Frequency Modulated Parametric Oscillation for Antenna Powered Wireless Transmission of Voltage Sensing Signals

Abstract

Wireless transmission of voltage signals are particularly useful for sensors embedded inside a closed environment where long-term operation without internal batteries is desirable. For this purpose, voltage tuning resonators can be used, because their voltage-dependent frequency responses can be contactlessly characterized by loop antennas connected to the output and input ports of a network analyzer. However, such passive sensors have limited remote detectability and temporal resolution, especially for smaller frequency shifts that would require repetitive averaging for acceptable measurement accuracy. To overcome these limitations, a double frequency parametric resonator is inductively coupled with a voltage tuning resonator to convert resonance frequency shifts of the passive sensor into frequency encoded oscillation signals that can be instantaneously detected over larger distance separations. This antenna powered FM transmitter has a compact design to achieve good voltage sensitivity and linearity, making it potentially useful for multiple applications from PH sensing to electrophysiological recording.

I. Introduction

Many physiological parameters, such as PH [1], metabolite concentrations [2–7], temperature [8] and neuronal activities [9], can be converted to or directly measured as voltage signals. When signal transducers are embedded inside closed body cavities for long-term monitoring, it is advantageous to convert locally measured voltage signals into resonance frequency shifts of an LC resonator that can be readout wirelessly. By applying voltage signals across nonlinear capacitors used inside this LC resonator, a pair of loop antennas connected to the output and input ports of a network analyzer can identify resonance frequency shifts as displacement of transmission (S₂₁) curves. Because such passive resonators require no internal power source, they are particularly suitable for long-term operation inside a closed environment, where neither wired connections nor battery replacement are desirable. Although passive sensors are simple to operate, they are difficult to characterize beyond the near-field region, when RF signals backscattered by the passive sensor are much smaller than residual signals directly coupled from the output to input ports of the network analyzer. Moreover, passive resonators cannot effectively identify frequency shifts much smaller than their own bandwidths, unless repetitive averaging of S₂₁ curves is performed to improve the measurement accuracy over extended period.

To overcome these limitations, voltage-controlled oscillators have been used to encode voltage signals into frequency modulated oscillation signals that can be instantaneously detected over larger distance separations. Such active devices requires rectifiers to convert wirelessly harvested power [10] into direct current, thus takes up extra space and complicates circuit design. Alternatively, a double-frequency parametric oscillator [11] has been proposed as a compact and battery less signal transmitter that can directly convert wireless pumping power into sustained oscillation signals at half of the pumping frequency. Because the pumping frequency is far away from the measurement frequency, RF signals emitted from the parametric oscillator can be easily distinguished from the noisy detection background. Despite of its circuit simplicity, remote characterization of the double frequency oscillator is inherently slow because the resonator’s frequency response needs to be indirectly estimated from the frequency-dependent power curve (that is obtained by multiple measurements performed over a range of pumping frequencies). Therefore, it would be ideal to combine the advantages of a voltage-controlled oscillator with the parametric oscillator, if voltage signals can be frequency encoded into wirelessly powered oscillation signals and remotely detected by instantaneous measurements.

In this work, we are going to construct a triple frequency oscillator whose oscillation signal is frequency modulated by the locally measured voltage. The entire circuit consists of a voltage tuning resonator that is inductively coupled with a double frequency parametric resonator. The voltage tuning resonator contains a pair of varactors that are symmetrically distributed around the resonator’s conductor loop and connected in head-to-head configuration. The resonator’s virtual grounds are attached with a pair of voltage sensing electrodes to modulate varactors’ capacitance and thus the circuit’s resonance frequency. To improve the remote detectability of the voltage tuning resonator, it is inductively coupled with a parametric resonator that can be wirelessly activated by an external dipole antenna. Because the oscillation frequency of the coupled resonators is linearly dependent on the resonance frequency of the voltage tuning resonator, the oscillation frequency shift can indicate voltage variation across the sensing electrodes. In addition to static voltage measurements in the frequency domain, the coupled resonators can also convert dynamic voltages in the time domain into frequency-modulated oscillation signals that can be remotely decoded to recover locally applied waveforms.

II. OPERATION PRINCIPLES

A. Voltage sensing by a single-frequency resonator

Fig. 1a shows a passive sensor based on a single-frequency resonator. This resonator consists of a square conductor (shown in blue) that is symmetrically bridged by two varactor diodes (shown in red) connected in head-to-head configuration. In addition, a large resistor (shown in gray) bridges the gap of the center conductor to neutralize excessive charge across diodes. When a bias voltage V_s is applied across the sensing electrodes (shown in pink) that are connected to the resonator’s virtual grounds, it will change varactors’ capacitance and thus the resonance frequency ω₁:

$${\omega }_1=\sqrt{\frac{2}{L_1{C}_1}}=\sqrt{\frac{2}{L_1{C}_{10}{\left(1-V_s/{\varphi }_1\right)}^{-{\lambda }_1}}}\approx \sqrt{\frac{2}{L_1{C}_{10}}}\left(1-\frac{{\lambda }_1{V}_s}{2{\varphi }_1}\right)\equiv {\omega }_{10}\left(1-\frac{{\lambda }_1{V}_s}{2{\varphi }_1}\right)$$ (1)

where ω₁₀ is the circuit’s resonance frequency under zero-bias. Eq. (1) indicates that when the sensing voltage V_s is much smaller than the diode’s junction potential Φ₁, there is an approximate linear relation between the resonance frequency ω₁ and the sensing voltage V_s. The frequency response of this stand-alone resonator can be directly characterized by a pair of loop antennas connected to the output and input ports of a network analyzer. However, such direct measurement has low sensitivity and temporal resolution, especially when the resonator is distantly separated from the loop antennas.

