24

CHAPTER 2

ANALYSIS AND DESIGN OF INTERACTION CAVITY FOR HIGH POWER GYROTRON

25

2.1 Introduction

2.2 Synthesis of Interaction Cavity

2.2.1 Design parameter

2.2.2 Operating mode selection

2.3 Analysis of Interaction Cavity for 120 GHz, 1 MW Gyrotron

2.3.1 Eigen mode analysis

2.3.2 Beam-wave interaction analysis

2.3.3 Sensitivity analysis

2.4

2.5

Characterization of Interaction Cavity for 120 GHz Gyrotron

2.5 Conclusion

26

2.1 Introduction

Interaction structure is a very important component of a Gyro-device for RF generation. In

a gyro-device, the beam-wave interaction takes place in the interaction structure, also called as

cavity due to its resonant behavior. Different types of interaction structures are used in the high

power and high frequency gyrotrons.

In Gyrotron, a simple cylindrical resonator cavity is widely used as the interaction

structure due to its simple structure and low loss. The simple cylindrical interaction cavity is a

three section structure with an input taper, an uniform middle section and an output taper. The

input taper is a cut-off section, which prevents the propagation of RF power towards the electron

gun. The cold cavity analysis and beam wave interaction takes place at the uniform middle

section where the RF field exits in the form of standing wave [12]-[15]. The output taper section

converts the standing wave into the traveling wave. The reducing radius of the input taper section

provides higher cut-off frequency region to the signal resonating in the middle section and thus

the resonating signal is reflected back. The output taper section is also a tapered cylindrical

waveguide of increasing radius. This section behaves like a cylindrical horn antenna and the RF

resonating at the middle section can propagate through this section as a travelling wave [24],

[25]. The tapered angle of the input section is decided in such a way that 100 % RF reflection

can occur. Similarly, the output taper angle is decided on the basis of partial reflection around

25-35%. The reflections on both sides of the middle section are very necessary to make a

standing wave so that the oscillator operation of the gyrotron can take place. The efficiency and

the output power performance are considered as the main goals in the optimization of the cavity

parameters.

The Gyrotron interaction cavity consisting of three parts, the down-taper (L1), the middle

section (L), and the up-taper (L2) is shown in Fig. 2.1, where 1 is down-taper angle, 3 is up-

taper angle and Rc is cavity radius. The interaction structure or cavity is made as an open

resonator (both sides are open) to minimize the mode density and thus the mode competition

[26]-[28]. The reflected RF on each of the ends of mid section makes a standing wave in

Gaussian profile. This kind of profile is very useful because the ohmic wall loss is minimum for

this case. The length of the middle section is decided by the diffractive quality factor. So the

cylindrical cavity shown in Fig. 2.1 fulfills the entire desired requirement like a particular value

27

of the diffractive quality factor (in terms of standing wave), the minimum mode competition and

the minimum ohmic wall loss. Parabolic smoothing is sometimes used at the both ends of middle

section to improve the efficiency of the structure, which is not included in the study and analysis

here for the sake of convenience. The overall performance of the Gyrotron tube including the

interaction cavity performance highly depends on the operating mode. A particular operating

mode is selected on the basis of power and frequency requirement of a Gyrotron. As the power

and frequency of a Gyrotron increase, the high order TE modes become more important [29].

The design of interaction cavity for 120 GHz, 1 MW Gyrotron for plasma heating during

ITER type of experiment start-up is discussed in this chapter. The synthesis of interaction cavity

covering design parameters and formalism of mode selection parameters are presented in Section

2.2. The cold cavity analysis, beam-wave interaction, parametric analysis, thermal and structural

analysis and its effect on power and frequency are discussed in Section 2.3. An in-house

developed code GCOMS [44]-[49] is used for the mode selection and start oscillation current

calculations. The eigen mode analysis of the cylindrical cavity has been carried out using

commercially available software HFSS and CST-Microwave Studio (CST-MW). Particle-in-cell

(PIC) electromagnetic simulation code MAGIC is used for beam-wave interaction analysis and

sensitivity analysis.

