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

The key to efficient and effective operation and coherent output radiation of High-Power Microwave (HPM) Vacuum Electron Devices (VEDs) is synchronism between the electrons in the “beam” and EM wave on the circuit. There are many viable methods to achieve this synchronous beam–wave interaction. Perhaps the most common, tried and tested method is to employ a slow-wave structure (SWS) on the circuit. Such a structure serves to slow down the phase velocity of the traveling wave on the circuit so that it approximately matches the DC electron beam velocity (or at least an electron layer having this velocity in the case of crossed-field devices that have an electron flow with a velocity distribution, e.g., Brillouin flow; this beam-circuit synchronism criterion is called the Buneman–Hartree Condition for crossed-field devices and relates the necessary applied voltage with the magnetic field^1,2). The design of the SWS also affects the outputs such as the output frequency and bandwidth of the VED.^3–5 

In the case of crossed-field VEDs where the electron motion is dictated initially by orthogonal DC electric [provided by an applied anode–cathode (AK) gap voltage] and magnetic fields, there is the extra complexity of the magnetic insulation of the electrons in the diode interaction region. If the electrons emitted from the cathode are not insulated and are allowed to fill the AK-gap, the device will be shorted because a conducting path between the anode and cathode is created via the electron flow (the current of which has been intensely studied in, for example, Refs. 6 and 7). Conversely, a high magnetic field will contain and localize the electrons to near the cathode.^8,9 Between these two extreme cases, there must be a critical magnetic field such that the electrons just approach the anode (and their trajectories become asymptotic to the anode wall for the electron flow case). This critical magnetic field is the Hull Cutoff magnetic field that is the threshold between crossed-field device operation and non-operation. An illustration of these three cases using a single-particle cycloidal orbit model and Brillouin flow model (as are common in the literature) is shown in Fig. 1 along with a plot of the anode current as a function of applied axial DC magnetic field.¹⁰ We note that for the magnetically insulated line oscillator/transmission line (MILO/MITL) type of crossed-field device, the analog of the Hull Cutoff is the parapotential current; for simplicity and in being consistent with the test case in Sec. III, we will refer to the Hull Cutoff in magnetrons/crossed-field amplifiers in the rest of this paper with a note that analyses and conclusions here may apply to MILOs/MITLs as well.

The anode current–magnetic field plot in Fig. 1 is a theoretical idealization (blue). In experiments and simulations, a smoother sigmoidal transition has been observed (red), meaning that there is a finite anode current even when the magnetic field is high and there is theoretical magnetic insulation. This indicates a failure of the classical Hull Cutoff condition for magnetic insulation, and there has been a plethora of research attempting to explain this phenomenon including accounting for misaligned magnetic field¹¹ and cathode back-bombardment¹².¹³ While our research is related and may contribute to the explanation of this apparent contradiction between theory and experiments/simulations, we are more focused on the observation and characterization of crossed-field device operation when the critical magnetic field is below the theoretical Hull limit and accounting for this with a revised Hull Cutoff condition considering the previously neglected SWS in the formulation. It should be noted though that the physics of this situation is very rich. We do not claim to have solved this outstanding problem in crossed-field device physics; rather, we aim to provide an alternative explanation that is theoretical and, thus, under ideal conditions where external factors such as magnetic field misalignment are not an issue.

The derivation of this classical critical threshold bias magnetic field has long been established. It is based on the fundamental governing equations of motion and the conservation of energy for the electrons and has been derived for relativistic and non-relativistic electron velocities (voltages) and parallel-plate and coaxial geometries.^14–16 Alternatively, one may use the conservation of canonical angular momentum (Busch's theorem) along with one of the two governing equations to derive the other.¹⁷ Monoenergetic electron emission and electron emission according to a velocity distribution from the cathode have also been considered.¹⁸ However, it is interesting to note that this classic analytic theory assumes a smooth-bore anode and does not consider the effects of including a SWS.¹⁴ We shall consider such a case here (this possibility was alluded to in Ref. 19), following the classical Hull Cutoff derivation. In doing so, we find a geometric correction factor to the typical Hull Cutoff formula. This factor may be treated as a first-order perturbation to the zeroth-order Hull Cutoff equation and serves to widen the area of operation in B–V space. This potentially means that one can operate a crossed-field device at lower B-field and higher V values if so desired: the traditional Hull Cutoff is not a strict rule but more of a conservative guideline. Particle-in-Cell (PIC) computer simulations done in CST Particle Studio²⁰ corroborate the theoretical formulation here.

