Remote characterization of resonance frequency with a wirelessly powered parametric oscillator

Abstract

It is of both fundamental importance and practical value to measure the frequency of an LC resonator beyond the near-field region, especially when the resonator is used as a standalone capacitive sensor embedded inside a closed environment. To improve the coupling efficiency between the resonator and the external sniffer loop, we propose a novel method to integrate the LC resonator with a wirelessly-powered parametric resonator whose oscillation signal can be remotely identified in a noisy background. By measuring the minimum power level that is required for oscillation at different pumping frequencies, the resonator can be indirectly characterized by the frequency response curve. Starting from the basic principle of parametric oscillation, we will predict the measurable extremities in the frequency-dependent power curve under various circumstances that are classified based on the relative ratio between the lower and higher resonance frequencies. Our analytical models are validated by on-bench measurements performed on several parametric resonators with different circuit topologies. Their ability for remote characterization will make parametric resonators useful in structural health sensors or biomedical implants.

I. INTRODUCTION

It is of fundamental importance to measure the resonance frequency of LC resonators, the basic building blocks of RF electronic circuits. Conventionally, the resonance frequency of an LC resonator can be inductively measured by a pair of partially overlapped loops connected to the output and input ports of a network analyzer. By properly adjusting the overlapped area for minimized interactions between the excitation and receiving loops, peaks in the transmission curve (S₂₁) can be attributed to resonance frequencies of the LC resonator. Because the resonator’s frequency response can be characterized in a contactless way, it can operate as a batteryless transceiver [1–4] embedded inside a closed environment. When integrated with a sensing capacitor, the LC resonator can encode environmental signals as capacitance changes, which can be measured wirelessly as shift in resonance frequencies. This scheme for telemetric capacitive sensing can be useful in many applications, including structural mechanical monitoring [5–9], environmental sensing [10–18], food quality control [19–21] and physiology monitoring [22–27]. However, telemetric capacitive sensor is less effective when the distance separation [28–30] between the sniffer loops and the LC resonator is much larger than its own dimension, making it challenging to identify peak frequencies of the S₂₁ curve in a noisy background. This problem will be exacerbated when the resonator is embedded inside lossy biological tissues that may further reduce RF transmission efficiency and twist the S₂₁ baseline. Although magneto-inductive repeaters [31–34] or waveguides [35] can greatly increase the effective range of wireless sensors, the requirement for intermediate stage devices mandates the use of catheters [36], which is still different from the application scenarios of embedded sensors.

In this work, we are going to demonstrate a compact circuit to increase the effective distance range for resonance frequency measurement. Utilizing the principle of parametric oscillation [37], the pumping signal generates an oscillation signal at half of the frequency that can be easily identified from the noisy background. By adjusting the required pumping power for marginal oscillation over a range of frequencies, the optimal oscillation frequency can be utilized to characterize the resonator’s frequency response. Unlike transistor-based oscillators [38–40] that require rectifiers to convert electromagnetic [41] or acoustic energy [42] into DC power, the parametric oscillator has a much simpler circuitry to directly convert the wirelessly provided pumping power into the oscillation signal, enabling more possibilities for further miniaturization.

II. Methods

A. Parametric oscillation

Parametric oscillator can be implemented as a multi frequency resonator (Fig. 1) with at least one nonlinear component (varactor). This nonlinear resonator can utilize its highest resonance mode to receive wirelessly-provided pumping power and convert it into sustained oscillation signals at lower resonance modes. Under the influence of an electromotive force ζ₃(t) that is induced inside the resonator by an external pumping loop, there is a voltage V₃ induced across the varactor C₂ at ω₃. Due to frequency mixing, a much smaller signal V₁ at ω₁ will interact with the strong pumping signal V₃ at ω₃ across the nonlinear capacitor C₂ to generate an idler signal V₂ at the difference frequency ω₂ = ω₃ − ω₁. As a result, the time-dependent capacitance [43] is:

$$C_2\left(t\right)=C_{20}{\left(1-\frac{V\left(t\right)}{\varphi }\right)}^{-\lambda }\approx C_{20}\left(1+\frac{\lambda \left(V_1+V_2+V_3\right)}{\varphi }\right)$$ (1)

where Φ is the contact potential of diode junction. λ is normally between 1/3 and 2, which describes the abruptness of charge distribution variation across the diode’s depletion layer. As a result, the total current in the varactor C₂ is:

$$I_1+I_2+I_3=\frac{d\left(C_2\left(t\right)\left(V_1+V_2+V_3\right)\right)}{dt}$$ (2)

Fig. 1.

The schematic diagram of a parametric resonator containing a nonlinear varactor (C₂). If the circuit’s linear part consists of a Foster network with a frequency dependent impedance R+jS, multiple resonance modes will appear.

