Critical point–based wireless sensors enabling tiny perturbation detection

Abstract

High quality factor and sensitivity are critical to wireless sensors targeting small perturbation detection. Although introducing parity-time symmetric dynamics promises enhanced performance, the symmetric scheme often yields limited sensing behaviors. In particular, its implementation is hindered by the inherent difficulties in decoupling capacitance- and coupling strength–induced responses, the requirements of delicately matching and/or tuning both gain and loss, and the need of strong coupling strength. Here, we report a concept of critical point (CP)–based wireless sensors that do not rely on balanced gain-loss configurations, delivering ultrahigh quality factor with an extended interrogation distance. Owing to a sharp and deep reflection dip, the CP-based scheme can resolve the change in coupling coefficient down to 1.92 × 10⁻⁴ and features frequency-independent responses. Furthermore, the CP-based scheme allows for identification of tiny asymmetric capacitive perturbations as small as 2.5 × 10⁻⁵ without requiring active tuning of the other parameters under weak coupling.

Critical-point wireless sensors with unbalanced gain and loss enable reliable and sensitive detection under weak coupling.

INTRODUCTION

Wireless and accurate monitoring of tiny perturbations in physical, chemical, and biological quantities has long been desired (1–3) for many advanced and emerging technologies, including implantable biomedical sensors (4–6), industrial sensing devices deployed in harsh environments or enclosed spaces (7, 8), and wireless sensor networks for the Internet of Things (9, 10), for which physical connections are not allowed. The architecture of wireless sensing devices based on inductor-capacitor (LC) resonators has received considerable interest (11–15) because of the ability to simultaneously transmit power and data through inductive coupling, which opens up the possibility of designing compact and battery-free wireless sensors. Over the past two decades, advances in microelectromechanical systems have substantially propelled the development of wireless sensing technology (4, 11, 14–20) by using these LC electronic resonance systems, in which the measured quantity modulates the coupling strength or capacitive sensing element, resulting in a measurable shift in the system resonance frequency. However, achieving high resolution [i.e., high quality factor (Q-factor)] and/or high sensitivity remains a long-standing challenge because of the inherent losses associated with conventional wireless schemes (Fig. 1A and fig. S1A), which typically result in complex eigenfrequencies (Fig. 1B) and a limited response to perturbations (18, 21, 22).

Fig. 1. Wireless sensing scheme based on non-Hermitian CPs.

(A to F) Schematics of the concept of conventional wireless sensing scheme (A), PT symmetry–based wireless sensing scheme (C), and non-Hermitian CP–based wireless sensing scheme (E), where g and $\mathrm{\gamma }$ denote the gain and loss parameters, respectively, and their corresponding imaginary parts of the normalized eigenfrequencies ($\tilde{\mathrm{\omega }}=\mathrm{\omega }/{\mathrm{\omega }}_0-1$, with $\mathrm{\omega }$ and ${\mathrm{\omega }}_0$ being the eigenfrequency and the natural resonance frequency for the reader and sensor) as a function of the coupling coefficient $\mathrm{\kappa }$ [top of (B), (D) and (F); the two eigenfrequency branches are plotted in orange and blue] and the corresponding typical reflection spectra [bottom of (B), (D) and (F)], where $S_{11}$ denotes the reflection coefficient and the curves for the CP-based, PT symmetry–based, and conventional schemes are shown in red, blue, and gray, respectively. Type 1 ($\mathrm{\kappa }$) and type 2 (C) represent the wireless sensing schemes for coupling strength ($\mathrm{\kappa }$) sensing and capacitive ($C_{\mathrm{s}}$) sensing, respectively. (G) Equivalent circuit model for the CP-based wireless sensing scheme and its distinct features, where $R_{\mathrm{r}}/R_{\mathrm{s}}$, $C_{\mathrm{r}}/C_{\mathrm{s}}$, and $L_{\mathrm{r}}/L_{\mathrm{s}}$ are the resistance, capacitance, and inductance of the reader/sensor, respectively. (H) Illustration of the architecture of the CP-based wireless sensing scheme.

Recently, inspired by the unique parity-time (PT) symmetric dynamics rooted in quantum mechanics (23–25) and evidenced in optics and photonics (26–38), efforts have been directed toward extending the concept of PT symmetry into the realm of electronics (18, 39–50). In particular, for a wireless electronic sensing system, on the basis of the PT symmetry concept (Fig. 1C and fig. S1B), it has real eigenfrequencies ($\mathrm{\omega }$) in the exact symmetry phase (Fig. 1D), which can promise an enhanced Q-factor according to $Q=\text{Re}\left[\mathrm{\omega }\right]/\left(2\text{Im}\left[\mathrm{\omega }\right]\right)$, and the bifurcation of eigenfrequencies near the exceptional point (EP) of the PT-phase transition promotes a notable increase in sensitivity (18, 40). However, the PT-symmetric scheme featuring balanced gain and loss configurations (18) often yields limited sensing behaviors. In particular, implementing the PT-symmetric scheme in wireless electronic sensing faces severe challenges (18, 21, 40–42). One is to decouple the response induced by coupling strength and capacitance for PT-symmetric systems, given that the eigenfrequencies are usually sensitive to both the coupling strength and capacitance disturbances (18, 40, 41). The other is the challenge associated with exactly matching gain and loss parameters and/or simultaneously fine-tuning both gain and loss sides, along with requiring a strong coupling strength to preserve symmetry at or near the EP (18, 41, 42), which essentially hinders their practical application. Issues include limited flexibility in device design, reduced interrogation distance, and potentially inaccurate readings resulting from fine-tuning processes.

Here, different from the PT-symmetric schemes that ensure real eigenfrequencies in the exact symmetry phase by requiring strictly balanced gain and loss for the Hamiltonian, we directly seek critical points (CPs) that allow for real eigenfrequencies in a general non-Hermitian framework without the restraining condition of balanced gain and loss (Fig. 1, E and F, and fig. S1C). The CP operates in a phase where the eigenvalues are degenerate, resulting in a single eigenfrequency with a sharp, deep reflection dip. This configuration yields an ultrahigh Q-factor ($Q=4304$), which is more than 48 times and 215 times greater than the PT-symmetric and conventional counterparts, respectively. In addition, the CP-based scheme provides an extended interrogation distance that is more than twice as long as that of the PT-symmetric counterpart. The large magnitude of the reflection dip delivered by the CP-based wireless sensing scheme can be used for coupling strength ($\mathrm{\kappa }$) sensing with greatly enhanced performance, offering a sevenfold improvement over the PT-symmetric EP-based scheme in resolving changes. Its frequency-independent feature facilitates decoupling the capacitance-induced response. Moreover, the CP-based scheme can be used for detecting tiny asymmetric capacitive ($C_{\mathrm{s}}$) variations in two and even extended to three coupled series RLC tanks, affording greater design flexibility with a further enhanced response.

