For mobile ionospheric heaters, it is necessary to develop highly efficient RF sources capable of delivering radiation in the frequency range from 3 to 10 MHz with an average power at a megawatt level. A promising source, which is capable of offering these parameters, is a grid-less version of the inductive output tube (IOT), also known as a klystrode. In this paper, studies analyzing the efficiency of grid-less IOTs are described. The basic trade-offs needed to reach high efficiency are investigated. In particular, the trade-off between the peak current and the duration of the current micro-pulse is analyzed. A particle in the cell code is used to self-consistently calculate the distribution in axial and transverse momentum and in total electron energy from the cathode to the collector. The efficiency of IOTs with collectors of various configurations is examined. It is shown that the efficiency of IOTs can be in the 90% range even without using depressed collectors.

## I. INTRODUCTION

Heating the ionosphere with Radio Frequency (RF) waves is of interest for a variety of scientific and practical reasons (see, e.g., Ref. 1). The RF frequency needed for effective heating (3–10 MHz) corresponds to wavelengths of 30–100 m. Therefore, existing facilities employ large antenna arrays (300–400 m) achieving sufficient antenna gain to operate with the total power of the order of several megawatts. It is desired to have a transportable heating facility with a much smaller antenna array. To achieve the same radiated power density as in the fixed facilities, much higher power levels are required (20–30 MW), and consequently, the considered RF sources and systems should be highly efficient.

In the frequency range of interest, 3–10 MHz, a promising source, which is capable of both high power and high efficiency, is a grid-less version of the inductive output tube (IOT), also known as a klystrode.^{2–4} Beam modulation without a grid is important because electron beam interception by grid elements is one of the most severe factors limiting klystrode power in high average power regimes. In this device, whose basic operation is discussed in Sec. II, an electron beam is accelerated from a cathode and fully modulated by an electrode near the cathode. This produces an alternating current that excites a resonant circuit.

To date, a number of klystrodes operating at higher frequencies have been developed. Table I summarizes the state-of-the-art in this development.^{5–7}

Parameters . | 250 kW CW klystrode (Los Alamos)^{5}
. | 10TD2130 (E2V)^{6}
. | VKP-9050 IOT Amplifier (CPI) . | VKP-9130 IOT Amplifier (CPI) . | 100 kW CW Multi-beam Klystrode^{7}
. |
---|---|---|---|---|---|

Frequency (MHz) | 267 | 470–810 | 500 | 1300 | 1300 |

Output power (kW) | 250 CW | 80–60 | 90 CW | 30 CW or 90 pulsed | 100 CW |

Beam voltage (kV) | 65 | 36–38 | 40 | 35/42 | 24 |

Beam current (A) | 5.5 | 2.2–1.57 | 3.5 | 1.3/3.4 | 4 |

Efficiency (%) | >70 | 65–58 | >65% | >60% | $\u2245$ 80 |

Operating regime | Class B | Class B | Class B | Class B | Class C |

Parameters . | 250 kW CW klystrode (Los Alamos)^{5}
. | 10TD2130 (E2V)^{6}
. | VKP-9050 IOT Amplifier (CPI) . | VKP-9130 IOT Amplifier (CPI) . | 100 kW CW Multi-beam Klystrode^{7}
. |
---|---|---|---|---|---|

Frequency (MHz) | 267 | 470–810 | 500 | 1300 | 1300 |

Output power (kW) | 250 CW | 80–60 | 90 CW | 30 CW or 90 pulsed | 100 CW |

Beam voltage (kV) | 65 | 36–38 | 40 | 35/42 | 24 |

Beam current (A) | 5.5 | 2.2–1.57 | 3.5 | 1.3/3.4 | 4 |

Efficiency (%) | >70 | 65–58 | >65% | >60% | $\u2245$ 80 |

Operating regime | Class B | Class B | Class B | Class B | Class C |

Also, the U.S. Naval Research Laboratory investigated via numerical simulation the dynamics of an IOT with the aim of improving output power,^{8} and a 350 MHz multiple beam IOT capable of generating 200 kW with about 70% efficiency and 22 dB predicted gain was proposed in Ref. 9.