Fig. 1.

(a) A passive voltage tuning resonator consists of a square loop inductor (labelled in blue) and two varactors (labelled in red) that are connected in head-to-head configuration. The gap in the center conductor is filled by a large resistor R_c (labelled in gray). The sensing electrodes (labelled in pink) will change the junction capacitance C₁ of each varactor, thus the resonator’s transmission (S₂₁) curve that can be detected by a pair of loop antennas (labelled in cyan) placed at close enough distance, (b-d) To improve the remote detectability of the voltage tuning resonator, its sensing information is coupled to a double frequency parametric resonator (labelled in orange) and broadcasted over longer distance separations via frequency-modulated oscillation signals. Through inductive coupling, the pair of resonators create the lower-frequency coupled mode at ω_L when currents in both resonators have the same direction (Fig. b) and the higher-frequency coupled mode at ω_H when currents in both resonators have opposite directions (Fig. c). Because the parametric resonator has a continuous center conductor to bridge its virtual grounds, it has an additional butterfly mode (Fig. d) that doesn’t directly couple with the voltage tuning resonator but necessary for parametric signal mixing.

B. Voltage tuning in a pair of coupled resonators

To improve the remote detectability of the voltage tuning resonator, it is coaxially overlaid with a parametric resonator that operates as a local signal enhancer (Fig. 1b). To explain how the voltage tuning resonator interacts with the parametric resonator, let us analyze the relation between the electro-motive force ξ and the induced current I in each resonator:

$${\xi }_1=I_1{Z}_1+j{I}_2\omega M=I_1(R_1+j\omega L_1+\frac{1}{j\omega C_1})+j{I}_2\omega M$$ (2)

$${\xi }_2=I_2{Z}_2+j{I}_1\omega M=I_2(R_2+j\omega L_2+\frac{1}{j\omega C_2})+j{I}_2\omega M$$ (3)

where R, L and C are the effective resistance, inductance and capacitance of each resonator, M is the mutual inductance. Therefore, the induced current in the parametric resonator is:

$$I_2=\frac{Z_1{\xi }_2-jM\omega {\xi }_1}{Z_1{Z}_2+M^2{\omega }^2}=\frac{\left(R_1+j\omega L_1-j{\omega }_1^2{L}_1/\omega \right){\xi }_2-jM\omega {\xi }_1}{\begin{array}{l}\left(\begin{array}{l}{R}_1{R}_2+M^2{\omega }^2-L_1{L}_2\left({\omega }^2-{\omega }_1^2\right)\left({\omega }^2-{\omega }_2^2\right)/{\omega }^2\hfill \\ +j\left(L_1{R}_2\left({\omega }^2-{\omega }_1^2\right)+L_2{R}_1\left({\omega }^2-{\omega }_2^2\right)\right)/\omega \hfill \end{array}\right)\hfill \end{array}}$$ (4)

where ω₁ and ω₂ are the stand-alone resonance frequencies of the voltage tuning resonator and the parametric resonator respectively. When the resonance frequency of coupled resonators is far away from those of stand-alone resonators, the term R₁ in the numerator of Eq. (4) and R₁R₂ in the denominator can both be neglected. Because I₂ needs to be real-valued at resonance, i.e.

$$M^2{\omega }^2-L_1{L}_2\left({\omega }^2-{\omega }_1^2\right)\left({\omega }^2-{\omega }_2^2\right)/{\omega }^2=0$$ (5)

the lower and higher-valued solutions for Eq. (5) are:

$${\omega }_L^2=\frac{2{\omega }_2^2}{{\omega }_2^2/{\omega }_1^2+1+\sqrt{{\left({\omega }_2^2/{\omega }_1^2-1\right)}^2+4{\kappa }^2{\omega }_2^2/{\omega }_1^2}}$$ (6)

$${\omega }_H^2=\frac{2{\omega }_2^2}{{\omega }_2^2/{\omega }_1^2+1-\sqrt{{\left({\omega }_2^2/{\omega }_1^2-1\right)}^2+4{\kappa }^2{\omega }_2^2/{\omega }_1^2}}$$ (7)

where κ²=M²/(L₁L₂) describes the coupling efficiency between the parametric resonator and the voltage tuning resonator. The lower-valued solution ω_L corresponds to the frequency when currents in both resonators flow along the same direction (Fig. 1b), while the higher valued solution ω_H corresponds to the frequency when current directions are opposite (Fig. 1c). As we will show in the next section, the co-rotating mode will be useful for FM-encoding, while the counter-rotating mode will be useful for harvesting pumping power provided by the external dipole antenna.