RESULTS

Concept of CP-based wireless sensors

The wireless sensors based on non-Hermitian CPs can be constructed with two inductively coupled RLC series resonators: an active reader characterized by a gain parameter $g=R_{\mathrm{r}}\sqrt{C_{\mathrm{r}}/L_{\mathrm{r}}}$ and a passive sensor defined by a loss parameter $\mathrm{\gamma }=R_{\mathrm{s}}\sqrt{C_{\mathrm{s}}/L_{\mathrm{s}}}$. The sensor can be wirelessly interrogated by the reader via mutual inductance coupling with a coupling coefficient $\mathrm{\kappa }$, as illustrated in Fig. 1G. Using coupled-mode theory, the dynamics of this system can be described as $\mathrm{i}d\mathbf{a}/d\mathrm{\tau }={\hat{\mathcal{H}}}_0\mathbf{a}$ with the Hamiltonian ${\hat{\mathcal{H}}}_0$ being

$${\hat{\mathcal{H}}}_0=\left(\begin{array}{cc}1+\mathrm{i}\frac{g}{2}& \frac{\mathrm{\kappa }}{2}\\ \frac{\mathrm{\kappa }}{2}& 1-\mathrm{i}\frac{\mathrm{\gamma }}{2}\end{array}\right)$$ (1)

where $\mathbf{a}={(a_{\mathrm{r}},a_{\mathrm{s}})}^T$, with $a_{\mathrm{r}}$ and $a_{\mathrm{s}}$ being the amplitude of the reader and the sensor, respectively, and $\mathrm{\tau }={\mathrm{\omega }}_{0}t$, with ${\mathrm{\omega }}_0=1/\sqrt{\mathit{LC}}$ denoting the natural resonance frequency of the LC resonator, which corresponds to the resonance frequency of the reader and sensor. Let the normalized eigenfrequency be defined by $\tilde{\mathrm{\omega }}=\overline{\mathrm{\omega }}-1$ with $\overline{\mathrm{\omega }}=\mathrm{\omega }/{\mathrm{\omega }}_0$; then, the characteristic equation $\text{det}({\hat{\mathcal{H}}}_0-\overline{\mathrm{\omega }}\mathbf{I})=0$ based on Eq. 1 (where $\mathbf{I}$ denotes an identity matrix) can be expressed as

$${\tilde{\mathrm{\omega }}}^2+\frac{g\mathrm{\gamma }-{\mathrm{\kappa }}^2}{4}+\mathrm{i}\tilde{\mathrm{\omega }}\frac{\mathrm{\gamma }-g}{2}=0$$ (2)

In contrast to the common method of letting $\text{Im}\left[\text{det}({\hat{\mathcal{H}}}_0-\overline{\mathrm{\omega }}\mathbf{I})\right]=0$ (i.e., with the prerequisite constraint of $g=\mathrm{\gamma }$) to determine the real eigenfrequency that leads to the EP of ${\mathrm{\kappa }}^{\text{EP}}=g=\mathrm{\gamma }$ in the case of the PT-symmetric schemes (18, 21, 22, 42), we examine a general scenario within a non-Hermitian framework relaxing the strict constraint of $g=\mathrm{\gamma }$. This is achieved by directly solving Eq. 2, which can be expressed as ${\tilde{\mathrm{\omega }}}_{1,2}=[\mathrm{i}(g-\mathrm{\gamma })\pm \sqrt{4{\mathrm{\kappa }}^2-2g\mathrm{\gamma }-(g^2+{\mathrm{\gamma }}^2)}]/4$, and letting $\mid {\tilde{\mathrm{\omega }}}_1\mid =0$ or $\mid {\tilde{\mathrm{\omega }}}_2\mid =0$ to obtain the real eigenfrequency, which yields the CP of ${\mathrm{\kappa }}^{\text{CP}}=\sqrt{g\mathrm{\gamma }}$. When the parameter g is equal to $\mathrm{\gamma }$, the CP overlaps with EP (${\mathrm{\kappa }}^{\text{CP}}={\mathrm{\kappa }}^{\text{EP}}$). Unlike the standard PT-symmetric (21, 22) and generalized PT-symmetric (18, 42) schemes that require a balance between parameters g and $\mathrm{\gamma }$, our CP-based scheme provides substantial flexibility for the design of sensors and readers with unbalanced g and $\mathrm{\gamma }$.

It should be noted that our CP-based scheme reveals a single eigenfrequency with a sharp and deep reflection dip because the CP functions in the phase where the eigenvalues are degenerate (fig. S2), which is fundamentally different from the exact PT-symmetric phase (fig. S3) that typically exhibits two bifurcated eigenfrequencies (18). The architecture of our CP-based wireless electronic sensing system is illustrated in Fig. 1H, which can be constructed as coupling strength sensing and asymmetric capacitive sensing devices to measure a variety of quantities, including proximity, pressure, humidity, temperature, gas concentration, movement, speed, and acceleration (8, 16, 41, 51, 52).

A high Q-factor, characterized by a sharp and deep reflection dip, has long been a pursuit in the development of highly resolved wireless sensing devices, owing to the enhanced detection capacity and substantially improved noise robustness (18). The Q-factor can be quantitatively characterized by the full width at half maximum (FWHM) of the reflection spectra, i.e., $Q=f_{\text{dip}}/\text{FWHM}$, where $f_{\text{dip}}$ denotes the eigenfrequency of the system. As depicted in Fig. 2A and fig. S4A, the estimated Q-factor obtained from the conventional schemes is relatively low, specifically $Q\approx 20$ (12). Although the Q-factor can be improved to 90 by implementing the PT-symmetric scheme when operating in the exact phase (Fig. 2B and fig. S4B), the interrogation distance is decreased from 2 to 1.3 cm because of the strong coupling strength required to maintain the PT symmetry (18). In contrast, within the CP-based scheme, the reflection spectrum exhibits a single eigenfrequency with a sharp and deep reflection dip, providing an ultrahigh Q-factor of 406 without reducing the interrogation distance (Fig. 2C and fig. S4C). Notably, the Q-factor can be substantially enhanced ($Q=4304$) by further reducing $\mathrm{\gamma }$, which is ∼48 times and 215 times larger than that achieved by the PT-symmetric (18) and conventional (12) systems, respectively. Moreover, the interrogation distance can be extended to 3.6 cm (Fig. 2D and fig. S4D), which is more than twice longer than that of the PT-symmetric one (18) and more advantageous for practical applications, such as implantation scenarios. This improvement in the interrogation range arises from the inherently unbalanced gain-loss configuration of the CP-based scheme. Because it allows $\mathrm{\gamma }<g$ or $g<\mathrm{\gamma }$, the resulting coupling coefficient (${\mathrm{\kappa }}^{\text{CP}}$) is weaker than that required in the PT-symmetric scheme at the EP (${\mathrm{\kappa }}^{\text{EP}}=g=\mathrm{\gamma }$). According to Neumann’s formula, a smaller coupling coefficient yields a longer interrogation distance ($\mathrm{\kappa }\propto 1/d$).

Fig. 2. Experimental characterization of the Q-factor for different schemes.

(A and B) Characterization of the Q-factor for conventional (A) and the PT-symmetric schemes (B). (C and D) Characterization of the Q-factor for the CP-based scheme with $\mathrm{\gamma }=0.0508$ (C) and $\mathrm{\gamma }=0.0054$ (D).

In practical applications, noise plays a crucial role in determining the sensing performance, and thermal noise is generally the dominant contribution in wireless electronic sensing circuits (49). To quantitatively compare noise tolerance, we conducted additional measurements under identical environmental conditions and extracted signal-to-noise ratios from the reflection spectra (fig. S5). The CP-based sensing scheme exhibits considerably higher signal-to-noise ratios of 66.49 and 61.61 dB (for $\mathrm{\gamma }=0.0471$ and $\mathrm{\gamma }=0.4714$, respectively), compared to 54.32 dB for the PT-symmetric system (for $\mathrm{\gamma }=0.2215$), because of the intrinsically amplified thermal noise of the latter when biased at or near EPs.

