## 1. Introduction

Understanding electromagnetic wave phenomena is very important to be able to design RF cavities such as for atmospheric microwave plasma torch, microwave vacuum oscillator/amplifier, and charged-particle accelerator. This chapter deals with some electromagnetic wave equations to show applications to develop the analytic design formula for the cavity. For the initial and crude design parameter, equivalent circuit approximation of radial line cavity has been used. The properties of resonator, resonant frequency, quality factor, and the parallel-electrodes gap distance have been considered as design parameters. The rectangular cavity is introduced for atmospheric microwave plasma torch as a rectangular example, which has uniform electromagnetic wave distribution to produce wide area plasma in atmospheric pressure environment. The annular cavity for klystrode is introduced for a microwave vacuum oscillator as a circular example, which adapted the grid structure and the electron beam as an annular shape which gives high efficiency compared with conventional klystrode. Some simulation result using the commercial software such as HFSS and MAGIC is also introduced for the comparison with the analytical results.

## 2. Equivalent circuit approximation of radial-line cavity

Microwave circuits are built of resonators connected by waveguides and coaxial lines rather than of coils and condensers. Radiation losses are eliminated by the use of such closed elements and ohmic loss is reduced because of the large surface areas that are provided for the surface currents. Radio-frequency energy is stored in the resonator fields. The linear dimensions of the usual resonator are of the order of magnitude of the free-space wavelength corresponding to the frequency of excitation. A simple cavity completely enclosed by metallic walls can oscillate in any one of an infinite number of field configurations. The free oscillations are characterized by an infinite number of resonant frequencies corresponding to specific field patterns of modes of oscillation. Among these frequencies there is a smallest one,

, where the free-space wavelength is of the order of magnitude of the linear dimensions of the cavity, and the field pattern is unusually simple; for instance, there are no internal nodes in the electric field and only one surface node in the magnetic field.

The oscillations of such a cavity are damped by energy lost to the walls in the form of heat. This heat comes from the currents circulating in the walls and is due to the finite conductivity of the metal of the walls. The total energy of the oscillations is the integral over the volume of the cavity of the energy density,

, where

, where

The fields and currents decrease in time with the factor

Most klystrons and klystrodes are built with cavities of radial-line types. Several types of reentrant cavities are shown in Fig. 1. It is possible to give for the type of cylindrical reentrant cavities a crude but instructive mathematical description in terms of approximate solutions of Maxwell's equations.

The principle, or fundamental, mode of oscillation of such cavity, and the one with the longest free-space wavelength, has electric and magnetic fields that do not depend on the azimuthal angle defining the half plane though both the axis and the point at which the fields are being considered. In addition, the electric field is zero only at wall farthest apart from the gap and the magnetic field is zero only at the gap. In this mode the magnetic field is everywhere perpendicular to the plane passing through the axis the electric field lies in that plane. Lines of magnetic flux form circles about the axis and lines of electric flux pass from the inner to the outer surfaces.

In the principle mode of radial-line cavity only

The cavities in which RF interaction phenomena happens with charged particles almost always have a narrow gap, that is, the depth of the gap
_{0} of the post (

In a radial-line cavity the electric field outside the gap tends to remain parallel to the axis, aside from some distortion of the field that is caused by fringing near the gap; it is weaker than in the gap and tends to become zero as the outer circular wall is approached. The magnetic field, on the other hand, increases from its value at the edge of the gap and has its maximum value at the outer circular wall.

It is seen that, whereas the gap is a region of very large electric field and small magnetic field, the reentrant portion of the cavity is a region of large magnetic field and small electric field. The gap is the capacitive region of the circuit, and the reenetrant portion is the inductive region. Charge flows from the inner to the outer conducting surface of the gap by passing along the inner wall, across the outer end. The current links the magnetic flux and the magnetic flux links the current, as required by the laws of Faraday, Biot and Savart.

### 2.1. Capacitance in cylindrical cavity

If the gap is narrow, the electric field in the gap is practically space-constant. Thus the electric field

At the end of the cavity near to the gap both

Therefore,

### 2.2. Inductance in cylindrical cavity

The magnetic field

Therefore,

The total magnetic flux in the cylindrical cavity,

Comparing this with inductance definition,

### 2.3. Resonance frequency in cylindrical cavity

The resonant wavelength of a particular mode is found from a proper solution of Maxwell's equation, that is, one that satisfies the boundary conditions imposed by the cavity. When the walls of the cavity conduct perfectly, these conditions are that the electric field must be perpendicular to the walls and the magnetic field parallel to the walls over the entire surface, where these fields are not zero.