By plugging $u_L={\omega }_2^2/{\omega }_L^2$ and $u={\omega }_2^2/{\omega }_1^2$ into Eq. (6),

$$u+1+\sqrt{{(u-1)}^2+4{\kappa }^{2}u}=2u_L$$ (8)

Take derivative with respect to u on both sides of Eq. (8):

$$1+\frac{(u-1)+2{\kappa }^2}{\sqrt{{(u-1)}^2+4{\kappa }^{2}u}}=\frac{2d{u}_L}{du}=\frac{2{\omega }_1^{3}d{\omega }_L}{{\omega }_L^{3}d{\omega }_1}$$ (9)

where the last equality holds because $d{u}_L=-\left(2{\omega }_2^2/{\omega }_L^3\right)d{\omega }_L$ and $du=-\left(2{\omega }_2^2/{\omega }_1^3\right)d{\omega }_1$. Eq. (9) can thus be rearranged as:

$$\begin{gathered} \frac{d{\omega }_L}{d{\omega }_1}=\frac{{\omega }_L^3}{2{\omega }_1^3}\left(1\pm \frac{(u-1)+2{\kappa }^2}{\sqrt{{(u-1)}^2+4{\kappa }^{2}u}}\right) \\ =\left(\frac{{\omega }_L^3}{2{\omega }_1^3}\right)\left(1+\frac{{\omega }_2^2/{\omega }_1^2-1+2\left({\omega }_1^2/{\omega }_L^2-1\right)\left({\omega }_2^2/{\omega }_L^2-1\right)}{\left|2{\omega }_2^2/{\omega }_L^2-{\omega }_2^2/{\omega }_1^2-1\right|}\right) \end{gathered}$$ (10)

The last equality in Eq. (10) holds because the coupling efficiency can be solved from Eq. (6) to be:

$${\kappa }^2=\left({\omega }_1^2/{\omega }_L^2-1\right)\left({\omega }_2^2/{\omega }_L^2-1\right).$$ (11)

Eq. (10) describes how the lower resonance frequency ω_L of the coupled resonators changes with respect to the stand-alone resonance frequency ω₁ of the voltage tuning resonator, ω_L can also be correlated with the sensing voltage V_s by plugging the derivative of Eq. (1) into Eq. (10):

$$\frac{d{\omega }_L}{d{V}_s}=\frac{{\omega }_L^3}{2{\omega }_1^3}\left(1+\frac{{\omega }_2^2/{\omega }_1^2-1+2\left({\omega }_1^2/{\omega }_L^2-1\right)\left({\omega }_2^2/{\omega }_L^2-1\right)}{\left|2{\omega }_2^2/{\omega }_L^2-{\omega }_2^2/{\omega }_1^2-1\right|}\right)\left(\frac{-{\omega }_{10}{\lambda }_1}{2{\varphi }_1}\right)$$ (12)

C. Oscillation frequency affected by resonance frequency

The coupled resonators have better remote detectability because they can convert wireless pumping power into sustained oscillation signals that can be broadcasted over longer distance. Similar as the passive resonator, the parametric resonator consists of a square inductor and two head-to-head connected varactors (labelled in green in Figs. 1b–1d). Besides its dipole mode that couples with the passive resonator as shown in Figs. 1b and 1c, the parametric resonator also has a butterfly resonance mode at ω_br (Fig. 1d) that is created by bridging its virtual grounds with a continuous conductor in its center. This butterfly mode doesn’t directly interact with the passive resonator, but it is necessary to sustain multiband parametric frequency mixing process. By adjusting the distance separation between the coupled resonators, their mutual inductance M could have the proper value to make the sum of resonance frequencies of the co-rotating dipole mode and the butterfly mode approximately equal to the resonance frequency of the counter-rotating mode, i.e. ω_L + ω_br ~ ω_H Thus, the coupled resonators can effectively utilize a pumping signal at ω_p that is near ω_H to modulate the junction capacitance of varactor diodes (labelled in green), leading to sustained oscillation at the co-rotating and butterfly modes. By analyzing the varactor currents at each mode, we have concluded [12] that oscillation can occur at off-resonant frequencies when the ratio between the circuit reactance and resistance are equal:

$$\frac{X_d}{R_d}=\frac{2\left({\omega }_d-{\omega }_L\right)L_d}{R_d}=\frac{X_b}{R_b}=\frac{2\left({\omega }_b-{\omega }_{br}\right)L_b}{R_b}$$ (13)

where ω_d and ω_b are the oscillation frequencies of the co-rotating dipole mode and the butterfly mode, L_d and L_b are effective inductance of the half circuit at ω_d and ω_b (as shown in Fig. 2). By plugging ω_p = ω_d+ ω_b into Eq. (13), the oscillation frequency of the dipole mode can be expressed as:

$${\omega }_d=\frac{{\omega }_L{L}_d/R_d-{\omega }_{br}{L}_b/R_b+{\omega }_p{L}_b/R_b}{L_d/R_d+L_b/R_b}$$ (14)

Eq. (14) indicates that the oscillation frequency ω_d is affected by the circuit’s resonance frequencies (ω_L and ω_br) as well as the pumping frequency ω_p. Because only the co-rotating mode resonance frequency ω_L in Eq. (14) can be affected by the sensing voltage, the oscillation frequency shift ∂ω_d and the resonance frequency shift ∂ω_L are correlated by the partial derivative relation obtained from Eq. (14):

$$\frac{\partial {\omega }_d}{\partial {\omega }_L}=\frac{L_d/R_d}{L_d/R_d+L_b/R_b}=1-\partial {\omega }_d/\partial {\omega }_p$$ (15)

In Eq. (15), ∂ω_d / ∂ω_p can be experimentally measured from the slope of a calibration curve obtained for a zero-biased oscillator when the oscillation frequency ω_d is observed as a function of the pumping frequency ω_p that is swept over a range.

