Low Limit of Detection Gas Density Sensing With a Digitally PI-Controlled Microcantilever

Abstract

This work describes a new platform for sensing mass or rheological properties of gases with unprecedented responsivity and limits of detection. The system consists of a microcantilever working in a phase-locked loop (PLL) with an imposed phase between its excitation and deflection signals. The optically detected cantilever deflection is demodulated against digitally synthetized reference signals, and the quadrature component ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${Q}$ </tex-math></inline-formula> -signal) is used as the error parameter in a PI controller, which continuously tracks the oscillation frequency. The direct digital synthesis of the reference and actuation signals allows low-noise and fast-transient responses of the sensor for real-time detection of minute changes of any environmental parameter. A general analytical model is derived, used to understand the dynamical response of the platform, and validated against experiments using different gases and pressures. In particular, the responsivity of the sensor to density variations of the fluids and the stability of its frequency response are studied and measured. It is shown that the responsivity and the achieved limits of detection depend on the chosen phase imposed in the loop. A limit of detection for density variations of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$3.5\times 10^{-{4}}$ </tex-math></inline-formula> kg/ <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text{m}^{{3}}$ </tex-math></inline-formula> in air is measured, in agreement with the theoretical predictions, and one to two orders of magnitude lower than any reported value achieved with the same type of physical uncoated resonant sensors.