2. Microwave tubes with Cherenkov's radiation mechanism

## 2.1 General subjects

Microwave sources operating on basis of Cherenkov's radiation are a wide class of microwave tubes, including the magnetrons, backward-wave tubes (BWTs), resonant TWTs, and orotron [9]. As for now, microwave sources with Cherenkov's interaction have allowed to reach the maximum levels of peak capacity ˜3 GW in the 3-cm range of wavelength and more than 5 GW in the 8-cm range at a pulse duration of 1–10 nanoseconds including the generation of ultrashort pulses of electromagnetic radiation (the effect of superradiation) [11]. Let us consider the development of these tubes on the example of their most famous representatives, which are magnetrons and TWTs.

#### 2.2 Magnetron

Historically, the first microwave tube whose operation is based on Cherenkov's interaction was a magnetron [12]. The successful combination of properties of a multimode oscillating system of the magnetron and electronic processes occurring in its interaction space has allowed the magnetron to become one of the most effective microwave generators [13, 14]. The constructions of magnetrons developed nowadays generate electromagnetic oscillations in the frequency range from 300 MHz to 300 GHz. The output power of continuous magnetrons ranges from a few fractions of a watt to several tens of kilowatts, and the pulsed magnetrons generate the electromagnetic oscillations with output power from 10 W to 20 MW. Both pulsed and continuous magnetrons are widely used in different ground-based and on-board electronic systems, industrial and microwave household heating systems, physical experiments for plasma heating, and the acceleration of charged particles, as well as in the phased antenna grids and space systems of solar energy conversion to direct current energy (the system of wireless energy transmission to the earth). The miniaturization of the magnetron construction has allowed to reduce their weight to 100 g at the pulsed power of 1 kW and the efficiency of which achieves about 50% that is quite competitive with the best samples of modern transistor oscillators [10, 13–20].

Concerning the existing trends of investigations, it is necessary to emphasize the problems of improvement of the cathode (emitter) operation of magnetrons. It is found (see, e.g., [13, 14, 21]) that the changes and instability of the cathode

emission characteristics influence considerably the frequency stability of the magnetron. On the other hand, the disorder in the operation mode of the cathode connected with its overheat leads to essential reduction of durability of the cathode and the cathode node as a whole. In this regard, it is of great interest to study the problem of excess electronic and/or ionic bombing of the cathode and to clarify the role of turbulence of a cathode electronic cloud in the course of secondary-emission multiplication of an electronic beam [22, 23]. In order to improve the magnetron operation, there conducted active investigations of new effective cathode materials and cathode node constructions, providing high durability and emission stability. It especially concerns investigations connected with applying the cold secondaryemission cathodes providing virtually instant readiness and nonfilament ("cold") start of magnetrons [22, 24–28].

The scope of magnetrons continues to expand constantly that is primarily caused by their advantages such as high electronic efficiency (more than 80%), relatively low voltages (in particular, anode voltage), the high relation of power output level to the weight of the tube, the compactness of construction, the simplicity of production, and comparatively low cost [10, 15]. However, such drawbacks of magnetrons as the low stability of frequency, the increased noise levels, and spurious oscillations require carrying out additional investigations and analyzing the ways for improving its output characteristics, in particular, in a short-wave part of a millimeter range. Solving mentioned above problems will allow to increase competitiveness of magnetrons and to expand their functionality in comparison with other microwave tubes, such as one-beam and multibeam klystrons, klystrons, complexificated microwave sources on the basis of "the solid-state generator and TWT" chains, etc. [10, 13, 14, 21, 22, 24, 25].

#### 2.2.1 The surface wave magnetrons

The state-of-the-art evolution of magnetrons is associated with increasing the frequency (phase) stability and rising the lifetime, as well as enhancing reliability due to an application of the cold cathodes [21, 24, 26–28]. In particular, there is a significant interest in the development of magnetrons in the millimeter range. These magnetrons can be applied in radar systems [29]. Among possible designs of similar magnetrons, it is necessary to select the magnetrons working on higher space harmonics (for example, by using as operating a first negative (˜1) space harmonic or an oscillation does not – π mode [30]). The magnetrons operating in such modes are named the surface wave magnetrons [30, 33]. According to the approach described in [32], a method has been developed to calculate the parameters and the operation modes. It is necessary to note that a major feature concerning to the application of the surface wave magnetron is associated with the generation of electromagnetic waves in the millimeter range at a considerably low magnetic fields and increased sizes of an interaction space. The prototypes of the surface wave magnetrons have been built. The prototypes operate at the <sup>π</sup> – mode, <sup>2</sup> and they provide the following output parameters: wavelength band from 1.25 mm up to 6.8 mm, a level output pulsed power from 1 up to 150 kW, and efficiencies 0.8–20% [31].

