ЖУРНАЛ РАДИОЭЛЕКТРОНИКИ. eISSN 1684-1719. 2026. №6
Текст статьи (pdf)
DOI: https://doi.org/10.30898/1684-1719.2026.6.6
Математическая модель осциллятора Колпитца
для анализа электромагнитной восприимчивости
и фазовой устойчивости автогенераторных схем
П.С. Глазунов 1, А.И. Гринько 2, Л.С. Болотников 2, В.А. Вдовин 1, В.А. Черепенин 1
1 ИРЭ им. В.А. Котельникова РАН, 125009, Москва, ул. Моховая 11, корп.7.
2 МГТУ им. Н.Э. Баумана, 105005, Москва, 2-я Бауманская ул., 5
Статья поступила в редакцию 17 июня 2026 г.
Аннотация. Предложена математическая модель осциллятора Колпитца на основе системы дифференциальных уравнений, описывающих динамику автогенератора с учетом паразитных электродвижущих сил, наводимых внешним электромагнитным полем в различных контурах схемы. Актуальность исследования обусловлена усложнением современной электромагнитной обстановки и миниатюризацией электронных компонентов, приводящей к снижению их помехоустойчивости. Показано, что кратковременное импульсное воздействие способно вызывать фазовый сдвиг колебаний осциллятора. Установлено, что величина этого фазового сдвига зависит от амплитуды помехи, момента ее приложения и контура, в который она вводится. С помощью численных экспериментов выявлены наиболее уязвимые участки схемы, что может быть использовано при проектировании помехоустойчивых автогенераторных схем и оптимизации их топологии.
Ключевые слова: электромагнитная совместимость, осциллятор Колпитца, автогенератор, фазовый сдвиг, электромагнитная восприимчивость, импульсная помеха, численное моделирование.
Финансирование: Работа выполнена по государственному заданию ИРЭ им. В.А. Котельникова РАН.
Автор для переписки: Владимир Александрович Вдовин, vdv@cplire.ru
Литература
1. Abdelnaby K.M. et al. The Intelligent Home: A Systematic Review of Technological Pillars, Emerging Paradigms, and Future Directions // Symmetry. – 2026. https://doi.org/10.3390/sym18050718
2. Pink S. et al. (ed.). Everyday automation: Experiencing and anticipating emerging technologies. – Routledge, 2022. https://doi.org/10.4324/9781003170884
3. Tsallis C. et al. Industrial Wireless Networks in Industry 4.0: A Systematic Review // Journal of Sensor and Actuator Networks. – 2026. – Т. 15. – №. 1. – С. 7. https://doi.org/10.3390/jsan15010007
4. Underberg L. et al. Towards Wireless Communications in Automation: An Overview // 2024 IEEE 35th International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC). – IEEE, 2024. – С. 1-7. https://doi.org/10.1109/PIMRC59610.2024.10817208
5. Pahlavan K. Understanding of RF cloud interference measurement and modeling // International Journal of Wireless Information Networks. – 2022. – Т. 29. – №. 3. – С. 206-221. https://doi.org/10.1007/s10776-021-00541-8
6. Brunner H. et al. Understanding and mitigating the impact of Wi-Fi 6E interference on ultra-wideband communications and ranging // 2022 21st ACM/IEEE International Conference on Information Processing in Sensor Networks (IPSN). – IEEE, 2022. – С. 92-104. https://doi.org/10.1109/IPSN54338.2022.00015
7. Taha H., Vári P., Nagy S. On the Challenges of Mutual Interference between Cable Television Networks and Mobile Fixed Communication Networks in the Digital Dividend Bands // Infocommunications Journal. – 2022. – Т. 14. – №. 3. – С. 63-71. https://doi.org/10.36244/ICJ.2022.3.8
8. Adekogba G.V., Adedeji K.B., Olasoji Y.O. Interference Reduction Scheme for Femtocell Ultra-Dense-Network: Concept and Research Challenges // ITEGAM-JETIA. – 2024. – Т. 10. – №. 49. – С. 144-158 https://doi.org/10.5935/jetia.v10i49.1125
9. Liu Y.H., Ku C.P., Chiueh T.D. Design and Implementation of a 5G NR Transmitter With Wi-Fi Coexistence by Beamforming and Power Control // IEEE Open Journal of the Communications Society. – 2024. – Т. 5. – С. 2132-2144. https://doi.org/10.1109/OJCOMS.2024.3381978
