Abstract
Phase noise analysis of an oscillator is implemented with its periodic time-varying small signal state equations by perturbing the autonomous large signal state equations of the oscillator. In this paper, the time domain steady solutions of oscillators are perturbed with traditional regular method; the periodic time-varying Jocobian modulus matrices are decomposed with Sylvester theorem, and on the resulting space spanned by periodic vectors, the conditions under which the oscillator holds periodic steady states with any perturbations are analyzed. In this paper, stochastic calculus is applied to disclose the generation process of phase noise and calculate the phase jitter of the oscillator by injecting a pseudo sinusoidal signal in frequency domain, representing the white noise, and a δ correlation signal in time domain into the oscillator. Applying the principle of frequency modulation, we learned how the power-law and the Lorentzian spectrums are formed. Their relations and the Lorentzian spectrums of harmonics are also worked out. Based on the periodic Jacobian modulus matrix, the simple algorithms for Floquet exponents and phase noise are constructed, as well as a simple case is demonstrated. The analysis difficulties and the future directions for the phase noise of oscillators are also pointed out at the end.
Similar content being viewed by others
References
Edson W A. Noise in oscillators. Proc IRE, 1960, (8): 1454–1466
Korukawa K. Some basic characteristics of broadband negative resistance oscillator circuits. BELL System Tech J, 1969, (7): 1937–1955
Lax M. Classical noise V noise in self-sustained oscillators. Phys Review, 1967, 160(2): 290–307
Leeson D B. A simple model of feedback oscillator noise spectrum. Proc IEEE, 1966, 54(2): 329–330
Rizzoli V, Mastri F, Masotti D. General noise analysis of nonlinear microwave circuits by the piecewise harmonic-balance techniques. IEEE Trans Microwave Theory Tech, 1994, 42(5): 807–819
Hajimiri A, Lee T H. A general theory of phase noise in electrical oscillators. IEEE J Solid-State Circ, 1998, 33(2): 179–194
Lee T H, Hajimiri A. Oscillator phase noise: A tutorial. IEEE J Solid-State Circ, 2002, 35(3): 326–336
Demir A, Liu E W Y, Sangiovanni-Vincentelli A L. Time-domain non Monte-Carlo noise simulation for nonlinear dynamic circuits with arbitrary excitations. IEEE Trans Comput-Aided Design, 1996, 15(5): 493–505
Okumura M, Tanimoto H. A time-domain method for numerical noise analysis of oscillators. In: Proceedings of the Asia South Pacific Design Automation Conference, 2, 1997. 477–482
Smedth B D, Gielen G. Accurate simulation of phase noise in oscillators. In: Proceedings of the 23rd European Solid-State Circuits Conference, 1997. 208–211
Poore R. Accurate simulation of mixer noise and oscillators phase noise in large RFICs. In: Proceedings of the Asia Pacific Microwave Conference, 1997, 357–360
Rohde U L, Change C R, Gerber J. Design and optimization of low-noise oscillators using nonlinear CAD tools. In: IEEE International Frequency Control Symposium, 1994. 548–554
Kaertner F X. Determination of the correlation spectrum of oscillators with low noise. IEEE Trans Microwave Theory Tech, 1989, 37(1): 90–101
Kartner F X. Analysis of white and f a noise in oscillators. Int J Circuit Theory Appl, 1990, 18(5): 485–519
Demir A, Mehrotra A, Roychowdhury J. Phase noise in oscillators: A unifying theory and numerical methods for characterization. IEEE Trans Circuits Systems-I: Fundam Theory Appl, 2000, 47(5): 655–674
Anzill W, Russer P. A general method to simulate noise in oscillators based on frequency domain techniques. IEEE Trans Microwave Theory Tech, 1993, 41(12): 2256–2263
Anfelo H D. Linear Time-Varing Systems: Analysis and Synthesis. Boston: Allyn and Bacon, 1970
Rice S O. Mathematical analysis of random noise. Bell System Tech J, 1944, 23: 282–332; 1945, 24: 46–156
Demir A, Sangiovanni-Vincentelli A L. Analysis and Simulation of Noise in Nonlinear Electronic Circuits and Systems. Boston: Kluwer Academic Publishers, 1998
Fan J, Yang H, Wang H. Full time-varying phase noise analysis for MOS oscillators based on Floquet and Sylvester theorems. Analog Integ Circ Signal Proc, 2005, 45: 247–261
HSPICE User’s Manual. Cambell: Meta-Software Inc., 1996
IC5.0 Online Help Document: Spectre RF. Cadence Inc., 2002
Freidlin M I, Wentzell A D. Random Perturbations of Dynamical Systems. Berlin: Springer-Verlag, 1984
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported in part by the National Fundamental Research Project (Grant Nos. G1999032903 and 90307016), the National Natural Science Foundation of China (Grant No. 60025101), and the “863” Program (Grant No. 2003AA1Z1390)
Rights and permissions
About this article
Cite this article
Fan, J., Yang, H., Wang, H. et al. Phase noise analysis of oscillators with Sylvester representation for periodic time-varying modulus matrix by regular perturbations. SCI CHINA SER F 50, 587–599 (2007). https://doi.org/10.1007/s11432-007-0050-5
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11432-007-0050-5