In Sec. II, the semi-analytical theory and analytical formulation are laid out; the geometrical correction factor to the Hull Cutoff formula due to the presence of a SWS is given. Section III presents a test case of an A6 magnetron; the traditional Hull Cutoff and the revised Hull Cutoff are compared to predictions from the PIC simulation code CST. Last, Sec. IV will conclude and summarize the salient points of this paper.

The original classical theory of Hull for magnetic insulation in a crossed-field diode gives the threshold DC magnetic field before insulation is lost and relates this to the voltage and the 2D cross-sectional geometry.¹⁴ However, the 2D cross-sectional geometry considered is simplified by removing the corrugations or vanes/cavities that make up the SWS. This massively simplifies the derivation and makes it analytically tractable. In reality though, this is not completely accurate for slow-wave crossed-field devices. The Hull Cutoff does provide a good design equation that is conservative because magnetic insulation with the SWS can be achieved if it is achieved for the case with a smooth anode wall (assuming that wall is taken as the anode and not the vane backwall location, see, e.g., Fig. 5).

Adding in the SWS introduces a layer of complexity. Although the applied AK-gap voltage is the same (i.e., the anode with the SWS is charged to the same voltage as its smooth-wall variant), the voltage profile and the electric field that acts on an electron are not, to the first order. This is due to the extra distance introduced by the depth of the vane and the vane opening itself. An electron will experience a perturbation to its instantaneous trajectory compared to the smooth-wall variant depending on its location, whether in the AK-gap or near the vane openings, making this a full 2D as opposed to a 1.5D problem. How much of a perturbation is introduced depends on the ratio of the anode length without a vane to the width of the vane; the depth of the cavity indicates the intensity of the perturbation. The Hull Cutoff will track with this, but to get a ballpark usable value, one can take a weighted average over a SWS period to find a revised average Hull Cutoff threshold, which will generally be smaller than the classical conservative value for the same applied voltage.

To illustrate the above description analytically, we begin with a planar crossed-field gap with the cathode (K) located at x = 0 for all y and z and the anode (A) situated a distance x = D above; this is the same simple setup that the seminal paper by Lovelace and Ott¹⁵ considers and later generalizes to coaxial cylindrical geometry via a change from the true AK-gap distance to an equivalent gap width. The anode is loaded with vanes/cavities that have a depth h and a width w, with width w′ separating adjacent vanes. This setup on the anode constitutes a basic SWS with period L = w + w′. Thus, the distance from the cathode to the backwall of a vane on the anode is D + h. We assume for simplicity that the anode and cathode are perfect electrical conductors (PECs). The crossed fields in the gap originate from an applied DC voltage between the anode and cathode and an applied DC axial magnetic field (which has had enough time to penetrate the metal¹⁶). This setup is illustrated in Fig. 2.

The electron trajectory in this geometry will follow the above force law. If the applied DC B-field is strong enough, the trajectory will be bent back to the birthing cathode. Conversely, for small values of B, the electron will not be constrained relative to the AK-gap and will, thus, strike the anode. At the critical threshold value B = B_H, the electron trajectory will just touch (become asymptotic with, in the Brillouin flow case) the anode at x = D and $d x d y$ $= d x / d t d y / d t=0=> d x d t≡ v x=0$ (the electron motion will purely evolve in the y-direction only with the E × B drift dominating, $v y y ^= E → × B → B 2$⁠, for the Brillouin flow case).

For the conventional smooth-wall anode setup where there are no vanes/cavities, $h→0,w→0, w ′→L$⁠, and

The above analysis was carried out using a single-particle cycloidal orbit (double-stream) theory,²¹ in the same vane as the traditional Hull Cutoff derivation. This model, though simple, is more intuitive and lends itself to ease-of-use and understanding. However, the above derived equations may also be shown to hold true when using the Brillouin (steady-state, laminar, single-stream) flow model for the electrons in the AK-gap.^22–25 In that model, which has been shown to be the preferred equilibrium state of electron flow in crossed-field AK-gaps^26,27 and self-consistently includes the effects of the plasma hub,²² the Hull Cutoff magnetic field is given by setting the Brillouin hub height equal to the AK-gap distance and is the same for both planar and cylindrical geometries, compared to the single-particle orbit model.²² Nevertheless, there has been no singular consensus on a model, and both (same equation form) have been extensively used historically and currently. For this current analysis, we have chosen to follow Hull¹⁴ and Lovelace and Ott¹⁵ so that a more ready comparison can be made.