CPs for coupling strength ($\mathbf{\kappa }$) sensing

In wireless LC sensing devices, the coupling coefficient or coupling strength $\mathrm{\kappa }$ can be used for signal detection. A typical example is monitoring physiological activities in implantable scenarios, where the variations in the interrogation distance between RLC tanks (41, 44) can be used to detect the changes. The critical challenge in introducing the PT-symmetric scheme for such applications (Fig. 3A) lies in decoupling the frequency response induced by the coupling strength and the capacitance, because the eigenfrequencies are sensitive to both of their disturbances in the exact symmetry phase (Fig. 3B). In contrast, the CP-based system (Fig. 3C) functions in the phase where the eigenfrequency is degenerate and is insensitive to the coupling strength, thus providing the opportunity to decouple the frequency response induced by these two factors (Fig. 3D). The large magnitude of the reflection dip provided by the CP-based system, instead of the frequency shift introduced by the PT-symmetric ones, can be used for sensing the coupling strength with greatly enhanced performance.

Fig. 3. Coupling strength sensing performance of the CP-based wireless electronic system.

(A) Equivalent circuit model for the PT-symmetric wireless sensing scheme. (B) Illustration of the coupling strength response for the PT-symmetric system. (C) Equivalent circuit model for the CP-based wireless sensing scheme. (D) Illustration of the coupling strength response for the CP-based system. (E) Measured reflection spectra under different interrogation distances (left, inset: photograph of the experimental setup; scale bar, 4 cm) and the zoomed-in views (right, inset: the zoomed-in view). (F) Change in the magnitude of $S_{11}$ at the dip frequency ($\mathrm{\Delta }{S}_{11}=S_{11}-S_{11}^{\text{min}}$, with $S_{11}^{\text{min}}$ being the minimal $S_{11}$ at the dip frequency biased at ${\mathrm{\kappa }}^{\text{CP}}$) and the normalized dip frequency change ($\mathrm{\Delta }\tilde{\mathrm{\omega }}/{\tilde{\mathrm{\omega }}}_0=(\tilde{\mathrm{\omega }}-{\tilde{\mathrm{\omega }}}_0)/{\tilde{\mathrm{\omega }}}_0$, with ${\tilde{\mathrm{\omega }}}_0$ being the eigenfrequency biased at ${\mathrm{\kappa }}^{\text{CP}}$) with respect to the change of interrogation distance in the logarithmic scale $\text{log}\left(\mathrm{\Delta }d\right)$ ($\mathrm{\Delta }d=d-d_0$, with $d_0$ being the distance biased at ${\mathrm{\kappa }}^{\text{CP}}$). The gray square points denote the relationship between the change in the magnitude of $S_{11}$ at the dip frequency and $\mathrm{\Delta }d$ (41).

Figure 3E shows the experimental results of the measured reflection spectra with respect to the interrogation distance (or, equivalently, the coupling strength). The reflection dip frequency remains essentially independent of the increased interrogation distance, while the magnitude of $S_{11}$ at the dip frequency is markedly reduced because of the increase in the imaginary part of the dip frequency. This behavior enables enhanced coupling-strength sensing while decoupling the capacitance-induced frequency response. In addition, the magnitude response versus the change in the interrogation distance on the logarithmic scale exhibits good linearity ($R^2=0.998$) (Fig. 3F), which facilitates signal processing. For comparison, we further investigated the interrogation distance–dependent reflection spectra of a PT-symmetric sensing system under identical conditions, as shown in fig. S6. In contrast to the CP-based scheme, variations in interrogation distance in the PT-symmetric system also induce resonance-frequency shifts, which inherently couple interrogation distance–dependent effects with capacitance-induced frequency variations. Moreover, operation at or near the EP is accompanied by increased noise, which limits the resolvable variation of the distance between interrogations. In particular, owing to the sharp and deep resonance dip (i.e., ultrahigh Q-factor) and the resulting large sensing window in the magnitude of $S_{11}$ provided by the CP-based scheme, a small variation of 10 μm at the initial interrogation distance of 3.6 cm can be resolved. This corresponds to a much lower variation in the coupling coefficient of 1.92 × 10⁻⁴, more than seven times lower than the detection limit (1.4 × 10⁻³) demonstrated in the PT-symmetric EP-based system (41). Such high sensing performance with an extended interrogation distance is highly valuable for tiny physiological signal monitoring in implantable scenarios.

CPs for asymmetric capacitive sensing in two coupled RLC tanks

The capacity for asymmetric capacitive sensing with enhanced performance is highly desirable for tiny signal detection in wireless LC sensing devices. The PT-symmetric scheme is generally unsuitable for asymmetric capacitive sensing, primarily due to the large imaginary part of resonance frequency, which results in indistinguishable reflection dips (53). Although symmetric perturbation can deliver enhanced performance by maintaining the symmetry, it often requires delicate tuning of both gain and loss sides (18, 41, 42). Here, we theoretically examined the asymmetric sensing performance of the proposed CP-based wireless electronic system by introducing a capacitive perturbation Hamiltonian ${\hat{\mathcal{H}}}_{\mathrm{\epsilon }}$ only on the sensor side (see Materials and Methods for details) and implemented the CP-based scheme by using the radio-frequency circuit depicted in Fig. 1G for experimental validation. The theoretical (solid lines) and experimental (red dots) results of the normalized eigenfrequencies in response to perturbation $\mathrm{\epsilon }$ are shown to be in good agreement (Fig. 4A). The CP-based scheme provides enhanced sensitivity (that is, the slope of the curve of $\text{Re}\left[\tilde{\mathrm{\omega }}\right]$ versus $\mathrm{\epsilon }$), beyond the limit of the conventional scheme represented by the gray dashed lines. As $\mathrm{\gamma }$ increases, the sensitivity can be further enhanced (Fig. 4B) within the specified regime where $\mathrm{\gamma }<g$. Alternatively, in the regime where $\mathrm{\gamma }>g$, as shown in Fig. 4 (C and D), the sensitivity exhibits an inverse trend that decreases with increased $\mathrm{\gamma }$, and the eigenfrequency shifts in the opposite direction, which can be revealed by the reduced value of the imaginary part as shown in fig. S7.

Fig. 4. Asymmetric capacitive sensing performance of the CP-based wireless electronic sensing scheme in two RLC tanks.

(A to D) Theoretical (solid lines) and experimental (red dots) results of the real part of normalized eigenfrequencies ($\text{Re}\left[\tilde{\mathrm{\omega }}\right]$) versus perturbation ($\mathrm{\epsilon }$) for the CP-based wireless electronic system under a fixed $g=0.2215$ with increasing loss parameter $\mathrm{\gamma }=0.0330$ (A), $\mathrm{\gamma }=0.1272$ (B), $\mathrm{\gamma }=0.3204$ (C), and $\mathrm{\gamma }=0.4712$ (D). The gray dashed lines are theoretical results derived from the conventional wireless electronic scheme under the same parameter settings of $\mathrm{\gamma }$ and $\mathrm{\kappa }$. The legend of the two solid lines refers to the two eigenvalues obtained by the CP-based scheme, and the shaded area indicates the eigenvalue with a smaller imaginary part revealed in fig. S7. (E to H) Measured reflection spectra for the CP-based system (top, solid lines) and the conventional system (bottom, dashed lines) with the same coupling coefficient $\mathrm{\kappa }$ at $g=0.2215$ with the increasing loss parameter $\mathrm{\gamma }=0.0330$ (E), $\mathrm{\gamma }=0.1272$ (F), $\mathrm{\gamma }=0.3204$ (G), and $\mathrm{\gamma }=0.4712$ (H). The inset in (E) shows a photograph of the experimental setup. Scale bar, 3 cm. (I) Schematic showing the evolution of sensing performance of the CP-based system with respect to parameter $\mathrm{\gamma }$.

