Inductor-capacitor passive wireless sensors using nonlinear parity-time symmetric configurations | Nature Communications

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An inductor–capacitor passive wireless sensor is essential to physical, chemical, and biological sensing for scenarios where physical access is difficult. Exceptional points of parity-time symmetric inductor–capacitor systems featuring the linear loss and gain have been utilized for enhancing sensing. However, the exceptional point sensing scheme might bring about fundamental resolution limits and noise enhancement. Here we show, employing a nonlinear saturable gain, the responsivity has a cube-root singularity distinct from a square-root singularity of the linear exceptional point scheme. The saturable gain eliminates the imaginary part of the eigenfrequencies and significantly suppresses the noise. Through an example of inductor–capacitor wireless wearable temperature sensors, we demonstrate the high figure of merit for the nonlinear PT-symmetric configuration. Our results resolve a debate on the effectiveness of the exceptional point sensing scheme for inductor–capacitor sensors and provide a way of enhancing precision for these types of sensors.

An inductor–capacitor (LC) passive wireless sensor which utilizes a spiral inductor connected with a sensing capacitor was developed to measure the intraocular pressure inside the eye as early as in 19671. To wirelessly interrogate the LC sensor, a readout coil is inductively coupled with the sensor. The sensing capacitor changes in response to a small perturbation \(\delta\), causing its resonant frequency to linearly shift, i.e., \(\Delta f=\left|f-{f}_{0}\right|\propto \delta\), where \({f}_{0}\) is the unperturbed resonant frequency. Such wireless LC sensors have been expanded to monitor physical, chemical, or biological parameters2. For applications such as implantable, wearable, or other devices with geometrical constraints, however, small coils reduce the inductive coupling. Hence, enhancing the sensitivity of the LC sensors is desired.

Recent advances in the areas of parity-time (PT)-symmetry3,4 have showed that enhanced sensitivity can be achieved by biasing a sensor system at exceptional points (EPs)5,6,7,8. The EPs are non-Hermitian degeneracies where both eigenfrequencies and their corresponding eigenvectors merge. The systems operated at the EPs exhibit a strong response to a small perturbation. The eigenfrequency detuning follows \(\Delta f=\left|f-{f}_{{{\rm{EP}}}}\right|\propto \root N \of {\delta }\) for linear PT-symmetric systems, where \({f}_{{EP}}\) is the eigenfrequency at the EP and N is the order of the EP degeneracy, respectively. Based on a PT-symmetric LC platform9, the performance of wireless LC sensors has been improved by biasing them at the exact phase, EP, and broken phase, respectively10,11,12,13,14. Although increased sensitivity has been demonstrated in the EP sensing scheme, it has triggered a controversial debate over fundamental resolution limits and noise enhancement15,16,17,18,19. In the EP sensing scheme, asymmetric perturbations usually cause PT-symmetry to break, leading to complex eigenfrequencies15,16,17,18,19,20. The presence of imaginary part of the eigenfrequencies results in a spectrum broadening and sets the fundamental resolution limit. Hence, an approach to exert the perturbation on the coupling capacitor between loss and gain resonators, which is symmetric about both, has been proposed using linear21,22 or nonlinear23,24 PT-symmetric LC configurations. Unfortunately, this approach is not appropriate for the wireless LC sensor in which the perturbation is usually exerted on the loss resonator2. It is shown that the nonlinear PT-symmetric LC system with the gain saturation mechanism can automatically achieve a steady-state oscillation, eliminating the imaginary part of the eigenfrequencies and suppressing the noise25,26,27. Hence, the nonlinear PT-symmetric LC configurations for the LC sensor have been explored23,24,28.