We also note that this is a fully DC analysis intended to provide a better guideline when considering crossed-field device operation. For this purpose, following the literature, a DC analysis is sufficient because it captures the dominant, bulk state of the electrons in the AK-gap before AC perturbations and, hence, RF generation starts (in a small-signal/linear theory). In reality for magnetrons, there is a time-dependent Hull Cutoff effect in play as well, where the RF TE axial magnetic field can temporarily negate the externally applied DC axial magnetic field. When time-averaged, this causes a disruption in normal magnetron operation; electrons can impact the anode, damaging it and generating unwanted plasma that can shorten the pulse length and kill the RF.⁴ Effects such as this as well as turn-on (as may be seen in the initial portions of the blue curve in Fig. 6, where the electron hub and spokes from the hub begin to form) are fully electromagnetic and are captured, for example, in a small-signal/linear theory, in an AC analysis where the RF perturbations are superimposed on top of the unperturbed DC state.²⁸ 

An example to which we will apply our theory to is the typical cylindrical cavity magnetron. This oscillator has been a staple in High-Power Microwave (HPM) sources since its inception in the early 1900s. A popular and historical iteration of this device is the MIT A6 design by Palevski and Bekefi.²⁹ It has a conventional cylindrical layout with six cavities/vanes azimuthally and periodically placed on the outer anode coaxial cylinder. The inner coaxial solid cylinder with endcaps at the axial ends defines the cathode (stalk/shank). Radial extraction is used, and the output RF is coupled to a WR284 standard rectangular waveguide. A schematic diagram is shown in Fig. 3 along with a sample cross-sectional view of the electric field distribution for a DC voltage applied between the anode and cathode, indicating the variation in the presence of the SWS [in particular, for Hull Cutoff, at the anode radius where an electron is exposed to the reduced (relative to the smooth-bore sections, approximately by the ratio $D ∗ D ∗ + h$⁠) electric field at the cavities’ openings].³⁰ 

The major dimensions of such a device are tabulated in Table I.  Appendix A includes a theoretical characterization of this test case magnetron in terms of its cold and hot dispersion properties (estimated oscillation frequency and operating mode number, linear growth rates, and Brillouin flow profiles for the electron hub) to complement the DC analysis that is the subject of this paper. It is important to note that the CST Particle Studio simulation software used here is a fully 3D Electromagnetic Particle-in-Cell solver, so it may be useful (for completeness) to see the associated theoretical AC analysis, which typically acts as a perturbation to the DC case in the linear theory.

Theoretically, this magnetron should only oscillate when the B-field and applied V are in the space bounded by the Buneman–Hartree condition^1,2 on the bottom and the Hull Cutoff^14,15 on the top (Fig. 4).

We next set up and simulated this A6 magnetron for different values of the applied DC axial magnetic field B and input AK-gap voltage V in CST. B was imposed as a constant background magnetic field, and an excitation of voltage V was inputted via a discrete port in CST. For the voltage waveform, a linear ramp of 20 ns was used and then the waveform was held constant for the duration of the simulation (to mimic CW operation). In these simulations, an ideal voltage generator is assumed with no impedance.^31,32

Values of B and V were systematically changed to map out a large subset of B–V space to determine the Buneman–Hartree and Hull Cutoff conditions. It was in the process of finding the operational points of this A6 magnetron that deviations from the expected theory were found, in particular, for the Hull Cutoff. We repeatedly found points to the left of the traditional Hull Cutoff parabola that oscillated when in theory, the magnetron should have been electrically shorted. This phenomenon is not by any means new (see, for example, Ref. 33); we simply want to give a more quantitative analytical explanation of it that has not, to the authors' knowledge, been considered previously. What this means is that the magnetron can operate for values of B lower than the Hull Cutoff and high values of V (as low voltages are potentially ruled out due to the Buneman–Hartree curve) if so desired. The results of our mapping of B–V space for the A6 magnetron described above are shown in Fig. 5 along with the traditional [Eq. (13a)] and modified [Eq. (11)] Hull Cutoff curves. Furthermore, if instead of a weighted average for the modified Hull Cutoff we used the maximum (taking the radius to be the anode radius) and minimum (taking the radius to be the cavity backwall radius) values of the B-field, the resulting curves are shown with green and blue dashes, respectively. Notice that the maximum B-field values (green dashes) coincide with the traditional Hull Cutoff (red solid) but both fail to adequately contain the new regime of operating points (red dots) that is nicely bounded by our weighted average approach (red dashes). This weighted average approach is a direct consequence of our first-order approximation of the electric field values (as a square wave) at the anode radius (where an electron at Hull Cutoff would be).