Fig. 2.1: Gyrotron interaction structure with uniform middle section and linearly tapered end

sections at both sides (Rc: cavity radius, L1: length of input down-taper, L: length of mid section,

L2: length of output uptaper, 1 : down-taper angle and 3 : up-taper angle)

Parabolic Smoothening

Rc

L1 L L2

Θ1

Θ3

28

2.2 Synthesis of Interaction Cavity

Synthesis of Gyrotron interaction cavity is basically getting the first hand information

regarding operating mode profile in the cavity, study of operating mode related to space charge,

quality factor, wall loss, start oscillation current, etc. leading to selection of a proper operating

mode for a Gyrotron. The present Section 2.2 discusses the synthesis of the simple cylindrical

Gyrotron interaction structure or cavity, yielding the operating mode, the cavity dimension, etc.,

for the Gyrotron under discussion.

2.2.1 Design parameter

The main initial design parameters of interaction cavity are output power, frequency,

operating mode and efficiency. The power growth in the interaction cavity critically depends on

the operating mode which in turn depends on various parameters related to operating frequency,

electron beam radius, cavity radius, cavity material, magnetic field at cavity center, start

oscillation current, coupling coefficient, etc. Different parameters related to design of interaction

cavity are discussed in this section [9], [25]-[29]. This can be easily accomplished through the

study of operating mode index, quality factor, ohmic loss, space charge effect, etc. in relation to

the ratio between cavity and beam radii and discussed below [9].

Frequency of operation: Normally in a Gyrotron, TE mode is selected as the operating mode

which depends upon the frequency of operation. The first step is to find out all the modes near

the frequency of operation by finding cut-off frequency ( cf ) of each mode near the frequency of

operation. The expression of cut-off frequency is given as:

푓 =휒

2휋푅 √휇휀 (2.1)

where 휒 is the nth root of the derivative of mth order, first kind Bessel function. Rc is the

radius of the interaction structure (cavity), and are the permeability and permittivity of the

medium, respectively.

Cavity radius and beam radius: For the selected operating mode TEm,n the cavity radius (Rc) is

related to the free space wavelength (λ0) and can be expressed through the following expression:

29

', 2

oc m nR

(2.2)

In case of a particular TEm,n mode, the suffix m and n are the azimuthal and the radial index

numbers, respectively. The electron beam launching position is defined through the electron

beam radius or the guiding center radius. The formula of the beam radius (Rb) or the maxima of

electric field in radial direction of a TE mode is given as:

' ', ,'

, 2c o

b m s i m s im n

RR

(2.3)

where s is the harmonic number and i (radial index) is the number of radial maxima presenting

the radial location of the electron beam launching. The azimuthal indexes ms and m+s represent

the co-rotating and the counter-rotating electric fields respectively [31].

The beam voltage is initially selected through the estimation of the relativistic mass factor

( o ) in terms of Vo, electronic charge (e) and rest mass of electron (mo) and given as:

훾 = 1 +푒푉푚 푐 (2.4)

The beam current can be easily estimated from the desired output power (Po) of the device,

through beam voltage.

Magnetic field at cavity center: A superconducting magnet surrounding the cavity produces

the magnetic field required for the interaction whilst an electron gun magnet positioned at the

cathode, helps in the gyration and transmission of the electron beam from electron gun to

interaction region. The cavity magnetic field ( oB ) for desired frequency in terms of relativistic

mass factor ( o ) can be expressed as:

퐵 (푇) =푓 (퐺퐻푧)훾

28 푠 (2.5)

Many high-frequency gyrotrons operate at higher harmonics to allow for the use of a larger

cavity size and to decrease magnetic fields by a factor of the harmonic number s.

