Floquet Eigenvectors Theory of Pulsed Bias Phase and Quadrature Harmonic Oscillators

ABSTRACT

The paper presents an analytical derivation of Floquet eigenvalues and eigenvectors for a class of harmonic phase and quadrature oscillators. The derivation refers in particular to systems modeled by two parallel RLC resonators with pulsed energy restoring. Pulsed energy restoring is obtained through parallel current generators with an impulsive characteristic triggered by the resonators voltages. In performing calculation the initial hypothesis of the existence of stable oscillation is only made, then it is verified when both oscillation amplitude and eigenvalues/eigenvectors are deduced from symmetry conditions on oscillator space state. A detailed determination of the first eigenvector is obtained. Remaining eigenvectors are hence calculated with realistic approximations. Since Floquet eigenvectors are acknowledged to give the correct decomposition of noise perturbations superimposed to the oscillator space state along its limit cycle, an analytical and compact model of their behavior highlights the unique phase noise properties of this class of oscillators.

The paper presents an analytical derivation of Floquet eigenvalues and eigenvectors for a class of harmonic phase and quadrature oscillators. The derivation refers in particular to systems modeled by two parallel RLC resonators with pulsed energy restoring. Pulsed energy restoring is obtained through parallel current generators with an impulsive characteristic triggered by the resonators voltages. In performing calculation the initial hypothesis of the existence of stable oscillation is only made, then it is verified when both oscillation amplitude and eigenvalues/eigenvectors are deduced from symmetry conditions on oscillator space state. A detailed determination of the first eigenvector is obtained. Remaining eigenvectors are hence calculated with realistic approximations. Since Floquet eigenvectors are acknowledged to give the correct decomposition of noise perturbations superimposed to the oscillator space state along its limit cycle, an analytical and compact model of their behavior highlights the unique phase noise properties of this class of oscillators.

Cite this paper

F. Palma and S. Perticaroli, "Floquet Eigenvectors Theory of Pulsed Bias Phase and Quadrature Harmonic Oscillators,"*Circuits and Systems*, Vol. 3 No. 1, 2012, pp. 72-81. doi: 10.4236/cs.2012.31010.

F. Palma and S. Perticaroli, "Floquet Eigenvectors Theory of Pulsed Bias Phase and Quadrature Harmonic Oscillators,"

References

[1] P. Andreani and X. Wang, “On the Phase-Noise and Phase-Error Performances of Multiphase LC CMOS VCOs,” IEEE Journal of Solid State Circuits, Vol. 39, No. 11, 2004, pp. 1883-1893. doi:10.1109/JSSC.2004.835828

[2] S. Perticaroli and F. Palma, “Phase and Quadrature Pulsed Bias LC-CMOS VCO,” SCIRP Circuit and Systems, Vol. 2, No. 1, 2011, pp. 18-24. doi:10.4236/cs.2011.21004

[3] T. H. Lee and A. Hajimiri, “Oscillator Phase Noise: A Tutorial,” IEEE Journal of Solid-State Circuits, Vol. 35, No. 6, 2000, pp. 326-336. doi:10.1109/4.826814

[4] A. Demir, “Floquet Theory and Non-Linear Perturbation Analysis for Oscillators with Differential-Algebraic Equations,” International Journal of Circuit and Theory Applications, Vol. 28, No. 2, 2000, pp. 163-185. doi:10.1002/(SICI)1097-007X(200003/04)28:2<163::AID-CTA101>3.0.CO;2-K

[5] G. J. Coram, “A Simple 2-D Oscillator to Determine the Correct Decomposition of Perturbations into Amplitude and Phase Noise,” IEEE Transactions on Circuits and Systems—I: Fundamental Theory and Applications, Vol. 48, No. 7, 2001, pp. 896-898. doi:10.1109/81.933331

[6] S. Perticaroli and F. Palma, “Design Criteria Based on Floquet Eigenvectors for the Class of LC-CMOS Pulsed Bias Oscillators,” Microelectronics Journal, 2011, Article in Press. doi:10.1016/j.mejo.2011.07.018

[7] F. X. Kaertner, “Analysis of White and f-α Noise in Oscillators,” International Journal of Circuit and Theory Applications, Vol. 18, No. 5, 1990, pp. 485-519. doi:10.1002/cta.4490180505