Here, we theoretically propose and experimentally demonstrate enhanced figure of merit (FOM) of LC passive wireless sensors based PT-symmetric configurations of the linear loss and saturable gain. As shown in Fig. 1a, it consists of a pair of parallel RLC resonators in which a passive sensor operates as the loss resonator, while parameters to be sensed, i.e., perturbation \(\delta\), are exerted on the sensitive capacitor at the loss resonator. To wirelessly interrogate the sensor, a readout circuit composed of –RLC tanks operating as the gain resonator is inductively coupled to the sensor. The gain (–R) is realized by a nonlinear amplifier26. The coupling strength is adjusted so that our system is initially located at the exact symmetric phase close to EPs. The proposed sensing scheme shows that the eigenfrequency detuning follows \(\Delta f=\left|f-{f}_{{{\rm{EP}}}}\right|\propto \root 3 \of {\delta }\), which is distinct from \(\Delta f\propto \sqrt{\delta }\) for the linear counterpart, as shown in Fig. 1b. Obviously, in the small perturbation limit, the responsive singularity in our scheme signifies an enhanced sensing.

a Schematic of nonlinear PT-symmetric LC configurations. It consists of a pair of coupled RLC resonators. The gain resonator includes a nonlinear negative resistance. b Real and imaginary parts of eigenfrequency as a function of coupling strength \(k\) relative to loss \(\gamma\) under different perturbations \(\delta\) for the nonlinear and linear system, respectively. The gray plane of \(\gamma=0.968k\) is set as the operating point in our experiments and simulations. The red circles represent the exceptional points. c The gain \({g}^{{\prime} }\) relative to loss \(\gamma\) as a function of perturbation \(\delta\). The eigenfrequency \({\lambda }^{{\prime} }\) is also indicated as a gray scale.

To describe how PT-symmetric LC configurations of the linear loss and saturable gain lead to a cube-root singularity, we develop a model (see Methods) for the PT-symmetric LC resonators shown in Fig. 1a, in which \({C}_{1}={C}_{2}=C\) and \({L}_{1}={L}_{2}=L\). The system dynamics are described by

where \({{{\bf{v}}}}_{{{\bf{n}}}}\) (n = 1, 2) is the node voltage, \(k=M/L\) is the coupling strength between the gain and loss resonator, M is the mutual inductance, \(g={R}_{1}^{-1}\sqrt{L/C}\) is the gain coefficient, \(\gamma={R}_{2}^{-1}\sqrt{L/C}\) is the loss coefficient, \(\tau={\omega }_{0}t\) is the normalized time, and \({\omega }_{0}=1/\sqrt{{LC}}\) represents the natural resonant frequency of an isolated LC resonator, respectively. Taking \({{{\bf{v}}}}_{{{\bf{n}}}}={V}_{n}{e}^{i\omega t}={V}_{n}{e}^{i\lambda \tau }\)(n = 1, 2), where \({V}_{n}\) is the voltage amplitude, ω is the operation angular frequency, and \(\lambda=\omega /{\omega }_{0}\) is the normalized frequency, respectively, one obtains

It yields the characteristic equation from Eq. (2):

For the linear PT-symmetric resonators with the balanced gain and loss, \(g=\gamma\), where the gain is fixed. We obtain the eigenfrequency

The eigenfrequency as a function of the coupling strength relative to loss is given in Fig. 1b. It is obvious that the system operates in the exact phase for \(k > \gamma\), which supports two real eigenfrequencies. Meanwhile, the two modes possess equal amplitude. When \(k=\gamma\), the eigenfrequencies are merged into \({\lambda }_{\pm }={\lambda }_{{{\rm{EP}}}}=1\), i.e., EPs at which the eigenfrequencies and the corresponding eigenvectors coalesce.

