Stability and Noise Analysis of Two Wirelessly Locked Oscillators

Wirelessly locked oscillators have been proposed for compact, high-sensitivity, and low-consumption tag-reader communications. However, their analysis is, in general, too simplified to reliably predict the behavior of transistor-based oscillators. In this work, we present a new analysis method based on numerical nonlinear models of the oscillator circuits. These models are extracted from harmonic balance (HB) with the aid of an auxiliary generator (AG). They are introduced in a formulation of the coupled system, which can rely on an analytical or numerical description of the coupling effects. The number of independent variables is reduced by expressing one of the oscillator voltages in terms of the other by means of interpolation. The complex error function is solved through contour intersections, which enables an exhaustive calculation of all the coexisting solution curves. Distinguishing between stable and unstable solutions is essential since only stable solutions will be physically observed. We present a detailed stability analysis of this coupled system based on the perturbation of this system about each periodic solution. This involves the linearization, about each solution, of the nonlinear numerical models of the oscillator circuits. Also, the phase noise is analyzed by introducing equivalent noise sources in the perturbed system and by obtaining the carrier modulation. The investigation is extended to the case in which one of the oscillators acts as a self-injection-locked tag for motion sensing. Its signal injection locks the second oscillator, acting as a receiver. The formulation addressing this case makes use of a numerical description of the coupling and reflection effects. The system has been experimentally characterized with very good results, confirming the analysis methods.