Journal of the Korean Physical Society, Vol. 44, No. 5, May 2004, pp. 1261∼1264
Operation of a Delayed-Feedback Oscillator Using an Electron Beam and a Traveling Wave in a Folded Waveguide S. T. Han, K. H. Jang, J. K. So and G. S. Park∗ School of Physics, Seoul National University, Seoul 151-742
N. M. Ryskin Saratov State University, Saratov, Russia (Received 10 November 2003) A delayed-feedback scheme is employed in a Ka-band folded waveguide traveling-wave tube (FWTWT) amplifier. The threshold feedback strength for an oscillation is measured to be −28 dB from the output signal. The optimum value of the feedback strength for a stable single-frequency oscillation at 32.7 GHz is about −20 dB with a net efficiency of 4 %, which shows reasonable agreement with the prediction of a one-dimensional non-stationary simulation. PACS numbers: 84.40.Fe, 07.57.Hm, 52.75.Va Keywords: Folded waveguide, Traveling-wave tube (TWT), Delayed feedback
I. INTRODUCTION The interest in THz frequencies located between the boundaries of the electronic principle and the photonic principle has been growing [1–3]. There have been many efforts to generate THz radiation, from quantum cascade lasers (QCLs) [4] using the tiny sub-bands in the semiconductor quantum wells to free electron lasers (FELs) [5, 6] and to Gyrotrons [7] using the radiation caused by the periodic motion of the electron beam in vacuum. However, the semiconductor devices have fundamental limits, such as low operating temperature and low output power, and vacuum devices have restrictions in application due to the bulk magnets. Studies of miniature high-frequency high-power vacuum tubes using microfabrication technology have been conducted to overcome these problems [8–10]. Among them, the folded waveguide traveling-wave tube (FW-TWT), due to its simple structure, has advantages because it has a high powerhandling capacity at higher frequencies and is compatible with planar fabrication using MEMS technologies. Therefore, many researchers have focused on this device [10–12]. If a delayed-feedback FW-TWT oscillator operating at higher frequencies is to be realized by using planar micro-fabrication technology, a feedback loop should also be integrated with the circuit. As a result, it is very important to characterize the performance of the delayed feedback FW-TWT oscillator with respect to the feedback strength, in advance. However, there has been no experimental study of the optimum condi∗ E-mail:
[email protected]
tions for a stable single-frequency oscillation of a delayedfeedback oscillator using a FW-TWT amplifier. Hence, in this paper, we discuss a Ka-band FW-TWT amplifier developed to perform experiments on an enlarged scale (Section II) and its performance as a delayed-feedback oscillator (Section III).
II. FOLDED WAVEGUIDE TRAVELING-WAVE TUBE AMPLIFER A schematic diagram of the electric-field-plane (Eplane) bend folded waveguide circuit used in this study is shown in Fig. 1. The lowest transverse electric (TE) mode, TE10 , is assumed to propagate along the waveg-
Fig. 1. Schematic diagram of an E-plane bend folded waveguide. Note that the periodicity itself slows down the axial propagation velocity of the electromagnetic wave along the beam line and generates space harmonics of a virtual TM mode.
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Fig. 2. Characteristics of a folded waveguide travelingwave tube circuit. The S21 parameter was measured with a vector network analyzer. The interaction impedance was estimated by considering the finite cross-section of the electron beam and the radial variation of the electric field around beam tunnel.
Fig. 3. Experimental layout of a Ka-band folded waveguide traveling-wave tube amplifier.
uide. The electron beam sees the transverse magnetic (TM) mode when it enters the beam holes. We designed a Ka-band FW TWT circuit [13] with a period of 1.4 mm and a 12-kV, 150-mA, 0.5-mm-radius electron beam. Figure 2 shows the cold characteristics, the wave transmission (S21 ) and interaction impedance, of the folded waveguide circuit with beam holes. The 3D code MWS, was used to estimate numerically the S21 parameter including the loss of the rectangular waveguide [14]. The measured S21 shows good agreement with the predicted values. The circuit was designed to minimize the reflection and the variation of the phase velocity at the bend portion [15]. The interaction impedance of a space harmonic was estimated by expanding the complex magnitude of the electric field along the center axis into a Fourier series, and the field-intensity profile was obtained from a high frequency structure simulator (HFSS) [16] simulation in the presence of a beam tunnel. The experimental layout of the FW-TWT amplifier is
Journal of the Korean Physical Society, Vol. 44, No. 5, May 2004
Fig. 4. Output power as a function of the drive frequency, where the input power was fixed to 20 dBm for all the frequencies swept. The instantaneous 3-dB bandwidths were 10 % and 9.6 % at 13.2 kV and 13 kV, respectively. Note that extraordinarily high output was obtained at 32.5 GHz when the anode voltage was 13.4 kV, which seems to correspond to the point where the slow space-charge wave of the electron beam crosses the dispersion curve of the folded waveguide circuit because of fabrication inaccuracies.
shown in Fig. 3. The E-plane bend folded waveguide consists of 70 periods. A conventional Pierce-type electron gun was used to produce a linear electron beam of 12 kV and 150 mA. The pulsed electron beam, with a repetition of 10 Hz and a duration of 10 µs, was focused by using a solenoid magnet. The two open waveguide ends (port1 and port2) of a standard Ka-band waveguide (WR-28) were connected to the interaction circuit and to the mica vacuum windows. A vacuum envelope with all rf circuits inserted was used for connecting to the electron gun and the rf ports. Figure 4 shows the output power as a function of the drive frequency, where the input power was fixed to 20 dBm for all the frequencies swept. The maximum gain was obtained at 13.4 kV, which differed from the designed value of 12 kV. The difference likely comes from the fabrication inaccuracy of the computer numerical control (CNC) milling. Figure 5 shows the drive curve (AM-AM conversion) of the FW-TWT amplifier for an input frequency of 35 GHz. The saturation at higher input powers than 25 dBm is attributed to a reduction of the collector current.