As indicated in Table I, IOTs typically operate in the frequency range of 200 MHz-1300 MHz. These devices all operate in class B and C mode and offer efficiencies ranging from 60% to 80%. The device we consider operates at much lower frequency, 3–10 MHz, which has two important consequences. First, the beam can be fully modulated by a solid-state driver allowing for class D operation. Second, the electron transit time through the device is much shorter than the wave period, and as a consequence, the time evolution of the device can be analyzed as a sequence of self-consistent equilibria. Previous results of our University of Maryland group in the development of such a device are described in Refs. 11–14. The present paper is focused on an exploration of the maximum achievable efficiency.

High efficiency operation can be achieved by operating in class D mode, in which the electron beam is pulsed with a period corresponding to that of the RF and with a duty factor that is sufficiently small to allow all electrons to see nearly the same decelerating field. If this is the case, selecting the duty factor thus involves a trade-off between minimizing the temporal variation of the decelerating field while the beam is on and minimizing the effects of space charge in the intense beam. The present paper analyzes this trade-off in detail.

The paper is organized as follows. Section II illustrates the device layout and presents the basic device trade-offs needed to reach high efficiency. Section III presents results of numerical simulations characterizing the electron momentum and energy distributions at various points in the device. In Sec. IV, the efficiency of IOTs with collectors of various configurations is examined. The results presented in Secs. III and IV are obtained by using the 2D version of the Particle-in-Cell (PIC) code Michelle.^{10} (Some of these results were verified later by using an alternative code, BOA.^{15}) Although Michelle can operate in the time domain, similar to standard PIC codes, because of the large disparity between the electron transit time and the RF period we operated in the so-called “Gun Mode” in which steady state solutions are sought. The time evolution of the device is then treated as a sequence of static equilibria. The driving RF voltage pulse at the modulation anode and the induced voltage across the deceleration gap are then imposed as given electrode voltages. The output circuit, which would consist of lumped reactive elements is not modeled in detail in this case, a sample circuit is illustrated in Sec. II. Analysis and optimization of the circuit will be treated in a future publication. Finally, results are discussed and summarized in Sec. V. In the Appendix, we describe the potential depression caused by the beam space charge and estimate the maximum current that can be realized for a nonzero thickness annular electron beam.

## II. DEVICE BASICS

The schematic arrangement of the basic device and its circuitry are shown in Fig. 1 [Fig. 1(a)] along with representative dimensions [Fig. 1(b)]. We note that in these initial studies a flat collector is modeled. This will be replaced by a tapered collector in Sec. IV.

The device has several features aimed at maximizing efficiency. Electrons are emitted from a cathode held at a potential –*V _{AK}* and accelerated towards a grounded anode. The beam is gated by a modulation anode that can turn the cathode emission on and off without intercepting any beam current. This feature eliminates the conventional semi-transparent grid and the problems associated with beam current intercepted by the grid. The beam is annular so that electrons passing through the device experience nearly the same accelerating and decelerating fields. Electrons pass a decelerating gap across which the RF field appears, and are collected by a collection surface. The collected current returns to ground through an energy extraction circuit.

The electron beam must propagate from the cathode to the collector with minimal spreading to insure that all electrons are collected at the lowest possible potential. For the sake of simplicity, we assumed at the first step that the collector surface, as shown in Figs. 1, 3(a) and 4, is perpendicular to the electron beam. Later, in Sec. IV, we consider collector profiles providing, in line with Ref. 16, conservative power density deposition of less than 0.5 kW/ $ cm 2 $.

The area of the cathode (8.32 cm^{2}) is selected to keep peak loading of the thermionic cathode below 5 A/cm^{2} (see, e.g., Refs. 16 and 17), the emitter width has been designed such that for a 30 A beam current the peak cathode loading is about 3.4 A/ $ cm 2 .\u2009$ The average loading will depend on the selected pulse width. It can be anticipated to be a factor of 4 – 8 smaller than the peak loading. The separation between the emitting surface and the mod-anode is 2 mm. With the emitting surface at −70 kV, for a 30 A beam current we will require the mod-anode to switch from −68.75 kV(on) to −71 kV(cut-off). The anode and collecting surface are separated by the decelerating gap (15 mm). The Kilpatrick breakdown limit^{18} for our frequency range 3–10 MHz varies from 40 kV/cm (for 3 MHz) to about 50 kV/cm. By increasing the radii of the rounded edges [Fig. 1(b)] from 2.5 mm to 5 mm, we can reduce the maximum electric field strength to well within the breakdown limit. The sharp edges also play a deciding factor in transverse electron beam energy, thus further optimizing of the design by rounding these edges also leads to reduction in transverse beam energy and a slight increase in the efficiency of the design.