Fig. 2.

(a) Because the co-rotating mode has zero voltage in the center, the coupled resonators can be equivalated to two half circuits sharing the same ground in the center. Each half circuit has an equivalent impedance of R_d+jX_d that also includes contribution from the voltage-tuning resonator due to inductive coupling, (b) For the butterfly mode of the coupled resonators, the circuit is equivalent to two half loops that are isolated in the center because the current flow in each half circuit is limited in its own mesh. The equivalent impedance R_b+jX_b of each half circuit contains impedance contribution only from the parametric resonator itself, (c) The counter-rotating mode can effectively interact with the external dipole antenna if its resonance frequency ω_H is close to the pumping frequency ω_p and if the counter-rotating currents in individual resonators doesn’t cancel.

D. Pumping current on the dipole antenna

To enable efficient oscillation [12], the magnitude of pumping voltage |V_p| across each varactor C₂ should be:

$$\left|V_P\right|\approx \left(C_{20}{\varphi }_2/{\lambda }_2\right)\sqrt{\left(R_d{R}_b+X_d{X}_b\right){\omega }_d{\omega }_b}$$ (16)

where C₂₀ is the varactor’s junction capacitance at zero-biased condition. Such a voltage corresponds to a current I_2P inside the parametric resonator

$$I_{2p}=j{\omega }_p{C}_{20}\left|V_p\right|\approx \left(j{\omega }_p{C}_{20}^2{\varphi }_2/{\lambda }_2\right)\sqrt{\left(R_d{R}_b+X_d{X}_b\right){\omega }_d{\omega }_b}$$ (17)

This level of current can be induced by an external dipole antenna that generates electromotive forces ξ₁ and ξ₂ in each resonator at the pumping frequency ω_p. When the pumping frequency is at resonance, i.e. ω_p = ω_H, by plugging Eq. (5) into Eq. (4) for the counter-rotating mode, the induced pumping current can also be expressed as:

$$I_{2p}\approx \frac{\left({\omega }_p^2{L}_1-{\omega }_1^2{L}_1\right){\xi }_2-{\xi }_1\sqrt{L_1{L}_2\left({\omega }_p^2-{\omega }_1^2\right)\left({\omega }_p^2-{\omega }_2^2\right)}}{L_1{R}_2\left({\omega }_p^2-{\omega }_1^2\right)+L_2{R}_1\left({\omega }_p^2-{\omega }_2^2\right)}$$ (18)

When the pair of coupled resonators have identical inductance L₁ = L₂, the electromotive forces induced by the external antenna are also identical:

$${\xi }_1={\xi }_2=j{I}_a{\omega }_p{M}_a=\frac{j{I}_a{\omega }_p{\mu }_{0}A}{2\pi }\text{ln}\left(1+\frac{A}{D}\right)\equiv \xi$$ (19)

where M_a is the mutual inductance between individual resonators and the antenna, and I_a is the required level of current in the external antenna, A is the side length of parametric resonator, D is the distance separation between the resonator and the antenna. Therefore, Eq. (18) can be further simplified to

$$I_{2p}=\frac{\xi -\xi \sqrt{{\omega }_p^2-{\omega }_2^2}/\sqrt{{\omega }_p^2-{\omega }_1^2}}{R_1\left({\omega }_p^2-{\omega }_2^2\right)/\left({\omega }_p^2-{\omega }_1^2\right)+R_2}$$ (20)

By plugging Eq. (19) into Eq. (20), the antenna efficiency can be expressed as the ratio between the current induced in the parametric resonator and the current applied on the antenna

$$\frac{I_{2p}}{I_a}=\frac{j{\omega }_p{\mu }_{0}A\text{ln}\left(1+\frac{A}{D}\right)\left(1-\sqrt{{\omega }_p^2-{\omega }_2^2}/\sqrt{{\omega }_p^2-{\omega }_1^2}\right)}{2\pi \left(R_1\left({\omega }_p^2-{\omega }_2^2\right)/\left({\omega }_p^2-{\omega }_1^2\right)+R_2\right)}$$ (21)

The required current on the dipole antenna can be estimated by plugging Eq. (17) into Eq. (21):

$$I_a=\frac{\left(\frac{R_1\left({\omega }_p^2-{\omega }_2^2\right)}{\left({\omega }_p^2-{\omega }_1^2\right)}+R_2\right)C_{20}^2\sqrt{\left(R_d{R}_b+X_d{X}_b\right){\omega }_d{\omega }_b}}{(1-\sqrt{{\omega }_p^2-{\omega }_2^2}/\sqrt{{\omega }_p^2-{\omega }_1^2})\left({\mu }_{0}A/2\pi \right)\text{ln}(1+A/D)}\left(\frac{{\varphi }_2}{{\lambda }_2}\right)$$ (22)

where the half circuit resistance of the butterfly mode can be approximated as R_b = R₁/2 = R₂/2 because this mode doesn’t interact with the voltage tuning resonator. Meanwhile, the half circuit resistance of dipole mode can be derived from the following simplified expression when the oscillation signal is on resonance, i.e. ω_d= ω_L:

$$2R_d=R_2+\frac{{\omega }_L^2{M}^2{R}_1}{R_1^2+{\left({\omega }_L{L}_1-1/\left({\omega }_L{C}_1\right)\right)}^2}\approx R_2+\frac{\left({\omega }_L^2-{\omega }_2^2\right)R_1}{{\omega }_L^2-{\omega }_1^2}$$ (23)

The last approximation in Eq. (23) is derived from Eq. (5).