The three voltage terms in Eq. (2) can be expressed in the complex forms:

$$\begin{gathered} V_1=\left(V_s{e}^{j{\omega }_{1}t}+V_s^{*}{e}^{-j{\omega }_{1}t}\right)/2 \\ {V}_2=\left(V_i{e}^{j{\omega }_{2}t}+V_i^{*}{e}^{-j{\omega }_{2}t}\right)/2 \\ {V}_3=\left(V_p{e}^{j{\omega }_{3}t}+V_p^{*}{e}^{-j{\omega }_{3}t}\right)/2 \end{gathered}$$ (3)

Similarly, current terms in Eq. (2) can be expressed as:

$$\begin{gathered} I_1=\left(I_s{e}^{j{\omega }_{1}t}+I_s^{*}{e}^{-j{\omega }_{1}t}\right)/2 \\ {I}_2=\left(I_i{e}^{j{\omega }_{2}t}+I_i^{*}{e}^{-j{\omega }_{2}t}\right)/2 \\ {I}_3=\left(I_p{e}^{j{\omega }_{3}t}+I_p^{*}{e}^{-j{\omega }_{3}t}\right)/2 \end{gathered}$$ (4)

By plugging Eqns (4) and (3) to both sides of Eq. (2) and retaining frequency components only at ω₁, ω₂ and ω₃:

$$\begin{gathered} I_s{e}^{j\omega t}+I_s^{*}{e}^{-j\omega t}+I_i{e}^{j{\omega }_{2}t}+I_i^{*}{e}^{-j{\omega }_{2}t}+I_p{e}^{j{\omega }_{3}t}+I_p^{*}{e}^{-j{\omega }_{3}t} \\ =j{C}_{20}{V}_s{\omega }_1{e}^{j\omega t}+\frac{j{C}_{20}{\omega }_1\lambda V_i^{*}{V}_p{e}^{j\omega t}}{\varphi }-j{C}_{20}{V}_s^{*}{\omega }_1{e}^{-j\omega t}-\frac{j{C}_{20}{\omega }_1\lambda V_i{V}_p^{*}{e}^{-j\omega t}}{\varphi } \\ +j{C}_{20}{V}_i{\omega }_2{e}^{j{\omega }_{2}t}+\frac{j{C}_{20}{\omega }_2\lambda V_s^{*}{V}_p{e}^{j{\omega }_{2}t}}{\varphi }-j{C}_{20}{V}_i^{*}{\omega }_2{e}^{-j{\phi }_{2}t}-\frac{j{C}_{20}{\omega }_2\lambda V_s{V}_p^{*}{e}^{-j{\omega }_{2}t}}{\varphi } \\ +j{C}_{20}{V}_p{\omega }_3{e}^{j{\omega }_{3}t}+\frac{j{C}_{20}{\omega }_3\lambda V_i{V}_s{e}^{j{\omega }_{3}t}}{\varphi }-j{C}_{20}{V}_p^{*}{\omega }_3{e}^{-j{\omega }_{3}t}-\frac{j{C}_{20}{\omega }_3\lambda V_i^{*}{V}_s^{*}{e}^{-j{\omega }_{3}t}}{\varphi } \\ +\frac{j{C}_{20}{\omega }_1\lambda V_s^2{e}^{j2at}}{\varphi }+\frac{j{C}_{20}{\omega }_2\lambda V_i^2{e}^{j2a_{1}t}}{\varphi }-\frac{j{C}_{20}{\omega }_1\lambda V_s^{*2}{e}^{-j2at}}{\varphi }-\frac{j{C}_{20}{\omega }_2\lambda V_i^{*2}{e}^{-j2a_{2}t}}{\varphi } \end{gathered}$$ (5)

By comparing the left and right sides of Eq. (5), the following admittance matrix can be obtained.

$$\left(\begin{array}{c}{I}_i^{*}\\ I_s\end{array}\right)=\left(\begin{array}{cc}-j{C}_{20}{\omega }_2& -j{C}_{20}{\omega }_2\lambda V_p^{*}/\varphi \\ j{C}_{20}{\omega }_1\lambda V_p/\varphi & j{C}_{20}{\omega }_1\end{array}\right)\left(\begin{array}{c}{V}_i^{*}\\ V_s\end{array}\right)$$ (6)

and subsequently the impedance matrix is

$$\left(\begin{array}{c}{V}_i^{*}\\ V_s\end{array}\right)=\frac{1}{C_{20}{\omega }_1{\omega }_2\left({\varphi }^2-{\lambda }^2{\left|V_p\right|}^2\right)}\left(\begin{array}{cc}j{\varphi }^2{\omega }_1& j\lambda \varphi V_p^{*}{\omega }_2\\ -j\lambda \varphi V_p{\omega }_1& -j{\varphi }^2{\omega }_2\end{array}\right)\left(\begin{array}{c}{I}_i^{*}\\ I_s\end{array}\right)$$ (7)

Using Kirchhoff’s Voltage Law, the equivalent voltage source ζ₁ applied at frequency ω₁ can be expressed as

$$\begin{gathered} {\xi }_1=V_s+I_s\left(R_1+j{\omega }_1{L}_1\right)=\frac{-j\lambda \varphi V_p{\omega }_1{I}_i^{*}-j{\varphi }^2{\omega }_2{I}_s}{C_{20}{\omega }_1{\omega }_2\left({\varphi }^2-{\lambda }^2{\left|V_p\right|}^2\right)}+I_s\left(R_1+j{\omega }_1{L}_1\right) \\ \equiv I_s\left(R_1+j{X}_1\right)-\frac{j\lambda \varphi V_p{\omega }_1{I}_i^{*}}{C_{20}{\omega }_1{\omega }_2\left({\varphi }^2-{\lambda }^2{\left|V_p\right|}^2\right)} \end{gathered}$$ (8)