30

Quality factor of interaction cavity (Q value): The middle section length (L) is decided on the

basis of the Q value, which is further defined in terms of diffractive Q value (Qdiff) and ohmic Q

value (Qohm). The diffractive quality factor (Qdiff) can be defined as the power loss from the

middle section (RF is generated in this section) of the cavity due to the RF propagation through

the output taper section. The time τ for energy to travel out of the cavity determines the

diffractive quality factor and is related to the minimum diffractive quality factor as:

푄 , = 휔휏 = 휔퐿휐 = 4π

퐿휆 (2.6)

where Qdiff,min, ω, vg, λo are the minimum diffractive quality factor, the angular frequency of RF,

the group velocity, and the RF wavelength, respectively. The interaction cavity middle section is

extended on both sides in terms of the input and the output tapers and thus the reflections occur

on both sides of the middle section. The actual diffractive quality factor becomes somewhat

different due to the reflections at both ends of the middle section and is modified as:

푄 =푄 ,

1 − 휌 =4휋

(1 − 휌) 퐿휆 (2.7)

Here ρ is the reflection at both the ends of the middle section (ρ = R1R2, R1 and R2 are the

reflections at both the ends). Another main RF loss in the interaction cavity is due to the finite

conductivity of the cavity material (oxygen free high conductivity copper) and expressed in

terms of the ohmic quality factor. The ohmic quality factor directly depends on the material

electrical properties (in terms of the skin depth) and the electric filed pattern (in terms of TE

mode). The expression for the ohmic quality factor is given as:

푄 =푅훿 1−

푚휒

(2.8)

where δ is the skin depth . Further, the total quality factor (Q) may be expressed as:

1 1 1

diff ohmQ Q Q (2.9)

31

Start oscillation current (SOC): Start oscillation current (SOC) in a gyrotron is the minimum

current for which gyrotron starts to oscillate. Thus, when the beam current exceeds the start

oscillation current, the self-excitation conditions are fulfilled, and this gives rise to oscillations.

The expressions of start oscillation current (Ist), obtained from self-consistent theory, can be

written as [9]:

'2 2 2 ', ,4 40

20 1

( )(1.68 10 ) m n m m nstN

stm b

m JI LIQ J k R

(2.10)

22

24

1

x

stNeIx

and 4x

',m n ck R (2.11)

where is the normalized transverse electron velocity, Jm(x) refers to the first kind Bessel

function of order m and argument x, is normalized interaction length and is detuning

parameter.

Coupling coefficent (CC): To avoid the excitation of competing mode, proper launching of the

electron beam is very necessary. The coupling coefficient curves provide information about the

beam launching position and its tolerance. The coupling coefficent should be maximum for the

selected mode, so that the desired mode can be excited during the gyrotron operation. The

expression for coupling coefficent (CC) is given as [9]:

21

'2 2 2 ', ,( )

m b

m n m m n

J k RCC

m J

(2.12)

Fresnel parameter: The Fresnel parameter (CF) is used in the estimation of the interaction

cavity middle section length (L). This parameter defines the oscillation of a particular RF mode

along the cavity length. The parameter is determined by the diffractions of radiation at the taper

transitions and given by the following formula [9]:

32

퐶 ≅휋4

퐿휆

휒 −푚 (2.13)

It has been studied that 1FC is required for the growth of a particular mode. Thus, by using

(2.1) and keeping the value of CF = 1, the minimum cavity length (L) can be estimated for every

competing mode, which may be called as cut-off cavity length for the excitation of the operating

mode. For lower values of the cavity length (L), the performance figures are better as the ratio of

transverse to axial velocities, , increases. For high values of L, these are somewhat better at

lower values but become worse at high velocity ratios, whereas for an intermediate value of L,

both the output power and the efficiency are fairly high throughout the range of the beam

parameters chosen. It is evident that a large L would result in over bunching of the electrons

thereby reducing the power and efficiency as increases, whereas a shorter cavity has smaller

quality factor, thereby reducing power and efficiency at lower values.

Ohmic wall loss: Some of the electromagnetic power in the interaction cavity is dissipated in

the form of ohmic heating in the cavity material. The loss in the electromagnetic power is

defined in terms of the ohmic wall loss (heat dissipated per unit area) dP/ dA and given as [9],

[11], [38]:

푑푃푑퐴 = 2휋

1휋푍 휎

푃 푄

퐿 휆

1휒 − 푚

(2.14)

where σ is the electrical conductivity of cavity material, Z0 is the free space impedance, 푃 is the

output power, L is length of middle section of the cavity and Qdiff is the diffractive quality factor

of the interaction cavity.