III. MATERIALS AND METHODS

The voltage tuning resonator shown in Fig. 3a was fabricated by printing a 1-cm square pattern on a copper-clad polyimide film, leading to an effective inductance of 25.2 nH. The top and bottom conductor legs had split gaps that were filled by varactor diodes (MA27V0400L, Panasonic, Japan). These two diodes were connected in head-to-head configuration, so that a bias voltage applied across the diode pair would modulate the circuit resonance frequency. As a passive sensor, the bias voltage can be supplied by a pair of electrodes that were connected to the resonator’s virtual grounds thus minimize their interference with RF currents. To stabilize the circuit’s resonance, excessive charges accumulated across the varactors were neutralized by a 237-kOhm shunting resistor connected between the sensing electrodes. The resonator’s frequency response was then characterized by S₂₁ measurement, using a pair of partially overlapped loop antennas connected to the output and input ports of a network analyzer. These two antenna loops were respectively used to send and to receive backscattered signals from the passive voltage sensor separated by less than 2.5 cm distance. Under zero-bias condition, the sensor was measured to have a resonance frequency at 343.9 MHz (Q = 68). Subsequently, the resonance frequency was measured for each bias voltage applied on the sensing electrodes through a pair of 620-nH RF chokes that were optionally introduced.

Fig. 3.

(a) The picture of the passive resonator that includes two varactor diodes C₁ (MA27V0400L, Panasonic, Japan) filling the gaps in the top and bottom conductors and one resistor R_c=237 kOhm filling the gap in the center conductor. A pair of voltage tuning electrodes are connected to the left and right ends of the resonator, (b) The picture of the parametric resonator that includes two varactor diodes C₂ (MA2737600L, Panasonic, Japan) filling the gaps in the top and bottom conductors and one continuous center conductor, (c) When these two resonators are coaxially overlaid, they create a parametric oscillator with three resonance modes. Powered by an external dipole antenna, the oscillator can convert the low-frequency bias voltages (applied through the RF chokes) into frequency-modulated oscillation signals that can be detected by loop antennas that are separated over larger distances. After passing through a low-pass filter, oscillation signals were observed in the frequency domain on a network analyzer. Alternatively, oscillation signals were mixed down to below 2 MHz and observed in the time domain on an oscilloscope.

The parametric resonator shown in Fig. 3b had an almost identical conductor pattern. But unlike the passive resonator described above, the parametric resonator had a continuous center conductor, with its top and bottom conductor gaps filled by varactor diodes with larger junction capacitance (MA2737600L, Panasonic, Japan). As a result, the parametric resonator had one dipole mode resonance at 292.5 MHz (Q = 58) and another butterfly mode resonance at 230.1 MHz (Q = 74). Subsequently, the parametric resonator was coaxially overlaid with the passive resonator and separated from the later by a 0.9-mm thick polyimide substrate so that the resonance frequencies of the co-rotating and the counter-rotating modes were 251.4 MHz and 481 MHz respectively. A dipole antenna was then placed approximately coplanar with the sensor to provide pumping power. As shown in Fig. 3c, when the dipole antenna was separated from the oscillator’s circuit edge by ~5 cm, about 16 mW of pumping power was required at 481.5 MHz to make the oscillation signal observable as a sharp peak standing above the noise floor on a network analyzer. The oscillation frequency was then measured for each bias voltage applied.

In addition to frequency-domain measurements, the sensor’s oscillation signal was also observed in the time domain when a series of square pulses were applied on the sensing electrodes. Each square pulse lasted for 2 ms with a repetition rate of 250 Hz. The oscillation signal at ~251.4 MHz that was received by the loop antenna was down-converted by a 251.2-MHz reference signal through a mixer (ZFY-2, Minicircuits, Brooklyn, NY). The down-converted signal was then fed into an oscilloscope via a low-pass filter (BLP-1.9+, Minicircuits, Brooklyn, NY) before being digitally sampled at 3.125 MHz and exported to Matlab (Mathworks, Natick, MA) for post processing. Using the resample function remove high frequency signals above 625 kHz, the down-sampled signal was combined with its Hilbert transformation to create a complex signal in the time domain. This complex signal was then low pass filtered by the Hamming window before its phase angle was retrieved by the angle and unwrap functions. After linear baseline correction using the detrend function, the time-dependent phase angle was derivatized by the differential filter (differentiatorfir). In the decoded waveform, the measurement noise was then estimated as the standard deviation of the baseline.

IV. SIMULATION

The circuit’s frequency response was simulated by S-parameters solver in the Advanced Design System (Agilent, CA). To model the voltage tuning resonator (labelled in green) with a stand-alone resonance frequency at 343.9 MHz, the resonator was represented as serial combination of two 12.6-nH inductors and two varactor diodes (labelled by MA27V0400L). Based on its datasheet value, the voltage-dependent junction capacitance of each diode was C=17.0/(1+V/1.55)^0.96. These two varactors were biased by a DC voltage (V_dc) that was connected to the resonator via RF chokes (labelled in gray). The internal resistance of each half circuit was set to 0.4 Ohm to make the resonator’s quality factor Q = 68, as experimentally measured at 343.9 MHz. To model the parametric resonator (labelled in red) with a stand-alone dipole mode resonance at 292.5 MHz, the resonator was represented as a symmetric circuit, where each half consisted of a 12.6-nH inductor placed in series with a varactor diode (labeled by MA2730600L). Based on its datasheet value, the voltage-dependent junction capacitance of this diode was C=23.5/(1+V/2.20)^1.32. To create the butterfly mode at 230.1 MHz, an effective inductance of 3.9 nH was placed in the center. Two 0.4-Ohm resistors were again incorporated into each half circuit to emulate circuit loss.