A sample output circuit for this device is shown in Fig. 2. It is a pi-type circuit with reactive elements, driven by a current source representing the beam, and feeding a resistive load representing a transmission line or tuned antenna.^{19} The pi-circuit is highly resonant and presents the input impedance needed to decelerate a 70 kV, 30 A beam with a rectangular temporal current profile. The main components of the pi-circuit are the input capacitor C_{1}, the output capacitor C_{2}, and the inductor L_{12} connecting the two capacitors. The circuit feeds an output impedance, R_{o}. The circuit parameters are selected so that the circuit presents a real impedance to the current source I_{Beam} at the operating frequency. The broad frequency range requires the circuit to be tunable, and the need for a constant decelerating voltage requires constant impedance, the study for engineering such a constant impedance pi circuit will be discussed in future publication. We note that because the circuit is highly resonant the voltage that develops across decelerating gap will always vary sinusoidally in time. That is, it will have a low harmonic content. Similarly the voltage delivered to the load will also be free of harmonics.

We have investigated three different configurations shown in Fig. 3: Model A, which has a uniform magnetic field (1 kG) and no beam compression; Model B, which has a tapered guiding magnetic field (peak field of 1 kG) and thus provides beam compression, and Model C, which has no guiding magnetic field.

A preliminary analysis of beam transport and efficiency was carried out for each model. It was found that the efficiency of Model A can exceed 90%, in Model B it can be at about 90% level, while in Model C the efficiency can be in the range of 85%–90%. In what follows, we will focus attention on model A, which at present is most promising.

In general, the energy extracted from the electrons and the efficiency can be estimated as follows. It is important to note that the transit time of electrons through the device (moreover, through the decelerating gap) is much shorter than the RF period. For example, when the voltage is in the range of 30–70 kV, the time for an electron to travel from the emitter to the collector ( $ T t r $) is on the order of 1 ns, while the 3–10 MHz frequency corresponds to a 100–300 ns RF period. Since

the RF field can be considered to be steady during the transit of individual electrons. However, electrons entering the device at different times during the pulse will experience different phases of the RF field and produce different efficiencies. Electrons emitted from the cathode at potential $ \u2212 V A K $ are accelerated to a potential close to the anode and are then decelerated and collected at a potential $ \u2212 V R F ( t ) $. (The difference between the beam potential and $ \u2212 V A K $ is caused by the potential depression due to the beam space charge; this issue is analyzed in Appendix.)

During this process the electrons have been given an energy $ e V A K $ by the power supply and have given an energy $ e V R F $ to the oscillating RF field on their return through the extraction circuit. Thus, for each electron the efficiency of energy conversion is

We refer to this as the instantaneous efficiency. Since the RF field varies in time as $ V R F ( t ) = V R F , max \u2009 sin \u2009 \omega t $, the average device efficiency is determined by averaging over the phases of the RF field during which the beam is on. For a flat top beam current micro-pulse of duration T centered in time on the peak of the RF field the average efficiency is given by

where $ \eta I , max = V R F , max / V A K $ is the peak instantaneous efficiency and

represents the efficiency reduction due to time averaging. In Eq. (4), we introduced $\theta $ describing the pulse duration: $ 2 \theta = \omega T $ is the duration of the beam micro-pulse in radians.

The average efficiency (3) is the product of two factors. Maximizing efficiency requires making each of these factors as large as possible. The maximum value of the RF voltage $ V R F , max $ in Eq. (2) is limited by a combination of space charge effects and transverse acceleration. In particular, as a result the maximum RF field is limited to a value less than the *V _{AK}*. Exceeding this limiting value causes reflection of the slowest electrons back to the cathode. This maximum value of the RF voltage depends on the beam current, the proximity of a thin annular electron beam to the metallic walls, and on the value of the focusing magnetic field suppressing electron transverse acceleration. In the case of the efficiency reduction (4), maximizing this factor means making the micro-pulse duration T as small as possible. If the peak current is fixed, shortening the pulse duration leads to a reduction in average beam power. Thus, it is desired to operate with the highest peak current possible. An increase in peak current, however, reduces the instantaneous efficiency and makes the evaluation of both factors in Eq. (3) necessary to obtain an optimum operating point.