where R₁+jω₁L₁ is the effective circuit impedance excluding the varactor at frequency ω₁, R₁+jX₁ is the effective circuit impedance including the varactor. Similarly, the equivalent voltage source ζ₂ at frequency ω₂ is

$$\begin{gathered} {\xi }_2^{*}=V_i^{*}+I_i^{*}\left(R_2-j{\omega }_2{L}_2\right)=\frac{j{\varphi }^2{\omega }_1{I}_i^{*}+j\lambda \varphi V_p^{*}{\omega }_2{I}_s}{C_{20}{\omega }_1{\omega }_2\left({\varphi }^2-{\lambda }^2{\left|V_p\right|}^2\right)}+I_i^{*}\left(R_2-j{\omega }_2{L}_2\right) \\ \equiv \frac{j\lambda \varphi V_p^{*}{\omega }_2{I}_s}{C_{20}{\omega }_1{\omega }_2\left({\varphi }^2-{\lambda }^2{\left|V_p\right|}^2\right)}+I_i^{*}\left(R_2-j{X}_2\right) \end{gathered}$$ (9)

where R₂+jω₂L₂ is the effective circuit impedance excluding the varactor at frequency ω₂, R₂+jX₂ is the effective circuit impedance containing the varactor at frequency ω₂. Sustained oscillation requires I_s and $I_i^{*}$ to be non-zero when ξ₁ = ξ₂ = 0. Therefore, the following relation can be derived from Eqns. (8) and (9):

$$\frac{{\lambda }^2{\varphi }^2{\left|V_P\right|}^2{\omega }_1{\omega }_2}{C_{20}^2{\omega }_1^2{\omega }_2^2{\left({\varphi }^2-{\lambda }^2{\left|V_p\right|}^2\right)}^2}-\left(R_1+j{X}_1\right)\left(R_2-j{X}_2\right)=0$$ (10)

Eq. (10) describes the general requirement of a multi-frequency parametric resonator that can effectively receive the external pumping signal at ω₃ and convert it to sustained oscillation at the signal and the idler frequencies (ω₁ and ω₂). To simplify circuit design, ω₁ and ω₂ can share the same resonance mode, i.e. ω₁ = ω₂ = ω₃/2, R₁ = R₂, X₁ = X₂. When the varactor is undergoing small signal modulation, i.e. λV_p/Φ ≪ 1, Eq (10) is simplified to:

$${\left|V_P\right|}^2\approx \left(R_1^2+X_1^2\right)C_{20}^2{\omega }_1^2{\varphi }^2/{\lambda }^2$$ (11)

Similar to the derivation of Eq. (6), by comparing terms with ω₃ frequency components on the left and right sides of Eq. (5),

$$I_p=j{C}_{20}{V}_p{\omega }_3+C_{20}{\omega }_1\lambda {\left(V_i+V_s\right)}^2/\varphi$$ (12)

The equivalent voltage source ζ₃ at frequency ω₃ is

$$\begin{gathered} {\xi }_3=V_p+I_p\left(R_3+j{\omega }_3{L}_3\right)=V_p+\left(j{C}_{20}{V}_p{\omega }_3+C_{20}{\omega }_1\lambda {\left(V_i+V_z\right)}^2/\varphi \right)\left(R_3+j{\omega }_3{L}_3\right) \\ \equiv j{C}_{20}{V}_p{\omega }_3\left(R_3+j{X}_3\right)+C_{20}{\omega }_1\lambda {\left(V_i+V_z\right)}^2\left(R_3+j{\omega }_3{L}_3\right)/\varphi \end{gathered}$$ (13)

where R₃+jω₃L₃ is the effective circuit impedance excluding the varactor at frequency ω₃, R₃+jX₃ ≡ R₃+jω₃L₃+1/(jω₃C₃) is the effective circuit impedance containing the varactor. ζ₃ is electromotive force induced by a current I_L in the pumping loop whose mutual inductance with the parametric resonator is m. Therefore, ξ₃=jω₃mI_L=j2ω₁mI_L and Eq. (13) is equivalated to:

$$I_L=\frac{C_{20}{V}_p\left(R_3+j{X}_3\right)}{m}+\frac{C_{20}\lambda {\left(V_i+V_s\right)}^2\left(R_3+j{\omega }_3{L}_3\right)}{2jm\varphi }$$ (14)

The second term in Eq. (13) originates from sustained oscillation signal, which is normally much smaller than the first term originating from the pumping signal. By neglecting the second term and plugging Eq. (11) into Eq. (14),

$$\begin{gathered} {\left|I_L\right|}^2\approx \frac{{\omega }_1^2{\varphi }^2{C}_{20}^4{R}_1^2{R}_3^2}{{\lambda }^2{m}^2}\left(1+\frac{X_1^2}{R_1^2}\right)\left(1+\frac{X_3^2}{R_3^2}\right) \\ =\frac{{\omega }_3^2{\varphi }^2{C}_{20}^4{R}_1^2{R}_3^2}{4{\lambda }^2{m}^2}\left(1+4Q_1^2{\left(\frac{{\omega }_1}{{\omega }_{10}}-1\right)}^2\right)\left(1+4Q_3^2{\left(\frac{{\omega }_3}{{\omega }_{30}}-1\right)}^2\right) \end{gathered}$$ (15)

