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Phase-domain Macromodeling of Oscillators for the analysis of Noise, Interferences and

Phase-domain Macromodeling of Oscillators for the analysis of Noise, Interferences and Synchronization effects. Paolo Maffezzoni. Dipartimento di Elettronica, Informazione e Bioingegneria Politecnico di Milano, Milan, Italy. MIT, Cambridge, MA, 23-27 Sep. 2013. Presentation Outline.

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Phase-domain Macromodeling of Oscillators for the analysis of Noise, Interferences and

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  1. Phase-domain Macromodeling of Oscillators for the analysis of Noise, Interferences and Synchronization effects Paolo Maffezzoni Dipartimento di Elettronica, Informazione e Bioingegneria Politecnico di Milano, Milan, Italy MIT, Cambridge, MA, 23-27 Sep. 2013

  2. Presentation Outline • Mathematical/Theoretical formalization • Computational issues • Pulling effects due to interferences • Phase-noise analysis Phase-domain Macromodeling of Oscillators

  3. Presentation Outline • Mathematical/Theoretical formalization • Computational issues • Pulling effects due to interferences • Phase-noise analysis Phase-domain Macromodeling of Oscillators

  4. Free-Running Oscillator State variables Vector-valued nonlinear function Scalar output response Vector solution Limit cycle

  5. Perturbed Oscillator s(t) small-amplitude perturbation -a Franz Kaertner, “Analysis of white and f noise in oscillators”, International Journal of Circuit Theory and Applications, vol. 18, 1990. a(t) is the time-shift of the perturbed response with respect to free-running one Tangential variation, Phase modulation (PM) Transversal variation Amplitude modulation (AM)

  6. Pulse Perturbation Small-amplitude pulse perturbation at

  7. Floquet theory of linear time-periodic ODEs Linearization around the limit cycle Direct ODE Right eigenvector N Solutions Floquet exponent Adjoint ODE Left eigenvector

  8. Phase and Amplitude Modulations Tangential variation is governed by: Perturbation-Projection Vector (PPV) Transversal variation is governed by

  9. Small-Amplitude Perturbations • Limit cycle is stable: small-amplitude signals give negligible transversal deviations from the orbit • Phase is a neutrally stable variable: weak signals induce large phase deviations that dominate the oscillator dynamics Scalar output response Excess Phase

  10. Pulse Perturbation Response (1) At time

  11. Pulse Perturbed Response (2) At time at

  12. Pulse Perturbed Response (3) At time at

  13. Phase-Sensitivity Response (PSR) (intuitive viewpoint) • Relation between α(t)and s(t) is described by the • periodic scalar function Γ(t) Scalar Differential Equation

  14. Presentation Outline • Mathematical/Theoretical formalization • Computational issues • Pulling effects due to interferences • Phase-noise analysis Phase-domain Macromodeling of Oscillators

  15. How the PPV and PSR can be computed -a • Franz Kaertner, “Analysis of white and f noise in oscillators,” • International Journal of Circuit Theory and Applications, vol. 18, 1990. Eigenvalue/eigenvector expansion of the Monodromy matrix • A. Demir, J. Roychowdhury, “A reliable and efficient procedure for oscillator PPV computation, with phase noise macromodeling applications,” IEEE Trans. CAD, vol. 22, 2003. Exploits the Jacobian matrix of PSS within a simulator

  16. (i) Monodromy Matrix State Transition Matrix: Monodromy matrix: Integration of direct and adjoint ODE : Eigenvalue/eigenvector Expansion:

  17. (ii) With the PSS in a simulator MNA variables Charges and Fluxes Resistive term Perturbed Equations

  18. Periodic Steady State (PSS) Initial guess supplied by Transient/Envelope (very close to PSS final solution) • The (initial) period T is discretized into a grid of M+1 points • Integration (BE) at tk gives the equation (dimension D): where for is replaced by

  19. Periodic Steady State (PSS) • DxM+1 unknowns and • DxM equations, thus we add • an extra constraint • Jacobian of the system

  20. Newton-Raphson Iteration Variables update • At convergence, we find a linearization around the • PSS response

  21. Computing Γ(t) Transient Problem A) IF: Periodic Steady State Problem Controllably Periodically Perturbed Problem: Miklos Farkas, Periodic Motion, Springer-Verlag 1994. B)

