**Journal of Electromagnetic Analysis and Applications** Vol.5 No.1(2013), Article ID:27029,4 pages DOI:10.4236/jemaa.2013.51001

Calculation of Start-Oscillation-Current for Lossy Gyrotron Traveling-Wave Tube (Gyro-TWT) Using Linear Traveling-Wave Tube (TWT) Parameter Conversions

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Department of Electrical and Computer Engineering, University of Colorado, Colorado Springs, USA.

Email: hsong@uccs.edu

Received November 13^{th}, 2012; revised December 13^{th}, 2012; accepted December 25^{th}, 2012

**Keywords:** Start-Oscillation-Current; Gyro-TWT; TWT; Lossy; Stable; Competing Mode

ABSTRACT

The start-oscillation-current of a gyro-TWT (gyrotron traveling-wave tube) determines the stable operating current level of the device. The amplifier is susceptible to oscillations when the operating current level is higher than the start-oscillation current. There are several ways of calculating the start-oscillation current, including using the linear and nonlinear theory of a gyro-TWT. In this paper, a simple way of determining the start-oscillation current of lossy gyro-TWT is introduced. The linear TWT parameters that include the effects of synchronism, loss, and gain, were converted to gyro-TWT parameters to calculate the start-oscillation-current. The dependence on magnetic field, loss, and beam alpha was investigated. Calculations were carried out for a V-band gyro-TWT for both operating and competing modes. The proposed method of calculating the start-oscillation current provides a simple and fast way to estimate the oscillation conditions and can be used for the design process of a gyro-TWT.

1. Introduction

The gyro-TWT (gyrotron traveling-wave tube) has long been viewed as an extremely promising device due to its high-power and broadband capabilities. Potential applications include radar, communication, surveillance, and scientific research [1]. However, in order for gyro-TWT to work properly, the interaction with competing mode must be suppressed. The beam current level where the unwanted oscillation takes place is called the “start-oscillation-current” for gyro-TWT. Therefore in gyroTWT, it is critical to operate the amplifier below I_{s} to ensure stability of the device. One way to increase I_{s} is to apply loss to the gyro-TWT circuit. Calculation results of I_{s} employing linear theory [2-4] and nonlinear theory [5] were reported for lossy gyro-TWT. However, using these methods require in-depth analysis on linear and nonlinear theories of gyro-TWT. In this paper, a simple method of obtaining I_{s} for lossy gyro-TWT by using the linear-TWT parameters is introduced. By using the linear TWT parameter conversions, the expression for I_{s} for the gyroTWT was obtained. The parameter conversion process and the calculation results for I_{s} for gyro-TWT are presented.

2. Conversion of Linear TWT Parameters to Gyro-TWT

The gain parameter of a linear lossy TWT that corresponds to the start-oscillation condition can be expressed by Equation (1) [6]. For gyro-TWT, the gain parameter is described as Equation (2) [7]. By combining Equation (1) used in a linear TWT and Equation (2) used in a gyro-TWT (by setting C_{st} = C_{g}), the start oscillation current for lossy gyro-TWT can be expressed by Equation (3).

(1)

(2)

(3)

Here L_{dB} is the total loss of the circuit in dB, N is the circuit length in wavelength, k_{c} is the cutoff wavenumber, k_{b} is the beam wavenumber, I_{b} is the beam current, F_{mn} is defined in Equation (4), and ε_{v} is defined in Equation (5).

(4)

(5)

Here J_{m} is the Bessel function of order m, k_{mn} is the m^{th} Bessel root defined by, n is the radial mode number, m is the azimuthal mode number, R_{a} is the guiding center radius, a is the waveguide radius, β_{z} is the axial velocity normalized by the speed of light, β_{^} is the transverse velocity normalized by the speed of light, ω_{c} is the cutoff frequency of the waveguide, and Ω_{c} is the relativistic cyclotron frequency. The comparison between critical parameters of linear and gyro-TWT is shown in Table 1.

3. Calculated Results and Discussion

In order to validate the I_{s} calculation method proposed above, a V-band (60 GHz) TE_{11} gyro-TWT was chosen to evaluate the I_{s} values. Figure 1 shows the dispersion diagram of the V-band TE_{11} gyro-TWT for α = 0.85, V_{b} = 100 kV, and. The waveguide mode expressed as is shown in parabolas and the beam mode which can be described as ω = sΩ_{c} + k_{z}v_{z} is shown in straight lines up to fourth harmonic. Here, ω is the frequency, k_{z} is the axial wavenumber, c is the speed of light, s is the harmonic number, and v_{z} is the axial velocity of the beam. The operating point is where the TE_{11} waveguide mode grazes with the s = 1 beam mode. The possible competing mode interactions occur when the waveguide mode intersects with the beam mode. These include TE_{11} and TE_{21} with s = 2, TE_{01} with s = 3, and TE_{02} with s = 4 beam modes. The specification

Table 1. Conversion of linear TWT parameters to gyroTWT [7-8].

