching ion densities above the space charge limit through the co-confinement of electrons. The experimental apparatus and suite of diagnostics are being developed to measure the critical parameters, such as plasma density, particle energy, and fusion rate for high-energy, non-thermal plasma conditions in the Orbitron. Initial results from experimental and computational efforts have revealed the need for cathode voltages on the order of 100–300 kV, leading to the development of a custom high voltage, ultra-high vacuum bushing rated for 300 kV.

On the two-dimensional Brillouin flow

The Brillouin flow is a rectilinear, sheared electron fluid flow in a crossed electric field (E) and magnetic field (B), in the E × B direction with zero flow velocity and zero electric field at the surface with which the flow is in contact. It is broadly considered as the equilibrium electron flow in high power crossed-field devices including the magnetron and magnetically insulated transmission line oscillator. This paper provides an examination of Brillouin flow in two dimensions, in a cylindrical geometry where the anode radius changes abruptly at a single axial location, while the cathode surface has a constant radius. Our simulation confirms the proof that there is no equilibrium Brillouin flow solution for such a geometry. It further reveals that this change in the anode radius introduces novel bunching of the electrons within the Brillouin hub. This bunching occurs at low frequencies and is very pronounced if the Brillouin flow is from the small gap region to the large gap region, but is minimal if the Brillouin flow is from the large gap region to the small gap region. New insights are provided into the physical processes that initiate and sustain the bunching processes that are unique for a crossed-field diode, as compared with a non-magnetized diode. We argue that this enhanced bunching, and its concomitant formation of strong vortices, is not restricted to an abrupt change in the anode–cathode gap spacing.

Experimental study of flake graphite large-area explosion electron emission cathode performance based on particle size discrepancy

Graphite is the most commonly used large-area cathode material in the high-power vacuum electron device for its long life and good repetitive-frequency performance, but its particle size selection scheme has not been elucidated. Therefore, in the present work, the explosive electron emission properties of flake graphite large area explosive electron emission cathodes with varied particle sizes are studied. The 320 mesh flake graphite has a current delay reduction of 0.2–0.9 ns compared to other samples at the same peak voltage. According to the Murphy–Good equation, the emission performance of each sample during the priming stage was calculated, and the threshold characteristics of 320 mesh flake graphite were significantly better than the other samples. After the space charge limited current is formed, the plasma expansion rate of all samples is 1.2±0.2cm/μs with a difference not exceeding 4% between samples. Our research provides a strong basis for the selection of raw materials for graphite based large area explosive electron emission cathodes and contributes to the development of cross-field high-power vacuum devices.

The Hull Cutoff condition for magnetic insulation in crossed-field electron devices in the presence of a slow-wave structure

Crossed-Field Vacuum Electron Devices are ubiquitous in the High-Power Microwave field in either an oscillator/source or amplifier variant. A typical configuration consists of a magnetically insulated laminar electron flow in an anode–cathode gap with crossed electric (∝V, voltage) and magnetic (B-) fields and a series of open resonant cavities/vanes located on the anode block that serve as a slow-wave structure (SWS). The SWS slows the phase velocity of the electromagnetic signal down so that the wave becomes synchronous with a layer of the electron flow but is often neglected when calculating the Hull magnetic field necessary to insulate the electrons. In particular, the guiding design equation for the critical cutoff B-field assumes a smooth anode wall. In this paper, we show that such an assumption severely narrows the operating regime in B–V space and that upon inclusion of a revised Hull Cutoff condition taking into account the SWS, operation at lower B and higher V is possible. This revised Hull Cutoff criterion for magnetic insulation in crossed-field devices is corroborated by Particle-in-Cell simulations using CST Particle Studio.

