Physics of efficient gridless tetrodes with intense electron beams

Recently, the Department of Defense initiated a new program to develop the technology base for transportable ionospheric heating facilities. Existing stationary facilities, located, as a rule, at high latitudes (see, e.g., Refs. 1 and 2 and references therein) are limited, due to their location, to a certain set of applications. However, a transportable heater would open the door for new applications.³ To create such a transportable heater, operating in the 3–10 MHz frequency range, with capabilities close to those of facilities such as HAARP (High-Frequency Active Auroral Research Program),⁴ it is necessary to reduce the size of the antenna array, which is 300 m × 400 m in HAARP, to a size on the order of 30 m × 40 m. This implies that the radiated power must increase dramatically to achieve the same effective power density in the ionosphere. Thus, it is incumbent to develop highly efficient, relatively compact RF sources.

Our group has studied the possibility of developing a source capable of operating in the 3–10 MHz frequency range at about the 1 MW level with high efficiency (90%).^5,6 The device described in this manuscript shares features of both the tetrode and the Inductive Output Tube (IOT). High efficiency is achieved due to a combination of features. The device is intended to operate in class D mode, where the beam is “on” for less than ¼ of an RF cycle and “off” for the remainder. We have found previously that the electron beam can be fully modulated by a solid-state driver.⁷ The driver is electrically connected to the modulating anode, instead of an intercepting grid. Power is then extracted through a low-loss resonant circuit.

The basic physical issues for this device were analyzed in Ref. 8. The present paper is a continuation of that study and focuses primarily on two issues: (a) the determination, in Sec. II, of the optimal dimensions of the interaction space in the configuration and (b) the determination, in Sec. III, that secondary electrons decrease the efficiency of operation, particularly with regard to electron collection. Our study is a numerical one, with all simulations carried out using the particle-in-cell (PIC) code Michelle.^9,10 We find in Sec. II that in order to achieve high efficiency, the anode-cathode gap and length of the anode must be sufficiently large so as to efficiently recover the bulk of the beam energy at the collector. In Sec. III where the dynamics of secondary electrons is considered, we find that during the “on” phase of the beam, the space charge of the primary beam electrons is sufficient to contain the secondary electrons in the collector region. However, once the beam is “off,” secondary electrons can stream back across the decelerating gap. These electrons must be confined by the combination of a strong magnetic field and the cathode potential long enough for the RF voltage to reverse sign and drive the secondary electrons into the collector.

The schematic arrangement of the device and its circuitry are shown in Fig. 1. The device has several features aimed at maximizing efficiency. Electrons are emitted from a cathode held at a steady potential −V_AK (V_AK = 70 kV for the studies presented here) and gated by a modulation anode that can turn the cathode emission on and off without intercepting any beam current. This feature eliminates the conventional semitransparent grid and the problems associated with beam current intercepted by the grid. (We note that Fig. 1 is schematic and that in a real device, the cathode and modulation anode would be supported on a stalk, thus maintaining an annular opening for the beam to pass through the modulation anode.) The voltage on the modulation anode swings between −V_AK + 1.25 kV and −V_AK − 1.25 kV to achieve class D operation. In previous work, we showed that the required fast voltage swings could be supplied using an inductive adder.⁷ The gated beam electrons are then accelerated toward a grounded anode. The beam is annular and thin so that electrons passing through the device experience nearly the same accelerating and decelerating fields. Electrons then pass a decelerating gap, across which the RF field is excited, and are collected by a collection surface. The DC potential of the collector is the same as that of the anode (i.e., it is at 0 kV). However, the instantaneous potential on the collector varies sinusoidally in time with a peak value, V_RF,max, that is close in magnitude to V_AK. The collected current returns to ground through a resonant energy extraction circuit. Because the frequency of operation is low (3–10 MHz) and the electron transit time is short (less than 2 ns), the device can be basically understood as cycling at the RF period through a sequence of static equilibria. Thus, when the beam is “on,” the collector potential is large and negative, and nearly all the beam kinetic energy is recovered by the collector. When the beam is “off,” the collector potential swings positive so that the time average potential at the collector is at ground. Since the collector potential oscillates at the RF voltage, the capacitance of the collector must be accounted for in designing the resonant extraction circuit. In Sec. III, we will simulate the time evolution of a single cycle including a treatment of secondary electrons.

