Phase Locked Loop Design

🔹 1. Exact vs. Approximate Models: Similar Math, Different Scope Both models reduce oscillator phase dynamics to a 1D equation driven by an input perturbation: Exact (Demir et al.) : Includes oscillator phase in its own evolution → nonlinear, more rigorous Approximate (Hajimiri & Lee) : Simpler integral form using ISF; easier to apply, but omits feedback For small perturbations and stationary noise , they yield identical phase noise predictions 🔹 2. Stationary Noise: Both Models Succeed Equally When noise input is stationary (e.g., thermal or 1/f noise) , the slow-phase behavior can be captured using averaging Averaging filters out high-frequency effects and isolates the core phase dynamics Both models reduce to the same stochastic differential equation, leading to the same phase noise growth (∝ √t) 🔹 3. Injection Locking: Only the Exact Model Captures It When driven by a non-stationary input (e.g., sine wave), only the exact model predicts phase and ...

🔹 1. Time-Variant Model Enables Accurate Noise Prediction Traditional LTI models fail for ring oscillators; phase sensitivity depends on when noise is injected. A linear time-variant (LTV) model characterizes phase noise using the Impulse Sensitivity Function (ISF) , a periodic function that quantifies how susceptible each point in the waveform is to noise. The phase noise is directly related to the RMS value of the ISF (Γ), and low-frequency 1/f noise upconversion is tied to its DC value (Γ). 🔹 2. Γ ∝ 1/N^1.5 Gives a Predictive Handle A closed-form expression for Γ shows it scales as 3⁄√N³ , where N is the number of stages. This ISF behavior is independent of frequency, amplitude, and supply voltage , making it broadly useful. While increasing stages lowers Γ, phase noise benefits plateau due to more noise sources and reduced swing. 🔹 3. Lower Bound on Phase Noise Identified Derived a theoretical minimum phase noise based on MOSFET noise physics, power, an...

To design low-phase-noise oscillators, maximize signal amplitude and resonator Q , minimize sensitivity to noise injection timing , and exploit waveform symmetry . Classical models fall short— linear time-varying (LTV) theory offers actionable guidance. 🔹 1. LTV Model > LTI Model for Predictive Accuracy Traditional linear time-invariant (LTI) models like Leeson's equation offer qualitative insights but fail to predict measured phase noise. Oscillators are inherently linear but time-varying (LTV) systems—impulse response varies with time due to periodicity. The Impulse Sensitivity Function (ISF) quantifies how susceptible the oscillator phase is to noise at each point in time. 🔹 2. Symmetry Suppresses 1/f Noise Upconversion Close-in phase noise is worsened by upconversion of 1/f noise; this can be greatly reduced by minimizing the ISF’s DC value . This requires symmetry - ensure rise/fall times are matched. Oscillators like the Colpitts and symmetrical negativ...

From "A General Theory of Phase Noise In Electrical Oscillators" Asymmetries in rise/fall times increase Γ and worsen 1/f³ noise. Experiments show symmetric nodes suppress phase noise by over 10 dB compared to asymmetric ones. Differential signaling alone is insufficient— per-node symmetry is what matters. Phase noise is minimized by maximizing charge swing across the oscillation node and engineering waveform shape to flatten Γ.

https://www.youtube.com/watch?v=QM-hVV8GUgA VCO : 840 mW Charge-pump current : 13 mA