Exceptional Points in Gyrator-Based Circuit and Nonlinear High-Sensitivity Oscillator

TL;DR

The paper demonstrates an exceptional point of degeneracy (EPD) in a gyrator-coupled pair of LC resonators, with one resonator composed of negative reactive elements. It develops both lossless and lossy analyses, showing that real-valued EPD frequencies can occur in the lossless case and predicting Puiseux-series–based frequency shifts under small perturbations. The authors validate the theory experimentally by building a practical gyrator-based oscillator that saturates into a nonlinear self-oscillating regime, achieving a narrow spectral linewidth (~10 Hz) and high sensitivity to small capacitor perturbations (e.g., ΔC≈0.625%), with frequency shifts readily detectable and competitive with prior EPD-based sensors. The work highlights the potential of EPD-enabled nonlinear oscillators for ultrasensitive sensing, offering a pathway to measure minute environmental changes in physical, chemical, or biological quantities while leveraging the saturation nonlinearity to stabilize operation.

Abstract

We present a scheme for high-sensitive oscillators based on an exceptional point of degeneracy (EPD) in a circuit made of two LC resonators coupled by a gyrator. The frequency of oscillation is very sensitive to perturbations of a circuit element, like a capacitor. We show conditions that lead to an EPD, assuming one of the two resonators is composed of an inductor and a capacitor of negative values. The EPD occurrence and sensitivity to perturbations in the linear case are demonstrated by showing that the eigenfrequency bifurcation around the EPD is described by the relevant Puiseux (fractional power) series expansion. We also investigate the effect of small losses in the system and show that they lead to instability. We fabricate the circuit, and exploit its instability and nonlinearity, observing experimentally stable self-oscillations under the saturated regime. We measure the circuit's sensitivity to a small capacitor perturbation. A shift in frequency of oscillation after saturation is well detectable with very distinct spectral peaks with 10 Hz linewidth, clean until -70 dB from the peak value. The sensitivity is (i) higher than the one of a comparable simple LC linear resonator, (ii) comparable or better than other published EPD circuits, and (iii) applicable to both negative and positive values of the capacitance perturbation, contrary to what happens in PT-symmetric circuits. The proposed scheme can pave the way for a new generation of high-sensitive sensors to measure slight variations in physical, chemical or biological quantities.

Figures (14)

• Figure 1: Schematic view of the lossy parallel-parallel configuration including losses in each resonator. Inductance and capacitance are negative in the right resonator.
• Figure 2: Variation of the (a) real and (b) imaginary parts of the two eigenfrequencies to a gyration resistance perturbation in the lossless parallel-parallel configuration. The bifurcation in the real part is observed for $R_{\mathrm{g}}>R_{\mathrm{g,e}}$. Voltage $v_{1}$ under the EPD condition in the (c) time domain, and (d) frequency domain. The frequency domain result is calculated from $150\mathrm{\:kHz}$ to $200\:\mathrm{kHz}$ performing an FFT with $10^{6}$ samples in the time window between $0\:\mathrm{ms}$ to $0.2\:\mathrm{ms}$.
• Figure 3: Variation of (a) real and (b) imaginary parts of the angular eigenfrequencies to a resistor perturbation on the left resonator. In these plots, $\gamma_{1}$ is varied whereas we assume $\gamma_{2}=0$. Variation of (c) real and (d) imaginary parts of the angular eigenfrequencies to a resistor perturbation on the right resonator. In these plots, $-\gamma_{2}$ is varied whereas we assume $\gamma_{1}=0$. In these plots, blue curves show stable branches with positive imaginary parts and red curves show unstable branches with negative imaginary parts. In addition, the right half of each plot demonstrates the variation in eigenfrequencies due to varying positive resistance, whereas the left half demonstrates the variation in eigenfrequencies due to varying negative resistance.
• Figure 4: High sensitivity of the (a) real and (b) imaginary parts of the eigenfrequencies to relative capacitance perturbation $\Delta_{\mathrm{C}}=(C_{\mathrm{1}}-C_{\mathrm{1,e}})/C_{\mathrm{1,e}}$. The two perturbed frequencies are real for $\Delta_{\mathrm{C}}<0$. High sensitivity of the (c) real and (d) imaginary parts of the eigenfrequencies to relative inductance perturbation $\Delta_{\mathrm{L}}=\left(L_{\mathrm{1}}-L_{\mathrm{1,e}}\right)/L_{\mathrm{1,e}}$. The two perturbed frequencies are real for $\Delta_{\mathrm{L}}<0$.
• Figure 5: (a) Schematic view of the lossless parallel-parallel configuration. (b) Root locus of zeros of $Y_{\mathrm{total}}$ shows the real and imaginary parts of resonance frequencies of the parallel configuration when varying gyration resistance. The EPD frequency corresponds to a double zero of the admittance $Y_{\mathrm{total}}$.