The measured reflection spectra of the CP-based system are presented in Fig. 4 [E and F (top, solid lines)], obtained by using settings identical to those in Fig. 4 (A to D), in comparison with those obtained by the conventional system (bottom, dashed lines) with the same coupling coefficient $\mathrm{\kappa }$. In the CP-based scheme, when operating in the regime of $\mathrm{\gamma }<g$, although an increase in $\mathrm{\gamma }$ leads to a reduced Q-factor with respect to the perturbation, the change in the reflection dip remains distinctly identifiable. Under the same parameter settings of $\mathrm{\gamma }$ and $\mathrm{\kappa }$, the conventional scheme exhibits a more pronounced reduction in the Q-factor as $\mathrm{\gamma }$ increases, yielding a broadened resonant linewidth with indistinguishable changes of the dips in response to perturbations. In the regime of $\mathrm{\gamma }>g$ (Fig. 4, G and H), the CP-based scheme still produces reasonably pronounced dips with distinguished variations in response to perturbations, and the sharpness of the dips, or the Q-factor, is improved and the shift direction of reflection dips is reversed. By contrast, under the same parameter settings of $\mathrm{\gamma }$ and $\mathrm{\kappa }$, reflection dips in the conventional scheme become increasingly difficult to discern.

The evolution of the sensing performance in the CP-based wireless electronic system with increasing $\mathrm{\gamma }$ is illustrated in Fig. 4I. In general, a tunable response can be obtained with a high resolution and/or high sensitivity beyond the limits achieved in the conventional systems. Specifically, in the regime of $\mathrm{\gamma }<g$, when $\mathrm{\gamma }$ increases, the sensitivity increases but the resolution decreases, while in the regime of $\mathrm{\gamma }>g$, the sensitivity and resolution exhibit opposite trends with increasing $\mathrm{\gamma }$. It is important to note that in the extreme case of $\mathrm{\gamma }\to 0$ for the CP-based scheme, there is no fundamental limit in the resolution and the Q-factor enhancement, as the imaginary part of the eigenfrequency approaches zero ($\text{Im}\left[\tilde{\mathrm{\omega }}\left(\mathrm{\epsilon }\right)\right]\to 0$) (fig. S8A), and the sensitivity still remains 1/2 ($\mathrm{\Delta }\text{Re}\left[\tilde{\mathrm{\omega }}\left(\mathrm{\epsilon }\right)\right]\to 1/2$) (fig. S8B). Moreover, for $\mathrm{\gamma }\to 0$, because ${\mathrm{\kappa }}^{\text{CP}}=\sqrt{g\mathrm{\gamma }}\to 0$, the interrogation distance between the reader and the sensor can be greatly extended, which is highly desirable for the practical implementation of wireless sensing systems.

CPs for asymmetric capacitive sensing in three coupled RLC tanks

Although it is known that higher-order (e.g., third-order) PT-symmetric schemes can offer higher sensitivity to small perturbations near the EP compared to the second-order scheme (42), the challenges in implementing high-order symmetric schemes are substantially increased because of the need to manage more parameters. Therefore, extending the concept of the CP-based scheme with relaxed constraints to three coupled RLC tanks would provide substantial design flexibility and practical implementation convenience, making it highly valuable. The equivalent circuit model of the CP-based scheme in three RLC tanks is shown in Fig. 5A. It consists of three inductively coupled RLC resonators: an active reader with a gain parameter g, a passive sensor with a loss parameter $\mathrm{\gamma }$, and an additional loss component with the parameter ${\mathrm{\gamma }}_{\mathrm{m}}=R_{\mathrm{m}}\sqrt{C_{\mathrm{m}}/L_{\mathrm{m}}}$. The coupling coefficients between the reader and the sensor and between the sensor and the additional loss component are denoted as $\mathrm{\kappa }$ and ${\mathrm{\kappa }}_{\mathrm{m}}$, respectively. Analogous to the CP-based scheme in two RLC tanks, on the basis of coupled-mode theory, the Hamiltonian operator ${\hat{\mathcal{H}}}_0^3$ in three RLC tanks can lead to the characteristic equation

$$\begin{array}{c}{\tilde{\mathrm{\omega }}}^3+\tilde{\mathrm{\omega }}\frac{g\mathrm{\gamma }+g{\mathrm{\gamma }}_{\mathrm{m}}-{\text{γγ}}_{\mathrm{m}}-{\mathrm{\kappa }}^2-{\mathrm{\kappa }}_{\mathrm{m}}^2}{4}\hfill \\ +\mathrm{i}({\tilde{\mathrm{\omega }}}^2\frac{\mathrm{\gamma }+{\mathrm{\gamma }}_{\mathrm{m}}-g}{2}+\frac{g{\text{γγ}}_{\mathrm{m}}+g{\mathrm{\kappa }}_{\mathrm{m}}^2-{\mathrm{\gamma }}_{\mathrm{m}}{\mathrm{\kappa }}^2}{8})=0\hfill \end{array}$$ (3)

Fig. 5. Asymmetric capacitive sensing performance of the CP-based wireless electronic sensing scheme in three RLC tanks.

(A) Equivalent circuit model for the CP-based wireless electronic sensing system in three RLC tanks. (B) Coupling coefficient ${\mathrm{\kappa }}_{\mathrm{m}}$ versus $\mathrm{\kappa }$ within the CP scheme in three RLC tanks when Eq. 4 is ensured, and a range of CPs can be obtained. Here, ${\mathrm{\kappa }}_1^{\text{CP}3}$, ${\mathrm{\kappa }}_2^{\text{CP}3}$, and ${\mathrm{\kappa }}_3^{\text{CP}3}$ denote the CPs within the range of $[{\mathrm{\kappa }}^{\text{CP}},{\mathrm{\kappa }}^{\text{GEP}3}]$ and ${\mathrm{\kappa }}_1^{\text{CP}3}<{\mathrm{\kappa }}_2^{\text{CP}3}<{\mathrm{\kappa }}_3^{\text{CP}3}$. (C) Imaginary parts of the normalized eigenfrequencies as a function of the coupling coefficient ${\mathrm{\kappa }}_{\mathrm{m}}$ at $\mathrm{\kappa }={\mathrm{\kappa }}^{\text{CP}}$ and $\mathrm{\kappa }={\mathrm{\kappa }}^{\text{GEP}3}$. (D) Theoretical (solid lines) and experimental (symbols) results of real parts of the normalized eigenfrequencies versus a small perturbation ($\mathrm{\epsilon }$) for the CP-based wireless electronic sensing system in three RLC tanks at CPs of ${\mathrm{\kappa }}_1^{\text{CP}3}$, ${\mathrm{\kappa }}_2^{\text{CP}3}$, ${\mathrm{\kappa }}_3^{\text{CP}3}$, and ${\mathrm{\kappa }}_4^{\text{CP}3}$ (operating near ${\mathrm{\kappa }}^{\text{GEP}3}$). The inset in (D) shows a photograph of the experimental setup. Scale bar, 3 cm. (E to G) Experimental results of the reflection spectra of our CP-based system in three RLC tanks at $\mathrm{\kappa }={\mathrm{\kappa }}_1^{\text{CP}3}$ (E), $\mathrm{\kappa }={\mathrm{\kappa }}_2^{\text{CP}3}$ (F), and $\mathrm{\kappa }={\mathrm{\kappa }}_3^{\text{CP}3}$ (G).