For our nonlinear PT-symmetric system, PT symmetry emerges from the nonlinear gain saturation mechanism, which allows the system to achieve PT-symmetry at steady state25 (see Supplementary S2). When we apply a perturbation \(\Delta C\) on the loss capacitance \({C}_{2}\), where \({C}_{2}=C\) and the relative perturbation \(\delta=\Delta C/C\), the system settles into a steady state oscillation within a few cycles, it yields

where \(0 < \delta \ll 1\), \({{V}_{n}}^{{\prime} }\) (n = 1, 2) refers to the voltage amplitude at steady-state oscillation, \({g}^{{\prime} }\) represents the saturated gain after re-establishing steady-state oscillation, respectively. \({g}^{{\prime} }\) is a nonlinear function of \({{V}_{1}}^{{\prime} }\) 25. The characteristic equation becomes

For the steady-state oscillation, we set the imaginary part of Eq. (6) to zero, then the saturation gain is given by

Substituting Eq. (7) into Eq. (6) and letting the real part be zero, it yields

This is the cubic equation about \(\lambda\). We can analytically solve Eq. (8) (see Supplementary S1). There are one real eigenfrequency and a pair of conjugate complex eigenfrequencies. For the steady-state oscillation, the eigenfrequency must be a real. Because our system is initially biased at the exact phase close to the EP, k is slightly greater than \(\gamma\). We expand the eigenfrequency \(\lambda\) at \(\delta=0\) according to Puiseux series. The real eigenfrequency of Eq. (8) is given by

It is clear from Eq. (9) that the PT-symmetric LC configurations of the linear loss and saturable gain has the cube-root singularity. Our subsequent experiments have validated the cube-root singularity.

Meanwhile, from a mathematical perspective, we notice that a pair of conjugate complexes also fulfil the Eq. (6), shown below

which is obtained that setting the imaginary part of Eq. (6) to be zero is a necessary but insufficient condition for the system to have a real eigenfrequency. It has known that the real eigenfrequency \({\lambda }^{{\prime} }\) corresponds exactly to steady-state oscillation, while the imaginary part of a pair of conjugate complex eigenfrequencies \({\lambda }_{{{\rm{com}}}}^{{\prime} }\) correspond to the exponential decay and growth of the node voltage. Figure 1b shows the variation of the eigenfrequencies before and after the perturbation. For the linear PT-symmetric system, it has two eigenfrequencies at the exact phase. While for the nonlinear PT-symmetric system, the system chooses the eigenfrequency that possesses the lowest gain to grow and reach steady state, suppressing the other mode (See Supplementary S1)25. Once a perturbation is applied to the capacitor at the loss resonator, it causes PT-symmetry to break. The presence of imaginary part of the eigenfrequencies results in a spectrum broadening and sets a fundamental resolution limit for the linear PT-symmetric system. On the other hand, our case is different from the scheme for the nonlinear PT-symmetric LC configurations23,24, where the perturbation is applied to the coupling capacitor between loss and gain resonators, it causes the system to go into the exact phase. While in our case, the nonlinear gain keeps pure real eigenfrequency which jumps from the lower branch to the higher branch \({\lambda }^{{\prime} }\). The eigenfrequency jump reflects the transition from the symmetric phase to the broken phase. Figure 1c shows the gain as a function of perturbation for our PT-symmetric LC configurations, indicating the nonlinear relationship between the perturbation and gain. While in the linear PT-symmetric system, the gain keeps unchanged during the perturbation, which results in the square-root of singularity.