As can be seen in Fig. 5, there are three potential scenarios: (1) oscillation in the traditional regime (blue squares), (2) oscillation in the new expanded regime (red dots), and (3) no oscillation (green x’s). This classification does not take into account which mode is excited and is oscillating, just that there is operation as defined by output frequency (-ies) and power.

There is a considerable set of (B, V) points to the left of the traditional Hull Cutoff where the A6 magnetron considered here does oscillate and demonstrate fairly good operation at those input parameters. These points are NOT captured in the regime bounded on the left by the traditional Hull Cutoff but are captured in the new expanded regime bounded on the left by the revised Hull Cutoff presented here. As one moves further to the left, further away from the traditional and revised Hull Cutoff parabolas, the effectiveness of the operation severely drops. That is, there is not a clean FFT frequency output and the generated output power is orders of magnitude lower than operation at the blue and red (oscillating) B, V points; these (B, V) points are represented as green x's. This is expected as the magnetron in this state is far away from the Buneman–Hartree curve where a maximum output power is expected and is on its way to being electrically shorted if it is not already. We refer to these points as semi-oscillating as there may be frequency content and the output power spikes near start up but settles to a much lower level (if not zero). It is not recommended to operate a magnetron here. Examples of this A6 magnetron operating at 325 kV in terms of output frequency and power for applied DC axial B-fields in the traditional operating regime above the traditional Hull Cutoff (a), in the revised operating regime below traditional Hull Cutoff but above the revised Hull Cutoff (b), and below the revised Hull Cutoff (c) are shown in Fig. 6. The first two sets of plots (a) and (b) indicate normal and fair operation of the A6 magnetron and are referenced to as blue squares and red dots, respectively, in the B, V space above. The last set of plots (c) in green is representative of green x's in the B, V space above and is referred to as semi-oscillating/pulsed operation. Again, the simulations suggest that one can operate the magnetron in the new expanded regime bounded on the left by the revised Hull Cutoff developed here but not in the regime to the left of the new Hull Cutoff, which has become the new electrically shorted regime that has no magnetic insulation.

In this paper, we have proposed a geometric correction factor to the ubiquitous Hull Cutoff criterion for magnetic insulation in crossed-field VEDs to account for the effects of a normally utilized SWS on the anode. For the simple textbook case of a planar parallel-plate diode with rectangular corrugations on the anode constituting cavities/vanes, we have derived this revised Hull Cutoff equation for relativistic and non-relativistic electrons and voltages from first principles and have shown that it reduces to the seminal equations provided by Hull¹⁴ and Lovelace and Ott¹⁵ for the smooth anode case. Relativistic and non-relativistic versions of the equation for the commonly encountered coaxial diode (with SWS on the anode) setup have also been given. The effects of including the SWS in the Hull Cutoff formulation can be taken as a first-order perturbation to the smooth-bore case, with the width of the opening of a cavity (relative to the period of the SWS) being an indication of the amount of perturbation introduced. The height of the cavity is another parameter that affects the perturbation, albeit in not as direct of a way as the cavity opening.

To test the viability of the proposed theory, we examined the test case of an A6 magnetron (with radial extraction from one of the cavities). The applied DC axial magnetic field and AK-gap voltage were varied in CST PIC simulations of this canonical A6 to map out the B–V operational parameter space, including the Buneman–Hartree line and Hull Cutoff parabola. In doing so with particular emphasis on the Hull Cutoff bound of the oscillating region, a substantial region of B–V space was found to oscillate with clean output frequencies and (average) RF power below the theoretical classical Hull Cutoff (where the magnetron should be electrically shorted). However, this additional region of B–V space was very well bounded on the left upon inclusion of the analytical revised Hull Cutoff found here, corroborating the validity of our formulation and the influence of the SWS.