Unlike the common strategy of letting $\text{Im}\left[\text{det}({\hat{\mathcal{H}}}^3-\overline{\mathrm{\omega }}\mathbf{I})\right]=0$ that requires $g=\mathrm{\gamma }+{\mathrm{\gamma }}_{\mathrm{m}}$ (22) or $g={\mathrm{\gamma }}_{\mathrm{m}}$ (when $\mathrm{\gamma }=0$) (32, 42) to achieve the real eigenfrequency, Eq. 3 clearly indicates that without the restrictive condition of $g=\mathrm{\gamma }+{\mathrm{\gamma }}_{\mathrm{m}}$, we can obtain real eigenfrequencies by only ensuring that

$$g{\text{γγ}}_{\mathrm{m}}+g{\mathrm{\kappa }}_{\mathrm{m}}^2={\mathrm{\gamma }}_{\mathrm{m}}{\mathrm{\kappa }}^2$$ (4)

which indicates that a variety of CPs can be obtained, as shown in Fig. 5B. When ${\mathrm{\kappa }}_{\mathrm{m}}=0$, the CP in three RLC tanks ${\mathrm{\kappa }}^{\text{CP}3}$ coincides with that of the two RLC tanks, that is, ${\mathrm{\kappa }}^{\text{CP}3}={\mathrm{\kappa }}^{\text{CP}}=\sqrt{g\mathrm{\gamma }}$. By substituting Eq. 4 into Eq. 3, the eigenvalues can be expressed as ${\tilde{\mathrm{\omega }}}_{1,2}=\frac{1}{4}[\mathrm{i}(g-\mathrm{\gamma }-{\mathrm{\gamma }}_{\mathrm{m}})\pm \sqrt{4{\mathrm{\kappa }}^2+4{\mathrm{\kappa }}_{\mathrm{m}}^2-(g^2+2g\mathrm{\gamma }+{\mathrm{\gamma }}^2+2g{\mathrm{\gamma }}_{\mathrm{m}}+{\mathrm{\gamma }}_{\mathrm{m}}^2)+2{\text{γγ}}_{\mathrm{m}}}]$. ${\tilde{\mathrm{\omega }}}_{1,2}=\frac{1}{4}[\mathrm{i}(g-\mathrm{\gamma }-{\mathrm{\gamma }}_{\mathrm{m}})\pm \sqrt{4{\mathrm{\kappa }}^2+4{\mathrm{\kappa }}_{\mathrm{m}}^2-(g^2+2g\mathrm{\gamma }+{\mathrm{\gamma }}^2+2g{\mathrm{\gamma }}_{\mathrm{m}}+{\mathrm{\gamma }}_{\mathrm{m}}^2)+2{\text{γγ}}_{\mathrm{m}}}]$. By letting $\mid {\tilde{\mathrm{\omega }}}_1\mid =0$ or $\mid {\tilde{\mathrm{\omega }}}_2\mid =0$, a general EP in the CP-based framework can be derived, i.e., ${\mathrm{\kappa }}^{\text{CP}3}={\mathrm{\kappa }}^{\text{GEP}3}=g\sqrt{\mathrm{\gamma }+{\mathrm{\gamma }}_{\mathrm{m}}}/\sqrt{g+{\mathrm{\gamma }}_{\mathrm{m}}}$, at which the eigenvalues coalesce, in contrast to the specific case ${\mathrm{\kappa }}^{\text{EP}3}=g\sqrt{1+\mathrm{\alpha }}/\sqrt{2+\mathrm{\alpha }}$ with $\mathrm{\alpha }=\mathrm{\gamma }/{\mathrm{\gamma }}_{\mathrm{m}}$ in the pseudo-Hermitian framework at $g=\mathrm{\gamma }+{\mathrm{\gamma }}_{\mathrm{m}}$ (22), which becomes ${\mathrm{\kappa }}^{\text{EP}3}=g/\sqrt{2}$ in the PT-symmetric framework at $g={\mathrm{\gamma }}_{\mathrm{m}}$ (when $\mathrm{\gamma }=0$) (32, 42). The corresponding real and imaginary parts of the normalized eigenfrequencies as a function of the coupling coefficient ${\mathrm{\kappa }}_{\mathrm{m}}$ at $\mathrm{\kappa }={\mathrm{\kappa }}^{\text{CP}}$ and $\mathrm{\kappa }={\mathrm{\kappa }}^{\text{GEP}3}$ are revealed in fig. S9 and Fig. 5C, which show that one real eigenvalue can be obtained at $\mathrm{\kappa }={\mathrm{\kappa }}^{\text{CP}}$ and three real eigenvalues can be obtained and coalesced at $\mathrm{\kappa }={\mathrm{\kappa }}^{\text{GEP}3}$.

We note that there is much potential for exploration of the physical dynamics within the CP-based scheme in three RLC tanks owing to the presence of numerous CPs and the involvement of more parameters. Here, our analysis emphasizes its effectiveness, examining a particular example of $g=0.2215$, $\mathrm{\gamma }=0.0330$, ${\mathrm{\gamma }}_{\mathrm{m}}=0.2969$, and $\mathrm{\kappa }\in [{\mathrm{\kappa }}^{\text{CP}},{\mathrm{\kappa }}^{\text{GEP}3}]$. The theoretical results (solid lines) of the perturbation behavior of normalized eigenfrequencies for our CP-based wireless electronic system in three RLC tanks at different CPs are shown in Fig. 5D, which is validated by the experimental data (symbols). It can be observed that as $\mathrm{\kappa }$ increases within the specified range of $\mathrm{\kappa }\in [{\mathrm{\kappa }}^{\text{CP}},{\mathrm{\kappa }}^{\text{GEP}3}]$, the sensitivity is beyond that of the CP-based scheme in two RLC tanks and can be further enhanced as $\mathrm{\kappa }$ increases, which is particularly substantial when $\mathrm{\kappa }$ approaches ${\mathrm{\kappa }}^{\text{GEP}3}$. Additional experimental results near ${\mathrm{\kappa }}^{\text{GEP}3}$ (fig. S10) further confirm this enhanced sensitivity behavior. The experimental results of the reflection spectra (Fig. 5, E to G) indicate that as $\mathrm{\kappa }$ increases, the resolution is slightly reduced because of the increase in $\text{Im}\left[\tilde{\mathrm{\omega }}\right]$ (fig. S11). The enhanced response and design flexibility afforded by a range of CPs in the three RLC tanks are expected to provide a compelling platform to investigate non-Hermitian physical dynamics for designing high-performance wireless sensing devices.

Demonstration of CP-based wireless capacitive proximity and pressure sensors

To illustrate the asymmetric capacitive sensing capability of our CP-based scheme (Fig. 6A), we integrated interdigitated electrodes into a CP-based wireless sensing system in two RLC tanks serving as the variable capacitor in the RLC resonator of the sensor, wherein these electrodes can perceive the proximity of objects through field-induced capacitance changes. Specifically, a decrease in the distance between a metallic object and interdigitated electrodes, which is precisely controlled by a linear translation stage, leads to a decrease in capacitance because of the mutual capacitance (54). The initial separation distance between the metallic object and the interdigitated electrodes is around 10 mm and decreases at a step of 500 μm. The experimental results of the reflection spectra with decreasing distance for the CP-based scheme are shown at the top of Fig. 6B compared to those for the conventional system at the bottom. The CP-based approach distinctly enables sharper and deeper reflection dips (more than 30-fold in the magnitude of $S_{11}$ at the eigenfrequency compared to that of conventional systems), resulting in distinguishable eigenfrequency changes (top of Fig. 5C), while the flattened reflection spectra with a broader linewidth in the conventional system render the change in the eigenfrequency indiscernible (bottom of Fig. 6C).