We performed the simulation of PT-symmetric LC configurations of the linear loss and saturable gain using ADS (Advanced Design System) (see Supplementary S2). Figure 2a shows the response of node voltage in the time-domain before and after perturbation is applied. Taking the Fourier transform of the response leads to the response amplitude as a function of frequency in the frequency-domain, as indicated in Fig. 2b. It shows that the node voltage manifests as the steady-state oscillation before the perturbation, the unsteady-state oscillation once the perturbation is applied, and the steady-state oscillation after a few circles. During the unsteady-state, it displays the dual frequency superposition (\({\lambda }^{{\prime} }\) and the real part of \({\lambda }_{{{\rm{com}}}}^{{\prime} }\)) oscillation in the form of the exponential decay and growth (time-domain), while the dual frequency can be clearly observed in the frequency domain. The unsteady-state oscillation in the form of the exponential decay and growth exhibits the same frequency because \({\lambda }_{{{\rm{com}}}}^{{\prime} }\) is complex conjugate pairs with identical real parts. As indicated in Fig. 2c, parameters for the resonant frequency and decay/growth factors extracted from Fig. 2a, b are generally consistent with those predicted by Eqs. (9) and (10). Consequently, it is suggested that the system oscillates as a steady state with frequency \({\lambda }_{{{\rm{EP}}}}\) before perturbation, the perturbation induces three distinct modes: the mode 1 manifests as the steady-state oscillation with frequency \({\lambda }^{{\prime} }\), the mode 2 does as the unsteady-state oscillation (the exponential decay) with frequency being the real part of \({\lambda }_{{{\rm{com}}}}^{{\prime} }\), and the mode 3 does as the unsteady-state oscillation (the exponential growth) with frequency being the real part of \({\lambda }_{{{\rm{com}}}}^{{\prime} }\). The gain that provides energy to the system varies with the oscillation amplitude. Inserting \({\lambda }^{{\prime} }\) and the real part of \({\lambda }_{{{\rm{com}}}}^{{\prime} }\) into Eq. (7) leads to the gain for the three distinct modes. It shows that the gain required for mode 1 is lowest (see Fig. S2). According to the principle of minimum energy in physics, a system always needs to adjust itself to minimize its total energy and maintain a stable equilibrium state. As a result, the mode 1 requiring the lowest gain will grow to reach its steady state and saturate out the gain, preventing other two modes from accessing the gain level they need to reach steady-state oscillation. On the basis of our theoretical predications and simulations, we have accounted for the unsteady-state phenomena due to perturbation. In order to obtain a steady-state solution, we initially assume that the eigenfrequency is a real number, and indeed we have obtained the real eigenfrequency as shown in Eq. (9). Interestingly, however, the real eigenfrequency forces the system to produce two complex eigenfrequencies as shown in Eq. (10). Our simulation results show that these two complex eigenfrequencies have an evident physical meaning discussed above, nevertheless, they need further demonstrations as a result of our initial hypothesis of real-valued \(\lambda\). Accordingly, a strict theoretical analysis for the unsteady-state phenomena remains an open area of investigation.

a Time-domain response of node voltage when the perturbation is applied on the loss resonator. The system undergoes attenuation, re-oscillation, and establishment of the steady state. b Frequency-domain response of node voltage after the Fourier transform of the time-domain waveform in Fig. 2a, where \(\lambda\) denotes the unperturbed eigenfrequency, \({\lambda }^{{\prime} }\) and \({\lambda }_{{{\rm{com}}}}^{{\prime} }\) denote the eigenfrequencies after perturbation. c Three eigenfrequencies as a function of perturbation. The blue curve represents the pure real eigenfrequency and the two green dashed lines indicate a pair of conjugate complex frequencies of theoretical derivation. The dots derived from simulation data are generally consistent with the theoretical predications. The gray projection curves are the real and imaginary parts of the eigenfrequency.

To validate the proposed scheme, we built a wireless wearable sensor for temperature monitoring, as shown in Fig. 3. The precise temperature measurement of the human body or skin can serve as the basis for a noninvasive and quantitative characterization of dermatological health and physiological status29,30. The LC wireless wearable sensor allows for real-time and continuous monitoring without long and laborious steps. The temperature sensing capacitor was implemented here by an interdigital capacitor coated with polyethylene glycol (PEG 6000) (see Methods). The temperature sensor as a loss resonator consists of the inductor coil and sensing capacitor fabricated on a fully flexible printed circuit (FPC). The capacitance of the sensing capacitor increases as temperature (36–60 °C)31, leading to a perturbation of 0 ~ 2% to the loss side, shown as the green dot curve in Fig. 3a. An inductor coil in a readout circuit is identical with that in the loss side. The inductor coil and discrete variable capacitor as the gain resonator were coupled with the sensor, forming PT-symmetric LC configurations, as shown in Fig. 3b. A negative resistance consisting of an operation amplifier (THS4304) and an adjustable resistor (BOCHEN) was integrated on the readout circuit board. We tuned the components (inductance and capacitance) on the loss and gain side so as to achieve PT-symmetric LC configurations. We then set the nonlinear gain slightly larger than the loss to make the system unstable in the initial state. By precisely adjusting the coupling distance d between the two coils, the system can be operated at the exact phase close to EPs, as shown in Fig. 3c.

