Stability Analysis of 2 Mode Operation in the Beam-wave Interaction Process for a Three-gap Hughes-type Coupled Cavity Chain

Jirun Luo1, Jian Cui 1, 2, Min Zhu 1, Wei Guo 1 1. Institute of Electronics, Chinese Academy of Sciences, Beijing 100190, China

2. Graduate University of Chinese Academy of Sciences, Beijing 100049, China

Abstract: The reciprocal of the beam quality factor 1 bQ is

used to determine the stability of the beam-wave interaction in a multi-gap Hughes-type coupled cavity chain. The stability of the beam-wave interaction in a three-gap Hughes-type coupled cavity chain was discussed with the CST code calculations for the 2π mode operation. The results show that the stable operation of the 2π mode may be reached in an extended-interaction klystron with the three-gap Hughes-type coupled cavity chain.

Keywords: Three-gap Hughes-type coupled cavity chain, mode stability, beam quality factor.

I. Introduction Extended-interaction klystrons (EIK), based on an

extended interaction structure, are a way to achieve high power, wide bandwidth and high gain in the millimeter/sub-millimeter wave length range [1-4]. The extended interaction structure is composed of several coupled-cavities that strongly couple to each other by the slots on the common walls between the adjoining cavities. The enlarged inner surface area of the multi-gap coupled cavity chain can improve the power capacity and the higher coupling impedance also can compensate for the lower beam conductance, keeping high efficiency in millimeter-scale circuits. In this paper, based on the formulae of the beam-wave coupling coefficient and the beam-loaded conductance in a multi-gap Hughes-type coupled cavity chain with electron beam and the some electrical parameters obtained with CST code, the stability of the beam-wave interaction in a three-gap Hughes-type coupled cavity chain will be discussed with the reciprocal of the beam quality factor 1 bQ

in order to better describe the effect of the gap electric field on the stability.

II. Description of stability In an N-gap coupled cavity chain, there are N cavity modes and N-1 slot modes [5], so it is very important to keep the stability of operating mode and avoid the oscillation of parasitic modes. Usually, the beam-loaded conductance (Gb) is used to characterize the stability, which can only reflect the energy exchange between the beam and the gap electric field

[6]. In this article, the reciprocal of the beam quality factor 2

1 ( )2

bb b

V GPQ G R Q

W W is used to describe the stability,

which can show the energy coupling between the beam and the gap electric field (Gb) and the effect of the gap electric field on the stability ( /R Q , characteristic impedance) at the

same time. The beam-loaded conductance and the beam-wave coupling coefficient can be written as [7]

2 2

0

1( ( ) ( ) )

4b e q e qG M MZ

(1)

1,

,1

1

( )( )

n

Nj l

g n nn

N

nn

M V eM

V

(2)

1,

1,

1,

2 20 1

/22( )0

, 1,

/2

( ( ) ) ( ( ) )

( ( ) )( , )

n n

n

n n

b bl d

j z lag n n n a

n l d

r r

rM E z l r e dz

V

1,2l

sd

t1cavity 2cavity cavity n

cr ar

dr

sr

1,nl

III. Stability analysis of 2 mode operation in a three –gap coupled cavity chain There are 5 modes in a three-gap coupled cavity chain

(figure 1), which are respectively mode, /2 mode, 2 mode, /2 slot mode and slot mode. As an application example, the calculations will be performed in the operating frequency range of Ku band with the same distance between the adjacent gaps l, the corresponding geometrical parameters are listed in table 1. Table 2 gives the “cold” parameters of the 5 modes calculated with CST code for the three gap coupled cavity chain, such as resonant frequencies, gap voltages and characteristic impedance.

It can be seen from the table 2 that first and fifth mode are π mode in which the electric field intensity on the middle gap is obviously stronger than that on the other gaps, the direction of the electric field between the adjacent gaps is opposite, the characteristic impedances of the modes are respectively 192.48 and 34.76. The electric field intensity of the π mode on the gap is higher than that of the π slot mode. The electric field on the middle gap is almost zero for the second and fourth mode, the amplitudes of the electric field on the first and third gap are equal and their direction is opposite, the characteristic impedances of the modes are respectively 155.23 and 16.46. The second mode is called π/2 mode and the fourth mode is called π/2 slot mode. The electric field

Fig. 1 Model of N-gap Hughes-type coupled

281978-1-4673-0369-9/12/$31.00 © 2012 IEEE

intensity of the π/2 mode on the gap is higher than that of the π/2 slot mode. For the third mode, the electric field on three gaps is almost equal and their direction is same, the characteristic impedance of the mode is 256.08, usually it is called 2π mode, which has maximum characteristic impedance in the five modes.