In spite of the intensive research over the years and getting experimental results including the constructions of the surface wave magnetrons, up to now, we have no necessary and full information about physical processes occurring in the interaction space of given magnetrons. Therefore, for studying and understanding the features of a mechanism of nonlinear interaction into an interaction space of the magnetron, it is necessary to carry out additional computer modeling, a phase

### Vacuum Microwave Sources of Electromagnetic Radiation DOI: http://dx.doi.org/10.5772/intechopen.83734

bunching process of an electron beam under its interaction with surface wave by using the Particle-in-Cell Method.

For studying, we used the design of a 3-mm range magnetron. Schematically, this design presents in Figure 2. The essential geometry sizes of an interaction space of this magnetron are presented in Table 1. As the operation mode, the mode other than the <sup>π</sup>˜ mode was taken, namely, the oscillation <sup>π</sup> – mode. As cathode of the <sup>2</sup> magnetron, an indirectly heated oxide cathode, which produced an emission current density of °2.0 A/sm<sup>2</sup> , was used [37].

For computer modeling, we used the 2-D mathematical model of the magnetron described in [34]. The basis of this model is the self-consistent set of equations including the motion equation, the equation of excitation, and Poisson's equation for calculation of space charge forces.

As already mentioned above, the theoretical basis of the surface wave magnetrons was described in the works [30–32]. It is necessary to note that the distinctive feature of electron-wave interaction in given magnetrons (for example, as distinguished from the classical magnetrons [14], using the π˜ mode as the operation one) is the distribution of an electromagnetic wave in the neighborhood of a surface of the RF structure and its interaction with electrons on a top of the space charge hub. In order to understand the mechanism of interaction of an electromagnetic field with an re-entrant electron beam, it is very important to define the features of the radial and azimuthal distributions of the electromagnetic wave in an interaction space (between a cathode and inside surface of an anode block (see Figure 2), as well as to provide clearer understanding about behavior of electrons and their motion trajectories.

An interaction space of the magnetron is shown in Figure 2, schematically. In order to determine the electromagnetic field in the resonance RF structure (anode

#### Figure 2.

Schematically image of a surface wave magnetron.


#### Table 1.

The main parameters of surface wave magnetron.

block), we used the decision obtained from Maxwell's equations for free space without charged particles [14]. As a result, the expressions for components electromagnetic field in the interaction space may be written as

$$E\_{\phi}(r,\phi) = E\_m \frac{N\theta}{\pi} \sum\_{m=-\infty}^{\infty} \left(\frac{\sin\gamma\theta}{\gamma\theta}\right) \times \frac{Z\_{\gamma}^{'}(kr)}{Z\_{\gamma}^{'}(kr\_a)} \cdot e^{jr\phi};\tag{1}$$

$$E\_r(r, \phi) = -jE\_m \frac{N\theta}{\pi kr} \sum\_{m=-\infty}^{\infty} \gamma \cdot \frac{\sin \gamma \theta}{\gamma \theta} \times \frac{Z\_\gamma(kr)}{Z\_\gamma'(kr\_a)} \cdot e^{j\gamma \phi},\tag{2}$$

where k ¼ 2π=λ is the propagation factor in free space; λ ¼ c=f the wavelength of the magnetron; 2θ—the angle subtended by a space between segments of the anode block;

$$Z\_{\gamma}(kr) = J\_{\gamma}(kr) - \frac{J\_{\gamma}^{\prime}(kr)}{N\_{\gamma}^{\prime}(kr\_{a})} \cdot N\_{\gamma}(kr) \tag{3}$$

and

$$Z\_{\gamma}^{\prime}(kr) = J\_{\gamma}^{\prime}(kr) - \frac{J\_{\gamma}^{\prime}(kr\_{\varepsilon})}{N\_{\gamma}^{\prime}(kr\_{a})} \cdot N\_{\gamma}^{\prime}(kr) \tag{4}$$

are the combinations of the well-known Bessel J<sup>γ</sup> ð Þ kr and Neumann N<sup>γ</sup> ð Þ kr functions; γ is zero or any positive or negative integer.