a The measured sensitive capacitance as a function of temperature, leading to a perturbation on the sensor. b A wearable wireless LC sensor consists of the gain and loss resonators. We apply the perturbation on the loss side by using the temperature sensitive interdigital capacitor coated with polyethylene glycol (PEG). c The state and eigenfrequency of the nonlinear PT-symmetric LC system as a function of the coupling distance d between the two coils. By precisely adjusting d, the system can be operated at the exact phase close to exceptional points.

In our experiments, the parameters of the gain and loss resonators are \({L}_{1}={L}_{2}=1.2\) µH, \({C}_{1}={C}_{2}=200\) pF, \({R}_{2}=800\) Ω, and \(-{R}_{1}=-780\) Ω, respectively. The resulting loss coefficient \(\gamma={R}_{2}^{-1}\sqrt{{L}_{2}/{C}_{2}}=0.097\). The coupling strength was fixed at \(k=0.1\). We used the DC source to supply power for the negative resistance circuit, and the oscilloscope to monitor the time-domain signal of the readout circuit (see Supplementary S4).

Figure 4a shows the frequency shift as a function of perturbation. We captured the node voltage in time-domain of the readout circuit and analyzed the spectrum by the fast Fourier transform (FFT). Obviously, due to the elimination of the imaginary part of oscillation frequency, the perturbations do not induce a broadening of the frequency spectrum, as shown in Fig. 4a. At the initial PT-symmetric state, the resonators chose the lower frequency to achieve a steady-state oscillation. Once we changed the ambient temperature, the sensitive capacitance in the sensor introduced a small perturbation into the loss resonator, leading to a shift from the lower branch to the higher branch \({\lambda }^{{\prime} }\). The normalized frequency \({\lambda }^{{\prime} }\) as a function of the perturbation \(\delta\) is given in Fig. 4b. The experimental results are in good agreement with our theoretical predictions. For sufficiently small perturbations, the response exhibits a slope of 1/3 through logarithmic coordinates, as shown in the inset of Fig. 4b, confirming that the perturbations lead to the cube-root singularity, though the PT-symmetric LC configurations of the linear loss and saturable gain has the second-order of EP degeneracy. We define here the normalized sensitivity of the sensor as \(\partial \Delta \lambda /\partial \delta\). The sensitivity of our scheme follows a scale of \({\delta }^{-2/3}\), which is higher than the linear EP scheme of \({\delta }^{-1/2}\). When the perturbation strength is small enough (\(\delta \ll 1\)), the sensitivity of our scheme is much higher than that of the linear EP sensing10,11 (See Supplementary S1 and S2). In our nonlinear PT-symmetric configurations, the temperature-sensitive capacitance increases with temperature, providing only positive perturbations relative to the initial PT-symmetric state. While the operational amplifier is operated in the negative saturation region under the negative perturbations, the gain cannot meet the demand for further growth, causing the system to be insensitive to negative perturbations (See Supplementary S2).

a Frequency shift as a function of the applied perturbations from 0 to 2%, which is obtained by applying a fast Fourier transform for the readout circuit voltage in time-domain after the system reaches saturation. b The normalized frequency \({\lambda }^{{\prime} }\) as a function of the perturbation \(\delta\). The blue dashed line represents the theoretical results, and the red circles are the experimental results. The inset is the comparison of logarithmic behavior between the linear EP system (1/2-slope, purple line) and our nonlinear scheme (1/3-slope, red line). The error bars indicate the standard deviation.