The mutual coupling between the voltage tuning resonator (labelled in green) and the parametric resonator (labelled in red) was defined by Symbolically Defined Devices (labelled in blue). In each of these two-node devices, the induced voltage in node 1 was expressed as F[1,0]=(_v1). This induced voltage was equal to the first-order time derivative of current in node 2 multiplied by mutual inductance, i.e. F[1,1]=(_i2)*M. Similar expressions were used to describe the equality between node 2 voltage F[2,0]=(_v2) and first-order time derivative of current in node 1 multiplied by mutual inductance, i.e. F[2,1]=(_i2) *M. The mutual inductance M was numerically adjusted to 6.99 nH until the coupled resonators had resonance frequencies at 251.4 MHz and 480.9 MHz.

To provide pumping power, a dipole antenna was modeled as a 50-Ohm matched port (labelled by P_nTone) that was inductively coupled to both resonators. The antenna’s mutual inductance with each resonator was estimated to be (μ₀A/2π)ln(1+A/D) = 0.36 nH, where the side length A of the parametric resonator was 1 cm and the distance separation D between the resonator and the antenna was 5 cm. Their mutual inductances were again defined by Symbolically Defined Devices labelled in purple and orange. (Each half circuit of the parametric resonator had a mutual inductance of 0.18 nH with the dipole antenna to account for 0.36 nH in total.) Harmonic balancing solver was then used to simulate the parametric frequency mixing process. By empirically adjusting the input power level to 12 dBm for pumping signal at 481.5 MHz, the current flow inside the parametric resonator reached a maximum at 251.4 MHz. At this time, the induced current on the parametric resonator and the pumping current on the dipole antenna were simulated to be 2 mA and 25 mA, corresponding to a current transfer efficiency of 8%. These simulated current values were in good agreement with the value predicted by Eqns. (21) and (17), given λ₂/Φ₂ = 0.60, ω₁ = ω₁₀= 292.5 MHz, ω₂ = 343.9 MHz, ω_L= 251.4 MHz, R_b≈ 0.4 Ohm and R_d ≈ 0.56 Ohm as estimated from Eq. (22). The optimal pumping power of 12 dBm as obtained by simulation was also very close to the actual pumping power (~16 mW) used in our experiment, leading to an estimated level of power consumption of −28 dBm in the coupled resonators.

V. RESULTS AND DISCUSSION

The linear relation between the resonance frequency shift and the bias voltage was first verified in numerical simulation. As shown by the red curve in Fig. 5a, every 1-mV increase in bias voltage would lead to 0.1077 MHz decrease in resonance frequency of the voltage tuning resonator in its stand-alone configuration. This linear relation was also experimentally observed from the peak response shifts of S₂₁ curves obtained by a pair of loop antennas that were placed at a close enough distance (i.e. 2.5 cm) from the resonator and that were connected to the output and input ports of a network analyzer. As shown by the orange curve in Fig. 5b, the resonance frequency of the stand-alone resonator decreased at a rate of 0.1083 MHz/mV. To obtain reasonable measurement accuracy, a large enough bias voltage of 10 mV was applied across the sensing electrodes to induce a 1.08-MHz resonance frequency shift that was large enough to identify within a single acquisition. The standard deviation for each measurement point was 0.18 MHz, corresponding to a coefficient variation of 17% for a 1.08-MHz shift induced by a 10-mV bias voltage.

Fig. 5.

The red curve describes the simulated relation between the forward bias voltage and the resonance frequency shift of the voltage tuning resonator in its stand-alone configuration. As shown by the blue curve, the voltage-dependent frequency response of the stand-alone resonator was also experimentally validated by the peak response shifts of S₂₁ curves that were obtained by a pair of loop antennas separated by 2.5-cm distance. The resonance frequency shift for each bias voltage had an average error bar of 0.18 MHz.