0 ð Þ kr <sup>γ</sup>Z<sup>γ</sup> ð Þ <sup>0</sup> kr In the expressions of Eqs. (1) and (2), we have the ratio of <sup>Z</sup><sup>γ</sup> and , <sup>Z</sup><sup>γ</sup> 0 ðkraÞ krZ<sup>γ</sup> 0 ðkraÞ which are the structure functions of the electromagnetic field. From these expres- ! sions, we have seen that their values depend on a radius-vector r and bring about changing the magnitude of the γ-th mode in the radial direction of the interaction space. It is necessary to note that when kra ≪ γ (so-called long-wave approximation), the expressions can be simplified as [14]:

$$\Psi\_r^{\nu}(r) = \frac{Z\_{\gamma}^{\prime}(kr)}{Z\_{\gamma}^{\prime}(kr\_a)} \approx \left(\frac{r}{r\_a}\right)^{\gamma - 1} \cdot \left[\frac{1 - \left(\frac{r\_c}{r}\right)^{2\gamma}}{1 - \left(\frac{r\_c}{r\_a}\right)^{2\gamma}}\right];\tag{5}$$

and

$$\Psi\_{\phi}^{\mathcal{V}}(r) = \frac{\eta Z\_{\mathcal{V}}^{\prime}(kr)}{krZ\_{\mathcal{V}}^{\prime}(kr\_{a})} \approx \left(\frac{r}{r\_{c}}\right)^{r-1} \cdot \left[\frac{1 + \left(\frac{r\_{c}}{r}\right)^{2\gamma}}{1 - \left(\frac{r\_{c}}{r\_{a}}\right)^{2\gamma}}\right].\tag{6}$$

Thus, the expressions obtained for components of an electromagnetic field of (1) and (2) in terms of (5) and (6) allow getting an electromagnetic field for the γ – mode of a cold resonance anode block of a magnetron as

$$
\overrightarrow{E}\ \left(\overrightarrow{r},t\right) = \text{Re}\left\{\overrightarrow{E}\ \left(\overrightarrow{r}\right)\cdot e^{-j\omega\_{r}t}\right\},\tag{7}
$$

! ( ' ! ! where ! <sup>0</sup> <sup>00</sup> E r <sup>¼</sup> Erðr; <sup>ϕ</sup>Þ • <sup>r</sup> <sup>þ</sup> <sup>E</sup>ϕðr; <sup>ϕ</sup>Þ • <sup>ϕ</sup>0, ωγ <sup>¼</sup> ωγ <sup>0</sup> - jωγ —the complex cold frequency of the γ – mode; ωγ <sup>0</sup> ¼ 2π • f—the angular frequency of the γ – mode; ωγ —the coefficient of attenuation. <sup>00</sup>

Vacuum Microwave Sources of Electromagnetic Radiation DOI: http://dx.doi.org/10.5772/intechopen.83734

Each mode excited in the anode block of the magnetron is characterized by <sup>0</sup> certain distribution of the electromagnetic field and its frequency ωγ ¼ ωγ in an approximation ωγ <sup>00</sup> ¼ 0. In the general case, the electromagnetic field in the interaction space is not sinusoidal and may be presented as a sum of the space harmonics, each of which corresponds to the wave rotating with an angular velocity Ω<sup>γ</sup> and containing along the azimuthal length of the interaction space of a magnetron the whole number of the complete periods

$$
\gamma = n + mN,\tag{8}
$$

where n ¼ 0, 1, 2, …, N=2 is the number of the fundamental mode (m ¼ 0); m ¼ �1, � 2, � 3, … is the integers corresponding to the high-order space harmonics. It is known (see, example, [13, 14]) that the excitation condition of the resonant system of the magnetron (or so-called a condition synchronism) may be written as

$$
\Omega\_{\epsilon} = \Omega\_{\mathcal{V}}.\tag{9}
$$

where Ω<sup>e</sup> is the angular velocity of rotating electron spokes (or an electron beam closed on itself—re-entrant electron beam). On the other hand, we have an electron cloud, which circles in the interaction space around the cathode [9, 14], i.e., there is an additional condition, which is associated with a re-entrant electron beam and may be written as

$$
\chi = \frac{a\_{\bar{\gamma}}}{\Omega\_{\text{e}}}.\tag{10}
$$

The fundamentalresults oftheoretical analysis are shown in Figure 3. The comparison of the radial functional dependences ofthe structural functions, Eq. (5) and Eq. (6), fortwo cases at using the higherspace harmonics (for example,space harmonic—1 and γ ¼ 18, curve 1) and at a classical π�mode (m ¼ 0 and γ ¼ 12, curve 2), showsthat in the first case for effective interaction between electrons and electromagnetic wave, it is necessary to form the electronic hub having a height more than 0.85.

