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.

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).

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).

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.

In order to validate the I_s calculation method proposed above, a V-band (60 GHz) TE₁₁ gyro-TWT was chosen to evaluate the I_s values. Figure 1 shows the dispersion diagram of the V-band TE₁₁ 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_zv_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 TE₁₁ 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 TE₁₁ and TE₂₁ with s = 2, TE₀₁ with s = 3, and TE₀₂ with s = 4 beam modes. The specification

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

Z₀: Circuit impedance; v_p: Phase velocity; V_b: Beam voltage; L: Loss per wavelength; u₀: Beam velocity; k_a: Waveguide wavenumber.

of the V-band TE₁₁ 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 TE₁₁ 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 TE₁₁ 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 TE₁₁ gyro-TWT.

r_c: Guiding center radius.

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

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: TE₁₁, TE₂₁, TE₀₁, and TE₀₂. Fixed values of α

= 0.85, V_b = 100 kV, and were assumed. The lowest I_s occurs for the TE₂₁ and the TE₀₁ mode exhibits the highest I_s value. As can be seen in Figure 5, this is due to TE₂₁ mode having the lowest |k_z| value and the TE₀₁ mode having the highest |k_z| value at the intersection of the beam-wave dispersion diagram. The I_s of the TE₀₁ mode is the most sensitive to L_dB variation.

In this paper, an expression for I_s for lossy gyro-TWT was derived using linear TWT parameter conversions. For V-band TE₁₁ 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.