Initially, in Ref. 8, three possible configurations were considered: a device with constant guiding magnetic field from the emitter to the collector (Model A), a device, in which the magnet has a finite length and, hence, the guiding magnetic field is nonuniform along the axis (Model B), and a device with no guiding magnetic field (Model C). It was found that in the absence of a guiding magnetic field, it is impossible to transport the intense electron beam (the nominal voltage and current are 70 kV and 30 A, respectively). In this paper, we analyze the device operation assuming a constant guiding field (Model A). In the future, we will address the issue of inhomogeneous magnetic fields (Model B).

The geometry used in Michelle simulations described in this section is shown in Fig. 2 and is similar to the schematic arrangement shown in Fig. 1. However, the collector is shaped differently. The collector surface is positioned at a small angle with respect to the device axis in order to reduce the power density of beam deposition to manageable limits of about 0.47 kW/$cm2$⁠. At the cathode, the annular electron beam has inner and outer radii of 36 and 40 mm, respectively, and the uniform guiding magnetic field is equal to 1 kG on axis. The distance between the cathode (more exactly, the modulating anode) and the anode, $LAK$⁠, and the anode length, $LAnode$⁠, will be determined to maximize efficiency. The RF voltage across the decelerating gap varies in time as $VRF(t)=VRF,max sin ωt$⁠. We determine the dimensions that result in the highest value of V_RF,max, while still collecting virtually all the beam electrons on the collector. Although the RF voltage varies in time, in this section, we describe the results of a sequence of simulations with a constant $VRF$⁠. This simplification is justified due to the fact that the electron transit time from the cathode to the collector (less than 2 ns) is much shorter than the RF period (exceeding 100 ns at frequencies less than 10 MHz). In Sec. III, when secondary electrons are included in the simulation, time-dependent simulations will be performed. We find that the optimal dimensions will be slightly different from those found based on static simulations.

Since the electron transit time is much shorter than the RF period, it is possible to carry out studies assuming that the RF voltage remains constant during the electron transit time (so-called “static simulations”). A sequence of static simulations, with varying V_RF, is then carried out to determine the performance of the device as a function of time. In static simulations, for electrons emitted from the cathode at the potential $−VAK$⁠, collected at collector at potential $−VRF$⁠, and passing through the power extraction circuit to ground, the ratio $VRF/VAK$ defines the instantaneous electronic efficiency of the device. The electronic efficiency is then determined by averaging the instantaneous efficiency over the period of an RF cycle for which the beam is on.

Since the ratio $VRF/VAK$ defines the instantaneous efficiency, the closer $VRF$ is to $VAK,$ the higher the instantaneous efficiency. However, the RF voltage is limited by the fact that we want all beam electrons to reach the collector. Electrons will be reflected if they acquire too much transverse energy on their passage from the cathode through the gap or if their space charge lowers the potential near the collector by a large amount. The electrode geometry determines the maximum achievable $VRF$ and instantaneous efficiency. Depending on the geometry, it is possible to reduce the transverse energy gained by the beam electrons and shield their space charge and thus maximize the RF voltage.

Two different anode geometries were considered: (a) an anode with sharp corners and (b) an anode with rounded corners (7.5 mm radius of the rounding). An example of the configuration with a rounded anode is shown in Fig. 2. The collector geometry is kept fixed, while the length of the A-K gap and the length of the anode were varied.

Table I displays the maximum instantaneous efficiencies for devices with different dimensions. In Table I, the anode length is varied with the AK gap length fixed at 37.5 mm. In Table II, the AK gap length is varied with the anode length fixed at 27.5 mm. Our goal is to decrease the overall length of the device while maintaining high efficiency. After multiple simulations, it was concluded that an anode length of 27.5 mm and an AK gap length of 32.5 mm are the optimal lengths. The maximum RF-voltage (⁠$VRF$⁠) for such a device is 65.75 kV with a 94% efficiency. The reason for the high efficiency is that the electrons are in a thin annular beam and propagate with a small radial distance (4 mm) of the anode. Thus, they gain little transverse energy while transiting the device and are collected at a potential close to that of the cathode.

It should also be noted that the maximum RF voltage is quite sensitive to modifications in the AK-gap length, especially when this length is shorter than 32.5 mm. This is due to the fact that reducing the AK-gap length leads to an increase in the transverse component of the electron momentum as a result of the radial component of the electric field from the edges of the mod-anode and anode. Adding transverse momentum to the beam electrons reduces their axial momentum and lowers the potential at which they can be collected. This is more pronounced in the case of sharp edges, as expected. The results obtained from the static-case simulation guide the computations presented in Sec. III when the role of secondary electron is studied, where we investigate the secondary electron dynamics for both the long model (AK-gap length = 37.5 mm and anode length = 52.5 mm) and the compact model (AK-gap length = 32.5 mm and anode length = 27.5 mm).

