Theoretical investigation of injection-locked differential oscillator
DOI:
https://doi.org/10.30837/rt.2022.4.211.11Keywords:
differential oscillator, synchronization, main tone, mathematical modelAbstract
A preliminary analysis of published works on this topic showed that at present there is no sufficiently substantiated theory of such devices, and the approximate approaches used are rough and do not always meet the requirements of practice. The proposed transition from a differential self-oscillator to an equivalent single-circuit oscillator has not
received a convincing justification.
This article presents a methodology for studying a synchronized differential oscillator using rigorous methods.
A mathematical model of such oscillator is presented in the form of nonlinear differential equations obtained on the basis of Kirchhoff's laws. Their analysis made it possible to substantiate the transition to the model of a single circuit LC oscillator, equivalent to a differential one. A technique for such a transition is proposed, including the procedure for determining the nonlinear characteristics of the amplifying element of this self-oscillator, based on the nonlinear characteristics of two amplifying elements of the differential oscillator.
The mathematical model of an equivalent oscillator is represented by a non-linear differential Van der Pol equation in a dimensionless form, it is simple and accurate. This form of representation made it possible to single out a small parameter and estimate its value. In the case of small values of the small parameter, as is known, traditional methods can be used for its analysis. Thus, the task of studying the synchronization process of a differential oscillator is reduced to the study of the synchronization process of a Van der Pol oscillator. The presented results can be useful in the development of various devices based on synchronized differential oscillators.
References
Ahmad Mirzaei, Mohammad E. Heidari, Rahim Bagheri. Saeed Chehrazi, Asad A. Abidi. The Quadrature LC Oscillator: A Complete Portrait Based on Injection Locking // IEEE Journal of Solid-State Circuits, vol. 42, no. 9, pp. 1916 – 1932, 2007, doi: 10.1109/JSSC.2007.903047.
Antonio Buonomo, Michael Peter Kennedy, Alessandro Lo Schiavo. On the Synchronization Condition for Superharmonic Coupled QVCO // IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 58, no. 7, pp. 1637 – 1646, 2011, doi: 10.1109/TCSI.2011.2123370.
Antonio Buonomo, Alessandro Lo Schiavo. Modeling, Analysis, and Experimental Validation of Frequency Dividers with Direct Injection // Journal of Electrical and Computer Engineering, article ID 365692, 7 p., 2013, doi:org/10.1155/2013/365692.
A. Buonomo, A. Lo Schiavo. Divide-by-Three Injection-Locked Frequency Dividers with Direct Forcing Signal // Hindawi Publishing Corporation Journal of Electrical and Computer Engineering, article ID 145314, 9 p., 2013, doi: org/10.1155/2013/145314.
Antonio Buonomo, Alessandro Lo Schiav. Analytical Approach to the Study of Injection-Locked Frequency Dividers // IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 60, no. 1, pp. 51 – 62, 2013, doi:10.1109/TCSI.2012.2215716.
Mihaela Izabela Ionita, David Cordeau, Jean Marie Paillot, Mihai Iordache LAII. Analysis and Design of an Array of Two Differential Oscillators Coupled Through a Resistive Network // 20th European Conference on Circuit Theory and Design, Linkцping : Sweden. 2012. doi: 10.1109/ECCTD.2011.6043612.
M. Ionita, D. Cordeau, J. Paillot, S. Bachir, M. Iordache. A CAD tool for an array of differential oscillators coupled through a broadband network // Physics International Journal of RF and Microwave Computer Aided Engineering, vol. 23, no. 2, pp. 178 – 187, 2012, doi:10.1002/mmce.20663.
Mihaela-Izabela Ionita, Mihai Iordache, Dumitriu Lucia, David Cordeau, Jean-Marie Paillot. Generation of the coupling circuit parameters for the coupled oscillators used in antenna arrays // Synthesis, Modeling, Analysis and Simulation Methods and Applications to Circuit Design (SMACD). 2012 International Conference on, Sep.. S_eville. Spain. pp. 237 – 240, 2012, doi: 10.1109/SMACD.2012.6339383.
David Cordeau, Mihaela-Izabela Ionita, Jean-Marie Paillot, Mihai Iordache. New Formulation of the Equations Describing the Locked States of Two Van der Pol Oscillators Coupled through a Broadband Network – Application to the Design of Two Differential Coupled VCOs // Frequenz, vol. 67, no. 7-8, pp. 237-247, 2013,doi:10.1515/freq-2012-0089.
Sanmitra Bharat Naik, R. K. Siddharth, Anirban Chatterjee, Y. B. Nithin Kumar, M. H. Vasantha, Ramnath Kini. A 1.8 V Quadrature Phase LC Oscillator for 5G Applications // 2020 IEEE International Symposium on Smart Electronic Systems (iSES) (Formerly iNiS), pp. 345 – 349, 2020, doi: 10.1109/iSES50453.2020.00068.
Hao-En Liu, Wei-Cheng Chen, Hong-Yeh Chang. A W-band Quadrature Voltage-Controlled Oscillator with an Injection-Locked Frequency Divider in 40-nm CMOS Process // 2021 IEEE International Symposium on Radio-Frequency Integration Technology (RFIT). Hualien, Taiwan, pp. 1 – 3, 2021, doi: 10.1109/RFIT52905.2021.9565268.
Kaouthar Djemel, Rahma Aloulou, David Cordeau, Hassene Mnif, Jean-Marie Paillot, Dorra Mellouli, Mourad Loulou. An original determination of the maximum phase shift range obtained for an array of N coupled oscillators // Analog Integrated Circuits and Signal Processing, vol. 106, no. 3, pp. 683 – 696, 2021, doi: org/10.1007/s10470-020-01791-x.
Рапин В. Теоретичне дослідження дифференційного автогенератора // Известия высших учебных заведений. Сер. Радиоэлектроника, т. 65. no. 5. c. 309 – 319, 2022, doi: org/10.20535/S0021347022050041
Рапин В. Решение укороченных уравнений синхронизированного автогенератора // Известия высших учебных заведений. Сер. Радиоэлектроника, т. 62, no. 6, c. 335 – 347, 2019, doi: 10.20535/S0021347019060037.
Downloads
Published
How to Cite
Issue
Section
License
Authors who publish with this journal agree to the following terms:
1. Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
