Piecewise Semi-Analytical Formulation for the Analysis of Coupled-Oscillator Systems

TL;DR

The paper addresses accurate prediction of synchronization ranges in arrays of coupled oscillators, where prior semi-analytical formulations linearized about a single operating point fail when tuning spans are large. It introduces a piecewise SAF that models each VCO with a piecewise linear input admittance $Y_i(V_i,\omega_i,\eta_i)$, deriving interval-specific derivatives from circuit-level HB via an auxiliary generator to capture the full VCO characteristic. The coupled system is assembled as a set of first-harmonic KCL equations with interval-dependent coefficients and solved under a constant phase-shift constraint $\Delta\varphi$, with stability assessed through a linearized envelope model yielding eigenvalues of $A(\Delta\varphi)$. The PW SAF demonstrates superior accuracy over the non-piecewise SAF when validated against circuit-level HB simulations for a Van der Pol-type three-oscillator array and a 5 GHz FET-based oscillator array, including injected and free-running regimes, and aligns well with measurements, underscoring its practical utility for beam-steering and synchronization engineering. The work provides a scalable, globally accurate tool for predicting synchronization ranges and injection-locking behavior in realistic oscillator networks.

Abstract

A new simulation technique to obtain the synchronized steady-state solutions existing in coupled oscillator systems is presented. The technique departs from a semi-analytical formulation presented in previous works. It extends the model of the admittance function describing each individual oscillator to a piecewise linear one. This provides a global formulation of the coupled system, considering the whole characteristic of each voltage-controlled oscillator (VCO) in the array. In comparison with the previous local formulation, the new formulation significantly improves the accuracy in the prediction of the system synchronization ranges. The technique has been tested by comparison with computationally demanding circuit-level Harmonic Balance simulations in an array of Van der Pol-type oscillators and then applied to a coupled system of FET based oscillators at 5 GHz, with very good agreement with measurements.

Figures (11)

• Figure 1: Procedure to extract the model of the individual $i-$th VCO. An auxiliary generator (AG) is connected to the output node $P$ to calculate the first-harmonic input admittance function $Y_i(V_i,\omega_i,\eta_i)$ and the derivatives $I_{iG_{\pm 1}}(V_i,\omega_i,\eta_i)$ providing the dependence of the input current $I_1^i$ on the injection source $i_s(t)$.
• Figure 2: (a) Schematic of the individual $i-$th VCO of the array, where $V_{dd}$ and $\eta_i$ are the drain and tuning voltages, respectively. The variables $L$ and $W$ represent the length and the width of each line in $mm$. (b) Photograph of the individual VCO.
• Figure 3: Evolution of the free-running steady state solution versus the tuning voltage $\eta$. Comparison of the results obtained with the non-PW SAF with the PW SAF. The PW SAF agrees with the circuit-level HB simulations at the points $\eta_i$ of the sampling interval. (a) First harmonic amplitude $V_o$. (b) Oscillation frequency $f_o$
• Figure 4: Schematic of the coupled oscillator array. The output port of each oscillator element is loaded with a resistor $R_L=50\ \Omega$ and coupled to the neighbor elements through a linear network.
• Figure 5: Components of the array of Van der Pol-type oscillators. (a) Schematic of each VCO, with $a=-0{.}023\ A/V$, $b=0{.}01\ A/V^3$ and $L=1{.}53\ nH$. The varicap is constituted by a p-n diode with $C_{jo}=0{.}72\ pF$ and $M={0.}5$. The capacitor $C_{out,i}$ models the parasitic components that may appear during the manufacturing process, producing differences between the oscillators in the array. (b) Coupling network $\mathcal{C}_I$, containing an ideal transmission line with $\psi_o=360\ deg$, $Z_o=50\ \Omega$, $f_o=5{.}2\ GHz$, and the resistances $R_s=1250\ \Omega$, $R_p=300\ \Omega$ (c) Load $\mathcal{C}_L$ connected to each of the outermost oscillators preserving the system symmetry.