Space charge effect: The space charge effect is created due to the potential of the transported

charge beam and critically depends on the charge transportation geometry and the structure

geometry through which the charge is moving. The space charge effect can be defined in terms

of the voltage depression (Vd) and the limiting current (IL). The voltage depression (Vd) and the

33

limiting current (IL) can be expressed in term of beam current, cavity radius and beam radius,

respectively as follow [9]:

(60 ) lnb cd

z b

I RVR

(2.15)

퐼 ≈ 8500 퐼∗ 푙푛⁄ 푅푅 (2.16)

3/ 21/ 3* 21 1o zoI

where βzo is the initial value of βz at cavity entrance.

2.2.3 Operating mode selection

On the basis of the various mode selection parameters expressed through (2.1)-(2.16), a

flow chart is developed and presented in Fig. 2.2 [44], [49]. Similarly, the computer program

GCOMS in MATLAB is made based upon the flow chart represented in Fig. 2.2 using (2.1) -

(2.16). The criteria for selection of operating mode with respect to different parameters such as

voltage depression (> 10% of operating beam voltage), limiting current (~ > 200% of operating

beam current), wall loss (< 1 kW/ cm2), etc. are also finalized. The operating beam voltage and

beam current are taken as 80 kV and 40 A, respectively. The indigenous mode selection code

GCOMS is used for the mode finalization for a typical high power gyrotron namely 1 MW, 120

GHz Gyrotron. The RF power is very high for 120 GHz Gyrotron compared to the low power

Gyrotron and thus the ohmic wall loss factor is very critical. To reduce the ohmic wall loading,

high volume cavity is required for the high power Gyrotron. A detailed study was made for a

large number of available RF modes as the possible operating modes of 120 GHz Gyrotron and

some typical results are presented in Table 2.1. Further, based upon the above-mentioned

typical selection criteria TE22, 6, TE24, 6 and TE26, 6 have been selected as the operating modes

for 120 GHz, 1 MW Gyrotron. However, due to the lowest azimuthal index, TE22, 6 is selected as

the operating mode for detailed study for 120 GHz, 1 MW Gyrotron.

34

Fig. 2.2: Flow chart of the mode selection program in a Gyrotron [49]

By using (2.10) through (2.11) the start oscillation currents are calculated for the modes

lying in the amplification band Δω= π/Ttr , where Ttr is the transient time of the electrons through

the resonator. The start oscillation currents (starting current) of the operating mode TE22, 6 along

with the neighboring competing modes are shown in Fig. 2.3. It is obvious from Fig. 2.3 that the

most competing modes for the operating mode are TE22,6+ are TE21,6+ , TE18,7- , TE19,7- , TE20,7-

and TE23,6+ (+sign for co-rotating and –sign for counter-rotating modes). The plus sign is omitted

generally to represent the co-rotating mode and it is also followed in this thesis. These competing

modes have approximately the similar start-up condition and may couple with the gyrating

electron beam. The maximum interaction efficiency has been achieved in the hard excitation

region, which lies on the lower magnetic field side. So the main competing modes for the

operating mode TE22,6+ are found as TE21,6+ and TE19,7-, respectively in the hard excitation region.

35

Table 2.1: Calculated parameters for few suitable modes for 120 GHz Gyrotron cavity

Mode 흌풎풏 Rc (mm) Rb (mm) Vd (kV) IL (A) dP/dA (kW/cm2) 풎흌풎풏

24,4 40.55 16.14 10.08 3.87 104.27 0.82 0.591

22,5 41.98 16.71 9.25 4.85 83.13 0.68 0.524

20,6 43.17 17.18 8.43 5.85 68.97 0.60 0.463

18,7 44.17 17.58 7.60 6.89 58.59 0.54 0.407

28,4 45.18 17.98 11.72 3.52 114.70 0.70 0.619

25,5 45.55 18.13 10.50 4.5 89.74 0.61 0.548

22,6 45.62 18.16 9.25 5.54 72.88 0.55 0.482

24,6 48.05 19.12 10.08 5.26 76.67 0.50 0.499

26,6 50.46 20.08 10.90 5.02 80.36 0.47 0.515

28,6 52.85 21.04 11.72 4.81 83.96 0.43 0.529

Fig. 2.3: Start oscillation current curves with respect to the cavity magnetic field for the

modes nearing to TE22,6 (B0=4.82 T, L=15 mm, Ib=40 A, Vb=80 kV, α=1.5)