The electron beam propagation through the device is illustrated in Fig. 4, where we display the results of a Michelle^{10} simulation for the Model A configuration of the device. Shown in the figure are the path of the beam electrons on top and the normalized momentum of the electrons ( $ p \u2032 = p / m c $) below, as functions of their position in the device as they pass from the cathode to the collector.

For this simulation, the RF voltage has been made as large as possible without reflecting any electrons. In this case *I* = 30 A, *V _{AK}* = 70 kV and

*V*

_{R}_{F}= 65.5 kV yielding an instantaneous efficiency of $ \eta I = 0.936 $. It can be seen that some electrons nearly come to rest at z = 0.14 m before striking the collector. The slight increase in the momentum in the vicinity of the collector is due to the suppression of the space charge field near the metallic surface of the collector.

The difference between the cathode potential and the RF voltage, $ V S C = V A K \u2212 V R F , max $, is due to space charge primarily in the region near the collector where the decelerated beam is slow. In Secs. III and IV, we consider several variations of the configuration shown in Fig. 4 aimed at reducing this potential $ V S C $. For now, we make the reasonable assumption that in high efficiency regimes the potential $ V s c $ is rather small and therefore this potential can be described by a collector “perveance” $ \mu c $ such that

The equation above helps us to relate the beam current to its limiting value given in Appendix by (A9)

where,

As follows from (6), when $ V S C $ is small ( $ V s c \u226a m c 2 / e = 511 k V $),

It must also be noted that the perveance relation given by Eq. (5) is posited on the basis of dimensional analysis. Assuming we have a valid solution to the Poisson equation and the equations of motion, we can find another valid scaled solution by lettering the potential scale as $\varphi \u2192\u2009\lambda \varphi \u2032 $. In non-relativistic cases, the velocities would then scale as $v\u2192 \lambda 1 / 2 v \u2032 $ in accordance with,

From the Poisson equation,

The charge density scales as $\rho \u2192\lambda \rho \u2032 ,$ thus the current density (j) scales as $ \lambda 3 / 2 $ as shown in Eq. (11).

The above argument assumes that all velocities scale with the potential to the ½ power. As we shall see later this is not strictly the case. In fact, if the transverse kinetic energy of electrons is significant and some other factors become important, this scaling relation would not hold true. We adopt this relation here for simplicity and for obtaining rough estimates. The MICHELLE simulations presented in Secs. III and IV will not rely on this assumption and this simplistic perveance relation.

The collector “perveance,” $ \mu c $, depends on the geometry of the beam: its thickness and proximity to conducting walls and on the strength of the confining magnetic field. For the device pictured in Fig. 4, *V _{SC}* = 4.5 kV, and the value of the collector “perveance” is $ \mu c = 99.4 \xd7 10 \u2212 6 $. For now, we will take this perveance to be a given constant and consider the consequences. We can introduce $ I \mu = \mu c V A K 3 / 2 $ and estimate dependence of the maximum instantaneous efficiency on current to be

We can now evaluate the average efficiency according to Eq. (3), and plot its level curves in the $ 2 \theta = \omega T $− $ I / I \mu = I / ( \mu c V A K 3 / 2 ) $ plane. Such a plot appears in Fig. 5.

We now adopt the approach that there is a minimum acceptable efficiency, and we wish to choose the “on” phase duration and peak beam current to maximize output power at fixed beam voltage. That is, we wish to move along a constant efficiency curve in Fig. 5 until we find the point at which power is maximized. In moving along the constant efficiency curve variations in peak current and “on” phase are related as follows:

Since we are moving along a constant efficiency curve at fixed beam voltage, maximizing power is equivalent to maximizing average beam current. Average beam current is proportional to the product of peak beam current and “on” phase. Thus, the maximum power occurs when