10. Li P., Liu D. Efficient beam selection and resource allocation scheme for WiFi and 5G coexistence at unlicensed millimetre‐wave bands // IET Communications. – 2020. – Т. 14. – №. 17. – С. 2944-2952. https://doi.org/10.1049/iet-com.2019.0746
11. Hoang T.M., Vahid A. Localization-Based 5G NR-Compliant Beam Management for Coexistence With Incumbents // IEEE Open Journal of Vehicular Technology. – 2026. https://doi.org/10.1109/OJVT.2026.3689802
12. Clavier L. et al. Experimental evidence for heavy tailed interference in the IoT // IEEE Communications Letters. – 2020. – Т. 25. – №. 3. – С. 692-695. https://doi.org/10.1109/LCOMM.2020.3034430
13. Hassan S. et al. Real-time investigation of cross-technology interference in heterogeneous IoT networks // IEEE Access. – 2023. – Т. 11. – С. 112223-112235. https://doi.org/10.1109/ACCESS.2023.3321221
14. Abass A.A. A., Anwar H., Alshaheen H.S. A Survey on Interference Mitigation for Wireless Body Area Networks // University of Thi-Qar Journal for Engineering Sciences. – 2024. – Т. 14. – №. 1. – С. 92-106. https://doi.org/10.31663/utjes.14.1.671
15. Kaleem M., Devarajan G.G. Energy-efficient classification strategy for detecting interference and malicious sensor nodes in wireless body area Networks // Cyber Security and Applications. – 2024. – Т. 2. – С. 100048. https://doi.org/10.1016/j.csa.2024.100048
16. Rostamikafaki Z., Chan F., D’amours C. Measurement-Based Analysis of 5G Cellular Network Interference on Radar Altimeters and Joint Power-Angle Control Mitigation Strategy // IEEE Access. – 2025. https://doi.org/10.1109/ACCESS.2025.3582024
17. Bai J., Yuan S., Duan Z. Research on the Interference Effects of 5G’s Key Parameters on Radio Altimeters // Aerospace. – 2024. – Т. 12. – №. 1. – С. 16. https://doi.org/10.3390/aerospace12010016
18. Watanabe K. et al. Electromagnetic interference with the mobile communication devices in unmanned aerial vehicles and its countermeasures // IEEE Access. – 2024. – Т. 12. – С. 11642-11652. https://doi.org/10.1109/ACCESS.2024.3351216
19. Petrov G., Stancheva A. Problems related to EMC caused by low-altitude flying drones in urban environment // Electrotechnica & Electronica (E+ E). – 2020. – Т. 55.
20. Li J., Kong L., Chu M. Analysis of the impact of electromagnetic fields on UAV flight control systems in EHV–UHV DC overhead transmission lines // AIP Advances. – 2024. – Т. 14. – №. 8. https://doi.org/10.1063/5.0225258
21. Dionísio R., Lolić T., Torres P. Electromagnetic interference analysis of industrial IoT networks: From legacy systems to 5G // 2020 IEEE Microwave Theory and Techniques in Wireless Communications (MTTW). – IEEE, 2020. – Т. 1. – С. 41-46. https://doi.org/10.1109/MTTW51045.2020.9245057
22. Siddiqui M.U. A. et al. URLLC in beyond 5G and 6G networks: An interference management perspective // IEEe Access. – 2023. – Т. 11. – С. 54639-54663. https://doi.org/10.1109/ACCESS.2023.3282363
23. Banafaa M.K. et al. A comprehensive survey on 5G-and-beyond networks with UAVs: Applications, emerging technologies, regulatory aspects, research trends and challenges // IEEE access. – 2024. – Т. 12. – С. 7786-7826. https://doi.org/10.1109/ACCESS.2023.3349208
24. Wang Y.H., Li C.C. Highly effective EMI shielding composites for 5G Ka-band frequencies // Applied Materials Today. – 2024. – Т. 36. – С. 102041. https://doi.org/10.1016/j.apmt.2023.102041
25. Yuan L. et al. EMI challenges in modern power electronic-based converters: recent advances and mitigation techniques // Frontiers in Electronics. – 2023. – Т. 4. – С. 1274258. https://doi.org/10.3389/felec.2023.1274258
26. Liu G. et al. Electromagnetic immunity performance of intelligent electronic equipment in smart substation’s electromagnetic environment // Energies. – 2020. – Т. 13. – №. 5. – С. 1130. https://doi.org/10.3390/en13051130