Although a passive sensor can measure voltage conveniently, precise measurement of smaller frequency shift requires repetitive averaging of its S₂₁ curves, which would take extra time. Moreover, when the passive sensor was separated from the loop antennas by a larger distance, its back-scattered signals could be too weak to observe. For example, when the loop antennas were separated from the passive sensor by 5 cm (that was 5-fold the sensor’s own dimension), the S₂₁ curve was completely buried beneath the noise floor (as shown in Fig. 6a). But when the parametric oscillator was used for voltage sensing, its oscillation signals could be clearly identified as sharp peaks standing above the noise floor (Fig. 6b). In addition to improved remote detectability, the sharp linewidths of oscillation peaks would also improve the measurement precision of small frequency shifts. For example, the line width of each peak in Fig. 6b is much smaller than the ~18.5-kHz frequency shift induced by every 1.2-mV voltage change. The coupled resonators were also characterized at 10 cm (a larger distance separation that was about 10-fold the device’s own dimension). As shown in Fig. 7a, every 1-mV increase in forward bias voltage would lead to 0.0154-MHz decrease in oscillation frequency. The standard deviation for each measurement point was 0.002 MHz, corresponding to a coefficient variation of 1.3% for a 0.154-MHz shift induced by a 10-mV bias voltage. This level of coefficient variation is already 13-fold smaller than that of the passive sensor. The voltage dependent frequency shift shown in Fig. 7a is consistent with the relation obtained in Fig. 6b, demonstrating the oscillator’s improved applicability and detectability over passive sensors, especially for larger distance separations. To experimentally correlate oscillation frequency shift with resonance frequency change, we calibrated the zero-biased oscillator by sweeping its pumping frequency. As shown by the calibration curve in Fig. 7b, every 1-MHz increase in pumping frequency would lead to 0.482-MHz increase in oscillation frequency. According to Eq. (15), the oscillation frequency shift ∂ω_d and the resonance frequency change ∂ω_L were correlated by ∂ω_d / ∂ω_L = 1 – ∂ω_d / ∂ω_p = 0.518. After dividing the curve’s slope in Fig. 7a by ∂ω_d / ∂ω_L, we could indirectly estimate the resonance frequency of the coupled resonators to decrease at a rate of 0.0297 MHz/mV, which was in close agreement with the value predicted by Eq. (12), given λ₁/Φ₁=0.62, ω₁ = ω₁₀ = 343.9 MHz, ω₂ = 292.5 MHz, ω_L= 251.4 MHz.

Fig. 6.

(a) The stand-alone voltage tuning resonator had non-detectable backscattered signals when it was separated from the loop antenna by 5 cm (that was 5-fold the resonator’s own dimension), (b) When the voltage tuning resonator was coupled with a parametric resonator to create a triple-frequency parametric oscillator, they could be wirelessly activated by a dipole antenna to emit frequency-encoded oscillation signals that could clearly stand above the measurement baseline. As a result, frequency shifts induced by small bias voltages could be sensitively discerned.

Fig. 7.

(a) The linear relation between the oscillation frequency shift and the bias voltage was validated at 10-cm distance separation, (b) The oscillation frequency was also measured as a function of pumping frequency when the oscillator was zero-biased, so that the linear correlation between oscillation frequency shift and resonance frequency change could be estimated from Eq. (15). The average error bar for each oscillation frequency shift was 2 kHz, which was primarily determined by the frequency resolution of the network analyzer over a 0.6-MHz sweep range.

In addition to improved remote detectability of static voltages in the frequency domain, the coupled resonators could also improve the temporal resolution for dynamic voltages in the time domain. Fig. 8 shows the waveforms decoded from frequency modulated oscillation signals when square pulses of different amplitudes were applied on the sensing electrodes. As expected, every 2-fold increase in applied voltage would lead to 2-fold increase in the plateau height, enabling reliable recovery of the pulse shapes at 10-cm distance separation. The standard deviation of the fluctuating baseline during the pulse intervals was ~72 Hz, corresponding to a minimum detectable voltage of 4.7 μV applied on the sensing electrodes. Compared to frequency domain measurements (in Fig. 7a) whose frequency resolution is predetermined by the limited number of sampling points in frequency sweeps, time domain measurements could reduce the coefficient variation by another factor of 28 due to improved frequency resolution at higher sampling rate. This improved measurement precision and speed would enable potential applications for sensitive detection of extracellular neuronal spikes [13]. Using an effective sampling rate of 625 kHz, the oscillator could detect voltage amplitudes as large as 20.3 mV, when the frequency deviation was 1.54 MHz (i.e. about half of the sampling rate at 3.125 MHz). Higher sampling rate would further increase the maximum detectable voltages, but at the cost of larger equivalent noise that would compromise the SNR for smaller voltage signals.

Fig. 8.

Voltage wave forms obtained by digitally demodulating oscillation signals emitted by the parametric oscillator at 10-cm distance separation, when the pulsed voltages applied on the sensing electrodes were (a) 73 μV(b) 147 μV (c) 294 μV (d) 588 μV (e) 1.176 mV (f) 2.352 mV(g) 4.703 mV(h) 9.406 mV.

In this work, we have fabricated a triple-frequency parametric oscillator that can operate as a wirelessly powered FM transmitter to improve the remote detectability and temporal resolution for wireless voltage sensing. This is an important improvement over our previous design where the oscillation frequency of mechanically deformable resonators was utilized only for static pressure estimation [12]. Unlike amplitude based voltage measurements [14–16], FM-encoding is more immune to amplitude fluctuations that may arise from positional variations of the voltage sensor during environmental perturbation or physiological motion. Because parametric oscillation doesn’t involve on-off switching of diode junctions, it will require smaller pumping power to sustain circuit oscillation, compared to transistor-based oscillators that often use rectifiers for wireless power harvesting and conversion.

The triple frequency oscillator has a simple design. It consists of a double-frequency parametric resonator that is coaxially overlaid with a single-frequency voltage tuning resonator. Each resonator was fabricated by soldering two varactor diodes on a square circuit pattern for a cost of less than $0.5. Through mutual inductive coupling, the linear frequency shift of the voltage tuning resonator will linearly modulate the oscillation frequency of the coupled resonators, so that the frequency-encoded oscillation signal can be detected by loop antennas over larger distance separations. For a certain bias voltage applied across the sensing electrodes, the oscillation frequency shift is scaled by ~14% with respect to the resonance frequency shift of the voltage tuning resonator, based on the slope ratio between Fig. 7a and Fig.5. Despite of this scaling factor, the greatly reduced linewidth of oscillation signal in Fig. 6b compared to the broad S₂₁ curve of the passive resonator will enable precise measurement of small frequency shift instantaneously, and without the need for repetitive averaging that is normally used to improve measurement precision for passive sensors.

