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Generalized Parity-Time Symmetry Condition for Enhanced Sensor Telemetry
Pai-Yen Chen, Maryam Sakhdari, Mehdi Hajizadegan, Qingsong Cui, Mark Cheng, Ramy El-Ganainy, Andrea Alù
TL;DR
The paper addresses circuit behavior relevant to applications requiring sharp resonances. It applies Kirchhoff’s laws to PTX-symmetric circuits and reports theoretical–experimental agreement, with potential use across circuit configurations and bandwidth-demanding communication.
Problem
Applications requiring sharp resonances motivate studying PTX-symmetric circuit behavior.
Method
The paper applies Kirchhoff’s laws to PTX-symmetric circuits and considers their extension to different series and parallel circuit configurations.
Results
Experimental results agree well with theory, including measured and simulated reflection responses.
Takeaways & Limitations
PTX symmetry can in principle extend across circuit configurations and may support applications requiring large bandwidth, such as high-speed communication.
Takeaways & Limitations
Reflection gain cannot be infinitely large in experiments.
Abstract
from arXiv · showhide
Wireless sensors based on micro-machined tunable resonators are important in a variety of applications, ranging from medical diagnosis to industrial and environmental monitoring.The sensitivity of these devices is, however, often limited by their low quality (Q) factor.Here, we introduce the concept of isospectral party time reciprocal scaling (PTX) symmetry and show that it can be used to build a new family of radiofrequency wireless microsensors exhibiting ultrasensitive responses and ultrahigh resolution, which are well beyond the limitations of conventional passive sensors. We show theoretically, and demonstrate experimentally using microelectromechanical based wireless pressure sensors, that PTXsymmetric electronic systems share the same eigenfrequencies as their parity time (PT)-symmetric counterparts, but crucially have different circuit profiles and eigenmodes. This simplifies the electronic circuit design and enables further enhancements to the extrinsic Q factor of the sensors.
Methods
The PTX-symmetric circuit is analyzed through Kirchhoff-derived eigenvalue equations, revealing parameter-dependent phases and mode behavior. A wireless MEMS pressure-sensor setup measures pressure by tracking resonance-frequency changes.
- Eigenvalue analysis: The PTX-symmetric circuit’s eigenfrequencies are computed from (H−ω_kI)Ψ_k=0 and depend on the non-Hermiticity parameter γ and coupling strength κ.The analysis considers four eigenfrequencies, k=1,2,3,4.
- Exceptional point and phase transitions: Complex eigenfrequencies produce three regimes, with bifurcation into the complex plane at the exceptional point where PTX symmetry breaks.The exact phase has real eigenfrequencies, whereas the broken phase has complex eigenfrequencies.
- Exceptional point and phase transitions: Below the lower critical region, eigenfrequencies become purely imaginary and modes exponentially grow or decay without oscillation.These overdamped modes are of limited interest for sensors requiring sharp resonances.
S1.1 Design of MEMS-Actuated Capacitive Pressure Sensor
The MEMS pressure sensor uses a deformable parallel-plate varactor whose pressure-dependent displacement changes capacitance. Theoretical and measured displacement and capacitance agree well, supporting contactless parameter extraction.
- Sensor architecture: The sensor combines a pressure-tuned LC resonator with a parallel-plate capacitor and planar micro-coil inductor.A movable upper electrode and stationary lower electrode form a variable air gap.
- Pressure transduction: Applied internal pressure bends the upper electrode upward, enlarging the air gap and reducing the varactor’s total capacitance.The displacement is modeled using Euler–Bernoulli theory, and capacitance is obtained from a surface integral over the metallic disk.
- Validation: Measured maximum electrode displacement agrees well with theory, and displacement increases as applied pressure increases.Scanning white-light interferometry is used to measure the movable electrode displacement.
- Contactless characterization: Reflection responses from an external coil are fitted to an equivalent circuit to retrieve capacitance as a function of applied pressure.The effective sensor resistance is nearly pressure-invariant at approximately 150 Ω.
- Validation: Capacitance measurements also agree well with theory, with small differences attributed to fringing effects and microfabrication imperfections.The capacitance decreases with pressure because the air gap enlarges.
S1.2 Design of Microcoil Inductor
The microcoil design uses analytical expressions for self- and mutual inductance, followed by simulation checks. Reader–sensor coupling coefficients in the designs range from 0 to 0.5.
- Microcoil design: Analytical formulas are used to design reader and sensor planar microcoils from their geometric and material parameters.The formulas include turn number, coil diameter, trace width, spacing, and filling ratio.
- Mutual inductance: The total mutual inductance of multiturn coils is calculated by summing mutual inductances between individual reader and sensor turns.The calculation uses a shape factor and the distance between filamentary loops.
- Coupling characterization: The reader–sensor coupling coefficient is defined as κ=M/√(L_RL_S), using the mutual inductance and the two coil self-inductances.L_R and L_S denote reader and sensor coil inductances, respectively.
- Validation: Self-inductance calculations are verified with full-wave simulation, while mutual-inductance calculations are likewise confirmed by simulation.The coupling coefficient κ in the designs ranges from 0 to 0.5.
