Sensors Based on Oscillators Loaded With Coupled LC Resonators Operating Near a Cusp Point

Oscillators loaded with two coupled LC resonators can exhibit an exceptional point of degeneracy (EPD), which enables the implementation of high-sensitivity sensors. We obtain two distinct oscillation curves (for positive and negative variations of the sensing parameter), whose amplitude tends to zero at the EPD. The goal of this work is to achieve similarly high sensitivity at a suitably large oscillation amplitude, which both reduces phase-noise spectral density and improves reliability. This is accomplished by operating the sensor near a cusp point, or a limit point of hysteresis. We derive mathematical conditions to directly determine the cusp point and the curve that defines the envelope of these points when varying the oscillation amplitude through appropriate changes in the circuit parameters. We demonstrate that the cusp-point locus tends to the EPD, identified by other researchers, as the oscillation amplitude approaches zero. First, we derive analytical conditions for hysteresis, cusp bifurcation, and the EPD. Then, we introduce a graphical procedure to synthesize a cusp point at any desired oscillation amplitude in practical oscillators. This is achieved using harmonic balance (HB) with as many harmonic terms as necessary. The sensor is implemented with two cross-coupled transistors and capacitors formed by wide microstrip stubs. Our experimental results show strong agreement with the theoretical analysis.