5. Turkey Conf

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Procedia - Social and Behavioral Sciences 195 (2015) 2548 2555

Available online at www.sciencedirect.com

1877-0428 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of Istanbul Univeristy.

doi:10.1016/j.sbspro.2015.06.444

ScienceDirect

World Conference on Technology, Innovation and Entrepreneurship

Numerical Study of the Dispersion Properties of an X-band

Backward Wave Oscillator with Rectangularly Rippled Wall

Resonator

Ruhul Amina*, Ghulam Sabera, Rakibul Hasan Sagora

aDepartment of Electrical and Electronic Engineering, Islamic University of Technology (IUT), Board Bazar, Gazipur 1704, Bangladesh.

Abstract

Backward Wave Oscillators (BWOs) are devices that transform the electron beam energy into electromagnetic radiation at

microwave frequencies. O-type BWOs consist of an axial electron beam propagating through a resonant cavity comprising of a

slow wave structure (SWS). The electron beam is guided by a strong magnetic field while propagating through the cavity. The

SWSs are used in BWOs in order to slow down the phase velocity of the electromagnetic wave so that the electron beam can

resonantly interact with the wave which is the prerequisite of microwave generation. Dielectric loaded periodic structures were

used as SWSs during the early stage of development of BWOs. However, this method of slowing down the wave suffers from

dielectric breakdown as it cannot support high electric field. In order to prevent the dielectric breakdown, periodically corrugated

metallic hollow waveguides are now being used as SWSs in BWOs. The sinusoidally corrugated SWS (SCSWS) has received the

greatest attention among different corrugation profile. However, the fabrication process of this profile is complex and requiressophisticated tools. In an attempt to mitigate the fabrication problem, SWSs with simpler geometry have been proposed by the

researchers. In this work, an earlier work on rectangularly corrugated SWS (RCSWS), which was investigated for non-relativisticelectron beam has been extended for relativistic BWO in the X-band frequency range. The dispersion properties and the temporal

growth rate of the axisymmetric transverse magnetic (TM) modes of have been analyzed numerically. In order to avoid the

complicated boundary condition at the discontinuous boundaries of the rectangular radial profile, Fourier series has been used to

approximate the axial profile of the SWS and the linear Rayleigh-Fourier theory has been utilized to determine the dispersion

properties. The study shows that the growth rate of microwave for the RCSWS is somewhat lower than that for the case of

SCSWS; however, a design tradeoff can be made to obtain a comparable performance.

2015 The Authors. Published by Elsevier Ltd.

Peer-review under responsibility of Istanbul University.

Keywords: Backward Wave Oscillator, Rectangular Corrugation, Slow Wave Structure

* Corresponding author.E-mail address:[email protected]

2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).Peer-review under responsibility of Istanbul Univeristy.
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2549Ruhul Amin et al. / Procedia - Social and Behavioral Sciences 195 (2015) 2548 2555

1. Introduction

The demand of Cherenkov type amplifiers and oscillators is growing due their possible applications in varieties

of civil and military fields such as high energy particle accelerators, industrial plasma heating, and high power radar

(Benford, Swegle, & Schamiloglu, 2007; Blyakhman et al., 2007; Prater, 2004; Tantawi et al., 2005). With the

evolution of the high power microwave (HPM) devices, BWOs and travelling wave tubes (TWTs) have drawn the

attention of the researcher community as effective devices for the conversion of electron beam energy intomicrowave radiation. SWSs perform the task of reducing the velocity of the structure electromagnetic modes below

the velocity of light so that annular electron beam can resonantly interact to produce microwave radiation.

Periodically corrugated metallic walls which are currently used as SWS in BWOs and TWTs provide stability of

operation, compact structure and better efficiency and output power (Bratman et al., 2010; Chen et al., 2002; Song et

al., 2011; Xiao et al., 2010). SWSs can support waves both in forward and backward directions and are used in

cross-field HPM devices as well (Dong, Dai-Bing, Fen, & Zhi-Kai, 2009; Lemke, Calico, & Clark, 1997). Recently,

many researchers have performed extensive theoretical and experimental study on the BWOs with SCSWS (M. R.