III. DELAYED FEEDBACK OSCILLATOR In order to realize a delayed-feedback FW-TWT oscillator, we introduced an external feedback loop that re-circulated the controlled fraction of the electromag-
Operation of a Delayed-Feedback Oscillator Using an Electron Beam· · · – S. T. Han et al.
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Fig. 5. Drive curve of an FW-TWT amplifier for an input frequency of 35 GHz.
Fig. 6. Output power and generated frequency as functions of the feedback strength.
netic wave radiation from the cathode-side port (port2) into the collector-side port (port1). The external feedback loop consisted of a broadband coupler, a variable attenuator, and a phase shifter, where the attenuator and the phase shifter were used to control the level and the phase of feedback, respectively. To predict the operation of the delayed-feedback oscillator, we performed a one-dimensional non-stationary calculation [17], which consisted of three main equations: # 3 " X In ejnθ ∂2θ 1 ∂θ 2 jθ = −L 1 + Re F e + jq , ∂ξ 2 2πN ∂ξ n n
Here, R = ρ exp(jψ) is the complex feedback parameter, 0 < ρ < 1, and δ is the dimensionless delay time. Up to ten current harmonics were taken into account to calculate the space-charge forces. The normalized parameters corresponding to the TWT under consideration were L ≈ 7.0 and q ≈ 2.67. Figure 6 depicts the output power and the generated frequency as functions of the feedback strength in the single-frequency region at a relative phase of ψ = 0. Except for the feedback range for a single-frequency oscillation being shifted by 2 dB, the measured results show reasonable agreement with those from the simulation. The discrepancy between the measurement and the simulation in the generated frequencies seems to be attributed to the fabrication inaccuracies of folded waveguide circuit. A non-stationary simulation was conducted with the theoretical dispersion to which the slow space-charge wave of the 12-kV electron beam was synchronized at 35 GHz. The measured carrier frequency of 32.7 GHz in the single-frequency region corresponds to the frequency that shows maximum gain in the amplifier operating at 13.4 kV (see Fig. 5) and seems to correspond to the points at which the slow space-charge wave of the electron beam crosses the dispersion curve of the folded waveguide circuit used in this experiment. In contrast to a backward wave oscillator under external feedback, the output power does not changed with respect to the level of feedback because there is no energy-recovery process in a delayed-feedback travelingwave tube oscillator [18]. Note that the oscillation occurs above a threshold of −28-dB feedback; then, the net efficiency is maintained at about 4 %. Above a feedback strength of −16 dB and −14 dB in the measurement and the simulation, respectively, the self-modulation process in which multiple frequencies are generated occurs [19]. These results lead us to the conclusion that the optimum value of the feedback strength for a stable singlefrequency oscillation in a delayed-feedback FW-TWT os-
(1) ∂F ∂F + = −LI, ∂τ ∂ξ I=
1 π
Z
(2)
2π
exp(jθ)dθ0 .
(3)
0
Here, θ = ω0 (t − x/v0 ) is the phase of an electron, θ0 is the initial phase, ω0 is the frequency of synchronism, v0 is the unperturbed beam velocity, and τ and ξ are dimensionless time and space coordinates, respectively. F is the normalized slowly varying amplitude of the electromagnetic wave, In is n-th current harmonic, L = 2πCN , where C is the Pierce gain parameter, N is the phase length of the slow-wave structure, and q = 4QC is the Pierce space-charge parameter. The equations were solved by using a finite-difference numerical method of second-order accuracy with appropriate initial conditions and the boundary conditions for a delayed feedback: ∂θ θ|ξ=0 = θ0 , = 0, (4) ∂ξ ξ=0 F (ξ = 0, τ ) = RF (ξ = 1, τ − δ).
(5)
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Journal of the Korean Physical Society, Vol. 44, No. 5, May 2004
cillator is about −20 dB.
IV. SUMMARY AND DISCUSSION In this work, we investigated the characteristics of a delayed-feedback oscillator using a Ka-band FW-TWT amplifier. Particularly, the optimum value of the feedback strength for a stable single-frequency oscillation was estimated. It was shown that a stable single-frequency oscillation occurred above a feedback threshold of −28 dB. The optimum value of the feedback strength for single-frequency oscillation at 32.7 GHz is about −20 dB with a net efficiency of 4 %. Throughout the comparison between the simulation and the experiment on an enlarged scale, we clearly saw that a delayed-feedback FW-TWT oscillator was a potential candidate for highfrequency radiation sources manufactured by using planar micro-fabrication [20].
ACKNOWLEDGMENTS This work is supported by the Ministry of Science and Technology of the Republic of Korea through the National Research Laboratory Program.
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