where ω₁₀ and ω₃₀ are the circuit’s lower and higher resonance frequencies, ω₃ is the externally-provided pumping frequency that is twice the oscillation frequency ω_1. For operating frequencies ω₁ and ω₃ that are deviated from resonance frequencies ω₁₀ and ω₃₀, the reactance is X₁ ≈ 2R₁Q₁(ω₁ / ω₁₀ −1) and X₃ ≈ 2R₃Q₃(ω₃ / ω₃₀ −1), where Q₁ and Q₃ are circuit’s quality factor near resonance at ω₁₀ and ω₃₀. By defining the relative pumping frequency as f ≡ ω₃ / ω₃₀ and the resonance ratio deviation as k ≡ 2ω₁₀ / ω₃₀ −1, Eq. (15) is rearranged as

$$\frac{{\left|I_L\right|}^2}{f^2}\approx \frac{{\varphi }^2{C}_{20}^4{R}_1^2{R}_3^2{\omega }_{30}^2}{4{\lambda }^2{m}^2}\left(1+4Q_1^2{\left(\frac{f}{\left(1+k\right)}-1\right)}^2\right)\left(1+4Q_3^2{\left(f-1\right)}^2\right)$$ (16)

Quite often, we can construct a double frequency resonator whose higher resonance frequency ω₃₀ is known a prior, Eq. (16) describes the quantitative relation to estimate the lower resonance frequency ω₁₀ (that contains sensing information) from the frequency sweep. At each externally applied pumping frequency ω₃, we can scale the required amount of pumping power (that is proportional to |I_L|²) by the relative pumping frequency f ≡ ω₃ / ω₃₀. By plotting |I_L|²/f² with respect to f, we can utilize Eq. (16) to estimate k, the resonance ratio deviation that correlates the lower resonance frequency ω₁₀ with the higher resonance frequency ω₃₀ via ω₁₀ = (k + 1)ω₃₀/2.

B. Extreme value analysis

It is possible to perform a complete fitting of Eq. (16), but in the following, we will exploit the possibility to estimate k from the curve’s local extremities by calculating the derivative of the second and third terms on the r.h.s. with respect to f:

$$8\left(f-1\right)\left(1+4Q_1^2{\left(\frac{f}{\left(1+k\right)}-1\right)}^2\right)Q_3^2+8\left(\frac{f}{{\left(1+k\right)}^2}-\frac{1}{1+k}\right)\left(1+4Q_3^2{\left(f-1\right)}^2\right)Q_1^2$$ (17)

To make the derivative zero, f_i − 1 ≡ S_i has three possible values.

$$S_1=\frac{1}{24}\left(12k+\frac{\left(1+\sqrt{3}j\right)6^{2/3}A}{{\left(9B+\sqrt{81B^2+6A^3}\right)}^{1/3}}+6^{1/3}\left(-1+\sqrt{3}j\right){\left(9B+\sqrt{81B^2+6A^3}\right)}^{1/3}\right)$$ (18)

$$S_2=\frac{1}{24}\left(12k-\frac{\left(-1+\sqrt{3}j\right)6^{2/3}A}{{\left(9B+\sqrt{81B^2+6A^3}\right)}^{1/3}}+6^{1/3}\left(-1-\sqrt{3}j\right){\left(9B+\sqrt{81B^2+6A^3}\right)}^{1/3}\right)$$ (19)

$$S_3=\frac{1}{12}\left(6k-\frac{6^{2/3}A}{{\left(9B+\sqrt{81B^2+6A^3}\right)}^{1/3}}+6^{1/3}{\left(9B+\sqrt{81B^2+6A^3}\right)}^{1/3}\right)$$ (20)

where

$$\text{A}\equiv \left(1/Q_3^2+{\left(k+1\right)}^2/Q_1^2-2k^2\right),\text{B}\equiv k\left(1/Q_3^2-{\left(k+1\right)}^2/Q_1^2\right).$$ (21)

By calculating the average of Eqns. (18) to (20)

$$\left(f_1+f_2+f_3-3\right)/3=k/2$$ (22)

where f₁, f₂ and f₃ correspond to the three local extremities on the right-hand-side of Eq. (16). Eq. (22) provides an important relation to estimate the resonance ratio deviation k from local extremities that are directly measurable from frequency sweep. If the higher resonance frequency ω₃₀ is known a prior and is made to be independent of the sensing capacitor, the lower resonance frequency (that reflects the sensing capacitance) can be estimated from

$${\omega }_{10}\equiv {\omega }_{30}\left(1+k\right)/2={\omega }_{30}\left(\left(f_1+f_2+f_3\right)/3-1/2\right).$$ (23)

