Hajimiri and Limotyrakis - Make your Ring Oscillator as Good as It Can Be


🔹 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, and symmetry.

  • Even well-optimized oscillators cannot beat this limit, which validates the predictive value of the LTV model.

  • Unlike inverter chains, differential ring oscillators show stronger dependence on N due to how tail currents scale. So, for the differential RO, you're better off with fewer stages!

🔹 4. Design Guidelines for Ring Oscillators

  • Symmetry is critical: A balanced rise/fall edge minimizes Γ, reducing 1/f noise upconversion.

  • Use resistive loads (e.g. polysilicon) to shape waveforms for better symmetry.

  • Phase noise improves when energy is delivered consistently at ISF minima — aligning with LC oscillator insights.

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