Comparison of the output spectrum for four models of devices with crossed fields

This paper considers four models of crossed-field devices: cylindrical magnetron, cylindrical magnetron diode, planar magnetron diode and planar magnetron. As is known, in crossed-field devices electrons move by cycloid-like trajectories and induced current arises. The spectral components of the induced current for every of the models are calculated. It has been found that the components of the induced current spectrum contain from one (planar magnetron diode) to eight (cylindrical magnetron diode) components. The calculated spectral components have been compared to the spectral components of the generation spectrum of the real device.
 Only one component whose frequency coincides with the cyclotron frequency was found in the induced current spectrum of the planar magnetron diode.
 The induced current spectrum in a planar magnetron has three components. These frequencies have a large difference with the frequency of magnetron generation for a microwave oven. Therefore, the models of the planar magnetron diode and the planar magnetron are not very suitable for simulating the operation of crossed-field devices.
 The induced current spectrum in a cylindrical magnetron diode contains eight components. The third harmonic is closest to the fundamental frequency of the microwave magnetron. In this case, the difference between the frequencies being compared is near 10 %. Therefore, the model of a cylindrical magnetron diode is not very suitable for modeling the operation of crossed-fields devices.
 Two components were found in the induced current spectrum of a cylindrical magnetron. The first harmonic is the closest to the fundamental frequency of the microwave magnetron. In this case, the difference between the frequencies being compared does not exceed 3 %. Such a small discrepancy between the calculated and real frequencies can be explained by the presence of a spatially periodic distribution of the electrostatic field. Thus, the cylindrical magnetron model is well suited for simulating the operation of crossed-fields devices.

Manipulating plasma turbulence in cross-field plasma sources using unsteady electrostatic forcing

Unsteady electrostatic forcing is investigated as a method for manipulating turbulent plasma behaviour within Hall-effect thrusters and similar cross-field plasma devices using a simplified one-dimensional three-velocity azimuthal electrostatic particle-in-cell simulation. A wide range of axial electric field forcing frequencies from 1 MHz up to 10 GHz at amplitudes of 10 V cm−1, 50 V cm−1 and 100 V cm−1 are applied to the plasma and the response is evaluated against a baseline case defined by the community benchmark LANDMARK Test Case 1. ‘Tailoring’ of plasma parameters, such as the electron cross-field mobility, is demonstrated via manipulation of the electron drift instability using unsteady forcing. Excitation of the unstable electron cyclotron modes by the electron drift instability is shown to be able to produce a reduction of the resultant electron cross-field mobility of the plasma by up to 50% compared to the baseline value. Additionally, forcing at the electron cyclotron frequency appears to be capable of increasing cross-field mobility by up to 2000%. Implications of the results for direct drive electric propulsion systems and improved current utilization efficiencies for Hall-effect thrusters are discussed.

Theoretical Linkage of Thermionic, Field, and Space-Charge Limited Emission for a Vacuum Crossed-Field Gap

Crossed-field devices, such as magnetrons and crossed-field amplifiers, where an external magnetic field is orthogonal to the electric field across the gap, are used in radars, communications systems, and material processing. Improving the efficiency and power density of crossed-field devices requires a robust understanding of electron emission, particularly when multiple emission mechanisms may contribute, such as for thermionic and field emission. Previous studies of nonmagnetic diodes have theoretically identified the transitions, or nexuses, between multiple electron emission mechanisms by examining the relevant equations defining each individual mechanism and then derived exact theoretical solutions that combined these mechanisms and approached the standard emission equations under appropriate limits. This work extends those studies by incorporating the orthogonal magnetic field and assessing the transitions between the Richardson–Laue–Dushman (RLD) formulation for thermionic emission, the Fowler–Nordheim (FN) equation for field emission, the space–charge limit given by both the general (for nonzero electron injection velocity) and (zero electron injection velocity) Child–Langmuir (CL) (generalized CL (GCL) and CL, respectively) laws for space–charge limited current, and the space–charge limited current for a noninsulated (magnetic field below the Hull cutoff) and insulated (magnetic field above the Hull cutoff) diode. We derive exact solutions for the current density for these mechanisms and demonstrate their asymptotic behavior under relevant conditions. Furthermore, we plot the exact solution of current density <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$J$ </tex-math></inline-formula> as a function of applied voltage <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$V$ </tex-math></inline-formula> by incorporating the general thermo-field emission theory for the current source and demonstrate the transition between each emission mechanisms with increasing <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$V$ </tex-math></inline-formula> for various temperatures and fixed ratios of magnetic field to the Hull cutoff. We also construct nexus plots to identify transitions between the various regimes for different conditions.