Performing the indicated differentiations in Eq. (13a) on the functions defined in Eq. (4) and (12), and using Eq. (13b) allows one to find the optimum peak beam current as a function of the “on” phase,

where $ \Psi ( \theta ) = ( 3 / 2 ) [ 1 \u2212 ( \theta ) cot ( \theta ) ] $. This curve is plotted in Fig. 5 as a solid green line that cuts across the level curves of efficiency. For a device with average efficiency equal to 90%, the optimum operating point is $ \omega T / ( 2 \pi ) = 0.15 $ and $ I / I \mu = 0.017 $. For the device pictured in Fig. 4 with operating voltage 70 kV and collector perveance $ \mu c = 99.4 \xd7 10 \u2212 6 $, the optimum peak current is then 31.3 A, and the average output power is $ P \xaf $= 300 kW. We can also use these numbers to determine the gap resistance required to provide the correct decelerating potential, $ R g = | V R F | 2 / 2 P \xaf = 7.15 \u2009 k \Omega $. Note that the emitter area is such that in the case of a 30 A beam current the cathode loading, as mentioned above, is about 3.4 A/cm^{2}. So, when the desire to shorten the current pulses requires utilizing higher peak currents this can be realized to some extent within the conservative limit of 5 A/ $ cm 2 $ or lower.^{16} In the case of approaching this limit, we have the freedom of enlarging the emitter cross-section, i.e., the beam thickness. The studies on efficiency values for beam currents ranging from 25 A to 60 A have been conducted while keeping the limit of cathode loading in mind.

## III. BEAM CHARACTERIZATION

In order to better understand how to increase the RF decelerating voltage while avoiding reflecting electrons, we examine the distribution of momenta for electrons at several axial positions for Model A. The electron momentum distribution is computed using the code Michelle^{10} and momentum values are recorded at the axial position values labeled in Fig. 6.

Using the data, we made histograms of the total energy, $ E T / m c 2 = 1 + p z 2 + p \u22a5 2 \u2212 1 $ and surrogates for the axial and transverse contributions to the energy, $ E z / m c 2 = 1 + p z 2 \u2212 1 $ and $ E \u22a5 / m c 2 = 1 + p \u22a5 2 \u2212 1 $. Here the momenta are normalized to *mc*, and we note that only in the nonrelativistic limit does $ E T = E \u22a5 + E z $. These histograms are displayed in Fig. 7, for three of the locations designated in Fig. 6.

The histogram data in Fig. 7 show that before the gap the energy associated with the axial motion greatly exceeds the energy associated with transverse motion as expected. The deceleration of the beam electrons by the RF field in the gap greatly reduces the axial energy, while increasing the transverse kinetic energy. So, after deceleration, the average transverse kinetic energy exceeds the axial energy by almost a factor of three. This increase in transverse energy is responsible for reducing the maximum value of the RF voltage that does not cause electron reflection. The fact that the energy associated with the transverse motion greatly exceeds the axial energy complicates the use of the perveance introduced in Sec. II. A constant perveance assumes that as current and potential are varied, electrons follow the same trajectories through space, only changing their speed. This is a good approximation if the magnetic field is strong, such that the product of the cyclotron frequency and the transit time from the gap to the collector is greater than unity, but it does not hold as we will show when the magnetic field is weaker.

The histograms of the total kinetic energy near the collector are shown in Fig. 8. The mean kinetic energy is about 4.2 keV consistent with the efficiency of 94%.

Figure 9 displays numbered simulation beamlets as they emerge from the cathode. These numbers are also displayed on the histograms in Fig. 7.

As one can see in Fig. 7, the innermost electrons 1 and 2 have the lowest axial energy before the gap because for them the clearance from the wall is maximal and, hence, the potential depression is the strongest. However, after the gap, the outer electrons, which were decelerated stronger than others, have the lowest axial energy, while the innermost electrons have the highest axial energy and, at the same time, the lowest transverse energy because their motion was perturbed by the space charge field not so strongly.

Let us now return to Fig. 6 and consider radial distribution of the potential within the electron beam and its vicinity in 4 out of 6 cross-sections shown in Fig. 6, i.e., in the cross-sections after the gap. This distribution is shown in Fig. 10. This distribution becomes more and more uniform as electrons approach the metallic collector wall.