The trajectories of moving electrons as a result of interacting with electromagnetic field of the (�1) space harmonic are shown in Figure 4. Besides, here for comparison, we can see the trajectories of the electrons in the static mode of magnetron operation (dashed curves). It is seen that the phase focusing of the electron beam takes place in the range of the proper phases of the RF wave. In the range of the improper phases, we observe the multiplication secondary electrons process, increasing the density of space charge in the electron hub.

Figure 3. The radial distributions of the structural functions.

Figure 4. The trajectories of electrons in the interaction space [35].

#### Figure 5.

The radial distributions of space charge density in the interaction spaces of the surface wave magnetron (1) and the classical (π˜mode) magnetron (2).

Figure 5 shows the steady-state radial distributions of space charge densities in the interaction spaces of a surface wave magnetron and a classical magnetron. As can be seen, there is a fundamental difference between the two processes of the phase focusing. It is associated with available maximum of the space charge density in the immediate neighborhood of the surface of the RF structure (anode block). The availability of a second maximum of the space charge density allows the double stream state to be established in the electron hub.

It is also important to note that the operation mode on higher spatial harmonics <sup>π</sup> applied in the magnetron, namely on ˜<sup>1</sup> spatial harmonic or –mode of oscillation, <sup>2</sup> was rather successfully used for creating a relativistic prototype of the high-power magnetron with pulsed power of °1 MW at the frequency of 37.5 GHz [34].

Vacuum Microwave Sources of Electromagnetic Radiation DOI: http://dx.doi.org/10.5772/intechopen.83734

On the basis of the design of the 3-mm surface wave magnetron, there is a possibility to design the amplifying variant of a new microwave tube. Figure 6 presents a design of a 3-mm surface wave magnetron amplifier (amplitron). The main difference of the amplitron from magnetron lies in the fact that a anode block of the amplitron is nonresonant slow-wave structure in which an electromagnetic wave propagates from a RF input to a RF output, i.e., in an interaction space of the amplitron, there is a process exchange of electromagnetic energy between a reentrant electron beam and traveling wave that is propagated from RF input to RF output and then to a matched load.

Figure 7 shows the experimental dispersion characteristics of a comb-shape slow-wave structure of the 3-mm surface wave amplitron.

Figure 6. A scheme of interaction space of an amplitron.

Figure 7. A scheme of interaction space of an amplitron.

#### 2.2.2 Magnetrons with two energy outputs

Recently, there have been functional problems of different electronic systems, which became all more complex [15, 16, 18]. In particular, multifrequency radar operating in various frequency ranges for monitoring the clouds and precipitation (meteorological radar) or observing the water area and the movement of vessels in port services are increasingly applied [16]. Wherever such radars are required, there is a great quantity of interfering factors, and the multifrequency systems allow solving these problems. The operation of such radars requires a new functional element base (vacuum tubes) capable to give a functionally simple solution to a problem of multifrequency generation with high-operating characteristics. As an example of the multifrequency generator it can be a magnetron implementing the mode of an electron frequency tuning (including frequency tuning from pulse to pulse) and its application in electronic systems of different functional purpose [35, 36, 38, 39]. The practice shows [see, for example, 38, 39], that in this case in a design of the magnetron we can use two RF outputs of energy: one as active output and other a reactive one which is used for tuning a frequency.

The anode block of the magnetron with different possible variants of arrangement of the second RF output is presented in Figure 8. As may be seen, the second RF output of energy can be placed both symmetrically (1<sup>0</sup> ) and antisymmetrically (2 и 2<sup>0</sup> ) to the active RF output (1<sup>0</sup> ). The main constructional and electrical parameters of the basic design of the magnetron are given in Table 2.

Using an existing design method of the magnetrons, described in [41], a code for computer aided design of geometry and electrical parameters of the magnetrons was developed. The computation being made with the help of this code allowed defining all parameters of the magnetron provided that a maximum current from cathode was not more than 1 А/sм<sup>2</sup> .

Figure 8. The possible variants of arrangement of the second RF output of energy.

Vacuum Microwave Sources of Electromagnetic Radiation DOI: http://dx.doi.org/10.5772/intechopen.83734


Table 2.

The parameters of a magnetron.

