On the Theory of Quadrature Oscillations Obtained Through Parallel $LC$ VCOs

This paper presents a theory of parallel quadrature voltage-controlled oscillators, which are widely used to generate in-phase and quadrature signals in communication systems. The theory is developed without making any hypothesis on the circuit topology and allows us to thoroughly analyze the synchronized oscillations, as well as the effect of unavoidable parameter mismatches, by means of closed-form expressions for the amplitude and phase of oscillations and their orbital stability. The stability of oscillations is assessed analyzing the relevant eigenvalues, which are obtained in a closed form. The validity of the presented results has been verified through circuit simulations and experiments.

[1]  M. Tiebout,et al.  Low-power low-phase-noise differentially tuned quadrature VCO design in standard CMOS , 2001, IEEE J. Solid State Circuits.

[2]  J. Morrison,et al.  Comparison of the Modified Method of Averaging and the Two Variable Expansion Procedure , 1966 .

[3]  Sanjay Raman,et al.  A Comprehensive Analysis of Quadrature Signal Synthesis in Cross-Coupled RF VCOs , 2007, IEEE Transactions on Circuits and Systems I: Regular Papers.

[4]  Alessandro Lo Schiavo,et al.  The effect of parameter mismatches on the output waveform of an LC‐VCO , 2010, Int. J. Circuit Theory Appl..

[5]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[6]  Jürgen Kurths,et al.  Synchronization: Phase locking and frequency entrainment , 2001 .

[7]  Pietro Andreani,et al.  A Time-Variant Analysis of Fundamental $1/f^{3}$ Phase Noise in CMOS Parallel $LC$ -Tank Quadrature Oscillators , 2009, IEEE Transactions on Circuits and Systems I: Regular Papers.

[8]  Zhipeng Ye,et al.  Accurate modeling and experimental validation of an injection-locked frequency divider , 2005, Proceedings of the 2005 European Conference on Circuit Theory and Design, 2005..

[9]  J. Cole,et al.  Multiple Scale and Singular Perturbation Methods , 1996 .

[10]  A. Bonfanti,et al.  Analysis and design of a 1.8-GHz CMOS LC quadrature VCO , 2002, IEEE J. Solid State Circuits.

[11]  A.A. Abidi,et al.  The Quadrature LC Oscillator: A Complete Portrait Based on Injection Locking , 2007, IEEE Journal of Solid-State Circuits.

[12]  A. Nayfeh.  Introduction To Perturbation Techniques , 1981 .

[13]  Alessandro Lo Schiavo,et al.  Modelling and analysis of differential VCOs , 2004, Int. J. Circuit Theory Appl..

[14]  Alessandro Lo Schiavo,et al.  Finding the Tuning Curve of a CMOS—LC VCO , 2008, IEEE Transactions on Circuits and Systems II: Express Briefs.

[15]  Jian Wang,et al.  Theory of cross-coupled oscillator system for RF quadrature generation , 2002, ICCSC'02. 1st IEEE International Conference on Circuits and Systems for Communications. Proceedings (IEEE Cat. No.02EX605).

[16]  A. Lo Schiavo,et al.  Modelling and analysis of differential VCOs: Research Articles , 2004 .

[17]  S. Wiggins.  Introduction to Applied Nonlinear Dynamical Systems and Chaos , 1989 .

[18]  Sebastian Magierowski,et al.  CMOS LC-oscillator phase-noise analysis using nonlinear models , 2004, IEEE Transactions on Circuits and Systems I: Regular Papers.