## IV. COLLECTOR DESIGN

There are two issues that must be addressed with regard to the configuration of the collector in our device. The first is the impact of collector design on instantaneous efficiency. As the electron beam is decelerated and approaches the collector, some electrons may be reflected due to the space charge electric field and the buildup of transverse energy. Avoiding this determines the maximum decelerating voltage and efficiency. We will address this by looking at several different collector configurations. The second issue is the potential problem of an excessive local heat load on the collector due to the distribution of electron impacts on the collector. This will be examined by considering a collector with a tapered wall radius to spread out the region where the beam strikes the collector.

As shown in the configurations in Figs. 4 and 6, electrons propagate a distance of 27.5 mm after the gap before striking the collector. We therefore varied this distance by moving the collector plate closer to the gap. It is shown in Table II that the device efficiency (2) increases as this distance is shortened by more than 2%.

Length of separation (mm) . | $ V RF $ (kV) . | Efficiency (%) . |
---|---|---|

27.5 | 65.15 | 93.07 |

22.5 | 65.5 | 93.57 |

17.5 | 65.75 | 93.92 |

12.5 | 65.95 | 94.21 |

7.5 | 66.125 | 94.46 |

0 | 66.775 | 95.39 |

Length of separation (mm) . | $ V RF $ (kV) . | Efficiency (%) . |
---|---|---|

27.5 | 65.15 | 93.07 |

22.5 | 65.5 | 93.57 |

17.5 | 65.75 | 93.92 |

12.5 | 65.95 | 94.21 |

7.5 | 66.125 | 94.46 |

0 | 66.775 | 95.39 |

Figure 11 illustrates the electron motion and axial dependence of the electron momentum in the limiting case where the collector plate is located in the same cross-section as the end of the gap. In this case, there is no electron re-acceleration in the vicinity of the collector as seen in Fig. 4. There is still transverse motion in the beam as it approaches the collector due to the fact that the beam has experienced some transverse perturbations on its trip from the cathode to the collector. This particular design uses a simple flat wall collector to study the effect of space-charge on electron deceleration in the gap region and on electron motion in the vicinity of the collector. A more realistic collector configuration is discussed below.

Effect of the guiding magnetic field. The simulations performed for the configuration shown in Fig. 11 (where electron trajectories are shown for a 1.0 kG magnetic field) were also performed for 1.5 kG. This increase in the focusing field resulted in significant suppression of electron transverse motion. Correspondingly, the maximum RF voltage in the case of a 30 A beam increased from 66.775 kV (for 1.0 kG) to 69.35 kV (for 1.5 kG). This corresponds to the increase of the instantaneous efficiency from 95.39% to 99.07%. It was also found that with this higher magnetic field the current dependence of the instantaneous efficiency was consistent with a constant perveance as assumed in Eq. (6).

Note that we limit our current study by an ideal case of a constant magnetic field that extends up to the collector surface, while ignoring possible effects of reflected and secondary generated electrons at the collecting surface. The detailed study describing collectors with reflected and secondary electrons taken into account will be presented in the next publication.

The downside of this simple design is the power density striking the collecting surface. In the absence of an RF field, the peak power of a 70 kV, 30 A electron beam is 2.1 MW, while the cross-section area of the annular beam is about 9 cm^{2}. When the beam pulse duration is a quarter of the RF period the mean value of the beam power density is 55 kW/cm^{2}. Even in the case of operation with 90% efficiency, the power density is still above 5 kW/cm^{2} that is unacceptable for high average power regimes.

To reduce the beam power density to an acceptable level [less than 500 W/cm^{2} (Ref. 15)], we considered a slanted collector surface with an angle of 6° or less with respect to the device axis. An example is shown in Fig. 12 where the angle is 4.3°, the resulting efficiency is 93.9%, and the beam power density is less than 500 W/cm^{2}.

We further explored the dependence of the efficiency on the beam current in this design. The current was varied from 25 A to 60 A, while the efficiency varied from 94% to 93% only. Presumably in this design the build-up of transverse motion (not space charge fields) is the cause of reflected particles. Such a design could operate at 60 A with a micro-pulse duration T equal to 1/8 of an RF period giving the same average current of 7.5 A as a 30 A device with T equal to 1/4 of an RF period. The average efficiencies in the two cases as given by Eq. (3) are 90.6% and 84.5%, respectively, that favors a shorter beam pulse.