[8] A. Demir, A. Mehrotra and J. S. Roychowdhury, “Phase Noise in Oscillators: A Unifying Theory and Numerical Methods for Characterization,” IEEE Transactions on Circuits and Systems—I: Fundamental Theory and Applications, Vol. 47, No. 5, 2000, pp. 655-674. doi:10.1109/81.847872

[9] A. Carbone, A. Brambilla and F. Palma, “Using Floquet Eigenvectors in the Design of Electronic Oscillators,” Emerging Technologies: Circuits and Systems for 4G Mobile Wireless Communications, 2005. ETW’05. 2005 IEEE 7th CAS Symposium, 23-24 June 2005, pp. 100-103. doi:10.1109/EMRTW.2005.195690

[10] A. Carbone and F. Palma, “Considering Orbital Deviations on the Evaluation of Power Density Spectrum of Oscillators,” IEEE Transactions on Circuits and Systems —II: Express Briefs, Vol. 53, No. 6, 2006, pp. 438-442. doi:10.1109/TCSII.2006.873527

[11] A. Carbone and F. Palma, “Discontinuity Correction in Piecewise-Linear Models of Oscillators for Phase Noise Characterization,” International Journal of Circuit Theory and Applications, Vol. 35, No. 1, 2007, pp. 93-104.

[1] P. Andreani and X. Wang, “On the Phase-Noise and Phase-Error Performances of Multiphase LC CMOS VCOs,” IEEE Journal of Solid State Circuits, Vol. 39, No. 11, 2004, pp. 1883-1893. doi:10.1109/JSSC.2004.835828

[2] S. Perticaroli and F. Palma, “Phase and Quadrature Pulsed Bias LC-CMOS VCO,” SCIRP Circuit and Systems, Vol. 2, No. 1, 2011, pp. 18-24. doi:10.4236/cs.2011.21004

[3] T. H. Lee and A. Hajimiri, “Oscillator Phase Noise: A Tutorial,” IEEE Journal of Solid-State Circuits, Vol. 35, No. 6, 2000, pp. 326-336. doi:10.1109/4.826814

[4] A. Demir, “Floquet Theory and Non-Linear Perturbation Analysis for Oscillators with Differential-Algebraic Equations,” International Journal of Circuit and Theory Applications, Vol. 28, No. 2, 2000, pp. 163-185. doi:10.1002/(SICI)1097-007X(200003/04)28:2<163::AID-CTA101>3.0.CO;2-K

[5] G. J. Coram, “A Simple 2-D Oscillator to Determine the Correct Decomposition of Perturbations into Amplitude and Phase Noise,” IEEE Transactions on Circuits and Systems—I: Fundamental Theory and Applications, Vol. 48, No. 7, 2001, pp. 896-898. doi:10.1109/81.933331

[6] S. Perticaroli and F. Palma, “Design Criteria Based on Floquet Eigenvectors for the Class of LC-CMOS Pulsed Bias Oscillators,” Microelectronics Journal, 2011, Article in Press. doi:10.1016/j.mejo.2011.07.018

[7] F. X. Kaertner, “Analysis of White and f-α Noise in Oscillators,” International Journal of Circuit and Theory Applications, Vol. 18, No. 5, 1990, pp. 485-519. doi:10.1002/cta.4490180505

[8] A. Demir, A. Mehrotra and J. S. Roychowdhury, “Phase Noise in Oscillators: A Unifying Theory and Numerical Methods for Characterization,” IEEE Transactions on Circuits and Systems—I: Fundamental Theory and Applications, Vol. 47, No. 5, 2000, pp. 655-674. doi:10.1109/81.847872

[9] A. Carbone, A. Brambilla and F. Palma, “Using Floquet Eigenvectors in the Design of Electronic Oscillators,” Emerging Technologies: Circuits and Systems for 4G Mobile Wireless Communications, 2005. ETW’05. 2005 IEEE 7th CAS Symposium, 23-24 June 2005, pp. 100-103. doi:10.1109/EMRTW.2005.195690

[10] A. Carbone and F. Palma, “Considering Orbital Deviations on the Evaluation of Power Density Spectrum of Oscillators,” IEEE Transactions on Circuits and Systems —II: Express Briefs, Vol. 53, No. 6, 2006, pp. 438-442. doi:10.1109/TCSII.2006.873527

[11] A. Carbone and F. Palma, “Discontinuity Correction in Piecewise-Linear Models of Oscillators for Phase Noise Characterization,” International Journal of Circuit Theory and Applications, Vol. 35, No. 1, 2007, pp. 93-104.