Free running VCO based on an unstable transistor circuit system stability optimization under delayed electromagnetic interferences and parasitic effects and engineering applications

  • Ofer Aluf Netanya, Israel
Keywords: Free running VCO, PLL, WBFM, Delay Differential Equations (DDEs), Bifurcation, Stability, Integrated circuit, CMOS, Bi-CMOS, Clock generation, ASIC, FPGA


IN this article, Very Crucial subject discussed in free running VCO based on an unstable transistor circuit system stability optimization under delayed electromagnetic interferences and parasitic effects. Additionally we discuss Free running VCO integrated circuit applications (PLLs, DLL, clock generation, etc.). There are many techniques to generate a Wideband Frequency Modulation (WBFM) signal: analog based, digitally based and hybrid based techniques. The VCO is a very low cost method of generating WBFM signals, such as chirp signals. The VCO has some important properties that are common to all frequency sources. These properties are frequency range, settling time, post-tuning drift, sensitivity and Maximum Sensitivity Ratio (MSR), frequency total accuracy, frequency modulation span, and modulation frequency bandwidth. The VCO frequency of oscillation depends on the resonance frequency set by its equivalent capacitance and inductance. By applying variable bias voltage to a Varactor diode, the capacitance is changed and the oscillation frequency is changed accordingly. The first delay line in our circuit (1) represents the electromagnetic interference in the Varactor diode (D1). We neglect the voltage on the first delay line (V1→ε) and the delay is on the current which flows through Varactor diode. The second and third delay lines (2 and 3) represent the circuit microstrip line's parasitic effects before and after the matching circuit. We neglect the voltages on the second and third delay lines (Vk→ε ; k=2, 3) and the delays are only on the current which flow through the microstrip lines. The free running VCO circuit can represent as delayed differential equations which, depending on variable parameters and delays. There is a practical guideline which combines graphical information with analytical work to effectively study the local stability of models involving delay dependent parameters. The stability of a given steady state is determined by the graphs of some function of τ1, τ2, τ3.


(1) Muhammad Taher Abuelma’Atti, Parametric amplification/mixing using Varactor diode, Active and Passive Elec. Comp, Vol.19, pp. 177–187, 1966.

(2) O. Aluf, Microwave RF antennas and circuits, Nonlinearity applications in Engineering, Springer (2016).

(3) Sean V. Hum, Michael Okoniewski, and Robert J. Davies, Realizing an electronically tunable reflectarray using Varactor Diode-Tuned elements, IEEE Microwave and Wireless components letters, Vol. 15, No. 6, 2005.

(4) K. Buisman, L. C. N. de Vreede, L. E. Larson, …, “Distortion-Free” Varactor diode topologies for RF additivity, Microwave Symposium Digest, 2005 IEEE MTT-S International.

(5) E. Beretta, Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal. Vol. 33, No. 5, pp. 1144-1165, 2002.

(6) Kuang, Y., 1993. Delay Differential Equations with Applications in Population Dynamics. Academic Press, Boston.

(7) Jiaoxun Kuang & Yuhao cong., 2007. Stability of Numerical methods for Delay Differential Equations. Elsevier Science.

(8) Balakumar Balachandran & Tamás Kalmár-Nagy & David E. Gilsinn., Delay Differential Equations: Recent Advances and New Directions (Hardcover). Springer; 1 edition (March 5, 2009).

(9) Jack K. Hale. Dynamics and Bifurcations. Texts in Applied Mathematics, Vol. 3, Springer-Verlag; 1996.

(10) Steven H. Strogatz, Nonlinear Dynamics and Chaos, CRC Press; 1 edition (December 29, 2000)

(11) John Guckenheimer, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences Vol 42, Springer; 1st ed. 1983. Corr. 6th printing 2002 edition (February 8, 2002)

(12) Stephen Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Text in Applied Mathematics (Hardcover), Springer; 2nd edition (October 1, 2003)

(13) Guillermo Gonzalez, Microwave transistor amplifiers: Analysis and design (2nd Edition), Pearson, 1996.

(14) Ali Behagi and Maou Ghanevati, Fundamentals of RF and Microwave circuit design: Practical analysis and design tools, RFPTA, 2017.

(15) David M. Pozar, Mocrowave engineering, Wiley; 4 th edition 1901.

(16) Arrl Inc, Antenna physics; An introduction, Amer radio relay League, 2016.

(17) ARRL Inc, The ARRL Handbook for radio communications 2018 Hardcover, Amer radio relay League; 95th, 2017.

(18) ARRL Inc, ARRL’s wire antenna classics, ARRL the national association for Amateur radio, 2006.

(19) L. Chiu, T. Y. Yum, Q. Xue, and C. H. Chan, A wideband compact parallel-strip 1800 wilkinson power divider for push-pull circuitries, IEEE microwave and wireless components letters, Vol. 16, No. 1, 2006.

(20) Jianpeng Wang, Jia Ni, Yong-Xin Guo, and Dagang Fang, Miniaturized microstrip Wilkinson power divider with Harmonic suppression, IEEE microwave and wireless components letters, Vol. 19, No. 7, 2009.