662 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 55, NO. 2, FEBRUARY 2008 Suppressing Self-Modulation Instability in a Delayed Feedback Traveling Wave Tube Oscillator Using Controlling Chaos Technique Nikita M. Ryskin, Member, IEEE, and Oleg S. Khavroshin Abstract—Self-modulation instability is typical for traveling wave tube (TWT) oscillators with time-delayed feedback. This in- stability results in the generation of multiple frequencies or even spread spectrum chaotic signal. In this paper, we propose a method for suppressing the self-modulation, that is, a modification of the well-known time-delayed controlling chaos technique. Numerical results are presented, showing that application of the method allows stabilization of unstable single frequency regimes and improves performance of the oscillator. The same method can be used for suppressing the gain ripples arising due to end reflections in the TWT amplifier. Index Terms—Controlling chaos, delayed feedback oscillator, gain ripples, self-modulation, traveling wave tube. I. INTRODUCTION T HE IDEA of controlling chaos proposed by Ott et al. in 1990 [1] usually means stabilization of unstable periodic orbits of a chaotic system using small controlling forces. Among many controlling chaos techniques, probably the most feasible and robust one is the time-delay autosynchronization (TDAS) introduced by Pyragas [2], which is based on adding of an exter- nal feedback loop with the delay time equal to the period of the motion to be stabilized. This method has been improved in many works, and its successful applications to mechanical oscillators, electric circuits, lasers, chemical reactions, and biomedical sys- tems have been demonstrated (see the review [3] for details). However, the TDAS method has some restrictions, in particular, it fails to stabilize fast, high-frequency oscillations. Recently, an interesting modification of TDAS scheme has been suggested by Dolov and Kuznetsov [4] for suppression of self-modulation in a vacuum microwave backward-wave oscillator (BWO). This method is based on electron beam current modulation by a con- trol signal applied to the modulating gate electrode through the external feedback circuit with the delay time close to one-half of the self-modulation period. It was shown that the application of the proposed control scheme could provide a substantial in- crease of electronic efficiency and output power of the BWO single-frequency generation. In this paper, we propose another modification of the TDAS method for controlling chaos in the microwave traveling wave tube (TWT) oscillator with time-delayed feedback. The possi- Manuscript received August 27, 2007; revised October 25, 2007. This work was supported by the Russian Foundation for Basic Research under Grant 06-02-16773. The review of this paper was arranged by Editor W. Menninger. The authors are with Saratov State University, Saratov, 410012 Russia (e-mail: Ryskinnm@info.sgu.ru; ktu@nonlin.sgu.ru). Digital Object Identifier 10.1109/TED.2007.912366 bility of chaos generation in the delayed feedback TWT oscil- lator has been known for many years [5]. Recently, microfabri- cated folded-waveguide TWT oscillators had been considered as promising sources of radiation with up to terahertz frequencies, and the first numerical and experimental studies have revealed that self-modulation and chaos are very typical for these types of oscillators [6], [7]. Therefore, it is important to know how to suppress this instability and provide stable single-frequency operation. The paper is structured as follows. The controlling chaos method is described in Section II. Section III contains the basic equations of nonstationary TWT theory. In Section IV, results of numerical simulations for the TWT oscillator are presented and successful suppression of self-modulation is demonstrated. In Section V, we investigate the possibility of application of the same method to suppress gain ripples in a TWT amplifier with end reflections. This paper is a significantly expanded version of the abstract [8] of the talk presented at the IEEE 2007 Pulsed Power and Plasma Science Conference. II. METHOD OF CONTROLLING CHAOS Consider the oscillator schematic that consists of the TWT amplifier and the external feedback loop [Fig. 1(a)]. The feed- back circuit contains a variable attenuator (VA) required to ad- just the amount of feedback, a phase shifter (PS) to control the phase of the feedback signal, which is also an important pa- rameter (see [9], [10] for details), and a delay line. It is well known that with the increase of either the TWT gain or the amount of feedback, the regime of single-frequency generation becomes unstable and changes first to multiple frequency self- modulation, and then, to a spread-spectrum chaotic one [9]–[11]. To suppress the self-modulation instability, the feedback loop is split into two paths, as shown in Fig. 1(b). The main idea is to choose the delay times δ 1, 2 and phases ψ 1, 2 , so that, after passing through the two feedback legs, the fundamental waves appear in the same phase, while the self-modulational sidebands appear in antiphase, and thus, suppress each other. Let k be the control parameter defining the relative power level of the signals passing through the first and the second feedback legs, ρ 1 be the attenuation produced by the additional VA shown in Fig. 1(b), and E (x, t) be the electric field of the electromagnetic wave propagating through the TWT. The boundary condition connecting the TWT input signal, E in = E (x = 0; t), with the output one, E out = E (x = l; t), where l is the length of the 0018-9383/$25.00 © 2008 IEEE