The resonant frequency

### 2.4. Unloaded Q in cylindrical cavity

In the cavity undergoing free oscillations, the fields and surface currents all vary linearly with the degree of excitation, that is, a change in one quantity is accompanied by a proportional change in the others. The stored energy and the energy losses to the walls vary quadratically with the degree of excitation.

Since the quality factor

The resonator losses per second, besides being proportional to the degree of excitation, are inversely proportional to the product of the effective depth of penetration of the fields and currents into the walls, the skin depth, and the conductivity of the metal of the walls. Since the skin depth is itself inversely proportional to the square root of the conductivity, the losses are inversely proportional to the square root of the conductivity. The losses are also roughly proportional to the total internal surface area of the cavity; and this area is proportional to the square of the resonant wavelength for geometrically similar resonators. The skin depth is proportional to the square root of the wavelength, and hence the losses per second are proportional to the three-halves power of the resonant wavelength.

The loss per cycle, which is the quantity that enters in

The surface current,

where the shunt resistance is

The surface current can be considered concentrated in a layer of resistive material of thickness. Surface resistance is that

As an example, the conductivity of copper is

The shunt conductance

is defined only when the voltage

### 2.5. Lumped-constant circuit representation

The main value of the analogy between resonators and lumped-constant circuits lies not in the extension of characteristic parameters to other geometries, in which the analogy is not very reliable, but in the fact that the equations for the forced excitation of resonators and lumped-constant circuits are of the same general form.

If, for example, it is assumed that the current i(t) passes into the shunt combination of

On taking the derivative and eliminating

In other word,

where

For a forced oscillation with the frequency

Thus, there is defined circuit admittance

These equations describe the excitation of the lumped-constant circuit.

## 3. Numerical analysis for the high frequency oscillator system with cylindrical cavity

In this section, we will meet an circular cavity example of a klystrode as a high frequency oscillator system with the knowledge which is described in previous sections.

Conventional klystrodes and klystrons often have toroidal resonators, i.e., reentrant cavity with a loop or rod output coupler for power extraction. These resonators commonly use solid-electron-beam which could limit the output power. One way to get away this limitation is to use the annular beam as was commonly done in TWTs. The main reason using reentrant cavities in most microwave tubes with circular cross sections is that the gap region should produce high electric field and thus high interaction impedance of the electron beam when the cavity is excited. In our design we assume a short cavity length,

The klystrodes consist of the gated triode electron gun, the resonator and the collector. The gated electron gun provides with the pre-modulated electron bunches at the fundamental frequency of the input resonator, where the voltage on the grid electrode is controlled by an external oscillator or feedback system. The other possible type of gated electron guns could be the field-emitter-array gun, RF gun, and photocathode. The electron bunches arrive at the output gap with constant kinetic energy but with the density pre-modulated. Here, we assumed the electron beam is operated on class B operation, that is, electron bunch length is equal to one half of the RF period. Through the interaction between electron beam and RF field, the kinetic energy is extracted from the pre-modulated electrons and converted into RF energy.

Figure 4 shows the schematics of the circular gridded resonator with center coupling mechanism for the easy and efficient power extraction. In this section, we will describe the design of annular beam klystrode in C-band.

### 3.1. RF interaction cavity design

As we have seen in previous section, using the lumped-circuit approach, the resonant frequency of this protuberance cavity with the annular beam is expressed as

Since this expression is an approximation which gives the tendency of frequency variation when we are adjusting design parameters, we can perform parameter tuning exercise using design tools such as HFSS. Fig. 5 shows an example of the detailed design using HFSS where the emission was introduced at the gap region between inner radial distances of 5.7 and 9.4mm. In the figure, the electric field is enhanced and fairly uniform due to the presence of resonator grid1 and resonator grid2. The grid structure in beam inlet and beam outlet make the electric field maintain fairly high intensity in the gap region through which the electron beam passes to interact with RF. Figure 6 also show scattering parameter plots where resonator grids of the klystrode are considered closed metal wall and the cavity has only output terminal as one port system. The bold line is the real value of S and the thin line is the imaginary one. The resonator frequency is 5.78 GHz in the absence of finite conductivity of cavity and electron beam.