In order to choose the operation mode of the magnetron and to apply computer modeling using its 3-D mathematical model, it is necessary to carry out the analytical calculations to define the Hull cutoff curve

$$U = 0,022 \cdot r\_a^2 \cdot B^2 \cdot \left(1 - \frac{r\_c^2}{r\_a^2}\right)^2$$

where rc and ra are the radiuses of the cathode and the anode in sm; B is the magnetic field, Gs. Also, Hartree's voltage that can be written as

$$U\_{Hartree} = U\_{\rm min} \cdot \left[\frac{2 \cdot B}{B\_0} - 1\right],$$

where

$$\begin{aligned} U\_{\min} &= 253 \cdot 10^3 \cdot \left[ \frac{2\pi r\_a}{n \cdot \lambda} \right]^2, \\ B\_0 &= \frac{21200}{n \cdot \lambda \cdot \left[ 1 - \left( r\_c / r\_a \right)^2 \right]}, \end{aligned}$$

<sup>n</sup>� <sup>a</sup> mode of oscillation (for <sup>π</sup>� mode <sup>n</sup> <sup>¼</sup> <sup>N</sup>) and <sup>λ</sup>� wavelength in <sup>a</sup> free space. <sup>2</sup>

As illustrated in Figure 8, as an example of the electrodynamics system of the basic design of the magnetron, we used an anode block with having the double twosided straps. We have investigated the electrodynamics of the anode block by applying boundary conditions on an FDTD simulation.

The theoretical dependence of dispersion characteristic of the trapezoidal anode block with double two-sided straps for <sup>π</sup>� mode (<sup>n</sup> <sup>¼</sup> <sup>N</sup>) and three nearest to the <sup>2</sup> � � � � � � �� N N <sup>N</sup> spurious modes <sup>n</sup> ¼ � 1 , � <sup>2</sup> , and � <sup>3</sup> is shown in Figure <sup>9</sup>. 2 2 <sup>2</sup>

� � �� As we can see from Figure 9, the separation between the main operative mode <sup>N</sup> (π� mode, when <sup>n</sup> <sup>¼</sup> <sup>N</sup>) and the nearest spurious modes <sup>n</sup> ¼ � <sup>1</sup> is more than <sup>2</sup> <sup>2</sup> 2200 MHz. Such separation between the nearest competing modes in the magnetrons allows effectively to solve a problem of the frequency tuning in wide frequency range. On the other hand, for understanding the general situation associated with the influence of the design and axial dimensions of end regions on the shift of resonance frequency of electrodynamics system, it is necessary to carry out an additional calculation and analysis.

Figure 10 shows a curve of shift of a resonance frequency of the anode block depending on the axial height of the end region between the vanes and end covers of electrodynamics system.

Figure 9. The results of computer modeling of the dispersion characteristic of an anode block.

An investigation of the electrodynamics parameters (dispersion) of the anode block was carried out experimentally. A panoramic measurer of VSWR of P2–65 type was used in the measurement. By using such approach, we viewed the resonance oscillation on an operating π� mode and nearest to the spurious modes of ˜ ° <sup>N</sup> oscillation when <sup>n</sup> <sup>¼</sup> <sup>2</sup> � <sup>1</sup> . The comparison of the theoretical computation with the data of the experiment is presented in Table 3.

A general view of the magnetron with two energy output ports is schematically illustrated in Figure 11. As may be seen, the active RF output 2 of the magnetron is matched with load 5 and its reactive (passive) RF output of energy 3 is connected with a length of waveguide containing a short-circuiting piston 4. By varying the distance

Figure 10. A resonance frequency of the "cold" anode block as a function of the distance between the vanes and end covers.

Vacuum Microwave Sources of Electromagnetic Radiation DOI: http://dx.doi.org/10.5772/intechopen.83734


Table 3.

Comparison theory with data of experiment.

from reactive RF output up to the short-circuiting piston, we are changing the input complex resistance of the waveguide 4 according to the following expression

$$Z\_{imp} = jZ\_0 \cdot \text{tg}\,\frac{2\pi L}{\lambda\_\text{g}},\tag{11}$$

where Z0—an input characteristic impedance of a waveguide and λg—wavelength into waveguide. As a result, a reactive component of a complex impedance of the anode block of the magnetron is changed and a resonant frequency of the anode block is retuned.

An experimental curve of the frequency tuning for a "cold" anode block of the 3-sm magnetron with two RF outputs of energy is presented in Figure 12. As may be seen, changing a length of line circuit (waveguide) L leads to a periodical changing a resonant frequency of the magnetron with a special period λg=2. As this

Figure 11.

Schematical image of a magnetron with two RF outputs of energy. 1—an anode block of the magnetron; 2—an active RF output of energy; 3—a reactive RF output of energy; 4—a waveguide including a short-circuiting piston; 5—a matched load.

Figure 12. Experimental curve of frequency tuning in the X-band magnetron with two RF outputs.

takes place, the full frequency tuning range is exceeded 200 MHz that is sufficient for practical application.

For comparison in Figure 13, we present the 3 D images and axial sections of a classical magnetron (a) and a magnetron with two RF outputs of energy (b).