Of course, estimation of ω₁₀ based on Eq. (23) requires Eqns. (18)–(20) to be real-valued. This will happen only if

$$81B^2+6A^3\le 0,$$ (24)

where A and B are defined in Eq. (21). As a result, the second and third terms in Eqns. (18)–(20) would cancel their imaginary parts. Because Eq. (24) also implicitly requires A < 0, Eq. (18) can be rearranged to identify the approximate linear relation between S_i and k under certain conditions:

$$\begin{gathered} S_1=\frac{1}{24}\left(12k-\frac{\left(1+\sqrt{3}j\right)6^{2/3}\left|A\right|}{{\left(9B+j\sqrt{6{\left|A\right|}^3-81B^2}\right)}^{1/3}}+6^{1/3}\left(-1+\sqrt{3}j\right){\left(9B+j\sqrt{6{\left|A\right|}^3-81B^2}\right)}^{1/3}\right) \\ =\frac{1}{24}\left(12k-\frac{\left(1+\sqrt{3}j\right)6^{2/3}\left|A\right|}{{\left({\left(6{\left|A\right|}^3\right)}^{1/2}{e}^{j\theta }\right)}^{1/3}}+6^{1/3}\left(-1+\sqrt{3}j\right){\left({\left(6{\left|A\right|}^3\right)}^{1/2}{e}^{j\theta }\right)}^{1/3}\right) \\ =\frac{1}{24}\left(12k-4\text{cos}\frac{\left(\pi -\theta \right)}{3}{6}^{1/2}{\left|A\right|}^{1/2}\right), \end{gathered}$$ (25)

where $\text{cos}\theta \equiv 9B/\sqrt{6{\left|A\right|}^3}$ and $\text{sin}\theta \equiv \sqrt{-81B^2+6{\left|A\right|}^3}/\sqrt{6{\left|A\right|}^3}$.

Similarly, Eq. (20) is rearranged as

$$\begin{gathered} S_2=\frac{1}{24}\left(12k-\frac{\left(+1+\sqrt{3}j\right)6^{2/3}\left|A\right|}{{\left(9B+j\sqrt{6{\left|A\right|}^3-81B^2}\right)}^{1/3}}+6^{1/3}\left(-1-\sqrt{3}j\right){\left(9B+j\sqrt{6{\left|A\right|}^3-81B^2}\right)}^{1/3}\right) \\ =\frac{1}{24}\left(12k-\left(1+\sqrt{3}j\right)6^{1/2}{\left|A\right|}^{1/2}{\left(e^{-j\theta }\right)}^{1/3}-6^{1/2}\left(1+\sqrt{3}j\right){\left|A\right|}^{1/2}{\left(e^{j\theta }\right)}^{1/3}\right) \\ =\frac{1}{24}\left(12k-4\text{cos}\frac{\left(\pi +\theta \right)}{3}{6}^{1/2}{\left|A\right|}^{1/2}\right). \end{gathered}$$ (26)

And Eq. (21) is rearranged as

$$\begin{gathered} S_3=\frac{1}{12}\left(6k+\frac{6^{2/3}\left|A\right|}{(9B+j{\sqrt{\left(6{\left|A\right|}^3-81B^2\right)}}^{1/3}}+6^{1/3}{\left(9B+j\sqrt{\left(6{\left|A\right|}^3-81B^2\right)}\right)}^{1/3}\right) \\ =\frac{1}{12}\left(6k+6^{1/3}{\left(9B-j\sqrt{\left(6{\left|A\right|}^3-81B^2\right)}\right)}^{1/3}+6^{1/3}{\left(9B+j\sqrt{\left(6{\left|A\right|}^3-81B^2\right)}\right)}^{1/3}\right) \\ =\frac{1}{12}\left(6k+6^{1/2}{\left|A\right|}^{1/2}\left(2\right)\text{cos}\left(\theta /3\right)\right). \end{gathered}$$ (27)

Further simplifications can be made when Q₁ and Q₃ are large enough and when 6|A³| ≫ 81B²,

$$\frac{6{\left|A\right|}^3}{81B^2}\approx \frac{16k^4}{27{\left(1/Q_3^2-{\left(k+1\right)}^2/Q_1^2\right)}^2}\gg 0$$ (28)

This also requires |k| to be sufficiently large to make θ close to π/2. As a result, Eq. (25) can be approximated as

$$\begin{gathered} S_1\approx \frac{1}{24}\left(12k-4\text{cos}\left(\frac{\pi }{6}\right)6^{1/2}{\left|A\right|}^{1/2}\right) \\ =\frac{1}{24}\left(12k-2\sqrt{18\left(2k^2-1/Q_3^2-{\left(k+1\right)}^2/Q_1^2\right)}\right)\approx \frac{1}{24}\left(12k-12\left|k\right|\right) \end{gathered}$$ (29)

The last approximation holds when

$$2k^2\gg 1/Q_3^2+{\left(k+1\right)}^2/Q_1^2,$$ (30)

which is a stronger condition than Eq. (28). Similarly, Eq. (27) can be simplified to

$$S_3\approx \frac{1}{12}\left(6k+\sqrt{18}{\left|2k^2-1/Q_3^2-{\left(k+1\right)}^2/Q_1^2\right|}^{1/2}\right)\approx \frac{1}{12}\left(6k+6\left|k\right|\right).$$ (31)

Therefore, when ω₁₀ = (k + 1)ω₃₀/2 is much deviated from ω₃₀/2, Eq. (29) is simplified to f₁ ≈ k + 1 (for k <0), and Eq. (31) is simplified to f₃ ≈ k + 1 (for k >0), both of which correspond to an optimal pumping frequency of 2ω₁₀. In another word, the lower resonance frequency ω₁₀ can be directly estimated from half of the optimal pumping frequency, provided that ω₁₀ is sufficiently different from ω₃₀/2.