To explicit the influence of noise, we measured and analyzed the phase noise spectrum of nonlinear PT symmetric LC passive wireless sensors by using spectrum analyzer (See Supplementary S5). Figure 5a illustrates the phase noise spectrum of 0∼2 kHz frequency offset with 0∼2% perturbations applied on the loss capacitance. And the inset shows the power spectrum at \(\delta=2\%\), under which the full width at half maximum (FWHM) of resonances is 0.2 kHz. The spectral linewidth due to phase noise is much lower than the frequency shift (\(\Delta f=0.54\) MHz), enhancing the resolution of temperature sensors. The phase noise at 1 kHz offset from the center frequency is all below −75 dB/Hz before and after the perturbations, indicating an excellent performance of nonlinear PT symmetric LC passive wireless sensors.

a The phase noise spectrum with 0∼2% perturbation applied on the loss capacitance. The inset shows that for \(\delta=2\%\) the measured FWHM is 0.2 kHz and the phase noise at 1 kHz offset is as low as −88.8 dB/Hz. b The measured FOM as a function of the applied perturbation up to 2%.

In the EP sensing scheme of a linear PT symmetric LC passive wireless sensor, asymmetric perturbations cause PT-symmetry to break, resulting in a spectrum broadening though its sensitivity is enhanced. In the nonlinear counterpart, however, the saturable gain allows the system to automatically achieve a steady-state oscillation, eliminating the imaginary part of the eigenfrequencies and significantly suppressing the noise with enhanced sensitivity. To quantitatively evaluate the overall performance of the sensors, the figure of merit (FOM) in optical sensing32 is utilized here, which is also referred as the signal-to-noise ratio24. It is defined as

where \(\partial \Delta f/\partial \delta\) is actually the sensitivity of the sensor. The FWHM of resonances can be obtained from the phase noise spectrum in Fig. 5a (see Supplementary Fig S6). It remains almost unchanged under asymmetric perturbations up to 2%. As illustrated in Fig. 5b, the FOM decrease slightly since the sensitivity of the sensor decreases as the perturbation increases. However, it is higher than 88 dB over the perturbations up to 2%.

The resolution of sensors is the minimal change of the input necessary to produce a detectable change at the output. It is approximately estimated as the three times ratio of noise to sensitivity33. For our LC passive wireless sensors, the resolution is estimated as \(3 \times {{\rm{FWHM}}}/(\Delta {f}_{{{\rm{FSO}}}}/\Delta T)\approx 0.027 \,{{\circ}}{{\rm{C}}}\), where \(\Delta {f}_{{{\rm{FSO}}}}\) is the total frequency shift (full-scale output, 0.54 MHz) measured with maximum input temperature (60\(\,{\circ}{{\rm{C}}}\)) and the lowest input temperature (36\(\,{{\circ}}{{\rm{C}}}\)), corresponding to \(\Delta T=24\,{{\circ}}{{\rm{C}}}\).

In summary, we have theoretically proposed, numerically simulated, and experimentally demonstrated that LC passive wireless sensors have an enhanced precision using PT-symmetric LC configurations of the linear loss and saturable gain. The nonlinear gain causes a cube-root frequency detuning from the EP induced by a small capacitance variation on the loss resonator. It also eliminates the imaginary part of the eigenfrequencies and significantly suppresses the noise. The results establish EP-sensing as an efficient platform with an enhanced precision. Our scheme can guide the design of EP-based high precision capacitive sensors for other physical, chemical, and biological sensing.

