A Novel First-Principles Model of Injection-Locked Oscillator Phase Noise

TL;DR

The paper tackles the need for accurate, topology-agnostic phase-noise modeling in injection-locked oscillators by introducing the ILO-PMM, a time-domain first-principles macro-model derived from coupled-oscillator Floquet theory. This framework delivers a closed-form spectral density L_{ILO}(ω_m), unifying topology, coupling, and noise sources within a single analytic interface and enabling integration with standard EDA tools. A reduced-order K-ILO model shows how the Kurokawa Q-SINUS results emerge as an FP sub-model, clarifying the range of validity and exposing scenarios where non-sinusoidal PSS or strong drive require the full FP treatment. Numerical experiments on a 0.9 GHz circuit validate the ILO-PMM against pnmx and reveal conditions under which traditional K-ILO models fail, underscoring the practical impact for designing low-noise injection-locked oscillators and for informing design rules.

Abstract

The paper documents the development of a novel time-domain model of injection-locked oscillator phase-noise response. The methodology follows a first-principle approach and applies to all circuit topologies, coupling configurations, parameter dependencies etc. The corresponding numerical algorithm is readily integrated into all major commercial simulation software suites. The model advances current state-of-the-art pertaining to analytical modelling of this class of circuits. Using this novel analytical framework, several important new insights are revealed which, in-turn, translate into useful design rules for synthesis of injection-locked oscillator circuits with optimal noise performance.

Figures (5)

• Figure 1: (a) : The limit-cycle $\gamma$ (blue orbit) is embedded in the $k$-dimensional, closed, phase-manifold, $\mathcal{M}$. All orbits on, and off, $\mathcal{M}$ approach this $1$-dimensional set asymptotically with time (red orbits). The tangent-space to $\mathcal{M}$ at a point $x\in \mathcal{M}$, $\mathsf{T}_x\mathcal{M}$, is an affine copy of $\mathbb{R}^k$. (b) : Illustrating the ILO circuit configuration. The free-running primary oscillator (P-OSC) is coupled unilaterally, through some buffer amplifier to the secondary oscillator (S-OSC). Assuming a synchronized PSS is reached, the S-OSC is locked to P-OSC injected signal.
• Figure 2: The figure shows the two oscillator units which are coupled, as shown in \ref{['sec1a:fig1']}.(b), to produce the ILO circuit discussed here. The unilateral buffer amplifier (see \ref{['sec1a:fig1']}.(b)), which connects to two circuits through the $v_{inj}$ ports, has a 4th order polynomial in-out characteristic $v_{out} = \sum_{k=0}^3 g_{ck}(v_{in})^k$. Component values are fixed to values shown unless otherwise stated. (a) : OSC1 (P-OSC) is a simple LC negative-resistance oscillator. Free-running oscillation frequency $f_0 = 900.9 \mathrm{MHz}$, corresponding to a period of $T_0 = 1.11\mathrm{ns}$. VCCS (negative resistor) parameters : $a_1 = -1.0\mathrm{mS}$, $a_3 = 100\mathrm{\mu A/V^3}$ and $a_i = 0$ for $i \neq 1,3$. The white-noise current source, $w(t)$, has the rms current strength $i_{\text{w,rms}} = 1\mathrm{pA/\sqrt{Hz}}$. (b) : OSC2 (S-OSC) is a CMOS cross-coupled LC-tank circuit. Free-running oscillation frequency $f_0 = 892.86 \mathrm{MHz}$, corresponding to a period of $T_0 = 1.12\mathrm{ns}$. The CMOS devices are considered noiseless. The rms current of white-noise source, $n(t)$, is given as $i_{\text{n,rms}} = 70.7\mathrm{pA/\sqrt{Hz}}$. CMOS model (all transistors) : $V_{\text{th0}}=0.5\mathrm{V}$, $\lambda = 0.05\mathrm{V^{-1}}$, $k_p = 120\mu \mathrm{A/V^2}$ which is also the model used in Maffezzoni13. The CMOS channel width-to-length ratios are listed next to the respective components.
• Figure 3: (a) : PSS solution of the ILO circuit shown in \ref{['sec3:fig1']}. (b) : phase-noise spectral densities for the ILO circuit in \ref{['sec3:fig1']}. The figure shows the PNOISE spectrum for the ILO circuit calculated using the novel ILO-PMM model developed herein along with the corresponding output of the pnmx routine, part of the Keysight-ADS© suite. The simulations are run for two parameter sets, PSET1 : $(C_r = 0.3035\text{pF},0.295\text{pF}, g_{c1} = 35 \mu\text{A/V})$ and PSET2 : $(g_{c1} = 40\mu\text{A/V},60\mu\text{A/V})$ with all other component values and circuit parameters fixed as listed in \ref{['sec3:fig1']}.
• Figure 4: (a) : figure shows the components of $\lambda_i(t) = v_i^{\top}(t)B$, $i=1,2$, corresponding to Floquet modes $\mu_1=0,\mu_2 < 0.0$, and representing the contributions due to noise sources $n(t),w(t) : \mathbb{R}\to \mathbb{R}$ (see \ref{['sec3:fig1']}) which dominate the response. (b) : the PNOISE spectrum calculated using ILO-PMM, K-ILO model and the pnmx routine. Due to the DC and even harmonic components of the contribution $\lambda_{2,j_n}(t)$ (see figure (a)) the reduced-order K-ILO model fails to predict the correct spectrum for higher offset frequencies.
• Figure 5: PNOISE spectrum of the modified circuit (see discussion in text). The spectrum is calculated using the ILO-PMM, K-ILO model and pnmx method. The modified circuit generates a solution which conforms to Q-SINUS/Kurokawa methodology and the reduced-order K-ILO equivalent model is hence able to correctly predict the ILO PNOISE response. This should be contrasted with the result reported in \ref{['sec3:fig3']}.(b) above.

Theorems & Definitions (4)

• lemma 3.1
• proof
• lemma 3.2: Kurokawa Model Range-of-Application
• proof