36

Fig. 2.4: Coupling coefficient for various competing modes with respect to the ratio of beam

radius to cavity radius

Further, these modes can be separated from the operating mode by the proper radial

positioning of the electron beam as shown in Fig. 2.4 which shows the coupling coefficients for

various modes, obtained through (2.12). Fig. 2.4 shows that TE19,7- can be separated very easily

on the basis of beam launching position because at the beam launching position of 9.25 mm, the

coupling coefficient of TE19,7- is very weak.

2.3 Analysis of Interaction Cavity for 120 GHz, 1 MW Gyrotron

The electrical design of RF interaction cavity for 120 GHz, 1MW Gyrotron is completed

through eigen mode analysis, beam-wave interaction analysis and parametric analysis, etc.

Eigen mode analysis has been carried out using CST-Microwave studio and Ansoft HFSS [76],

[77]. Particle-in-cell (PIC) electromagnetic simulation code MAGIC [78] is used for beam-wave

interaction analysis and sensitivity analysis. The design of interaction cavity for Gyrotron

requires the knowledge of the RF field profile, the resonator eigen frequency, and the quality

factor Q [79]. The need for high power, high frequency Gyrotron for electron cyclotron

resonance heating (ECRH) in magnetically confined plasma requires tens of megawatt of

electromagnetic power at high frequency. The device can generate several MW’s of

electromagnetic power by the efficient interaction of the gyrating electron beam of very high

37

electron beam power with RF in the interaction cavity. A cylindrical interaction cavity is quite

practical for 1 MW output power at 120 GHz operating frequency due to its easy fabrication and

thus used for the study and analysis. The main design parameters of the 120 GHz Gyrotron are

summarized in Table 2.2.

Table 2.2: The design parameters of the 120 GHz Gyrotron

Parameter Value

Frequency (fo) 120 GHz

Output power (Po) ≥ 1.0 MW

Efficiency (η) ≥ 30%

Beam voltage (Vb) 80 kV

Beam current (Ib) 40 A

Alpha (α) 1.5

Magnetic field (B0) 4.82 T

RF collection Radial

2.3.1 Eigen mode and eigen frequency analysis

In eigen mode analysis the resonance frequency of the structure has been obtained by

applying different boundary conditions and studying the electric field patterns for different

modes. Eigen mode analysis confirms the excitation of the operating modes in the cavity of the

Gyrotron [80], [81]. As mentioned above the eigen mode analysis of the cylindrical cavity has

been carried out using commercially available software CST-Microwave Studio (CST-MS) and

HFSS, which are 3D electromagnetic simulators [76],[77]. For high frequency (>100 GHz) and

above frequency it is necessary to assign the mesh refine operation in HFSS software.

The asymmetric mode TE22,6 is selected as the operating mode for 120 GHz, 1 MW

Gyrotron after going through different mode selection process as discussed in sub-section 2.2.2.

The interaction cavity radius, the beam radius and the middle section length calculated for the

operating mode as 18.16 mm, 9.25 mm and 15 mm, respectively through (2.1), (2.2) and (2.13).

It is of interest to mention here that 18.16 mm of cavity radius is rounded to 18.2 mm in

38

simulations in both HFSS and CST-MS codes to keep matching with the mesh size. For the

simulation, a complete three section interaction cavity having input taper, uniform mid section

and output taper is considered. The input taper angle and output taper angle are optimized

between the ranges of 2.8° to 3° with various lengths of all the three sections for getting the

optimized results for eigen mode and eigen frequency. Table 2.3 shows the optimized interaction

cavity geometry for the operating frequency 120.0 GHz. The Q value shown in Table 2.3 is

calculated by using (2.6)-(2.8) for the optimized interaction cavity geometry and the selected

operating mode. The calculated Q value for 120 GHz interaction cavity is 706.