As shown in Fig. 1a, according to Kirchhoff’s laws, the dynamics of the voltages at the gain and loss nodes follow the equations

where \({{{\bf{i}}}}_{{{\bf{n}}}}\) is the current through an inductor, \({{{\bf{v}}}}_{{{\bf{n}}}}\) is the node voltage, \(\dot{{{{\bf{v}}}}_{{{\bf{n}}}}}=d{{{\bf{v}}}}_{{{\bf{n}}}}/{dt}\) (n = 1, 2), respectively. Taking \({{{\bf{v}}}}_{{{\bf{n}}}}={V}_{n}{e}^{i\omega t}={V}_{n}{e}^{i\lambda \tau }\), where \(\tau={\omega }_{0}t\) is the normalized time, and \({\omega }_{0}=1/\sqrt{{LC}}\) is the natural resonant frequency of an isolated LC resonator, and ω is the operation angular frequency, respectively, we obtain \(\dot{{{{\bf{v}}}}_{{{\bf{n}}}}}=i\omega {{{\bf{v}}}}_{{{\bf{n}}}}\) and \(\ddot{{{{\bf{v}}}}_{{{\bf{n}}}}}=-{\omega }^{2}{{{\bf{v}}}}_{{{\bf{n}}}}\). On the other hand, we have

where \(k=M/L\) is the coupling strength between the gain and loss resonator and \(M\) is the mutual inductance, respectively. Inserting Eq. (12) into Eq. (13), it yields

where \(g={R}_{1}^{-1}\sqrt{L/C}\) is the gain coefficient and\(\gamma={R}_{2}^{-1}\sqrt{L/C}\) is the loss coefficient, respectively. Since we set the system at the exact PT-symmetric phase close to EPs, the weak coupling holds. We perform the Taylor’s expansion at \(\lambda=1\) and obtain \({\lambda }^{2}\cong 1+2\left(\lambda -1\right){{\mathscr{+}}}{{\mathscr{O}}}({\lambda -1})^{2}\) with \(0 < \gamma < k\ll 1\). We then have

By taking \({{{\bf{v}}}}_{{{\bf{n}}}}={V}_{n}{e}^{i\omega t}={V}_{n}{e}^{i\lambda \tau }\), Eq. (15) is recast into Eq. (1).

We printed the inductor and the electrodes of interdigital capacitors on a fully flexible printed circuit (FPC). The interdigital capacitors were then coated with the polyethylene glycol (PEG 6000) as sensing layers. In the experiment, the solid PEG was heated to 100 °C to melt. We dropped the liquid PEG to the interdigital capacitor plane, cooling down to room temperature. The PEG material completely covered all the interdigital capacitor plane. The measured parameters of the gain and loss resonators were \({L}_{1}={L}_{2}=1.2\) µH, \({C}_{1}={C}_{2}=200\) pF, and \({R}_{2}=800\) Ω, respectively. The resulting loss coefficient \(\gamma={R}_{2}^{-1}\sqrt{{L}_{2}/{C}_{2}}=0.097\).

The minimum dataset generated in this study have been deposited in the Zenodo repository database under accession code [https://doi.org/10.6084/m9.figshare.26964661].

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This work is supported by the National Natural Science Foundation of China, grant no. 62274030 (L.D.) and National Natural Science Foundation of China, grant no. 61727812 (Q.A.H.).

Key Laboratory of MEMS of the Ministry of Education, Southeast University, Nanjing, China

Dong-Yan Chen, Lei Dong & Qing-An Huang

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Q.A.H. and L.D. conceived and planned the research. D.Y.C. and L.D. performed the simulations and experiments. D.Y.C. and L.D. wrote the paper. Q.A.H. revised the paper.

Correspondence to Lei Dong or Qing-An Huang.

The authors declare no competing interests.

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Chen, DY., Dong, L. & Huang, QA. Inductor-capacitor passive wireless sensors using nonlinear parity-time symmetric configurations. Nat Commun 15, 9312 (2024). https://doi.org/10.1038/s41467-024-53655-x

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Received: 12 March 2024

Accepted: 18 October 2024

Published: 29 October 2024

DOI: https://doi.org/10.1038/s41467-024-53655-x

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