C. On-bench measurements

First, we use the conventional method to characterize the resonator’s frequency response, by measuring the transmission coefficient (S₂₁) between the excitation and receiving loops that are connected to the output and input ports of a network analyzer. The overlapped area of the loop pair is empirically adjusted until the S₂₁ reached minimum in the absence of resonator, thus minimizing direct interaction between the excitation and receiving loops. When the resonator is placed beneath the pair of loops to inductively couple with them, its frequency response can in principle be identified from the change in S₂₁ curve. But when the resonator is separated from the double pick-up loop by 4 cm, which is 13-fold larger than its own dimension, the change in S₂₁ curve is no larger than the network analyzer’s noise baseline (− 100 dBm for 1 kHz acquisition bandwidth). To improve the measurement sensitivity, a pumping loop is connected to a frequency synthesizer to provide the pumping signal. The pumping frequency ω₃ is swept over an anticipated range. At each pumping frequency, the required pumping power is empirically adjusted until marginal oscillation is observed on the network analyzer. To minimize direct interference from the pumping signal, a low-pass filter can be optionally incorporated between the receiving loop and the input port of a network analyzer. In the following, the same measurement method will be applied to three resonators with different design features. For each case, the resonator will be sufficiently separated from the receiving loop to emulate deep-lying capacitive sensors that would introduce unobservable change in S₂₁ curve by conventional measurement method. Subsequently, pumping power will be gradually increased until the resonator marginally oscillates as observed on the network analyzer. Because the observation frequency is half of the pumping frequency, the receiving signal can be easily separated from the excitation signal, enabling more sensitive characterization of the resonator’s frequency response beyond the near-field region.

III. RESULTS

A. Fabricated resonator #1 (One frequency extremity)

Fig. 3a shows the circuit diagram of a nonlinear double frequency resonator. It is constructed by wrapping a continuous copper wire around a 3.6-mm diameter cylindrical rod for two turns and around a rectangular rod with a 1.5 × 2.5 mm² cross-section for four turns (Fig. 3b). The cylindrical rod and the rectangular rod are placed perpendicular to each other, so that the two constituting inductors L₁ and L₂ can effectively couple to the external pumping field from different orientations. C₁ is a 2.4-pF chip capacitor, and C₂ is a varactor (BBY51, Infineon, Germany) to perform signal mixing. When the pumping frequency is swept over the range between 606 MHz and 634 MHz at 1-MHz interval, there is only one optimal pumping frequency at 622 MHz, which doesn’t correlate to 2ω₁₀ in a straightforward manner. This is because the resonator’s higher resonance frequency ω₃₀ is not sufficiently separated from twice the lower resonance frequency 2ω₁₀, which is evidenced by direct measurement of ω₁₀ to be 307.48 MHz (Q₁ = 57) and ω₃₀ to be 624.6 MHz (Q₃ = 67). Based on these measured values, the resonance deviation ratio is k = 2ω₁₀/ω₃₀-1=−0.0154, which will make $\text{A}\equiv \left(1/Q_3^2+{\left(k+1\right)}^2/Q_1^2-2k^2\right)>0$. Because the left-hand side of Eq. (24) is larger than 0, the resonator is indeed predicted to have only one optimal pumping frequency whose value is described by Eq. (20). This example is chosen to demonstrate the condition for single-valued frequency extremity. In the following, we will illustrate two more examples where ω₁₀ is easier to estimate based on its simpler relation with the measurable frequency extremities.

Fig. 3.

(a) The circuit diagram and (b) the picture of a double-frequency parametric resonator, where the varactor C₂ is placed in parallel with the inductor L₂. (c) For each integral frequency point swept between 606 MHz and 634 MHz, the required pumping power for marginal oscillation is scaled by f² (which is the square of the relative frequency) and displayed in units of dBw. The measured power levels (shown in dots) indeed correlate well with the simulated results (shown in continuous line) predicted by Eq. (16).

B. Fabricated resonator #2 (Three frequency extremities)

Fig. 4a is the circuit diagram of another implementation of a parametric resonator. This circuit consists of a parallel resonant circuit (L₁ and C₁) placed in series with L₂ and C₂, where C₂ is the varactor diode to perform frequency mixing. The resonator is constructed by wrapping a continuous copper wire around a rectangular rod with a 1.6 × 1.6 mm² cross-section for two turns and around a 1.8-mm diameter cylindrical rod for three turns (Fig. 4b). C₁ is a 5.1-pF chip capacitor and C₂ is a varactor (BBY52, Infineon, Germany). These component values are chosen so that twice the lower resonance frequency 2ω₁₀ is sufficiently separated from the higher resonance frequency ω₃₀. When the pumping frequency is swept, the required pumping power for marginal oscillation has two local minima at 1013 MHz and 1032 MHz and one local maxima at 1024 MHz (Fig. 4c). Therefore, the average of these three frequency extremities is calculated to be 1023 MHz. According to Eq. (23), the lower resonance and higher resonance frequencies should satisfy the relation ω₁₀ = 1023 − ω₃₀/2. This simple relation is indeed confirmed by direct measurement of ω₁₀ to be 517 MHz (Q = 84) and ω₃₀ to be 1012 MHz (Q = 103). In another word, it is possible to estimate the lower resonance frequency from the higher resonance frequency along with the three extremities of the frequency-dependent pumping curve.

Fig. 4.