27. Angelov G.V., Nikolov D.N., Hristov M.H. Technology and modeling of nonclassical transistor devices // Journal of Electrical and Computer Engineering. – 2019. – Т. 2019. – №. 1. – С. 4792461. https://doi.org/10.1155/2019/4792461
28. Chen X., Touba N.A. Fundamentals of CMOS design // Electronic Design Automation. – Morgan Kaufmann, 2009. – С. 39-95. https://doi.org/10.1016/B978-0-12-374364-0.50009-6
29. Bespalov V.A., Dyuzhev N.A., Kireev V.Y. Possibilities and limitations of CMOS Technology for the production of various Microelectronic systems and devices // Nanobiotechnology Reports. – 2022. – Т. 17. – №. 1. – С. 24-38. https://doi.org/10.1134/S2635167622010037
30. White M. Scaled CMOS Technology Reliability Users Guide. Pasadena, California: Jet Propulsion Laboratory, California Institute of Technology, 2010. JPL Publication 09-33. https://doi.org/10.13140/RG.2.1.1441.5525
31. Wang M. A review of reliability in gate-all-around nanosheet devices // Micromachines. – 2024. – Т. 15. – №. 2. – С. 269. https://doi.org/10.3390/mi15020269
32. Liu T.W. et al. Reliability challenges of gate dielectric materials in transistors // Information & Functional Materials. – 2025. – Т. 2. – №. 1. – С. 62-92. https://doi.org/10.1002/ifm2.31
33. Ghfiri C. Development and validation of a predictive model to ensure the long-term electromagnetic compatibility of embedded electronic systems: dis. – INSA de Toulouse, 2017. URL: https://theses.hal.science/tel-02062116/
34. Kumari N.A., Sreenivasulu V.B., Prithvi P. Impact of scaling on nanosheet FET and CMOS circuit applications // ECS Journal of Solid State Science and Technology. – 2023. – Т. 12. – №. 3. – С. 033001. https://doi.org/10.1149/2162-8777/acbcf2
35. National Transportation Safety Board. Aircraft Accident Report. URL:https://data.ntsb.gov/carol-repgen/api/Aviation/ReportMain/GenerateNewestReport/10750/pdf
36. Morant A. et al. Railway EMI impact on train operation and environment // International Symposium on Electromagnetic Compatibility-EMC EUROPE. – IEEE, 2012. – С. 1-7. https://doi.org/10.1109/EMCEurope.2012.6396847
37. North American Electric Reliability Corporation. 1989 Quebec Disturbance Report. URL: https://www.nerc.com/globalassets/programs/rapa/gmd/reference-documents/nerc_1989-quebec-disturbance_report.pdf
38. Bolduc L. GIC observations and studies in the Hydro-Québec power system // Journal of atmospheric and solar-terrestrial physics. – 2002. – Т. 64. – №. 16. – С. 1793-1802. https://doi.org/10.1016/S1364-6826(02)00128-1
39. Singapore Exchange. News Release BCOI Report. URL: https://links.sgx.com/FileOpen/20150624_News_Release_BCOI_report.ashx?App=ArchiveAnnouncement&FileID=357241&AnncID=5NDW88WZ0QVNLLAT
40. Pulkkinen A. et al. Geomagnetic storm of 29–31 October 2003: Geomagnetically induced currents and their relation to problems in the Swedish high‐voltage power transmission system // Space weather. – 2005. – Т. 3. – №. 8. https://doi.org/10.1029/2004SW000123
41. North American Electric Reliability Corporation. Transient-Induced Misoperation Approach II. URL: https://www.nerc.com/globalassets/programs/event-analysis/lessons-learned/ll20210204_transient_induced_misoperation_approach_ii.pdf
42. Lodwig S.G., Schuetz C.C. Coupling to control cables in HV substations // 2001 IEEE EMC International Symposium. Symposium Record. International Symposium on Electromagnetic Compatibility (Cat. No. 01CH37161). – IEEE, 2001. – Т. 1. – С. 249-253. https://doi.org/10.1109/ISEMC.2001.950621
43. de Medeiros L.H. A. et al. High frequency transients and electromagnetic interference within 69 kV substations // Electric power systems research. – 2011. – Т. 81. – №. 7. – С. 1534-1540. https://doi.org/10.1016/j.epsr.2011.03.009