Based on the formulae (1)-(2) and all parameters given in the tables 1 and 2, the 1 bQ can be calculated for the three-

gap coupled cavity chain with the electron beam of the radius rb=0.8mm and Pe r = 1.0P (2.0P). The calculating results are given in the table 3 and figure 3. It can be seen from the table 3 that, if the three-gap coupled cavity chain is used as input cavity or bunching cavity, 1 bQ of all modes is positive when

the beam voltage is selected as 20kV, so no self-excited oscillation may happen. Figure 3 shows that 1 bQ versus

beam voltage for 1.0erP P and 2.0μPerP . If it is used

as output cavity, 1 bQ of the operational mode (2π mode)

must be negative, 1 bQ of the other modes must be positive

and the amplitude of 1 bQ for the operational mode should be

as large as possible under the condition of the operational mode stabilization. When the beam voltage is selected as 29kV, 1 bQ of the operational mode is negative and its

amplitude reaches maximum. 1 bQ of the π mode always

keeps positive. 1 bQ of the π/2 mode always keeps positive

among the beam voltage range from 7.7kV to 32.5kV. Although 1 bQ of the π slot mode and the π/2 slot mode may

be negative near 29kV, their amplitudes are at least one order of magnitude less than that of the 2π mode, so their self-excited oscillation may easily be suppressed with distributed wall losses.

References [1]. T. Wessel-Berg, Stanford university, Stanford,

Microwave Laboratory Reports, pp376, 1957 [2]. Y. M. Shin, et al., J. Korean Physical Society, Vol. 44,

May, pp1239-1245, 2004. [3]. A. Roitman, et al., IEEE Trans. on Electron Devices，

Vol.52, No.5, pp895-898, 2005. [4]. K. T. Nguyen, et al., IEEE Trans. Plasma Sci, Vol.34,

No.3, pp576-583, 2006. [5]. Frank Kantrowitz, et al, IEEE Trans. on Electron Devices

，Vol.35, No.11, pp2018-2026, 1988 [6]. D. Perring, et al, AGARD Conference Proceedings,

No.197, 1976 [7]. Jian Cui, et al., Acta. Phys. Sin., vol.60, No.5, 2011,

051101

Table 1. Geometrical parameters of the three-gap cavity chain for the 2 mode operation ra/mm rc/mm rd/mm rs/mm r/mm d/mm ds/mm t/mm l/mm /

1.2 2.2 1.7 3.4 4.9 1.2 2.4 1.4 5.0 130

Table 2. “cold” parameters of the different modes for the 2 mode operation

No. Mode name f (GHz) Amplitude of the gap voltages Characteristic impedance () V1 V2 V3

1 π mode 13.86 -1.35 3.09 -1.35 192.48

2 π/2 mode 15.62 -2.76 0.00 2.76 155.23

3 2π mode 16.56 2.51 2.48 2.51 256.08

4 π/2 slot mode 23.83 -1.11 0.00 1.11 16.46

5 π slot mode 25.91 0.87 -1.62 0.87 34.67

Table 3. 1 bQ of five modes for the 2π mode operation

V / kV Pe r / P π mode π/2 2π π/2 slot π slot

20.00 1.00 5.45e- 9.98e-3 3.32e-3 6.15e-4 6.94e-4

29.00 1.00 1.56e- 8.30e-3 -1.70e-2 -5.46e-4 -1.00e-3

20.00 2.00 1.15e- 1.82e-2 5.28e-3 1.00e-3 1.04e-3

29.00 2.00 3.03e- 1.45e-2 -3.09e-2 -8.37e-4 -1.67e-3

10 20 30 40 50-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

V0 / kV

1./Q

b

mode/2 mode2 mode/2 slot mode slot mode

10 20 30 40 50

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

V0 / kV

1./Q

b

mode/2 mode

2 mode/2 slot mode slot mode

(a) 1.0μperP (b) 2.0μperP

Fig. 2 1 bQ versus beam voltage for different perveance for 2 mode operation with rb=0.8mm

282