(a) The circuit diagram and (b) the picture of a double-frequency parametric resonator, where the varactor C₂ is placed in series with the inductor L₂. (c) For each integral frequency point swept between 1010 MHz and 1035 MHz, the required pumping power for marginal oscillation is scaled by f² (which is the square of the relative frequency) and displayed in units of dBw. The measured power levels (shown in dots) indeed correlate well with the simulated results (shown in continuous line) predicted by Eq. (16).

C. Fabricated resonator #3 (Three frequency extremities with a higher resonance frequency ω₃₀ that remains invariant.)

Up till now, we have presented two examples to demonstrate the necessary conditions to observe three frequency extremities on the frequency response curve. We have also validated Eq. (23) that describes the quantitative relation between these three extremities and the lower resonance frequency ω₁₀. Here, we will present another circuit design that can be practically used for capacitive sensing, by making the higher resonance frequency ω₃₀ independent of changes in ω₁₀.

Figs. 5a shows the circuit diagram of a cylindrical parametric resonator mounted on a plastic cylinder. Its higher resonance mode responsible for receiving the pumping signal is created by the top end-ring split by a pair of identical varactors, so that the higher resonance frequency ω₃₀ is only determined by the diameter of the top end-ring and the zero-biased capacitance of varactors C₂. Meanwhile, the circuit’s lower resonance mode responsible for sustaining the oscillation signal is created by the vertical legs along with the top and bottom end-rings. If the capacitors C₁ on the bottom end-rings are sensing capacitors, changes in the lower resonance frequency ω₁₀ will be indicative of changes in C₁, without affecting the higher resonance frequency ω₃₀. Because ω₃₀ can be measured a prior and will remain constant, it can effectively function as an internal frequency reference based on which the lower resonance frequency ω₁₀ can be estimated from Eq. (23). Moreover, estimation of ω₁₀ will be even simpler if k ≡ 2ω₁₀ / ω₃₀ −1 is large enough to make Eq. (31) valid, so that ω₁₀ is approximately half of the frequency extremity value. This simpler relation is validated on the circuit implemented in Fig. 5b, where the cylindrical resonator has a diameter of 2.5 mm and a length of 5.9 mm. By splitting the resonator’s top end-ring with a pair of identical varactors (PMEG2005EB, NXP, Netherland) and the bottom end-ring with a pair of 39-pF chip capacitors, the resonator has a transverse resonance frequency at ω₁₀ = 299.7 MHz (Q = 60) and a longitudinal resonance frequency at ω₃₀ = 550.3 MHz (Q = 23). By sweeping the pumping frequency over the range between 592 MHz and 605 MHz, the optimal pumping frequency is 599 MHz (Fig. 8c), which is indeed very close to 2ω₁₀. In principle, we can also measure all the three frequency extremities. But because 2ω₁₀ is too far away from ω₃₀, the local maximum that is approximately half way between 2ω₁₀ and ω₃₀ would require an impractically high level of pumping power for oscillation.

Fig. 5.

(a) The circuit diagram and (b) the picture of a cylindrically-symmetric parametric resonator. In this design, a pair of varactors C₂ are placed around the top end-ring whose resonance frequency is independent of the capacitors at the bottom end-ring. (c) For each integral frequency swept between 592 MHz and 605 MHz, the required pumping power for marginal oscillation is scaled by f² (which is the square of the relative frequency) and displayed in units of dBw. The measured power levels (shown in dots) indeed correlate well with the simulated results (shown in continuous line) predicted by Eq. (16).

D. Measurements under different distance separations

Based on the above examples, it is easier to evaluate the resonance behavior of resonators #2 and #3 whose higher resonance frequencies ω₃₀ are sufficiently separated from twice their lower resonance frequencies 2ω₁₀. Here, we measure the frequency extremities of these two resonators when their distance separation from the external pumping loop are gradually increased. For each distance separation, we measure the frequency extremities five times and calculate the averages:

As shown in Table 1, highly consistent frequency extremities are obtained over a range of distance separations, even at a distance separation of 16 cm that is more than 60-fold larger than the resonator’s own width. This large distance separation is already larger than the half-thickness of human torso, indicating the possibility to use parametric resonators as physiological sensors deep inside the body.

Table 1.

Frequency extremities measured over several distance separations

D	Resonator #2 (MHz)			# 3 (MHz)
4 cm	1012.0±0.0	1024.0±0.0	1032.0±0.0	599.0±0.0
8 cm	1012.0±0.0	1024.0±0.0	1032.0±0.0	599.0±0.0
12 cm	1011.8±0.4	1023.8±0.4	1032.4±0.5	598.8±0.4
16 cm	1012.2±0.5	1023.6±0.5	1032.2±0.4	599.2±0.4

IV. Discussion

In this work, we demonstrate the use of parametric oscillation to increase the effective interaction between an LC resonator and the sniffer loop. Three versions of parametric oscillators are fabricated based on nonlinear multi-frequency resonators. Utilizing the nonlinearity of their constituting varactors, these resonators can convert wirelessly provided pumping signal at ω₃ into sustained oscillation signal at ω₃/2. By sweeping the frequency of the pumping signal, we can measure the minimal power for marginal oscillation and obtain local extremities along the frequency-dependent power curve. When the resonator is constructed in a way so that its lower resonance frequency ω₁₀ reflects sensing capacitance and its higher resonance frequency ω₃₀ remains invariant, it is possible to estimate ω₁₀ from ω₃₀ using Eq. (23), given that three extremities are measurable along the frequency-dependent curve. Estimation of lower resonance frequency ω₁₀ can be further simplified if 2ω₁₀ is distantly separated from ω₃₀ to make Eq. (31) valid, so that ω₁₀ can be approximated as half of the optimal pumping frequency.

