Derivation of Floquet Eigenvectors Displacement for Optimal Design of LC Tank Pulsed Bias Oscillators

ABSTRACT

The paper presents an approximated and compact derivation of the mutual displacement of Floquet eigenvectors in a class of LC tank oscillators with time varying bias. In particular it refers to parallel tank oscillators of which the energy restoring can be modeled through a train of current pulses. Since Floquet eigenvectors are acknowledged to give a correct decomposition of noise perturbations along the stable orbit in oscillator's space state, an analytical and compact model of their displacement can provide useful criteria for designers. The goal is to show, in a simplified case, the achievement of oscillators design oriented by eigenvectors. To this aim, minimization conditions of the effect of stationary and time varying noise as well as the contribution of jitter noise introduced by driving electronics are deduced from analytical expression of eigenvectors displacement.

The paper presents an approximated and compact derivation of the mutual displacement of Floquet eigenvectors in a class of LC tank oscillators with time varying bias. In particular it refers to parallel tank oscillators of which the energy restoring can be modeled through a train of current pulses. Since Floquet eigenvectors are acknowledged to give a correct decomposition of noise perturbations along the stable orbit in oscillator's space state, an analytical and compact model of their displacement can provide useful criteria for designers. The goal is to show, in a simplified case, the achievement of oscillators design oriented by eigenvectors. To this aim, minimization conditions of the effect of stationary and time varying noise as well as the contribution of jitter noise introduced by driving electronics are deduced from analytical expression of eigenvectors displacement.

Cite this paper

nullS. Perticaroli, N. Luli and F. Palma, "Derivation of Floquet Eigenvectors Displacement for Optimal Design of LC Tank Pulsed Bias Oscillators,"*Circuits and Systems*, Vol. 2 No. 4, 2011, pp. 311-319. doi: 10.4236/cs.2011.24043.

nullS. Perticaroli, N. Luli and F. Palma, "Derivation of Floquet Eigenvectors Displacement for Optimal Design of LC Tank Pulsed Bias Oscillators,"

References

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[3] F. O’Doherty and J. P. Gleeson, “Phase Diffusion Coefficient for Oscillators Perturbed by Colored Noise,” IEEE Transactions on Circuits and Sys-tems-II: Express Briefs, Vol. 54, No. 5, May 2007, pp. 435-439.

[4] T. Djurhuus, V. Krozer, J. Vidkjaer and T. K. Johansen, “Oscillator Phase Noise: A Geometrical Approach,” IEEE Transactions on Circuits and Systems-I: Regular Papers, Vol. 56, No. 7, July 2009, pp. 1373-1382. doi:10.1109/TCSI.2008.2006211

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

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

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

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

[9] A. Buonomo, “Nonlinear Analysis of Voltage-Controlled Oscillators: A Systematic Approach,” IEEE Transaction on Circuit and Systems-I: Regular Papers, Vol. 55, No. 6, June 2008, pp. 1659-1670.

[10] A. Carbone, A. Brambilla and F. Palma, “Using Floquet Eigenvectors in the Design of Electronic Oscillators,” 2005 IEEE 7th CAS Symposium on Emerging Technologies: Circuits and Systems for 4G Mobile Wireless Communications, St. Petersburg, 23-24 June 2005, pp. 100-103.

[11] A. Carbone and F. Palma, “Considering Orbital Deviations on the Evaluation of Power Density Spectrum of Oscillators,” IEEE Transac-tions on Circuits and Systems-II: Express Briefs, Vol. 53, No. 6, June 2006, pp. 438-442.

[12] B. Razavi, “A Study of Phase Noise in CMOS Oscillators,” IEEE Journal of Solid-State Circuits, Vol. 31, No. 3, March 1996, pp. 331-343. doi:10.1109/4.494195

[13] R. Aparicio and A. Hajimiri, “A Noise-Shifting Differential Colpitts VCO,” IEEE Journal of Solid-State Circuits, Vol. 37, No. 12, December 2002, pp. 1728-1736. doi:10.1109/JSSC.2002.804354

[14] S. K. Magierowski and S. Zukotynski, “CMOS LC-Oscillator Phase Noise Analysis Using Nonlinear Models,” IEEE Transactions on Circuits and Systems-I: Regular Papers, Vol. 51, No. 4, April 2004, pp. 664-677. doi:10.1109/TCSI.2004.826209

[15] T. H. Lee, “Design of CMOS Radio-Frequency Integrated Circuits,” Cam-bridge Press, Cambridge, 1998, pp. 669-677.