## V. DISCUSSION AND SUMMARY

The efficiency of inductive output tubes (IOTs) was considered. Maximizing average efficiency involves a trade-off between pulse duration and peak beam current as indicated in Eq. (3). The peak instantaneous efficiency depends on the beam current, magnetic field, and collector geometry. For sufficiently strong magnetic fields and flat collector surfaces the peak efficiency satisfies a scaling relation based on a geometry-dependent perveance as indicated in Eq. (6). However, for lower magnetic fields and for slanted collector surfaces, the efficiency depends less strongly on the current. In this case, it is advantageous to operate with a high peak current and as short a beam pulse as possible. For example, the configuration of Fig. 12 is capable of 90% average efficiency even without a depressed collector. Overall system efficiency will also depend on the efficiency of the other subsystems. It may be the case that these limit the system efficiency and that further optimization of the electronic efficiency of the source is not needed.

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 solenoids, which should generate the 1 kG or even higher magnetic field. Therefore, it makes sense also to study Model C which has no guiding magnetic field and Model B that can require 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. Thus, the study presented above was not aimed at describing the optimal design of the MW-class IOT, but at analyzing some issues important for making such a design.

## ACKNOWLEDGMENTS

This work was supported by the Air Force Office of Scientific Research under Grant No. FA95501410019.

#### APPENDIX: POTENTIAL DEPRESSION CAUSED BY THE BEAM SPACE CHARGE

The effect of the beam space charge on potential depression in a thin annular electron beam was analyzed in Ref. 11. In this Appendix, we briefly outline the same steps in the case of a beam finite thickness; a thorough analysis of limiting currents in electron beams with gyromotion propagating down a waveguide immersed in an external magnetic field has been done in Ref. 20. Our consideration describes the potential depression, first, in a uniform pipe and, then, in a region of the decelerating gap.

##### 1. Potential depression in a uniform pipe

Consider an annular beam propagating in a uniform metallic pipe (1D problem). By using the 1D Poisson equation

and matching its solutions in three regions: (A) inside the beam ( $ r < R b , i n $), (B) within the beam ( $ R b , i n < r < R b , o u t $) and (C) between the beam and the wall ( $ R b , o u t < r \u2264 R w $), one can readily find that the potential of the innermost electrons with $ r = R b , i n $ is determined by the sum of two potential drops

The potential of the electron at any given $ r \u0303 $ is found similarly

Note that in Ref. 11 the maximum potential drop was defined for a thin electron beam by the last term in the RHS of (A2) only. Now, Eqs. (A2) and (A3) contain two more terms accounting for the potential drop in the case of a finite beam thickness. In (A2), these terms are described by the function

This function is shown in Fig. 13. In other words, now the limiting current normalized to $ I A = 4 \pi \epsilon 0 m c 3 / e = 17.04 k A $ ( $ I \u2032 = I / I A $, $ I A $ is related to the Alfven current) is given by

Equation (A5) defines the maximum current, which is determined by the energy of the innermost electrons, which, due to the potential depression, may stop first in the process of electron deceleration in the gap. In general, in a thick beam, the electron energy normalized to the rest energy $ \gamma = \Epsilon / m c 2 = 1 / 1 \u2212 ( v / c ) 2 $ depends on the radial coordinate of an electron

where the electron potential is determined by Eq. (A3).

##### 2. Potential depression in a gap

Under the same assumptions as those made in Ref. 11 (the outer radius of the gap $ R w , g $ is much larger than all other transverse dimensions), we can match the potential and its derivative to a solution in the beam potential assuming that the outer beam radius is close to the radius of the uniform part of the pipe $ R w $ as shown in Fig. 1. Then, the condition for stable propagation of the beam of a finite thickness through the gap is

In the left-hand side of (A7)

In Ref. 11, the LHS of the corresponding condition is the same as in (A7). This function denoted as $ f \gamma ( z ) $ was shown in Fig. 3 of Ref. 11 for the voltage close to 70 kV. This function reaches its minimum value at $ z = L $. So, taking into account that $ g ( z = L ) = 0 $, we can define the maximum value of the normalized beam current $ I \u2032 $ as

In (A9),