Nonlinear enhancement of locking range of mutually injection-locked oscillators for resonant sensing applications

Abstract

Sensor architectures based on coupled resonators are receiving increased interest from the resonant sensing community. Certain output metrics of such sensors have an increased sensitivity to the measurand, compared to conventional resonant sensors with frequency-modulated outputs. In the present paper, we investigate the properties of a differential architecture based on mutually injection-locked oscillators beyond the linear theoretical framework, by driving the resonators higher than the critical Duffing amplitude. Our results show that a trade-off must be made between sensitivity and measurement (locking) range. Experimental results obtained with our MILO architecture and the amplitude ratio output metric are also reported for the first time.

Nonlinear Operation of Resonant Sensors Based On Weakly Coupled Resonators: Theory and Modeling

“…In this section, the dynamic properties of MILOs are illustrated: it is shown that their bandwidth is inversely commensurate with their sensitivity. We also show that the immunity of the amplitude ratio to the A-f effect reported in section III is limited to a narrow range of values of , close to = 0, irrespective of the increase of locking range resulting from nonlinear restoring forces [9], but depending on nonlinear damping forces. Finally, practical and fundamental limits of our analysis are discussed.…”

“…In [17], the FOM of a MILO with nonlinear restoring forces and linear damping is computed for several values of . These simulations show that the linear increase of the FOM of with the oscillation amplitude is only valid within a narrow range of values of , commensurate with the linear locking range of the architecture, in spite of the nonlinear increase of the locking range with resulting from nonlinear restoring forces [9]. These results can be extended to the case when nonlinear damping is present in the system.…”

“…On the other hand, still assuming that > 0, the equilibrium corresponding to 0 = −90° is stable regardless of the oscillation amplitude 1 . In this case, when ≫ , the sensitivity to mismatch (20) of both and decreases as 1/ 2 ; as illustrated in [9], this decrease of sensitivity is accompanied by an increase of the MILO's locking range. Furthermore, as shown by (21), the sensitivity to noise of then decreases as 1/ 3 , whereas that of decreases as 1/ , because of the A-f coupling term appearing in the square root on the right-hand side.…”

“…In "mode-localized" sensors (MOLOs), the ratio of the oscillation amplitudes of (two of) the resonators is the output metric of choice [1][2]5], but other amplitude-based metrics have also been proposed [7][8]. In "mutually injection-locked oscillators" (MILOs) based on two synchronized oscillators, the phase-difference between the two oscillating structures is usually preferred as output metric [3][4], although it was recently shown that, in some MILO architectures, amplitude ratio could also be used [9].…”

“…If only quasistatic parameter fluctuations are considered, as may be deemed sufficient for a rough sensitivity analysis, the time derivatives may be neglected in such a formulation, as in [10,17]. The stability of the steady state may also be trivially determined from the perturbed set of equations (9). This approach may be generalized to an arbitrary set of coupled resonators with little difficulty.…”