When primary beam electrons strike the collector surface, they are not all absorbed by the metal, and some fraction of electrons are scattered and reflected. Some electrons ejected from the collector due to collisions with primaries are known as secondary electrons, and their presence can affect device operation. Once the secondary and reflected electrons are included in the computations, we have to switch from static simulations to the time-domain simulations. The behavior of these secondary electrons depends on the time within the RF cycle. That is, the behavior depends on whether the primary beam current is on or off.

As in Sec. II, we assume that the magnetic field is uniform along the axis. So secondary and reflected primary electrons can propagate back into the interaction space during the positive cycle of the RF voltage. The time evolution of the RF voltage and the beam current are shown in Fig. 3. The beam current is switched on at t = 13 ns and switched off at t = 27 ns. The electrons take roughly 1.5 ns to travel from the emitter to the collector.

Figure 4 shows the axial momentum and total transverse momentum obtained at two different times during the on period of the beam current, 20 ns and 26 ns. The simulations are done for the AK gap distance of 37.5 mm and the anode length of 52.5 mm.

As seen in Figs. 4(a) and 4(c) (at t = 20 ns and 26 ns, respectively), the primary beam, emitted from the cathode (shown in green), is accelerated in the AK-gap. The beam then travels past the anode with a small change in axial momentum until it encounters the decelerating gap. The decelerating RF field, extending a distance on both sides of the gap region, reduces the axial momentum of the beam, which then travels to the collector. Figures 4(b) and 4(d) show the transverse component of the momentum plotted over the same axial distance. The transverse momentum increases as the beam passes through the gap region as a result of the transverse component of the RF field. The energy associated with this transverse motion then limits the maximum RF voltage. The figures are similar, indicating that the behavior of the electrons during the flat-top portion of the current pulse is also similar.

Note from Fig. 3 that the RF voltage is different at these two times. When the decelerated primary beam strikes the slanted collecting surface, it produces secondary electrons. Here, red, black, and pink colors indicate the first, second, and third generation secondaries. It is also important to mention that adding further secondary generations in the simulation does not change the axial and transverse momenta profile of the electrons. The secondary electrons are separated into different groups having different momenta and spatial locations. Further, there is a group of relatively low energy secondary electrons in addition to secondary electrons with higher energy and positive axial momentum taking them in the direction of the collector. The slow moving secondaries have almost zero axial and transverse momentum. These secondary electrons, produced close to the decelerating gap, are susceptible to the RF field, which could accelerate them back into the gap. However, these slow moving secondary electrons are unable to leave the collecting region when the primary beam is present due to the space charge of the primary electrons which creates a potential barrier keeping them near the collector. This will change later in the RF cycle, when the primary beam is off.

As follows from Fig. 4, the secondary electrons can be separated spatially into various groups. This separation is the result of the gyration of the primary electrons as they approach the collector. The primary electrons strike the collector at three different locations as shown in Fig. 5. The secondary electrons generated are thus also separated spatially into three different groups, as shown in Fig. 6.

The production and motion of secondary electrons were studied by using the PIC-code Michelle.¹⁰ A spent beam of primary electrons is represented by a set of energy-modulated macroparticles. For taking into account the role of energy and incident angle distribution of the primaries in the production of secondaries, a comprehensive secondary emission model¹¹ is included in the code. The high-energy secondaries are essentially primary electrons that have been scattered out of the metal collector. They have positive axial momenta and hence can generate new secondary electrons; each subsequent generation shifts to the right. However, this shift is quite small since they also have large transverse momenta. The low energy electrons are true secondaries that have been produced in the collector by collisions with the primaries. These electrons are produced in each generation. Only the third generation is visible in the figures as they are plotted last obscuring the earlier generations.

Once the primary beam current is switched off at t = 27 ns, the slow secondary electrons, which were previously confined in the collector area by the space charge field of the primaries, begin to escape from this region into the interaction space as shown in Fig. 7. As these electrons pass through the decelerating gap, they are now accelerated by the RF voltage in the negative z-direction until they reach the anode region and encounter the barrier created by the difference in potentials between the mod-anode and anode. These electrons are now trapped between two potential barriers and move back and forth in the interaction space. This is similar to electron motion in a device with a virtual cathode (see Ref. 12 and references therein).

