Dynamic Characteristics of a Biharmonic Self-Oscillator

Introduction. Modern methods of stabilizing a frequency of self-oscillations use an improvement of the stability of reactive parameters of the self-oscillators circuit and an increase in the quality factor of an oscillating system. It is also possible to improve the frequency stabilization based on the phenomenon of mutual synchronization of the self-oscillator modes using a multi-loop oscillation system. Previously, a method for reducing a phase noise of an auto-oscillator with synchronization of two modes in a biharmonic auto-oscillator with multiple frequencies was described. The method was developed under the assumption that an active element is inertialess. The idea of the method of synchronizing of the main oscillation with its 2-nd harmonic using an additional loop is based on the consideration that internal fluctuation processes in the active element modulate in-phase all current har-monics. Therefore, it is possible to use this "natural" cross-correlation of noise processes to neutralize their influence.

Aim. Building and analysis of a mathematical model of a biharmonic oscillator in order to analyze the operating modes of such generator and reduction of the phase noise of its output oscillation.

Materials and methods. The mathematical model was developed by the method of slowly changing amplitudes, and the analysis was performed by methods of numerical integration and differentiation.

Results. It was demonstrated that synchronization of two oscillations at multiple frequencies in the active element reduced the phase noise of the main oscillation.

Conclusion. In the paper dynamic modes of a biharmonic Colpitts oscillator operating in the phase synchronization mode of two waves were analyzed. It was shown that with an increase in an inertia of the active element, the synchronous mode was preserved. Shortened differential equations of the system for slowly changing amplitudes and phases of oscillatory modes were obtained. The study of nonlinear dynamics and of stationary synchronous mode of the system was carried out by the method of phase space in coordinates of "mode amplitude – phase difference". The conducted field experiment allows one to conclude that it is possible to reduce the phase noise in a stationary synchronous biharmonic mode. It can be used in the frequency stabilization task.

1. Nosal Z. M. Design of GaAs MMIC Transistors for the Low-Power Low Noise Applications. 2000 IEEE MTT-S Intern. Microwave Symposium. Boston, USA, 11–16 June 2000. Digest. Piscataway, IEEE, 2000. doi: 10.1109/MWSYM.2000.860872

2. Seif M., Pascal F., Sagnes B., Hoffmann A., Haendler S., Chevalier P., Gloria D. Low Frequency Noise Temperature Measurements in SiGe:C Heterojunction Bipolar Transistors. 2015 Intern. Conf. on Noise and Fluctuations. Xian, China, 2–6 June 2015. Piscataway, IEEE, 2000. doi: 10.1109/ICNF.2015.7288603

3. Ustinov A. B., Nikitin A. A., Kalinikos B. A. Magnetically Tunable Microwave Spin-Wave Photonic Oscillator. IEEE Magnetics Letters. 2015, vol. 6, art. 3500704. doi: 10.1109/LMAG.2015.2487238

4. Ustinov A. B., Kondrashov A. V., Nikitin A. A., Lebedev V. V., Petrov A. N., Shamrai A. V., Kalinikos B. A. A Tunable Spin Wave Photonic Generator with Improved Phase Noise Characteristics. J. of Phys.: Conf. Ser. 2019, vol. 1326, art. 012015. doi: 10.1088/1742-6596/1326/1/012015

5. Goryachev M., Galliou S., Imbaud J., Bourquin R., Dulmet B., Abbé P. Recent Investigations on BAW Resonators at Cryo-genic Temperatures. 2011 Joint Conf. of the IEEE Intern. Frequency Control and the European Frequency and Time Forum (FCS) Proc., San Francisco, USA, 2–5 May 2011. Piscataway, IEEE, 2000. doi: 10.1109/FCS.2011.5977293

6. Tsarapkin D. P., Chichvarin M. I., Isakov I. A. Experimental Verification of Compensation Phenomena in Oscillators with Two Multiple Modes. Proc. of the 2000 IEEE/EIA Intern. Frequency Control Symposium and Exhibition, Kansas City, USA, 7–9 June 2000, pp. 463–470. doi: 10.1109/FREQ.2000.887401

7. Utkin G. M. self-Oscillating systems and volnew amplifiers. M., Sov. radio, 1978, 272 p. (In Russ.)

8. Karachev A. A., Levchenkov O. I., Carapkin D. P. Experimental study of the two-frequency response of the Gann generator. Tr. MEI. Radio transmitting and radio receiving devices, Moscow, MEI publishing House, 1977, iss. 317, pp. 36-38. (In Russ.)

9. Mitrofanov A. A., Safin A. R., Udalov N. N., Kapranov M. V. Theory of Spin Torque Nano-Oscillator-Based Phase-Locked Loop. J. of applied physics. 2017, vol. 122, iss. 12, art. 123903. doi: 10.1063/1.5004117

10. Mitrofanov A. A., Safin A. R., Udalov N. N. Amplitude and phase noise of a spin-transfer nanoscillator synchronized by a phase-locked frequency adjustment system. Letters to ZhTF. 2015, vol. 41, iss. 16, pp. 29-35 (In Russ.)

11. Carapkin D. P., Frolov D. A. To analyze the characteristics of a transistor autogenerator with two synchronous modes. Twenty-fourth int. scientific and technical Conf. of students and postgraduates. TEZ. Dokl. Moscow, March 15-16, 2018. M.: LLC "Center of printing services "Raduga", 2018, 19 p. (In Russ.)

12. http://www.nxp.com (accessed 06.08.2020)

13. Rohde U., Poddar A., Bock G. The Design of Modern Microwave Oscillators for Wireless Applications. Theory and Optimization. New York, John Wiley & Sons, 2005, 543 p. doi: 10.1002/0471727172.fmatter

14. Grebennikov A. RF and microwave transistor oscillator design. New York: John Wiley & Sons, 2007, 458 p. doi: 10.1002/9780470512098

15. Kapranov M. V., Tomashevsky A. I. Analysis of phase trajectories in the vicinity of special points of 2-D and 3-D nonlinear dynamical systemsю. Textbook. Moscow, Izd-vo MEI, 2003, 80 p. (In Russ.)

16. Alekseev E. R., Chesnokova O. V. Introduction to Octave for engineers and mathematicians. Moscow, AL Linux, 2012, 368 p. (In Russ.)

17. Alekseev E. R., Chesnokova O. V. Rudchenko E. A. Scilab: Solving engineering and mathematical problems. Moscow, ALT Linux; BINOM. Laboratory of knowledge, 2008, 260 p. (Bibl. ALT Linux). (In Russ.)