Z_{0}: Circuit impedance; v_{p}: Phase velocity; V_{b}: Beam voltage; L: Loss per wavelength; u_{0}: Beam velocity; k_{a}: Waveguide wavenumber.

Figure 1. Dispersion diagram of a TE_{11} 60 GHz gyro-TWT for α = 0.85, V_{b} = 100 kV, and. Waveguide modes (TE_{11}, TE_{21}, TE_{01}, TE_{02}) and beam modes (s = 1, 2, 3, 4) are shown.

of the V-band TE_{11} gyro-TWT is described in Table 2. The calculated I_{s} using Equation (3) is shown in Figures 2-5 under various conditions. Figure 2 shows dependence of I_{s} on circuit loss, L_{dB}, for several values of beam velocity ratio, α, for the operating TE_{11} mode with and V_{b} = 100 kV. The indicates operating magnetic field, B_{o}, normalized by the grazing magnetic field, B_{g}. As L_{dB} increases, I_{s} increases which indicates that with higher value of L_{dB}, the device becomes more stable. For fixed value of L_{dB}, I_{s} increases as α decreases. This indicates that the loss stabilizes the device and the gyro-TWT becomes unstable for higher values of α. Figure 3 describes I_{s} change with of the operating TE_{11} mode for several values of beam voltage, V_{b}, for fixed values of α = 0.85 and L_{dB} = 100 dB. For, I_{s} decreases as increases. For fixed value of, I_{s} is higher for higher V_{b} when. For, higher beam voltage makes the device unstable and as the operating magnetic field increases the device becomes more stable due to increasing |k_{z}|. Figure 4 shows I_{s} as a function for several values of α and fixed values of V_{b} = 100 kV and L_{dB} = 100 dB. For, I_{s} decreases as increases. For fixed value of, I_{s} in

Table 2. Specifications of the V-band TE_{11} gyro-TWT.

r_{c}: Guiding center radius.

Figure 2. Start-oscillation-current, I_{s}, for the operating TE_{11} mode as a function of loss, L_{dB}, for different values of beam velocity ratio, α, and the fixed value of the operating magnetic field normalized by the grazing magnetic field, B_{o}/B_{g} = 1.5, and beam voltage, V_{b} = 100 kV.

Figure 3. Start-oscillation-current, I_{s}, for the operating TE_{11} mode as a function of B_{o}/B_{g} for different values of beam voltage, V_{b}, and fixed values of α = 0.85 and L_{dB} = 100 dB.

Figure 4. Start-oscillation-current, I_{s}, for the operating TE_{11} mode as a function of B_{o}/B_{g} for different values of α and fixed values of V_{b} = 100 kV and L_{dB} = 100 dB.

creases as α increases for. For

, device is more stable for lower alpha because when the perpendicular component of the velocity decreases, the gyro-TWT beam-wave interaction becomes

Figure 5. Dispersion diagram of a TE_{11} 60 GHz gyro-TWT for α = 0.85, V_{b} = 100 kV, and B_{o}/B_{g} = 1.5. Waveguide modes (TE_{11}, TE_{21}, TE_{01}, TE_{02}) and beam modes (s = 1, 2, 3) are shown.

Figure 6. Start-oscillation-current, I_{s}, as a function of loss, L_{dB}, for different modes. Fixed values of α = 0.85, V_{b} = 100 kV, and B_{o}/B_{g} = 1.5 were assumed.

weaker. Figure 5 shows the dispersion diagram of the device for α = 0.85, V_{b} = 100 kV, and. Figure 6 describes I_{s} as a function of L_{dB} for four different modes: TE_{11}, TE_{21}, TE_{01}, and TE_{02}. Fixed values of α

= 0.85, V_{b} = 100 kV, and were assumed. The lowest I_{s} occurs for the TE_{21} and the TE_{01} mode exhibits the highest I_{s} value. As can be seen in Figure 5, this is due to TE_{21 }mode having the lowest |k_{z}| value and the TE_{01} mode having the highest |k_{z}| value at the intersection of the beam-wave dispersion diagram. The I_{s} of the TE_{01} mode is the most sensitive to L_{dB} variation.

4. Summary and Conclusion

In this paper, an expression for I_{s} for lossy gyro-TWT was derived using linear TWT parameter conversions. For V-band TE_{11} gyro-TWT, I_{s} was calculated for various parameters including loss, beam voltage, magnetic field, and beam velocity ratio. The method introduced in this paper can be used to quickly estimate the I_{s} values of a lossy gyro-TWT.

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