Amin et al., 1995; Minami et al., 1990). Efforts are being given by the BWO research community to enhance the

output power and conversion efficiency of BWOs with SCSWS. V. Bratman et al. reported a peak output power of

4.3-5.3 GW with conversion efficiency 31-22% at 9.4 GHz and ~20ns pulse width (Bratman et al., 2011). Pulse

compression technique has been proposed as a method of increasing the output power level. In some literatureswave propagation in cylindrical and rectangular SWS have been studied theoretically and experimentally (Amari,

Vahldieck, Bornemann, & Leuchtmann, 1998; Jerby & Bekefi, 1993; Kesari, Jain, & Basu, 2005a, 2005b; Shi,

Yuzheng, & Higo, 2001). Some of the researchers reported the dispersion characteristics of cold structures only that

is the resonant coupling of electron beam and structure EM modes have not been considered (Hu, Tang, Chen, Lin,

& Tong, 2001; Wang, Yang, Zhao, & Liang, 2005). Alternative radial profiles such as trapezoidal (M. R. Amin &

Ogura, 2013; M. R. Amin, Ogura, Kojima, & Sagor, 2014), semi-circular (Saber, Sagor, & Amin, 2015), doubly

rippled (M. R. Amin, 2013), oversized co-axial SWSs (Ogura, Yambe, Yamamoto, & Kobari, 2013) etc. are being

investigated by the researchers nowadays. In (M. R. Amin & Ogura, 2007), rectangularly periodic cylindrical SWS

was used with electron beam energy


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2550 Ruhul Amin et al. / Procedia - Social and Behavioral Sciences 195 (2015) 2548 2555

0

1

4 1( ) sin (2 1)( )

2 1 2o

n

hR z R n k z

n

.. (1)where,R0 is the average radius, h is the corrugation amplitude, 00 2k z andz0 is the period of corrugation. Theaxial variation of the radial function creates the periodic profile which as a consequence generates periodic

dispersion curves. The waveguide wall has been assumed to be perfectly conducting and the infinitely strongmagnetic field ensures the 1D motion of the electron beam in the axial direction. The intense relativistic electron

beam (IREB) has been assumed to be infinitesimally thin.

The waveguide size parameters have been chosen such that the X-band regime operation of the BWO can be

simulated. Based on the experimental results presented in (Butler, Wharton, & Furukawa, 1990; Carmel et al., 1992;

Main et al., 1994) the selected beam parameters would not cause any beam transport related problems. Theparameters are presented in table 1. A representative section of the RCSWS under investigation is depicted in fig.

1a. The axial profile of the SCSWS has also been given so that an analogy can be made. The structure parameters

are given inside the figure. A 3D representation of the proposed RCSWS is presented in fig. 1b.

0 1 2 3 4 50

0.5

1

1.5

2

Axial Coordinate (cm)

Radius(cm)

Rb=0.80 cm

R0=1.50 cm

z0=1.67 cm

h=0.28cmh=0.41cm

(a)

Fig. 1 (a) Axial profile of the proposed RCSWS and previously used SCSWS along with structure parameters. (b) 3D view of the proposedRCSWS.

For beam free vacuum structure, the EM field components of the TM modes of the structure can be expressed by aninfinite sum of the Floquet harmonics as;

zi(k

n

znzznerEr,z,tE

)()( .... (2)

where,0nKkk zzn , 2,1,0,n is the Floquet harmonic number and kz is the axial wavenumber.

Equation (2) signifies that an EM wave with angular frequency in the SWS has infinite number of displaced

spatial harmonic components with wavenumber0nKkk zzn .

Using boundary conditions of EM fields and the continuity equation of the annular beam, the mathematical steps for

deriving the dispersion relation of the BWO driven by an infinitesimally thin annular electron beam have been

described in (Amin & Ogura, 2013). With IREB (16) of (Amin & Ogura, 2013) is,

0)()(1,,

nm

nmnNmnn

Jmnn

nm

nn ADPLPKQnmA .... (3)

where,

2

0

n

znn

kKQ

, )]()(1[ Nmnn

J

mnnnmn PLPKQmnD ,

zKuduzRJumnP nJ

mn 0

0

0 and)()(cos

,


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00

cos ( ) ( )Nmn nP n m u N R z du

.