44. Keskar P.Y. Analysis of lightning-related damages to instrumentation and control systems for water and wastewater plants // ISA Transactions. – 1996. – Т. 35. – №. 1. – С. 9-15. https://doi.org/10.1016/0019-0578(96)00002-X
45. Prekodravac Filipovic J. et al. Electromagnetic Interference in the Modern Era: Concerns, Trends, and Nanomaterial-Based Solutions // Nanomaterials. – 2025. – Т. 15. – №. 20. – С. 1558. https://doi.org/10.3390/nano15201558
46. Pradip Pawar. Global EMI Shielding Market Size, Industry Share, Growth Trends & Forecast 2026-2034. URL: https://www.verifiedmarketreports.com/product/emi-shielding-market/
47. EMI Shielding Market (Material: Conductive Polymers, Conductive Coatings and Paints, EMC/EMI Filters, Metal Shielding Product, and Others) - Global Industry Analysis, Size, Share, Growth, Trends, and Forecast, 2022-2031. URL: https://www.transparencymarketresearch.com/emi-shielding-market.html
48. Williams T. EMC for product designers. – Newnes, 2016. https://doi.org/10.1016/B978-0-7506-8170-4.X5000-2
49. IEC/TR 62433-2-1-2010 EMC IC modelling – Part 2-1: Theory of black box modelling for conducted emission. URL: https://cdn.standards.iteh.ai/samples/17948/6ad242729241471db3f5b96db2e01f16/IEC-TR-62433-2-1-2010.pdf
50. SAE/EIA-STD-656-B. High Speed TDM Data Bus: HSTDB: стандарт / Society of Automotive Engineers, Electronic Industries Alliance. URL: https://ibis.org/ver4.2/ver4_2.pdf
51. IEC TS 62404:2007. Logic digital integrated circuits – Specification for I/O interface model for integrated circuit (IMIC version 1.3). Geneva: International Electrotechnical Commission, 2007. URL: https://cdn.standards.iteh.ai/samples/13353/a429219327ca44bd9459c743e7cb3f62/IEC-TS-62404-2007.pdf
52. IEC 62433-2:2017. EMC IC modelling – Part 2: Models of integrated circuits for EMI behavioural simulation – Conducted emissions modelling (ICEM-CE)]: международный стандарт. URL: https://cdn.standards.iteh.ai/samples/22575/7eb90dcf3388418c9cb890c1702de052/IEC-62433-2-2017.pdf
53. Ghfiri C. et al. A new methodology to build the internal activity block of ICEM-CE for complex integrated circuits // IEEE transactions on electromagnetic compatibility. – 2017. – Т. 60. – №. 5. – С. 1500-1509. https://doi.org/10.1109/TEMC.2017.2767084
54. IEC 62433-3:2017. EMC IC modelling–Part 3: Models of integrated circuits for EMI behavioural simulation – Radiated emissions modelling (ICEM-RE): международный стандарт. URL: https://cdn.standards.iteh.ai/samples/21777/deb4e85c97b34121a651e971f036e870/IEC-62433-3-2017.pdf
55. IEC/PAS 62433-4:2007. EMC IC modelling – Part 4: Models of integrated circuits for EMI behavioural simulation – Conducted immunity modelling (ICIM-CI): общедоступная спецификация. URL:https://cdn.standards.iteh.ai/samples/21778/77925e80d2664cbe80a74c9428a0ac0b/IEC-62433-4-2016.pdf
56. IEC/TS 62433-6:2013. EMC IC modelling – Part 6: Models of integrated circuits for EMI behavioural simulation – Conducted immunity modelling (ICIM-CI) – Pulse immunity: техническая спецификация. URL:https://cdn.standards.iteh.ai/samples/100202/3a353032f5d54136926c4e4b3acfb719/IEC-62433-6-2020.pdf
57. Ramdani M. et al. The electromagnetic compatibility of integrated circuits–Past, present, and future // IEEE Transactions on Electromagnetic Compatibility. – 2009. – Т. 51. – №. 1. – С. 78-100. https://doi.org/10.1109/TEMC.2008.2008907
58. Chen X., Xie S., Wei M. Conduction immunity modeling approach for nonlinear IB module based on multi-harmonic distortion theory in analog-digital integrated circuit (MADIC) // IEICE Electronics Express. – 2024. – Т. 21. – №. 19. – С. 20240324-20240324. https://doi.org/10.1587/elex.21.20240324