Fig. 6. Demonstration of the CP-based wireless sensors.

(A) Schematic diagram of the CP-based wireless sensors. Inset: Photograph of the pressure sensing element. Scale bar, 1 cm. (B and C) Measured reflection spectra (B) and their zoomed-in view (C) of the CP-based system (top, solid lines) and conventional system (bottom, dashed lines) when the distance between a metallic object and the interdigitated electrodes decreases. Coupling coefficients are identical, $\mathrm{\kappa }=0.0835$, in these two systems. The inset in (B) shows a schematic of the proximity sensor. (D) Theoretical (solid line) and experimental (dots) results of the real part of the normalized eigenfrequencies as a function of perturbation. (E and F) Measured reflection spectra (E) and their zoomed-in view (F) of the CP-based system (top, solid lines) and the conventional system (bottom, dashed lines) when pressure increases. The coupling coefficients are identical in these two systems ($\mathrm{\kappa }=0.1364$). The inset in (E) shows a schematic of the pressure sensor. The inset in (F) shows the measured reflection spectra to detect a pressure of ∼1 Pa. (G and H) Proximity response of a polymethyl methacrylate (PMMA) block (G) and pressure response (H) under VNA-induced vibration noise measured by the CP-based scheme and the PT-symmetric EP-based scheme. (I) Responses to pure airflow and exhalation recorded by the CP-based sensor, where $S_{11}^{\text{dip}}$ denotes the change in the magnitude of $S_{11}$ at the dip frequency. Inset: Photograph of the sensor component fabricated using a flexible polyimide (PI) film. Scale bar, 1 cm.

The experimental results (Fig. 6D, dots) of the real part of the normalized eigenfrequencies as a function of perturbation align well with our theoretical prediction (Fig. 6D, solid line). This finding reveals that an approximate change of one femtofarad in capacitance leads to a change in the eigenfrequency of ∼800 Hz, which is consistent with the simulation results (fig. S12). We note that our CP-based scheme can resolve an eigenfrequency change as small as 460 Hz, which corresponds to a capacitance change of ∼0.6 fF or a perturbation of around 2.5 × 10⁻⁵, far smaller than previous demonstrations, e.g., capacitance changes greater than subpicofarad (5, 15, 18, 51) or perturbations beyond 10⁻³ (22, 51). Such a small perturbation or subfemtofarad capacitance detection capability would be highly advantageous across various domains, including detection of tiny physiological activity and trace gases or chemicals in the environment.

The integration of a pressure sensor into the CP-based wireless system is schematically illustrated in fig. S13A. The pressure sensor consists of a floating electrode, an air-gap dielectric, and an array of interdigitated electrodes, in which the applied pressure depresses the air-gap dielectric, resulting in a smaller thickness and, therefore, an increase in capacitance (55). The experimental results of the reflection spectra as pressure increases for the CP-based system are shown at the top of Fig. 6E. For comparison, the results for the conventional system are displayed at the bottom of Fig. 6E. Compared with the conventional system (bottom of Fig. 6F), our CP-based system clearly provides substantially sharper and deeper reflection dips with distinguishable eigenfrequency changes (top of Fig. 6F) as the pressure increases by ∼5 Pa. We have demonstrated that our CP-based pressure sensor can resolve pressures as low as ∼1 Pa (inset of Fig. 6F and fig. S13B), which is more than an order of magnitude lower than those resolved in previous reports [∼133 Pa (4, 56), ∼67 Pa (11), ∼500 Pa (17), and ∼667 Pa (57)]. A better pressure resolution is expected with the introduction of more sensitive pressure elements (19, 58) because of the subfemtofarad capacitance detection capability of our CP-based scheme.

In addition to its superior tiny perturbation sensitivity, the CP-based scheme itself does not impose a fundamental limitation on large-range sensing. To evaluate its wide-range sensing capability, we further examined its response under extended capacitive perturbations (fig. S14). Although the sensitivity gradually decreases at larger perturbation levels, the CP-based scheme consistently outperforms the conventional design over an extended perturbation range of up to ∼0.1. These results indicate that the CP-based sensing architecture supports a wide sensing dynamic range while maintaining enhanced sensitivity.

Moreover, when mechanical vibration noise was intentionally introduced by directly connecting the reader to the measurement instrument of the vector network analyzer (VNA) port, the PT-symmetric scheme failed to reliably detect the tiny sensing signal because of resonance-frequency drift and substantially increased noise when biased at or near EPs, whereas the CP-based scheme clearly resolved tiny proximity signals (Fig. 6G) and pressure responses down to 1 Pa and below (Fig. 6H). This robustness originates from the reduced thermal noise level and the amplitude-only modulation behavior of the CP-based scheme, which avoids frequency drift even under interrogation-distance fluctuations. By leveraging the unique decoupled coupling strength– and capacitance-induced responses of the CP-based scheme, the sensor can clearly distinguish different types of physical stimuli, such as separating pure airflow from exhalation (Fig. 6I), in which pure airflow only induces mechanical vibration, yielding the resonance frequency nearly unchanged. Meanwhile, exhalation introduces not only mechanical vibration but also moisture and potential temperature variations, leading to a capacitance change and a corresponding resonance-frequency shift. Besides, the dielectric material sensing capability is demonstrated by using the CP-based sensor even under the vibration-induced noise from the VNA. The sensor is able to reliably record resonance frequency variations as an additional thin layer of biaxially oriented polypropylene films stacked on the sensing element (fig. S15A). Also, the CP-based sensor can discriminate different dielectric materials on the basis of their equivalent dielectric constants (fig. S15B). These results collectively demonstrate the superior noise tolerance and sensing fidelity of the CP-based wireless sensing architecture in realistic noisy environments.

To further shed light on the advancement of the CP-based scheme, we present a detailed comparison of key performance metrics between our CP-based scheme and the PT-symmetric counterpart, as summarized in table S1. In addition, the temperature dependence of the CP-based wireless sensing scheme was evaluated in a range of 25° to 65°C, where only relatively small drifts were observed in both asymmetric capacitance and interrogation-distance responses (fig. S16). Given that the dominant drift originates from the intrinsic temperature dependence of the resistance and capacitance within the sensing element, similar to conventional and PT-symmetric schemes, standard hardware-based temperature compensation strategies can be readily applied to further suppress temperature drift. The dynamic performance of the CP-based wireless sensor was further examined, where the sensor exhibits rapid response and relaxation times of ∼11 and 17 ms under an applied load of 18 g, respectively (fig. S17A). In addition, clear and stable cyclic pressure responses were recorded at different excitation frequencies (fig. S17B), confirming the reliable tracking capacity of the transient signal.