  22. Computing Γ(t)

  23. Presentation Outline • Mathematical/Theoretical formalization • Computational issues • Pulling effects due to interferences • Phase-noise analysis Phase-domain Macromodeling of Oscillators

  24. Analysis of Interferences • Signal leakage through the packaging and the substrate in ICs • Weak interferences (-60/-40 dB) may have tremendous effect on the oscillator response • This depends on the injection point and the frequency detuning • Purely numerical simulation is not suitable to explore all the potential injection points

  25. Examples • Injection from the • Power Amplifier B) Mutual Injection between Two Oscillators

  26. Synchronization Effect: Injection Pulling INPUT OUTPUT frequency detuning f frequency shift

  27. Synchronization Effect: Injection Locking Synchronization effect: Injection Locking Quasi-Lock Injection Locking

  28. Studying interference with PPV/PSR • Phase Sensitivity Response (PPV component) is To-Periodic: • For a perturbation with • the Scalar Differential Equation • transforms to:

  29. Averaging Method • The time derivative of a(t)is dominated by the “slowly-varying” term: , , Notation: • Similar to Adler’s equation but generally applicable

  30. Approximate Solution (1) • We make the following assumption: where: are unknown parameters • Substituting in

  31. Approximate Solution (2) • Expanding …

  32. Closed-Form Expressions: Frequency Shift

  33. Closed-Form Expressions: Amplitude Tones • For a Free-running response • The perturbed response becomes

  34. Example: Colpitts Oscillator (1) PPV component Current injection:

  35. Example: Colpitts Oscillator (2) • Variable Detuning • Excess Phase • Numerical integration of the • Scalar-Differential-Equation • The average slope of excess • phase waveform gives the • frequency shift

  36. Frequency shift vs. Detuning • Square marker: • Numerical solutions of • the Scalar Equation • Broken line: • Closed-form estimation

  37. Comparison to Simulations with Spice For detuning Quasi-Locking For detuning Injection Pulling

  38. Example: Relaxation Oscillator • Current injection into nodes E, D • PPV components: Injection in E causes no pulling !

  39. Spice simulations versus Closed-form prediction • Injection in D: Ain=25 mA • Dw = -1.8 rad/s • Injection in E: Ain=25 mA • Dw = -1.8 rad/s

  40. Mutual pulling (1) When decoupled: When coupled:

  41. Mutual pulling (2) Case A: Case B:

  42. Mutual pulling (3) Output Spectra Case A Case B

  43. Presentation Outline • Mathematical/Theoretical formalization • Computational issues • Pulling effects due to interferences • Phase-noise analysis Phase-domain Macromodeling of Oscillators

  44. Phase-Noise Analysis Noise source Stationary zero-mean Gaussian: White/Colored Autocorrelation function Power Spectral Density (PSD) • Asymptotically is a non-stationary Gaussian process variance mean value AlperDemir, “Phase Noise and Timing Jitter in Oscillators With Colored-Noise Sources,” IEEE Trans. on Circuits and Syst. I, vol. 49, no. 12, pp. 1782-1791, Dec. 2002.

  45. Averaged Stochastic Model • Nonlinear Stochastic Equation (2) Averaged Stochastic Equation Flicker noise source White noise source Solutions to (1) and (2) have the same

  46. Phase-Noise Spectrum Time Domain Frequency Domain Power Spectral Density

  47. Noise Macro-model Effect of All Noise Sources Equivalent Noise Sources • Phase- Noise parameters are derived • by fitting DCO Power Spectrum

  48. Application: Frequency Synthesis in Communication Systems • Phase-locked loop (PLL): VCO PD Filter • Evolution from Analog towards Digital PLLs

  49. Bang-Bang PLL (BBPLL) • BPD: single bit quantizer • DLF: Digital Loop Filter • DCO: Digitally-Controlled • Oscillator

  50. Digitally-Controlled Oscillator (DC0) Digital-to-Analog Converter (DAC) Free-running Period Period Gain Constant Analog Section: Ring Oscillator

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