59. Machado M.B. et al. Analysis and design of a fully-integrated Colpitts oscillator operating at ultra-low-voltages // Analog Integrated Circuits and Signal Processing. – 2015. – Т. 85. – №. 1. – С. 27-36. https://doi.org/10.1007/s10470-015-0581-4
60. Li L. et al. Low Phase Noise, Dual-Frequency Pierce MEMS Oscillators with Direct Print Additively Manufactured Amplifier Circuits // Micromachines. – 2025. – Т. 16. – №. 7. – С. 755. https://doi.org/10.3390/mi16070755
61. Naing T.L. et al. Low-power MEMS-based pierce oscillator using a 61-MHz capacitive-gap disk resonator // IEEE transactions on ultrasonics, ferroelectrics, and frequency control. – 2020. – Т. 67. – №. 7. – С. 1377-1391. https://doi.org/10.1109/TUFFC.2020.2969530
62. Wu Z.Z. et al. A low phase-noise Pierce oscillator using a piezoelectric-on-silica micromechanical resonator // 2013 Transducers & Eurosensors XXVII: The 17th International Conference on Solid-State Sensors, Actuators and Microsystems (TRANSDUCERS & EUROSENSORS XXVII). – IEEE, 2013. – С. 490-493. https://doi.org/10.1109/Transducers.2013.6626810
63. Zhang Z. et al. On the DC-settling process of the Pierce crystal oscillator in start-up // IEEE Transactions on Circuits and Systems II: Express Briefs. – 2022. – Т. 70. – №. 1. – С. 26-30. https://doi.org/10.1109/TCSII.2022.3209025
64. Cirjulina D. et al. Nonlinear Dynamics and Hybrid Synchronization of DC Biased Colpitts Chaotic Oscillators // Electronics. – 2025. – Т. 14. – №. 20. – С. 4005. https://doi.org/10.3390/electronics14204005
65. Cirjulina D. et al. Experimental study on Colpitts chaotic oscillator-based communication system application for the internet of things // Applied Sciences. – 2024. – Т. 14. – №. 3. – С. 1180. https://doi.org/10.3390/app14031180
66. Maggio G.M., De Feo O., Kennedy M.P. Nonlinear analysis of the Colpitts oscillator and applications to design // IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications. – 2002. – Т. 46. – №. 9. – С. 1118-1130. https://doi.org/10.1109/81.788813
67. Karimi-Bidhendi A., Pu H., Heydari P. Study and design of a fast start-up crystal oscillator using precise dithered injection and active inductance // IEEE Journal of Solid-State Circuits. – 2019. – Т. 54. – №. 9. – С. 2543-2554. https://doi.org/10.1109/JSSC.2019.2920084
68. Pankratz E., Sánchez‐Sinencio E. Survey of integrated‐circuit‐oscillator phase‐noise analysis // International Journal of Circuit Theory and Applications. – 2014. – Т. 42. – №. 9. – С. 871-938. https://doi.org/10.1002/cta.1890
69. Maffezzoni P., D'Amore D. Time‐domain analysis of phase‐noise and jitter in oscillators due to white and colored noise sources // International Journal of Circuit Theory and Applications. – 2012. – Т. 40. – №. 10. – С. 999-1018. https://doi.org/10.1002/cta.768
70. Demir A., Mehrotra A., Roychowdhury J. Phase noise in oscillators: A unifying theory and numerical methods for characterisation // Proceedings of the 35th annual Design Automation Conference. – 1998. – С. 26-31. https://doi.org/10.1145/277044.277050
71. Глазунов П.С., Салецкий А.М., Вдовин В.А. Исследование воздействия импульсных электромагнитных помех на устойчивость работы кольцевых генераторов // Радиотехника и электроника. – 2022. –Т. 67, № 8. – С. 816–824. https://doi.org/10.31857/S0033849422080058
72. Glazunov P. et al. Modeling the Influence of Electromagnetic Interference on Failures in the Operation of Clock Generators // 2020 7th All-Russian Microwave Conference (RMC). – IEEE, 2020. – С. 273-275. https://doi.org/10.1109/RMC50626.2020.9312353
Для цитирования:
Глазунов П.С., Гринько А.И., Болотников Л.С., Вдовин В.А., Черепенин В.А. Математическая модель осциллятора Колпитца для анализа электромагнитной восприимчивости и фазовой устойчивости автогенераторных схем // Журнал радиоэлектроники. – 2026. – №. 6. https://doi.org/10.30898/1684-1719.2026.6.6