Figs. 2.5 (a) and (b) present the results of eigen mode and eigen frequency obtained by

HFSS and CST-MS simulation tools, respectively. The electric field patterns shown in Figs. 2.5

(a) and (b) clearly show the excitation of TE22,6 mode at the resonant frequency of 120.01 GHz

in HFSS and 120.14 GHz in CST simulations, respectively. The radial position of the first

maxima of TE22,6 mode position equals to the beam radius (Rb) and this value is used as the

electron beam launching for better beam-wave coupling. For the maximum interaction efficiency

and the minimum ohmic wall loss, a Gaussian type standing wave profile at the middle section of

interaction cavity is required.

Table 2.3: Optimized cold cavity parameters for 120 GHz, 1 MW Gyrotron

Parameter Value

Middle section length (L) 15 mm

Input taper length (L1) 12 mm

Output taper length (L2) 20 mm

Cavity radius (Rc) 18.1 mm

Input taper angle (θ1) 2.8°

Output taper angle (θ3) 2.8°

Quality factor (Q) 706

39

The normalized electric field profile for the cavity geometry summarized in Table 2.3 is

presented in Fig. 2.6, which shows a cut-off region at the input taper side and a travelling wave

region at the output taper side. Fig. 2.6 clearly shows the required profile as a Gaussian profile

with the peak electric field value at the center of the middle section. The cold cavity analysis is

carried out to verify the oscillation of desired operating mode and axial electric field profile in

cold condition (without the electron beam).

(a)

(b)

Fig. 2.5: Electric field pattern for the operating mode for 120 GHz cavity

in simulations with (a) HFSS and (b) CST [76], [77]

40

Fig. 2.6: Normalized axial electric field profile

2.3.2 Beam-wave interaction analysis

In this section, beam-wave interaction analysis of the taper cavity for 120 GHz Gyrotron

has been investigated. MAGIC-3D electromagnetic simulator is used for the beam-wave

interaction analysis [78]. The interaction cavity geometry is decided on the basis of eigen mode,

eigen frequency and Q value.

The final design goal of any gyrotron interaction cavity is its output power and frequency

performance according to the requirement and thus small changes in the geometrical parameters

can be made during beam-wave interaction simulations for the optimization of output power.

The completely closed interaction cavity except the output taper mouth is made of fully

conducting wall for the beam-wave interaction simulations in MAGIC [82], [83]. The output

taper mouth is closed by a port for the observation of power, frequency and other parameters.

Fig. 2.7 shows the meshed interaction cavity with the gyrating electron beam.

A gyro-magnetic beam with a current of 40 A is introduced at the left end of the

confining cavity [84]. The beam energy is kept equal to 850 KeV. The guiding center radius is at

10.01 mm. A confining magnetic field of 4.82 T is applied along the axial direction. The detuned

41

cavity magnetic field (B0) is optimized as 4.82 T through (1.10). Further, for the maximum

power the electron beam velocity ratio (α) is found as 1.5 [85], [86].

Fig. 2.7: View of the interaction cavity with the meshing and the

electron beam lets trajectory (red color) in MAGIC

The electron beam emitted from the magnetron injection gun (MIG) is launched at the

input taper entrance. The electron beam properties like beam radius, larmor radius and velocity

ratio are defined at the input taper section entrance [87], [88]. The same electron beam moves

across the interaction cavity and interacts with the insignificant electric field of the operating

mode in the middle section. The bunching process takes place in the gyrating electron beam due

to the relativistic effect and a coherent radiation emission mechanism starts in the electrons,

which amplifies the weak operating mode signal in the resonant cavity.

The power transfer mechanism along the interaction cavity length is shown in Figs. 2.8-

2.12 in different ways. Fig. 2.8 shows the phase space diagram of the particles energy, which

indicates the efficient interaction as the density of the low energy particles (below the initial

energy value, 80 keV) increases along the cavity length. Figs. 2.9 and 2.10 indicate the loss in

beam energy and beam power along the cavity length. It is clear that maximum power transfer

takes place at the middle of the interaction cavity from the gyrating electrons to RF.