In (3)An is a column matrix represents the amplitude of Floquet harmonics. The dispersion relation results from the

becomes,

0det

mn

m.n

D . (4)

Equation (4) is the required generalized expression for the analysis of dispersion characteristics.Table 1. The dimension parameters of the SCSWS and RCSWS

Parameters R 0 (cm) z0 (cm) h (cm) K 0 (cm-1)

SCSWS 1.50 1.67 0.41 3.76

RCSWS 1.50 1.67 0.28 3.76

3. Numerical Results

We have performed numerical calculations for both zero and non-zero beam dispersion characteristics. The zerobeam dispersion characteristics can be determined from the real value of frequency and wavenumber while the non-

zero beam dispersion characteristics require measurements of imaginary frequency. From the imaginary frequency

TGR can be calculated which qualitatively signifies the microwave radiation strength.

3.1 Zero-beam dispersion characteristics

The operating frequency of BWOs can be roughly estimated from the intersection points of the dispersion curves

and the beam lines. The dispersion curves for TM 01, TM 02 and TM03 modes have been determined numerically and

presented in fig. 2a. Information regarding the dispersion properties of SWS is required in order to approximate the

operating frequency before real experiment is performed. Therefore, the cold structure dispersion characteristics

have been determined first. Fig. 2b presents the fundamental mode (TM01) dispersion curve for different corrugation

amplitude. The light line and beam lines are also drawn inside the figure in order to give an estimate of the probable

oscillation frequencies of the BWO. It can be observed from the figure that the oscillation frequency decreases as

the value of is decreased. Where

Bam Velocity

Light Velocity in Vacuum)Beam Voltage(kV)

511

2

11

1

.

Before comparing the TGR of two different SWSs, the zero-beam dispersion curves of the operating mode of both

the SWSs should be matched. We have found that the TM01 mode dispersion curve for RCSWS with h=0.28cm

matches well with the dispersion curve for SCSWS with h=0.41cm. Thus we have used h=0.28cm in rest of the

analysis. Fig. 2b depicts the agreement of the dispersion curves for RCSWS and SCSWS.

Information regarding the speed of axial energy transport can be obtained from the group velocity which is

calculated from the derivative of the dispersion relation. The group velocities for TM 01, TM02 and TM03 modes are

presented in fig. 2c. If we determine the group velocity at the wavenumber where the dispersion curve intersects the

beam lines, we would find that the value is negative. BWO has been named so because it operates in the negativegroup velocity regime. The energy transfer occurs in the opposite direction of the flow of electron beam.


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16

18

20

Frequency(GHz)

13.3

13.4

13.5

Frequency(GH

z)

0 1 2 37

8

9

Wavenumber (cm-1

)Frequency(GHz)

TM01

TM03

TM02

(a)

0 1 2 36.5

7

7.5

8

8.5

Wavenumber (cm-1

)

Frequency(G

Hz)

beam line

=0.8

beam line=0.6

light line

h=0.28cm

dashedredline->sinusoidal

(h=0.41cm)

h=0.40cm

h=0.35cm

(b)

0 1 2 3-15

-10

-5

0

5

10

15

Wavenumber (cm-1)

GroupVelocity(cm/nsec) TM

01

TM03

TM02

(c)

Fig. 2 (a) TM 01, TM02 and TM03 mode dispersion curves for the RCSWS. (b) TM 01 mode dispersion curves for different corrugation amplitudes.(c) Group velocities for the three TM modes.

3.2. Coupling of Beam Space Charge Wave with TM01 Mode

The average radius, R0 and the period of axial variation, z0 play the dominant role in determining the oscillation

frequency of the BWO. When the IREB is included in the analysis, the beam line split up into slow and fast

SCWMs. The fast SCWM propagates at a velocity higher than the beam velocity whereas the slow SCWM

propagates at lower velocity. The degree to which the slow and fast SCWM will go downward and upward from the

beam line depends on the beam current and energy. The slow and fast SCWMs for 0.70 and 0.60 keeping

beam current fixed at 0.5kA and beam radius at 0.8cm have been determined and depicted in fig. 3a and 3b. As the

value of is decreased from 0.70 to 0.60, the slow and fast SCWMs moves further away from their correspondingbeam line. The region of instability shrinks and the peak TGR reduces as well. As the beam line shifts in the right

direction with the decrease of , the oscillation frequency and the TGR reduces.