[16] 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. doi:10.1002/cta.383

[17] R. I. Leine, D. H. Van Campen and B. L. Van der Vrande, “Bifurcations in Nonlinear Discontinuous Systems,” Nonlinear Dynamics, Vol. 23, No. 2, October 2000, pp. 105-164. doi:10.1023/A:1008384928636

[18] M. Aleksic, N. Ne-dovic K. W. Current and V. G. Oklobdzija, “Jitter Analy-sis of Nonautonomous MOS Current-Mode Logic Cir-cuits,” IEEE Transaction on Circuit and Systems-I: Regular Papers, Vol. 55, No. 10, November 2008, pp. 3038-3049.

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

[1] 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, July 2001, pp. 896-898. doi:10.1109/81.933331

[2] Affirma? RF Simulator (SpectreRF?) User Guide, Product Version 4.4.6, June 2000, pp. 696-699.

[3] F. O’Doherty and J. P. Gleeson, “Phase Diffusion Coefficient for Oscillators Perturbed by Colored Noise,” IEEE Transactions on Circuits and Sys-tems-II: Express Briefs, Vol. 54, No. 5, May 2007, pp. 435-439.

[4] T. Djurhuus, V. Krozer, J. Vidkjaer and T. K. Johansen, “Oscillator Phase Noise: A Geometrical Approach,” IEEE Transactions on Circuits and Systems-I: Regular Papers, Vol. 56, No. 7, July 2009, pp. 1373-1382. doi:10.1109/TCSI.2008.2006211

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

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

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

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

[9] A. Buonomo, “Nonlinear Analysis of Voltage-Controlled Oscillators: A Systematic Approach,” IEEE Transaction on Circuit and Systems-I: Regular Papers, Vol. 55, No. 6, June 2008, pp. 1659-1670.

[10] A. Carbone, A. Brambilla and F. Palma, “Using Floquet Eigenvectors in the Design of Electronic Oscillators,” 2005 IEEE 7th CAS Symposium on Emerging Technologies: Circuits and Systems for 4G Mobile Wireless Communications, St. Petersburg, 23-24 June 2005, pp. 100-103.

[11] A. Carbone and F. Palma, “Considering Orbital Deviations on the Evaluation of Power Density Spectrum of Oscillators,” IEEE Transac-tions on Circuits and Systems-II: Express Briefs, Vol. 53, No. 6, June 2006, pp. 438-442.

[12] B. Razavi, “A Study of Phase Noise in CMOS Oscillators,” IEEE Journal of Solid-State Circuits, Vol. 31, No. 3, March 1996, pp. 331-343. doi:10.1109/4.494195

[13] R. Aparicio and A. Hajimiri, “A Noise-Shifting Differential Colpitts VCO,” IEEE Journal of Solid-State Circuits, Vol. 37, No. 12, December 2002, pp. 1728-1736. doi:10.1109/JSSC.2002.804354

[14] S. K. Magierowski and S. Zukotynski, “CMOS LC-Oscillator Phase Noise Analysis Using Nonlinear Models,” IEEE Transactions on Circuits and Systems-I: Regular Papers, Vol. 51, No. 4, April 2004, pp. 664-677. doi:10.1109/TCSI.2004.826209

[15] T. H. Lee, “Design of CMOS Radio-Frequency Integrated Circuits,” Cam-bridge Press, Cambridge, 1998, pp. 669-677.

[16] 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. doi:10.1002/cta.383

[17] R. I. Leine, D. H. Van Campen and B. L. Van der Vrande, “Bifurcations in Nonlinear Discontinuous Systems,” Nonlinear Dynamics, Vol. 23, No. 2, October 2000, pp. 105-164. doi:10.1023/A:1008384928636

[18] M. Aleksic, N. Ne-dovic K. W. Current and V. G. Oklobdzija, “Jitter Analy-sis of Nonautonomous MOS Current-Mode Logic Cir-cuits,” IEEE Transaction on Circuit and Systems-I: Regular Papers, Vol. 55, No. 10, November 2008, pp. 3038-3049.

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