Feature of statistical data processing by computer mathematics systems tools

Processes of error accumulation were analysed during arithmetic operations on statistical data obtained in the course of research on oscillations in cross-field electron vacuum devices.&#x0D; The features of statistical data processing obtained as a result of experimental research were investigated using the most widespread computer mathematical packages.&#x0D; The features of statistical data processing were investigated by processing a sample of 80 values of the magnetron generation frequency using popular mathematical packages Excel, Maple, Matlab and MathCad and comparing the results obtained with calculations using theoretical formulas. Calculation results for all packages give the same results for mean, variance and standard deviation. As for the coefficients of skewness and kurtosis, most of the results do not coincide.&#x0D; Analysis of the calculation results showed that the difference in the obtained values ​​of the skewness and kurtosis is due to different definitions of these indicators in mathematical packages Excel, Maple, Matlab and MathCad. It is proved that in Microsoft Excel we cannot correctly construct a histogram without using additional operations, because the interval limits are calculated with errors. It leads to an incorrect determination of the number of elements into these intervals.&#x0D; To build correctly a histogram using the Excel package, it is necessary to calculate the interval limits in advance.&#x0D; It is concluded that before using computer mathematical packages for processing statistical data, it is necessary to analyze first by what formulas the required parameters are calculated and take appropriate measures to eliminate possible discrepancies with the parameters calculated using theoretical formulas.

Space-charge limited current in planar and cylindrical crossed-field diodes using variational calculus

Crossed-field space-charge limited current (CF-SCLC) represents the maximum stable current that can be produced in crossed-field devices (CFDs); prominent CFDs include magnetrons and cyclotrons, critical technologies in vacuum electronics. While planar crossed-field geometries have the most developed theoretical models, cylindrical geometries are far more common and useful in practical applications. CFDs are characterized by the strength of the externally applied magnetic field B, orthogonal (crossed) to the electric field; this is often normalized to the magnetic insulation condition, the Hull cutoff field BH. We apply variational calculus to derive a new theoretical model for CF-SCLC for planar and cylindrical devices below and above BH. The variational model offers a concise derivation for planar results without transforming to the time domain and gives the first analytic results from first principles for cylindrical CF-SCLC. We implement a fully three-dimensional simulation in CST Particle Studio which, in addition to additional derived simple theoretical models, explains the often overlooked experimental current scaling ∝1−B/BH21/2 which decreases to zero current as B→BH−. These additional simple models reduce the maximum mismatch magnitude between theory and experiments or simulations by up to 68% compared to the variational model, with the most improvement at the critical limit B→BH−. Justification for the variational model and future applications are discussed.

Analysis of Injected Electron Beam Propagation in a Planar Crossed-Field Gap

This paper examines basic crossed-field device physics in a planar configuration, specifically electron beam perturbation and instability as a function of variation in magnetic field, and angle between magnetic and electric field. We perform a three-dimensional (3-D) simulation of electron perturbation in a planar crossed-field system using the full 3-D particle trajectory solver in CST Particle Studio (CST-PS). The structure has a length, height, width and anode-sole gap of 15 cm, 2 cm, 10 cm, and 2 cm, respectively. The anode to sole voltage is fixed at 3 kV, and the magnetic field and injected current varied from 0.01 T to 0.05 T and 1.5 mA to 1 A, respectively. The simulations show that applying a magnetic field of 0.05 T makes the beam stable for a critical current density of 94 mA/cm2 for an anode-sole gap of 20 mm. Above this current density, the beam was unstable, as predicted. Introducing a 1° tilt in the magnetic field destabilizes the beam at a current density of 23 mA/cm2, which is lower than the critical current density for no tilt, as predicted by our theory. The simulation results also agree well with prior one-dimensional (1-D) theory and simulations that predict stable bands of current density for a 5° tilt where the beam is stable at low current density (&lt;13.3 mA/cm2), unstable above this threshold, and then stable again at higher current density, (&gt;33 mA/cm2).

Innovative Crossed-Field Devices for the Generation of High Power Microwaves

Innovative Crossed-Field Devices for the Generation of High Power Microwaves

Full fluid moment model for low temperature magnetized plasmas

A one-dimensional full fluid moment model is developed to study shear-induced electron transport in low-temperature, partially magnetized plasmas. Ions and electrons are treated as two different fluids that interact with each other through collisions and electromagnetic fields. Conservation of mass, momentum, and energy is solved for both the fluids and the Poisson equation is solved to obtain the electrostatic electric field. A global Lax–Friedrichs flux splitting scheme with a monotonic upwind scheme for conservation laws is used for the inviscid flux calculation. Moments of a shifted Maxwellian distribution that is constructed from the fluid quantities are taken to calculate fluxes at the boundaries, henceforth called the kinetic fluxes. Two full fluid moment models are developed based on the treatment of the electron energy: (i) the total energy and (ii) the internal energy, i.e., temperature. A DC low-temperature magnetized plasma in a Hall effect thruster is modeled using the full fluid moment model and the results are compared with drift-diffusion models that are commonly used for low-temperature plasmas, providing insights into nonclassical electron transport driven by the electron shear in cross-field devices.