One additional design feature of our design is that the voltage tuning resonator can also improve the effectiveness of pumping power. When the voltage tuning resonator is separated from the parametric resonator by 0.9 mm, the resonance frequency of the counter-rotating mode is approximately equal to the sum of the resonance frequencies corresponding of the co-rotating mode and the butterfly mode. In addition to this frequency matching consideration, efficient excitation of the coupled resonators also requires them to have non-zero net magnetic dipole in their counter-rotating mode. (To make the resonators’ inductive coupling more effective, it is advantageous to bring their resonance frequencies ω₁ and ω₂ closer. But if ω₁ and ω₂ are too close, these two coupled resonators will have opposite current flows with almost identical magnitudes at the pumping frequency, thus reduce their effective interactions with the dipole antenna.) In our design, the resonance frequency of the voltage tuning resonator is about 1.18-fold larger than the parametric resonator, so that the coupled resonator pair can be effectively excited by ~16 mW of power applied on a dipole antenna that is separated by a distance about 5-fold the device’s own dimension. This level of pumping power is only ~20% of the power required for non-resonant pumping when the counter-rotating mode was not effectively used for wireless power harvesting [12], thus further reduce the SAR.

One practical consideration for the parametric oscillator is its achievable internal impedance. To encode sensing voltages as stable resonance frequency shifts, the varactor pair in the voltage tuning resonator are shunted by a large resistor that can neutralize excessive charges while minimizing internal voltage drop. It is beneficial to make the shunt resistor sufficiently large in order to reduce voltage attenuation, especially for real electrophysiological signals with limited drivability. However, because the bandwidth of an RC circuit is 1/(2πRC_e), too larger a center resistor will also reduce the detection bandwidth. In this work, a 237-kOhm resistor is connected in parallel with an effective capacitance of 34 pF, producing a detection bandwidth of ~20 kHz. This bandwidth is suitable for signals from DC to the audio frequency range. For higher frequency signals in the MHz range, it might also be possible to substitute the center resistor with a large valued inductor to create high impedance at a certain RF frequency. More work is going on towards this direction.

VI. CONCLUSION

We have fabricated a triple frequency parametric oscillator that can directly utilize antenna power to frequency encode locally detected voltages into sustained oscillation signals. Because the oscillation frequency is distinctly different from the pumping frequency, oscillation signals emitted from the parametric oscillator can be remotely identified in both the frequency and time domain. The device was fabricated by low-cost components and bio-inert polyimide substrate to operate without internal power source. Future work may involve the combined choice of higher operation frequencies and higher-valued varactors to further reduce the device dimension. Compared to other types of backscattering methods based on principles of multiband signal mixing [17–21], the triple-frequency parametric oscillator has combined the desirable features of previous designs (Table I). It can encode DC voltages as well as time-dependent voltages, thus pave way for multiple applications from humidity sensing [22] to electrophysiological monitoring [23].

TABLE I:

Summary of different types of wireless sensors.

	[1], [9], [14–20]*	[10]**	This work***
Type	Backscatter	Voltage controlled oscillator	FM parametric oscillator
Power consumption	Passive	−22 dBm (DC)	−28 dBm (RF)
Speed	< 3 kHz	--	~ 20 kHz
Resolution	20 μV	0.5%	4.7 μV
Sensor size	10 × 9 mm2	8 × 8 cm2	10 × 10 mm2
$\frac{\mathit{\text{Distance}}}{\mathit{\text{Sensor}}\mathit{\text{Size}}}$	< 1.5	3.8	10

^*Ref [19] reported the best voltage resolution and smallest dimension.

^**The VCO consumed −22 dBm of power wirelessly harvested via a rectifier. Its sensing resolution was determined by its 0.5% fluctuation in oscillation frequency. This VCO was characterized at a distance that was 3.8-fold its own dimension.

^***The sensor’s power consumption was calculated from Joule’s law. Its voltage resolution was estimated from the baseline fluctuation of time-domain signals.

Fig. 4.

The circuit diagram created inside Advanced Design System. The voltage tuning resonator (labelled in green) and the parametric resonator (labelled in red) were individually simulated by S-parameters solver to confirm their stand-alone resonance frequencies before they were inductively coupled together by the Symbolically Defined Devices (labelled in blue). These two coupled resonators were inductively excited by a pumping signal provided by a 50-Ohm matched dipole antenna (represented as P_nTone). Using Harmonic Balance solver, the power level at the pumping frequency was empirically adjusted until the signal current in the parametric resonator reached a maximum at the RF frequency (251.4 MHz).

Biographies

Wei Qian received the B.S. and M.S. degrees from Wuhan University (China) in 2003 and 2006, and the Ph.D. degree from Michigan State University, East Lansing, in 2011. She is currently an assistant professor at the Michigan State University with research focus on power electronics and energy harvesting.

Chunqi Qian received his BS degree in chemistry from Nanjing University (China) in 2002 and his PhD from the University of California, Berkeley in 2007. He is currently a principle investigator at the Michigan State University, focusing on the development of wireless sensors.