Analysis of a Beat-Frequency Sensor Operating Near the Locking Boundary

We present an in-depth investigation of a beat-frequency sensor based on an injected oscillator operating near its locking boundaries. Under these conditions, the beat frequency exhibits higher sensitivity to the material under test (MUT) than the free-running oscillation frequency. We will derive a general expression for the beat frequency as influenced by the MUT. This expression depends on an admittance function that can be extracted from harmonic-balance (HB) simulations, so it can be applied to oscillators of arbitrary complexity. In the new formulation, both the free-running solution and the locking bandwidth will vary with the sensing parameter. We will analyze in depth the beat-frequency curve relative to the parameter under test, as well as its dependence on the design elements. We will also present a new method to establish the selected locking boundary at a suitable value for the anticipated MUT variation range. Additionally, we will demonstrate the potential to sense at a multiple of the beat frequency, thereby increasing frequency sensitivity. For the first time to our knowledge, we will analyze the oscillator phase noise when operating near the locking boundaries by means of a perturbation formulation in the frequency domain. The methods will be illustrated through their application to a cubic-nonlinearity oscillator, enabling a deep theoretical insight, and to a realistic transistor-based oscillator with the MUT placed on top of a capacitive transmission line.