DISCUSSION

We have introduced a concept of a CP-based scheme featuring unbalanced gain and loss configurations, which fundamentally advances the performance, robustness, and practicality of wireless LC sensing systems. Unlike conventional and PT-symmetric architecture, the proposed CP framework operates under relaxed constraints while achieving an ultrahigh Q-factor with an extended interrogation distance, simultaneously overcoming the long-standing trade-off between sensitivity and coupling strength. The CP-based scheme enables reliable decoupling of capacitance- and coupling-induced responses and allows wireless sensors to resolve extremely small perturbations in the coupling coefficient and asymmetric capacitance, down to the subfemtofarad level, without requiring precise gain-loss tuning. Notably, the scheme with relaxed constraints allows the system to be even extended to three coupled series RLC tanks, offering enhanced sensitivity and substantial design flexibility and practical implementation convenience. These capabilities establish a practical route toward next-generation wireless sensing platforms and demonstrate the feasibility of exploiting non-Hermitian critical physics in real electronic systems. In addition, the CP-based scheme can be readily extended to other fields, such as wireless power transfer systems, to efficiently power the sensing systems (fig. S18). We note that the CP-based scheme provides unique and rich dynamics beyond the EPs associated with bifurcations, including ultrahigh Q-factor, weak coupling, suppressed thermal noise, asymmetric capacitive sensing capacity, and amplitude-only modulation behavior, enabling reliable detection of minute asymmetric capacitance variations even under noisy vibration conditions and real-time tracking of transient signals with extended interrogation distance, which is considerably valuable for real-time scenarios in realistic noisy environments. Overall, this work represents an important step forward in translating non-Hermitian dynamics into practical wireless sensing technologies and opens broad opportunities for the development of highly resolved, sensitive, and robust wireless electronic sensing devices and systems.

MATERIALS AND METHODS

Perturbation behavior of the CP-based wireless electronic system in two RLC tanks

To analyze the perturbation behavior of our CP-based scheme, we introduce an asymmetric perturbation, namely a capacitive perturbation Hamiltonian ${\hat{\mathcal{H}}}_{\mathrm{\epsilon }}$ on the sensor side, which can be expressed as

$${\hat{\mathcal{H}}}_{\mathrm{\epsilon }}=\left(\begin{array}{cc}0& 0\\ 0& \frac{\mathrm{\epsilon }}{2}\end{array}\right)$$ (5)

where the perturbation $\mathrm{\epsilon }=-\mathrm{\Delta }{C}_{\mathrm{s}}/(C_{\mathrm{s}}+\mathrm{\Delta }{C}_{\mathrm{s}})$, with $\mathrm{\Delta }{C}_{\mathrm{s}}$ being the change of capacitance. The perturbation behavior for the CP-based scheme is determined by the Hamiltonian $\hat{\mathcal{H}}={\hat{\mathcal{H}}}_0+{\hat{\mathcal{H}}}_{\mathrm{\epsilon }}$, which can be written as

$$\hat{\mathcal{H}}=\left(\begin{array}{cc}1+\mathrm{i}\frac{g}{2}& \frac{\mathrm{\kappa }}{2}\\ \frac{\mathrm{\kappa }}{2}& 1-\mathrm{i}\frac{\mathrm{\gamma }}{2}+\frac{\mathrm{\epsilon }}{2}\end{array}\right)$$ (6)

By solving $\text{det}(\hat{\mathcal{H}}-\overline{\mathrm{\omega }}\mathbf{I})=0$ at the CP (that is, $\mathrm{\kappa }=\sqrt{g\mathrm{\gamma }}$), we can obtain the perturbation characteristic equation

$${\tilde{\mathrm{\omega }}}^2-\tilde{\mathrm{\omega }}\frac{\mathrm{\epsilon }}{2}+\mathrm{i}(\tilde{\mathrm{\omega }}\frac{\mathrm{\gamma }-g}{2}+\frac{g\mathrm{\epsilon }}{4})=0$$ (7)

The perturbation behavior of the eigenfrequencies, including the real and imaginary parts, for the CP-based scheme can be derived from Eq. 7. Solving Eq. 7 yields

$${\tilde{\mathrm{\omega }}}_{1,2}=\frac{1}{4}(1\mp \frac{g+\mathrm{\gamma }}{\mid g-\mathrm{\gamma }\mid })\mathrm{\epsilon }+\mathrm{i}[\frac{\mid g-\mathrm{\gamma }\mid }{4}\pm \frac{\mid g-\mathrm{\gamma }\mid g\mathrm{\gamma }}{2{(g-\mathrm{\gamma })}^4}{\mathrm{\epsilon }}^2]+\mathcal{O}{\left(\mathrm{\epsilon }\right)}^3$$ (8)

By abandoning the higher-order terms, the real part of ${\mathrm{\omega }}_{1,2}$ can be approximated as

$$\text{Re}\left[{\tilde{\mathrm{\omega }}}_{1,2}\right]\approx \frac{1}{4}(1\mp \frac{g+\mathrm{\gamma }}{\mid g-\mathrm{\gamma }\mid })\mathrm{\epsilon }$$ (9)

whose slope $g/\left[2(g-\mathrm{\gamma })\right]$ is greater than 1/2 in the regime of $g>\mathrm{\gamma }$, indicating a better sensitivity than the conventional scheme (see Eq. 18). We note that when $\mathrm{\gamma }$ is further increased, while the sensitivity can be enhanced, the resolution is reduced because of the increased imaginary part, as depicted in Fig. 4. In particular, for the PT-symmetric configuration when $g=\mathrm{\gamma }$, the reflection dips become difficult to distinguish because of the large imaginary part of resonance frequency from asymmetric capacitive perturbations (53).

Perturbation behavior of the CP-based wireless electronic system in three RLC tanks

Analogous to the CP-based scheme in two RLC tanks, the Hamiltonian operator ${\hat{\mathcal{H}}}_0^3$ for the three RLC tanks can be expressed as

$${\hat{\mathcal{H}}}_0^3=\left(\begin{array}{ccc}1+\mathrm{i}\frac{g}{2}& \frac{\mathrm{\kappa }}{2}& 0\\ \frac{\mathrm{\kappa }}{2}& 1-\mathrm{i}\frac{\mathrm{\gamma }}{2}& \frac{{\mathrm{\kappa }}_{\mathrm{m}}}{2}\\ 0& \frac{{\mathrm{\kappa }}_{\mathrm{m}}}{2}& 1-\mathrm{i}\frac{{\mathrm{\gamma }}_{\mathrm{m}}}{2}\end{array}\right)$$ (10)

We introduce a capacitive perturbation Hamiltonian ${\hat{\mathcal{H}}}_{\mathrm{\epsilon }}^3$ on the sensor side, which reads

$${\hat{\mathcal{H}}}_{\mathrm{\epsilon }}^3=\left(\begin{array}{ccc}0& 0& 0\\ 0& \frac{\mathrm{\epsilon }}{2}& 0\\ 0& 0& 0\end{array}\right)$$ (11)

The perturbation behavior for the CP-based scheme in three RLC tanks is determined by the Hamiltonian ${\hat{\mathcal{H}}}^3={\hat{\mathcal{H}}}_0^3+{\hat{\mathcal{H}}}_{\mathrm{\epsilon }}^3$, which can be written as

$${\hat{\mathcal{H}}}^3=\left(\begin{array}{ccc}1+\mathrm{i}\frac{g}{2}& \frac{\mathrm{\kappa }}{2}& 0\\ \frac{\mathrm{\kappa }}{2}& 1-\mathrm{i}\frac{\mathrm{\gamma }}{2}+\frac{\mathrm{\epsilon }}{2}& \frac{{\mathrm{\kappa }}_{\mathrm{m}}}{2}\\ 0& \frac{{\mathrm{\kappa }}_{\mathrm{m}}}{2}& 1-\mathrm{i}\frac{{\mathrm{\gamma }}_{\mathrm{m}}}{2}\end{array}\right)$$ (12)