To oscillate a parametric resonator, it is energy-efficient to have the higher resonance frequency approximately twice the lower resonance frequency, i.e. k = 2ω₁₀/ω₃₀−1 ~ 0, so that both quadratic terms in the right-hand side of Eq. (16) become zero. On the other hand, when |k| is very large to make Eq. (31) valid, the optimal pumping frequency is approximately twice the lower resonance frequency, i.e. f_r–1=k, making the right-hand side of Eq. (16) (1+4Q₃²k²)-fold larger than the condition when k = 0. Therefore, the convenience to directly estimate ω₁₀ from half of the optimal pumping frequency is obtained at a cost of larger pumping power. In order to estimate the resonance frequency with better energy efficiency, it is also possible to fabricate the resonator in a way to make the left side of Eq. (24) not too smaller than 0, so that three local extremities can be obtained within a moderate frequency range. In this case, although estimation of ω₁₀ is contingent upon prior knowledge about ω₃₀, there are potential applications where the relative change of ω₁₀ is a direct indication of the relative change in capacitance, so that long-term variation of sensing parameters can be obtained without the need for measuring ω₃₀ a prior.

Conventional methods use a pair of loops to measure the resonator-induced perturbation on power transmission curve (S₂₁) between the excitation and receiving loops. If the mutual coupling between the loops and the resonator is much small than the direct coupling from the excitation to the receiving loops, it is very hard to measure the resonator’s frequency response, because the resonator’s impact on the S₂₁ curve is buried below the noise baseline. However, because parametric resonator can generate oscillation signal that is different from the input signal, emitted signals from the resonator can stand above the background, enabling resonance characterization at much larger distance separations. Of course, a larger distance separation necessitates a larger oscillation signal, which corresponds to a larger value for the second term in Eq. (14). But when the circuit has large enough quality factor, i.e. R₃ ≪ ω₃L₃, this second term in Eq. (14) only has a slow-varying frequency component jω₃L₃, making its contribution to the |I_L|²/f² curve almost frequency independent. In another word, even though the resonator is operating above marginal oscillation, as long as its oscillation signal is maintained at a constant level over the frequency sweep, the position of frequency extremities will not be affected by the marginal oscillation approximation. The validity of this approximation is evident from the negligible changes in frequency extremities measured under different distance separations (Table 1). Besides the marginal oscillation approximation, another possible source of systematic error is the assumption for small signal modulation of the varactor, i.e. λV/Φ ≪ 1, leading to the approximation described in Eq. (1) and (11). This assumption is self-consistent because the required pumping voltage is V_p ~ Φ/(λQ₃) according to Eq. (11). If the resonator’s quality factor Q₃ is sufficiently large, λV/Φ ≪ 1 can always be ensured.

Experimentally, a larger source of error may come from the positional variation of the resonator during measurement. Because the required pumping power is measured at individual frequency points, it is important to hold the resonator static with respect to the pumping loop throughout the frequency sweep. For real-world sensors operating in a moving environment (due to physiological motion for example), it is important develop external control apparatus to automatically adjust the pumping frequency and to simultaneously track the oscillation signal so that frequency sweep can be finished within a shorter period compared to the motion time scale. More work is going on towards this direction.

V. Conclusion

In this work, an analytical framework is proposed to improve the effective distance range for resonance frequency characterization based on wirelessly powered parametric oscillator. Three different circuits have been constructed to demonstrate the versatile design of double-frequency parametric resonators and to validate analytical predictions under different resonance conditions. Instead of using multiple excitation signals to interrogate the circuit’s resonance behavior [44–47], a parametric oscillator is simple to operate with a single signal source. Since the oscillation frequency is different from the pumping frequency, it is easy to measure the oscillation signal in both the time domain and frequency domain, thus alleviate special requirements on the external read-out circuitry [3, 48–51].

Fig. 2.

The frequency response of a parametric oscillator activated by an external pumping loop can be characterized beyond near-field regions, for example, at a distance separation of at least 4 cm, which is more than 13-fold larger the resonator’s width.

Biography

Wei Qian received her B.S. and M.S. degrees from Wuhan University, Wuhan, China, in 2003 and 2006, respectively, and a Ph.D. degree from Michigan State University, East Lansing, in 2011, all in electrical engineering. From 2011 to 2016, she was a Power Electronics Engineer in Magna Electronics (the former E-Car Systems). She is now an assistant professor at Michigan State University. Her research interests include dc-dc converters, dc-ac inverters, vehicle electrification and energy harvest.

Chunqi Qian received his BS degree in chemistry from Nanjing University and his PhD in physical chemistry from the University of California, Berkeley, in 2007. Following postdoctoral trainings at the National High Magnetic Field Laboratory and the National Institutes of Health, he became a principle investigator in the department of Radiology in Michigan State University. His research interest includes the development and application of RF sensor technology for biomedical research.