Spatial evolution of small wavelength fluctuations in a Hall Thruster

The spatial evolution of small wavelength (&amp;lt;1 cm) fluctuations in the Hall direction of a 9-kW class magnetically shielded Hall effect thruster is experimentally characterized. High-speed electrostatic probes are used to measure plasma density perturbations in the acceleration region and near field plume of this cross field device. Two types of waves are observed: broadband turbulence in the low hundreds of kilohertz and megahertz oscillations characterized by distinct peaks in the measured power spectrum. The lower frequency content is shown to be commensurate with linear ion acousticlike waves, while the higher frequency oscillations are indicative of cyclotron resonances. It is found that the higher frequency content dominates the power spectrum in the upstream acceleration zone and saturates at a fraction (&amp;lt;0.002) of the thermal energy. Downstream of this region, the lower frequency content becomes dominant, growing until the total wave energy again saturates. It is proposed that both the upstream and downstream modes are the same oscillation, the electron drift instability, but in different limits. The transition from upstream to downstream dispersion may be explained by the electron drift instability undergoing an inverse energy cascade as it forms and propagates with the ion drift in the discharge.

Map-Based Multipactor Theory for Cross-Field Devices

A novel theoretical model for multipactor discharge based on principles from nonlinear dynamics and chaos theory has been introduced and recently extended to two-surface multipactor in a parallel-plate geometry. A major advantage of this approach is that it is universal in that different systems can be studied by simply changing the underlying equations. In this paper, this is demonstrated for cross-field devices, where a dc magnetic field is transverse to the radio frequency (RF) electric field. As in the initial study, maps that relate the RF emission phase to the RF arrival phase are developed with no a priori assumptions on the electron trajectories (neglecting space-charge effects). By systematically applying the maps and scanning system parameters, bifurcation diagrams are constructed that summarize all possible multipactor modes. These scans recover conventional resonant multipactor in addition to high periodicity and nonresonant forms. By incorporating the secondary emission properties of the surface material and additionally scanning a realistic spread in the secondary emission energies and angles, the multipactor growth (or decay) rate is computed over a wide range of parameters. As in the initial study, it is found that the attractor form manifests in the exponential growth rate. These theoretical predictions are validated with particle-in-cell simulations.

Instabilities and Plasma Flares in Moderate-Current Confined Magnetron Sputtering in Three Dimensions

Plasma instabilities, spokes, and flares are important in low-temperature cross-field devices (think magnetron sputtering, or Hall thrusters for spacecraft), yet their roles in discharge maintenance and charged-particle transport are not fully understood. The authors use high-speed photography to investigate the formation of plasma flares, and low-frequency oscillations in the plasma potential. At high power there is a drastic change in plasma characteristics, and the oscillations become chaotic. The instabilities assist particle transport far from the cathode, which may be of considerable technological importance.

Space-charge and thermal effects in relativistic crossed-field devices

In this paper, a fully kinetic theory for the relativistic electron flow in a crossed-field device is developed and analyzed. The theory takes into account self-electric, self-magnetic, and thermal effects and allows determining the final stationary state achieved by the electrons in phase-space. A number of different possible stationary modes are identified and described in detail. Particular attention is given to the study of how space charge and thermal effects affect the magnetic insulation when the external magnetic field exceeds the Hull cutoff field. In the nonrelativistic limit, it is found that there is only a single mode transition that leads to the loss of the magnetic insulation. This transition is completely independent of the electron density and occurs for relatively large injection temperatures. On the other hand, in a moderate relativistic regime a much richer scenario is found with the onset of a series of stationary state mode transitions as both electron density and injection temperature are varied. In particular, it is found that the transitions and the consequent loss of magnetic insulation may occur even at very low injection temperatures. Self-consistent numerical simulation results are also presented and used to verify the theoretical findings.

Noise Reduction in Cross field Devices

Noise Reduction in Cross field Devices

Advances and present trends in magnetically insulated line oscillator

The magnetically insulated line oscillator is an attractive high-power microwave (HPM) source. It is a compact lightweight gigawatt-class coaxial crossed-field device requiring no externally applied magnetic field to insulate the electron flow between the anode–cathode gap. This paper is aimed at reviewing the work carried out in the area with emphasis on performance enhancement of the device, and its future in the wake of rapid development of HPM sources. A brief outline of the research pursued in the country also has been presented.

Stability of Brillouin flow in the presence of slow-wave structure

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.