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0

2

46

8

10

12

=0.70 and Ib=0.5 kA

Freque

ncy(GHz)

0 1 2 30

1

2

Wavenumber (cm-1

)

TGR

(nsec-1)

Unstable regionFast Space ChargeWave

Slow Space Charge Wave

Beam line for

=0.7

(a)

0

2

4

6

8

10

12=0.60 and I

b=0.5 kA

Frequency(GHz)

0 1 2 30

1

Wavenumber (cm-1

)TGR

(nsec-1)

Slow Space Charge Wave

Fast Space Charge

Wave

Beam line for

=0.6

Unstable region

(b)

Fig. 3 Slow and fast space charge wave modes for (a) 0.70 . (b) 0.60 . Other parameters are fixed.

3.2b Temporal Growth Rate Study of the RCSWS

Microwave generation occurs due to the wave-particle interaction inside the BWO. From zero-beam analysis, itis possible to qualitatively estimate the oscillation frequency; however, information regarding the strength of the

microwave radiation cannot be obtained. Inclusion of the IREB initiates the instability required for microwave

generation. The magnitude of the imaginary frequency is called the TGR which provides information about the

microwave signal strength. It has already been mentioned that the zero-beam dispersion curves of the operating

mode should match in order to compare the TGR of RCSWS and SCSWS. The TGR for different values of has

been presented in fig. 4a. As the value of is decreased the TGR shifts rightwards in the k-plane since the

intersecting point of dispersion curve and beam line shifts rightwards. Figures 4b and 4c present the peak TGR

(PTGR) for both RCSWS and SCSWS as functions of and beam current (Ib)). The PTGR keeps on increasing as

is increased and saturates at higher values of . Similar phenomena can be observed in fig. 4c also. The PTGR

of the proposed RCSWS can be increased by increasing the beam current; however, there is a certain limit abovewhich the beam current cannot be increased due to transportation problem by IREB. From both the fig. 4b and 4c, it

can be observed that the PTGR for the proposed RCSWS is very close to the PTGR for SCSWS. Therefore, RCSWS

can be an ideal candidate as the SWS in BWO for real experiments.

0 1 2 3

0.5

1

1.5

2

2.5

Wavenumber (cm-1

)

T

GR

(nsec-1)

=0.8

=0.85

=0.7

=0.6

=0.9

(a)

0.6 0.65 0.7 0.75 0.8 0.85 0.91.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

=Vb/c

P

TGR

(nsec-1)

Rectangular

Sinusoidal

(b)


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0.2 0.3 0.4 0.5 0.6 0.7 0.8

2.2

2.4

2.6

2.8

3

3.2

Beam Current (kA)

PTGR

(nse

c-1)

Rectangular

Sinusoidal

(c)

Fig. 4 (a) TGR for different values of . (b) PTGR as a function of for RCSWS and SCSWS. (c) PTGR as a function of beam current for

RCSWS and SCSWS.

The oscillation frequency estimated from the cold structure(Ib= 0) dispersion curve (fig. 2b) decreases due to the

space charge effect of the IREB in BWO operation. The percentage of reduction of oscillation frequency from theestimated value is presented in table 2. The frequency corresponding to the PTGR for a specific value of b is taken

as the oscillation frequency for non-zero beam current.

Table 2. Percentage of reduction in resonant frequency of the axial TM01 mode of RCSWS for different beam energy (h=0.28 cm).

Frequency (GHz) Percentage of reduction

in oscillation frequency ForIb=

0kA

ForIb=0.5kA

0.8 8.63 8.37 3.01

0.6 8.02 7.65 4.61

4. Discussion and Conclusion

A rigorous analysis on zero and non-zero beam dispersion characteristics of RCSWS has been presented. The

cold structure dispersion characteristics provide information regarding the expected oscillation frequency while the

non-zero beam dispersion characteristics give idea about the strength of the microwave signal as well as SCMWs.

The SCWMs and PTGR have been calculated by varying the beam current and voltages and the zero-beam

dispersion characteristics have been calculated for different corrugation amplitude. Results obtained from our

analysis exhibit that the RCSWS provides almost similar performance as the SCSWS. In addition, while the

fabrication of SCSWS requires sophisticated tools, the RCSWS can be fabricated easily using the sections of

metallic cylinders with different inner radii. Thus it can concluded that the proposed RCSWS will be useful to the

BWO research community.

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