The detailed tuning of beam parameters for efficient klystrode could be investigated using PIC code such as MAGIC. As an example, the current is assumed density-modulated in the input cavity and cut-off sinusoidal,

whose peak current, I_{peak}, is 3 amperes.

The tube is supposed of being operated in class B as shown in Fig. 7. A class B amplifier is one in which the grid bias is approximately equal to the cut-off value of the tube, so that the plate current is approximately zero when no exciting grid potential is applied, and such that plate current flows for approximately one-half of each cycle when an AC grid voltage is applied.

The fundamental mode (TM_{01}-mode) to be interacted with longitudinal traversing electron beam was adapted to our annular beam resonators for the high efficiency device.

Electron transit angle between electrodes gives limitation in the application of the conventional tubes at microwave frequencies. The electron transit angle is defined as

where

The transit angle was chosen to give that the transit time is much smaller than the period of oscillation for the efficient interaction between RF and electron beam, so that, the beam coupling coefficient,

### 3.2. RF interaction efficiency calculation

There are some computational design codes for the klystrode. But in this section, 1-dimensional but realistic electron beam and electric field shape are introduced to develop analytical calculations for the klystrode design, which results in easy formulas for the efficiency and electric field in the gap region of the klystrode in steady state.

Maxwell's equations for electron beams are followings,

and

As an approximation, the electron dynamics is in 1-D space. Then, the Maxwell's equations are simplified as the followings.

and

Therefore, this means that

From the Lorentz force equation,

for each particle with velocity

Define the snapshot time be

where

Therefore, we can say that the variables of electrons are denoted by

and

Let’s assume that

Fig. 10. Schematic representation for the definition of snapshot time (

Then, we have

and

By the way, from the above equation,

and

so that for

and

Figure 11 shows a typical case that electrons are decelerated due to the interaction between RF and electron beam. Extreme case would be 100 % donation of its kinetic energy to RF, which makes its velocity be zero at the resonator grid2. In that case electrons are delayed by 30 ps to the phase of RF field, where the half period of RF is 84 ps (6 GHz).

The resonator field theory is described by the equation

where

Define

Then, the time-averaged output power becomes

Because

Since

at steady state, we have

Therefore, the efficiency becomes that

As an example,

give us the followings

where

Therefore,

and

In the previous example, we have seen that the tube reveals

On the other hand, we can compare MAGIC analysis with analytical calculation as can be seen Fig. 12 which shows efficiency vs. peak current density by MAGIC PIC simulation and analytical calculation. The slope of the line fitted with the results from MAGIC analysis shows 0.266.

From the analytical equation of efficiency, we can get the slope of the line, m, as the followings,

Analytical calculation says the slope of the line be

## 4. Cavity design for the uniform atmospheric microwave plasma source

The atmospheric-pressure microwave-sustained plasma has aroused considerable interest for many application areas. The advantages of this plasma are electrode-less operation and efficient microwave-to-plasma coupling. It has the characteristics of high plasma density (

Atmospheric-pressure microwave plasma sources consist of a magnetron as a microwave source, an isolator to isolate the magnetron from the harmful reflected microwave, a directional coupler to monitor the reflected microwave power, a 3-stub tuner to match impedance from the magnetron to that from the plasma generator, and a plasma generator through which gases pass and are discharged by the injected microwave. The plasma in the discharge generator is sustained in a fused quartz tube which penetrates perpendicularly through the wide walls of a tapered and shorted WR-284 waveguide.

The plasma’s cross-sectional area of the atmospheric-pressure microwave plasma source is limited by the quartz tube’s diameter, which is also limited by the maximum intensity of the electric field profile sustainable by the injected microwave and the shorted waveguide structure, as shown in the Fig. 13. The diameter of the quartz tube is set around 30 mm to maintain a dischargeable electric field intensity in the discharge area. If the diameter of the quartz tube becomes much larger than that, no discharge occurs. This is the main reason that the cross-sectional area cannot be extended more widely in the atmospheric-pressure microwave-sustained plasma source.