S1.3 Design of Negative Resistance Converter (NRC)
The negative resistance converter provides an active reader modeled as a tunable negative-resistance RLC tank. Its measured and simulated responses agree over the relevant frequency range, under low-frequency assumptions that limit parasitic effects.
- NRC architecture: A Colpitts-type negative resistance converter supplies an active reader that can form a series or parallel negative-resistance tank.The NRC is represented as a negative resistance and equivalent capacitance, with a planar coil producing the −RLC tank.
- Circuit tuning: The NRC’s effective resistance is controlled by transistor transconductance and DC bias, while its effective capacitance can be tuned with varactors.The resulting input impedance is decomposed into a negative resistance and equivalent capacitance.
- Characterization: Measured reflection data are used to retrieve the NRC impedance by separating the coil inductance contribution.An isolated −RLC tank can produce reflection gain, with the reflected signal exceeding the incident signal in steady state.
- Practical limitations: Real components limit reflection gain and prevent the theoretically ever-growing eigenmode because of voltage/current limits, large-signal effects, and nonlinearities.These constraints distinguish practical circuits from the idealized model.
- Operating assumptions: Parasitic capacitance and inductance are minimized in the model when operation is moderately below the transistor cutoff and maximum frequencies.The negative resistance and capacitance are nearly constant at low frequencies, and parasitic effects are negligible there.
- Validation: Experimental and simulated reflection spectra agree well for the Colpitts-NRC across the tested DC bias conditions.The equivalent resistive and reactive values are extracted from measured reflection coefficients by numerical optimization.
S2.1 Fabrication of Wireless Pressure Sensors by the MEMS processes
The MEMS intraocular pressure sensor is fabricated through layered parylene and copper processing, ending in a flexible air cavity actuated by internal pressure.
- The fabrication flow begins with RCA-cleaned silicon, an 8 μm parylene layer, and an oxygen-RIE-etched 0.8 mm pressure access hole.
- A 3 μm copper film is deposited and patterned by photolithography to form the sensor’s metallic structures.
- A sacrificial photoresist layer and a second 4 μm parylene layer define the suspended sensor structure.
- The top and bottom metallic structures are connected by a copper interconnect, followed by an 8 μm protective parylene coating.
- Release in KOH and acetone with critical-point drying removes the sacrificial material and produces a flexible air cavity actuated by internal pressure.
S.2.2 Measurement Setup for the MEMS Wireless Pressure Sensors
The measurement setup combines MEMS pressure sensors with active or passive inductive readers to compare conventional and PT-symmetric wireless responses under pressure-driven capacitance changes.
- Wireless measurements use an active reader connected to a vector network analyzer, with controlled coupling strength κ between the reader and MEMS sensor.
- The MEMS pressure sensor uses a variable capacitor connected in series with a planar microcoil, forming an RLC tank whose natural frequency changes with pressure.
- The pressure sensor has a measured equivalent resistance of 150 Ω and a micro-coil inductance of about 300 nH.
- The active reader’s effective capacitance is tuned through varactor DC-offset control to maintain PT symmetry as sensor capacitance varies.
- PT-symmetric systems provide a larger resonance-frequency shift for pressure-driven capacitance variations, while PTX systems retain the same frequency response because the systems share an eigenspectrum.
S3.1 PTX-Symmetric Circuits in the Parallel Configuration
The parallel PTX circuit pairs active and passive RLC tanks and uses reciprocal scaling to preserve PT eigenfrequencies while allowing resonance linewidth and sensor Q-factor tuning.
- PTX symmetry can in principle extend to series, parallel, and complex combinations of circuits.
- The PTX-symmetric circuit consists of parallel −RLC and RLC tanks coupled inductively.
- PTX symmetry combines parity, time reversal, and reciprocal scaling transformations applied to the tank charges.
- The effective Hamiltonian is non-Hermitian, commutes with PTX, and connects to the PT counterpart through an invertible similarity transformation.
- The PTX and PT systems share eigenfrequencies, and the eigenfrequency expression is independent of the scaling coefficient x.
- The scaling coefficient x controls resonance linewidth, enabling PTX symmetry to improve the Q-factor and sensitivity of parallel wireless resonant sensors.
S3.2 Reflectionless Property and Impedance Matching
PTX circuits preserve the relevant resonant frequencies and reflectionless behavior while reciprocal scaling independently tunes resonance bandwidth, providing design flexibility for wireless telemetry.
- The input impedance and reflection coefficient are independent of the scaling parameter y in the corresponding circuit configuration.
- At real eigenfrequencies, perfect impedance matching Z_in = Z_0 produces zero reflection and reflection dips in the measured spectrum.
- For PTX circuits excited through the active tank, reflection remains zero at the eigenfrequencies while resonance linewidth is tuned by varying x.
- The Q-factor increases with increasing x because Q is inversely proportional to resonance bandwidth, consistent with telemetry experiments.
- A passive reader with xη ≤ can exhibit low reflection over a broad bandwidth, relevant to high-speed communication applications.
- When the RF port is connected to the passive tank, PTX and PT circuits share eigenfrequencies and achieve zero reflection at those frequencies.
- In the passive-port configuration, the resonant frequencies remain constant while bandwidth or Q-factor is tailored by varying x or y.