By solving $\text{det}({\hat{\mathcal{H}}}^3-\overline{\mathrm{\omega }}\mathbf{I})=0$ at the CP (that is, $g{\text{γγ}}_{\mathrm{m}}+g{\mathrm{\kappa }}_{\mathrm{m}}^3=$ ${\mathrm{\gamma }}_{\mathrm{m}}{\mathrm{\kappa }}^2$), we can obtain the perturbation characteristic equation as

$$\begin{array}{c}{\tilde{\mathrm{\omega }}}^3-\frac{\mathrm{\epsilon }}{2}{\tilde{\mathrm{\omega }}}^2-(\frac{{\mathrm{\kappa }}^2}{4}\frac{{\mathrm{\gamma }}_{\mathrm{m}}+g}{g}-\frac{g^2}{4}\frac{\mathrm{\gamma }+{\mathrm{\gamma }}_{\mathrm{m}}}{g})\tilde{\mathrm{\omega }}-\frac{g{\mathrm{\gamma }}_{\mathrm{m}}\mathrm{\epsilon }}{8}\hfill \\ -\mathrm{i}\tilde{\mathrm{\omega }}(\frac{g-\mathrm{\gamma }-{\mathrm{\gamma }}_{\mathrm{m}}}{2}\tilde{\mathrm{\omega }}+\frac{{\mathrm{\gamma }}_{\mathrm{m}}-g}{4}\mathrm{\epsilon })=0\hfill \end{array}$$ (13)

The perturbation behavior of the eigenfrequencies, including the real and imaginary parts, for the CP-based scheme in three RLC tanks can be obtained on the basis of Eq. 13.

Perturbation behavior of the conventional wireless electronic system

The Hamiltonian operator for the conventional wireless electronic system can be expressed as

$${\hat{\mathcal{H}}}_0^{\text{conv}}=\left(\begin{array}{cc}-\mathrm{i\eta }& -\mathrm{\kappa }\\ -\frac{\mathrm{\kappa }}{2}& 1+\mathrm{i}\frac{\mathrm{\gamma }}{2}\end{array}\right)$$ (14)

where $\mathrm{\eta }=Z_0\sqrt{C_{\mathrm{s}}/L_{\mathrm{r}}}$, with $Z_0$ being an intrinsic impedance of the source on the reader side. The perturbation behavior for the conventional scheme is determined by the Hamiltonian ${\hat{\mathcal{H}}}^{\text{conv}}={\hat{\mathcal{H}}}_0^{\text{conv}}+{\hat{\mathcal{H}}}_{\mathrm{\epsilon }}^{\text{conv}}$, which can be written as

$${\hat{\mathcal{H}}}^{\text{conv}}=\left(\begin{array}{cc}-\mathrm{i\eta }& -\mathrm{\kappa }\\ -\frac{\mathrm{\kappa }}{2}& 1+\mathrm{i}\frac{\mathrm{\gamma }}{2}+\frac{\mathrm{\epsilon }}{2}\end{array}\right)$$ (15)

By solving $\text{det}({\hat{\mathcal{H}}}^{\text{conv}}-\overline{\mathrm{\omega }}\mathbf{I})=0$, we can obtain the perturbation characteristic equation as

$${\tilde{\mathrm{\omega }}}^2+(1-\frac{\mathrm{\epsilon }}{2})\tilde{\mathrm{\omega }}+\frac{\text{γη}}{2}-\frac{{\mathrm{\kappa }}^2}{2}-\frac{\mathrm{\epsilon }}{2}+\mathrm{i}(\mathrm{\eta }\tilde{\mathrm{\omega }}-\frac{\mathrm{\gamma }}{2}\tilde{\mathrm{\omega }}-\frac{\mathrm{\gamma }}{2}-\frac{\text{εη}}{2})=0$$ (16)

The perturbation behavior of the imaginary part of the eigenfrequencies and the slope of the real part for the conventional CP-based scheme can be derived from Eq. 16. To obtain a clear analytical formula for the perturbation behavior of eigenfrequencies within the conventional scheme, we commence by introducing the perturbation of the capacitance change $\mathrm{\Delta } C_{\mathrm{s}}$, whereby the resonance frequency for the system can be expressed as

$${\mathrm{\omega }}_{\mathrm{c}}=(1+\frac{{\mathrm{\kappa }}^2}{4})\frac{1}{\sqrt{L_{\mathrm{s}}(C_{\mathrm{s}}+\mathrm{\Delta }{C}_{\mathrm{s}})}}=(1+\frac{1}{4}{\mathrm{\kappa }}^2)\frac{\sqrt{1+\mathrm{\epsilon }}}{\sqrt{L_{\mathrm{s}}{C}_{\mathrm{s}}}}$$ (17)

The normalized resonance frequency for the system ${\tilde{\mathrm{\omega }}}_{\mathrm{c}}=$ ${\mathrm{\omega }}_{\mathrm{c}}/{\mathrm{\omega }}_{\mathrm{c}0}-1$ with ${\mathrm{\omega }}_{\mathrm{c}0}=(1+{\mathrm{\kappa }}^2/4)/\sqrt{L_{\mathrm{s}}{C}_{\mathrm{s}}}$ can be rewritten as

$$\tilde{\mathrm{\omega }}=\sqrt{1+\mathrm{\epsilon }}-1=\underset{\text{Taylor series}}{\underbrace{1+\frac{1}{2}\mathrm{\epsilon }-\frac{1}{8}{\mathrm{\epsilon }}^2+\mathcal{O}{\left(\mathrm{\epsilon }\right)}^3}}-1\approx \frac{1}{2}\mathrm{\epsilon }-\frac{1}{8}{\mathrm{\epsilon }}^2$$ (18)

Equation 18 indicates that the slope of the real eigenfrequency versus the perturbation (that is, the sensitivity) is about $\frac{1}{2}-\frac{1}{8}\mathrm{\epsilon }$ for the conventional scheme.

Device fabrications

To verify wireless electronic systems, spiral planar inductors fabricated on printed circuit boards are connected in series with resistors and capacitors, where the negative resistance on the reader side is supplied by the intrinsic impedance of a VNA (Keysight E5063A). To achieve a small capacitive perturbation of 0.02 on the sensor side, an adjustable capacitor is connected in series with a 1-pF capacitor, which is subsequently connected in parallel with a 20-pF capacitor. The coupling coefficient $\mathrm{\kappa }$ is determined by the separation distance between the two inductors, which is precisely regulated on the xyz axes by a linear translation stage. The interdigitated electrodes, with a width of 120 μm, are combined for proximity sensing. A metallic object is placed directly above the interdigitated electrodes, maintaining a separation distance of 10 mm between them. For the pressure sensor, a polydimethylsiloxane spacer is used to provide an air gap dielectric, which is produced from an as-fabricated polydimethylsiloxane film around 500 μm thick, featuring a circular hole with a radius of 5 mm in the middle, which is achieved by laser cutting. The floating electrode is fabricated on a 10-μm-thick polyethylene terephthalate substrate, onto which a 20-nm/50-nm-thick Ti/Au layer is deposited by electron beam evaporation (DE Technology Inc.). The applied pressure is measured using a precise force gauge (M7-012, Mark-10 Inc.). The experimental setup is depicted in fig. S19. The simulations are performed with LTspice software.

Supplementary Materials

This PDF file includes:

Figs. S1 to S19

Table S1