In the long version of the device (⁠$LAK$ = 37.5 mm and $LAnode$= 52.5 mm), most of these electrons are intercepted by the grounded anode as shown in Fig. 8(a). This process continues until about t = 36 ns when the RF voltage falls so that the barrier on the right becomes very low. Then, the electrons that were not absorbed by the anode can escape from the interaction space back into the collecting region where they are absorbed back into the collector.

By the time the RF voltage changes sign at t = 40 ns (see Fig. 3), there are no more electrons (secondary or reflected) left in the device. The power contained in the secondary and reflected electrons accounts for roughly about 5% of the incoming beam power, the vast majority of which was absorbed into the anode as shown in Fig. 8(a). This is disadvantageous since this reduces the device efficiency from roughly 93% to 88%.

The effects of reducing the AK-gap and the anode length on RF voltage are presented in Sec. II. These simulations did not include secondary electrons and were static-case simulations. Then, we studied the relation between inclusion of secondary electrons and RF voltage in Sec. III, and thus, we can use the results shown in Table I while keeping in mind that inclusion of secondary electrons will likely reduce the RF voltage. Since the model being used for the time-domain simulation is the same compact model as that used in the static case simulation, the maximum RF voltage should likely still be within the range of 65 kV. Taking these factors into account, the AK-gap length was reduced from 37.5 mm to 32.5 mm and the anode length from 52.5 mm to 27.5 mm. A snapshot of the particle trajectory at t = 32 ns is shown in Fig. 8(b). There is no loss in efficiency due to absorption of secondary electrons at the anode as seen in the other longer model [Fig. 8(a)].

There are several other ways to address the issue of efficiency loss due to interception at the anode. By increasing the magnetic field confinement from 1 kG to 1.5 Kg, it is possible to avoid interception at the anode; however, this will also increase the power consumption of the solenoid. Similarly, we can also increase the clearance between the beam and the anode by increasing the anode radius. This, however, will make space charge effects stronger, thus reducing the maximum RF voltage and the device efficiency.

As shown in Fig. 9, the key features such as evolution of axial momentum, generation of slow and fast moving secondary electrons, and clumping of secondary electrons into groups remain unchanged. However, the total transverse momentum does not fall to zero unlike in the longer device. This is due to the short anode length. The stronger transverse electric fields generated at the decelerating gap amplify both radial and azimuthal components equally.

The effect of changing the AK-gap length, anode length, and anode geometry on the maximum RF voltage and device efficiency was studied. Reducing the AK-gap increases the radial electric field, which in turn increases the transverse energy of the beam current, thus reducing the device efficiency. Similarly, reducing the anode length reduces the maximum energy gained by the primary beam via acceleration. The device efficiency is more susceptible to changes in the AK-gap length and anode geometry than the anode length. Thus, it is possible to optimize the device geometry and make it much more compact while still operating with high efficiency.

The device efficiency (⁠$η$⁠) is dependent on the space charge potential (⁠$VSC$⁠) as given by Ref. 8,

However, the space charge potential increases due to the inclusion of secondary electrons. The effects of including secondary electrons were studied using particle in cell time-domain simulations. The axial and transverse momentum evolution of beams with and without secondary electrons was studied across different cross sections to understand the interaction between the primary and secondary beam currents. The secondary electron dynamics in the absence of primaries was also studied. No cathode bombardment was detected due to the presence of the strong repelling field in the AK-gap. There is a significant efficiency loss due to the absorption of secondary electrons at the grounded anode. However, this can be negated by optimizing the device geometry using the results obtained by changing the AK-gap length, anode length, and anode geometry. In summary, time dependent simulations revealed two important issues in operation of this device: (1) even in the absence of a primary electron beam, there is no cathode bombardment by the secondaries; (2) by the end of the positive half period of RF voltage, all secondaries disappear, and hence, there is no accumulation of electron clouds.

Although we limited our consideration to Model A, which has a constant focusing magnetic field and offers the highest efficiency, the choice of the model and optimization of all design parameters should be done having specific system requirements in mind. A number of factors should be taken into account such as weight and power consumption of the solenoids, which should generate the 1 kG or even higher magnetic field. Therefore, it makes sense to also study Model C, which has no guiding magnetic field, and Model B, which requires lighter solenoids (than Model A), and provides us with the ability to compress the beam. Also, the cathode area in Model B can be larger than in Model A that mitigates cathode loading, while an expansion of the beam in the collector region can mitigate collector loading.

This work was supported by MURI Grant No. FA95501410019. The simulation results were obtained using PIC-code MICHELLE.