Here is an example of a rectangular reentrant resonator, a single ridge cavity, to overcome this size-limited plasma discharge in an atmospheric microwave-sustained plasma source. The box-type 915 MHz reentrant resonator has gridded walls on both sides of the gap so that the plasma gases pass thorough the grid holes and are discharged by the induced electric gap field due to the injected microwave energy. As we have studied in last section, it could be calculated analytically the cavity design formula by using lumped-circuit approximation, then, we performed a RF simulation by using the HFSS (High Frequency Structure Simulator) code. The designed plasma cross-sectional area is 810 mm x5 mm. The cavity has a fundamental TE_{10}-like mode to discharge gases uniformly between the gridded walls and has a 915 MHz resonant frequency. This large cross-sectional area plasma torch can be effectively used in commercial processing and other environmental applications.

### 4.1. Numerical analysis of the plasma source rectangular cavity

A reentrant cavity in cylindrical symmetry has been widely used in vacuum tubes such as klystrodes and klystrons. However, in order to keep the resonant frequency constant whatever the cavity lengths for atmospheric microwave plasma torch, it would be better to consider a rectangular reentrant cavity, as shown in Fig. 14. The resonant frequency of the cavity for the atmospheric microwave plasma source has a fundamental TE_{10}-like mode.

The gap region between the two gridded walls in the reentrant cavity sustains a high electric field intensity and, thus, easily discharges the gases to the plasma state when it is excited by the microwave energy. With the grid planes surrounding the gap region, the TE_{10}-like mode is the dominant one, and 915 MHz is the cavity’s resonant frequency. In the simulation process using the HFSS code, the cavity grids were assumed to be smooth conducting surfaces to design the box-type reentrant cavity.

In this example, the cavity for the microwave plasma source was set to be resonated at 915 MHz, rather than 2.45 GHz, so as to have a much larger cross-sectional plasma area. Its design, setting a cavity to 915 MHz, is based on the design formula obtained by using a theoretical calculation based on the lumped-circuit approximation for the cavity. Then, we used a simulation tool, the HFSS code, to check the cavity frequency.

From Gauss’ law, the equivalent capacitance value of the reentrant cavity for a TE_{10}-like mode is

where
_{10}-like mode can be expressed as follows. Since the magnetic field flux density in the cavity is

Therefore, the cavity resonant frequency is given by

This resonant frequency formula shows that the resonant frequency is independent of the cavity length

Since we have a theoretically-approximated expression for the design parameters, we should use this formula to pick up the initial design parameters to get detailed parameters for the resonator reentrant cavity as a plasma source. Then we should investigate the RF characteristics using the design tool, HFSS.

The energy stored in the gap region can be expressed as

where

In other words, the cavity length for the atmospheric-pressure microwave-sustained plasma source is

where

Fig. 15 and Fig. 16. show a rectangular cavity design example using one of the available magnetrons, 915 MHz, 60 kW, as a RF source, and the geometrical parameters to

This uniformly distributed plasma area is

## 5. Conclusion

The annular beam cavity design was investigated analytically and simulated using the HFSS and MAGIC PIC codes to find the fine-tuned design parameters and optimum efficiency of the TM_{01}-mode operation in the klystrode with the reentrant interaction cavity. We also studied how to induce the governing efficiency formular for the microwave vacuum tube, klystrode. The efficiency of this exampled annular beam klystrode reveals 67 % at 6.004 GHz. The theoretical calculation anticipated that the efficiency would be 60% and the electric field intensity 4000V/m which is well matched with the results of numerical analysis. This concept could yield an economic sized device comparable to commercial magnetron devices.

A single ridge cavity was designed to show how to design uniform atmospheric microwave plasma source which does not need ignitors for the initial discharge. The single ridge cavity in the shape of resonator was built in the grid structures for the wide cavity torch of atmospheric pressure microwave plasma. This cavity design process has been studied by analytical calculation using lumped circuit approximation and simulation using commercial 3D HFSS code. A self-dischargeable study also has been investigated for the breakdown feasibility in the cavity gap region. These cavities show the uniform electromagnetic wave distribution between the grids through which large-sized microwave plasma could be generated.

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