42

Fig. 2.8: Phase space diagram of the emitted particles

(Vb=80 kV, Ib=40 A, α=1.5, B0= 4.82 T)

Fig. 2.9: Beam energy profile along the axial distance of cavity

(Vb=80 kV, Ib=40 A, α=1.5, B0= 4.82 T)

43

Fig. 2.10: Temporal growth of electric field

(Vb=80 kV, Ib=40 A, α=1.5, B0= 4.82)

Fig. 2.11: RF power growth with respect to time

(Vb=80 kV, Ib=40 A, α=1.5, B0= 4.82)

44

Fig. 2.12: Frequency spectrum (Vb=80 kV, Ib=40 A, α=1.5, B0= 4.82)

Figs. 2.11 and 2.12 present the typical plots of the time response of the output power

growth and frequency, respectively. The stability in the power and frequency growth occurs at

108 ns. Fig. 2.11 clearly shows 1.7 MW output power at 120.36 GHz frequency. Before the

stability time (108 ns), some noise oscillations start and power growth takes place at other

frequency (108 GHz). Fig. 2.12 shows the frequency spectrum curve. This result is obtained by

the Fast Fourier Transformation (FFT) of the time history of frequency evolution in the

interaction cavity. The gain is maximum for 120.36 GHz signal (sharp peak) which corresponds

to the TE22,6 mode. The absence of the frequency peaks around the 120.36 GHz peak indicates

the suppression of the spurious modes during the beam-wave interaction.

2.3.3 Sensitivity analysis

The sensitivity analysis becomes very important in case of high frequency Gyrotron. The

analysis gives the flexibility in actual fabrication of the device and establishes a range of cavity

geometrical and electron beam parameters. The sensitivity analysis always helps in the

optimization of design as well as decision of tolerance limits of the parameters of the device.

Keeping all these into consideration, a number of studies have been carried out with the help of

45

MAGIC code [78] for various parameters like beam current, beam voltage, magnetic field,

velocity ratio, etc. in different situations related to eigen frequency and output power.

Figs. 2.13- 2.17 show the results of output power and frequency with respect to the cavity

geometrical parameters. Fig. 2.13 shows the power and frequency variations with respect to the

interaction cavity middle section length (L). The output power reduces with the interaction cavity

middle section length which is matching with the aspect that the net output efficiency depends

inversely on the diffractive Q factor (Qdiff ~ (L/λ)2). Figs. 2.14- 2.16 demonstrate the results of

effects of beam parameters on the output power and frequency, which clearly illustrate that the

output power criticaly depends on the various beam parameters such as beam current, beam

voltage and magnetic field. The frequency variation with the beam parameters also called

frequency pulling is also shown in Figs. 2.14 to 2.16. Fig.2.17 show the results of effect of

electron velocity ratio (α) on the output power and frequency. Table 2.4 shows the tolerences in

various cavity and beam parameters according to the output power and frequency requirment.

Fig. 2.13: Output power and frequency with respect to middle section length (Vb= 80 kV,

Ib=40A, α=1.5, Rc= 18.2 mm, Rb= 10.01 mm, B0= 4.82 T, θ1= 2.8°, θ2= 2.8°)

46

Fig. 2.14: Output power and frequency with respect to beam current (Vb=80 kV, α=1.5,

Rc= 18.2 mm, Rb= 9.25 mm, L= 15 mm, B0= 4.82 T, θ1= 2.8°, θ2= 2.8°)

Fig. 2.15: Output power and frequency with respect to beam voltage (Ib=40 A, α=1.5,

Rc= 18.2 mm, Rb= 9.25 mm, L= 15 mm, B0= 4.82 T, θ1= 2.8°, θ2= 2.8°)

47

Fig. 2.16: Output power and frequency with respect to magnetic field (Vb= 80 kV,

Ib=40 A, α=1.5, Rc= 18.2 mm, Rb= 9.25 mm, L= 15 mm, θ1= 2.8°, θ2= 2.8°)

Fig. 2.17: Output power and frequency with respect to electron velocity ratio (Vb= 80 kV,

Ib=40 A, B0= 4.82 T)

48

Table 2.4: Tolerance values in cavity and electron beam parameters for 120 GHz Gyrotron

Parameter Tolerance

Cavity length 15 –18 mm

Beam current 38 –42 A

Beam voltage 78- 82 kV

Magnetic field 4.81 -4.83 T

Electron velocity ratio 1.4 –1.5

2.4 Characterization of Interaction Cavity for 120 GHz Gyrotron

The RF characterization of the 120 GHz Gyrotron cavity has been carried out and briefed

in this section. These cold cavity parameters directly depend on the interaction cavity

geometrical parameters. Fig.2.18 shows the fabricated cavity for 120 GHz Gyrotron.