In practice, the findings of this research indicate that there is a substantially expanded region of B–V parameter space to operate in and one can operate a crossed-field VED (e.g., a magnetron) at magnetic fields lower than the traditional Hull Cutoff (but above the revised Hull Cutoff, approximately calculated including the SWS) and moderate to high AK-gap voltages, if so desired. For instance, this may be useful when one has experimental and equipment limitations or when moderate output power is needed (recall that in this regime, we are not near the Buneman–Hartree line where maximum synchronism and output power are expected).^34,35 It is expected that the formulation here carries over to other crossed-field devices as well, such as the MILO/MITL, and is applicable for Brillouin (single-stream) flow theory (single-particle cycloidal orbit, double-stream theory was used here). This work may also aid in explaining the observed violation of Hull Cutoff found in the literature.^13,33

The authors acknowledge Professor Edl Schamiloglu from the University of New Mexico, Department of Electrical and Computer Engineering, for his indispensable advice.

The authors have no conflicts to disclose.

Patrick Y. Wong: Conceptualization (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Project administration (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). Artem Kuskov: Methodology (supporting); Validation (supporting); Visualization (supporting); Writing – review & editing (equal). Benjamin Tobias: Resources (equal); Writing – review & editing (equal). Jonathon Heinrich: Resources (equal); Writing – review & editing (equal).

The data that support the findings of the study are available from the corresponding author upon reasonable request.

We characterize the conventional cylindrical (coaxial) relativistic A6 magnetron considered a test case in this paper here using theory for reference. The formulation of the cold-tube and hot-tube dispersion relations are given in Refs. 22–24 and will not be repeated here for brevity. For the hot-tube dispersion relation, the electron flow is modeled by Brillouin (steady-state, laminar) flow theory as advocated by Davidson et al.^36,37 with (linear) electromagnetic perturbations governed by the theory set by Chernin and Lau.³⁸ 

The cold-tube eigenmode frequencies as a function of mode number were calculated from the cold-tube dispersion relation²² for the first and second passbands and shown in Fig. 7(a). A sample beam line using single-particle cycloidal orbit theory computed from an operating B = 0.43 T, V = 325 kV point (shown in Fig. 4) is included, and the resulting theoretical linear growth rates of the modes from the hot-tube dispersion relation²⁴ for this set up are shown in Fig. 7(b), with the Brillouin profiles for equilibrium flow velocity, electric field, and number density in Fig. 7(c). The sample beam line is calculated using the simple electron beam dispersion relation (in the absence of space-charge effects, assuming time-harmonic, uni-directional traveling-wave solutions to the linearized equations of motion for the electron beam): $ω= β 0 v 0$⁠, where $ω=2πf$ is the angular frequency corresponding to frequency f, $β 0$ is the phase advance per cavity, and $v 0= E B≈ V D ∗ B$ is the unperturbed (by the AC signal) DC beam velocity, which is just the E × B drift with the DC electric (from applied voltage) and magnetic fields. This, when plotted on the same graph as the cold-tube dispersion relation, yields a zeroth-order estimate of the oscillation frequency and operating (in the linear regime) phase advance/mode number based on the intersection between the two dispersion relations. Note that when considering a Brillouin flow model, there will be a continuum of beam lines each with a different velocity v₀ corresponding to a layer of the Brillouin hub. This leads to an excitation of many modes and, hence, mode competition in a magnetron. The mode with the highest growth rate usually dominates.

The modified Hull Cutoff relation as formulated in the main body of this paper yields the minimum applied DC axial magnetic field B for a given AK-gap voltage V and seeks to answer the question of the necessary B-field for magnetic insulation and, hence, crossed-field device operation. The inverse of this problem is also a valid one, i.e., given a certain DC axial B-field, what is the necessary DC voltage that must be applied to the AK-gap? This V(B) relation also better relates to B–V plots for magnetron operation (parameter bounds) normally seen in the literature, where B is on the abscissa and V is on the ordinate.

Last, for cylindrical systems, take $w→θ, w ′→ θ ′,L→p,$ $D→ D ∗$¹⁵ in the above equations.