In this case of 120 GHz interaction cavity, non-destructive method is used for resonant

frequency measurements. The power through the horn antenna is made incident on the cavity.

According to the resonant frequency of the cavity, RF signal is kept at or close to that frequency.

The reflected RF power from the resonator is used for the measurement of cavity parameters. In

case of non-destructive method, two WR-8 standard horn antennas (frequency range 92.30 GHz -

140 GHz), one as transmitter and the second as receiver, have been used. Both the antennas are

connected with the two ports of Agilent Vector Network Analyzer (VNA) and waveguide

transition (WR-8 to WR-6) has been used [89], [90]. The horn antenna is connected to a detector

and used as a receiving device to collect the power radiated by resonator. The interaction cavity

is placed at 450 with respect to both antennas as shown in Fig.2.19. The measurement is

performed in far field region of antenna field to avoid any disturbance. It is a simple method to

excite the cavity and to determine the resonance frequency and quality factor.

49

Fig. 2.18: Fabricated cavity for 120 GHz Gyrotron

The reflected RF signal of frequency range between 118 GHz-122 GHz is measured and

analyzed by VNA [89]. This reflected signal shows the resonating behavior of the interaction

cavity at a particular frequency. Fig. 2.20 shows the sharp peak of the reflected signal at 119.5

GHz, which is -57.95 dB. The measured frequency is 119.5 GHz, which is within the design

constraint, that is, 120 GHz ±0.5 GHz.

Fig. 2.19: Cold test set up of 120 GHz Gyrotron cavity

50

Fig. 2.20: Resonance curve of 120 GHz cavity

Further, with the help of the resonating curve shown in Fig.2.20, the Q value is calculated

by 3 dB method through the expression as:

푄 =푓∆푓

where Δf = 170 MHz and fo = 119.5 GHz, so calculated Q value is 702. Table 2.5 shows the

comparison between the calculated and measured values of quality factor and resonant

frequency. The results show very good agreement between the calculated and the measured

resonant frequencies and quality factors.

Table 2.5: Calculated and measured values of resonate frequency and quality factor

Calculated Measured Calculated Measured

f (GHz) f (GHz) Q Q

120.05 119.5 706 702

51

2.5 Conclusion

The most important parameter for beam-wave interaction is the operating mode and thus

this mode as well as criteria for the mode selection is first discussed. The mode is selected on the

basis of various parameters such as space charge parameters, ohmic wall loss factor, mode

competition, beam-wave coupling, etc. The selected operating mode shows small mode

competition and small wall loss. To check the stability of the operating mode, the eigen mode

and cold cavity analysis are carried out. 3D-PIC MAGIC code is used for the beam-wave

interaction simulations for operating mode, frequency and power estimations. The beam-wave

interaction computation shows more than 1MW of output power at 120.05 GHz of operating

frequency for the designed interaction cavity. Thus, the presented design of a 120 GHz Gyrotron

interaction cavity clearly shows the feasibility of 1 MW of output power growth when operating

with the TE22,6 mode. The sensitivity analyses of each of the electron beam parameters and

cavity geometry with respect to output power, frequency and interaction efficiency are also

carried out.

On the basis of design value, a test interaction cavity has been fabricated. The RF

measurement has been carried out for 120 GHz Gyrotron interaction cavity and the results are

also discussed in this Chapter 2. Non-destructive horn antenna method also called as non-

destructive external adapter method is used for the measurements of resonant frequency and

quality factor value. The experimental results show the good agreement between the theoretical

and the simulated results.