Non-Foster acoustic radiation from an active piezoelectric transducer

Significance

Many of the exciting wave phenomena found in acoustic metamaterials arise as a result of the collective response of subwavelength resonators. To widen the frequency range of these phenomena, it is necessary to understand the source of these bandwidth limitations and explore techniques that can overcome them. Here, we quantify the general tradeoff between size and bandwidth of passive acoustic resonators and show how this constraint can be overcome by actively loading them. As an example, we demonstrate a large bandwidth enhancement of a sonar transducer by connecting it to a specialized active non-Foster circuit. This important finding can have long-lasting impact in acoustic metamaterials for exotic sound interactions, paving the way to enhanced compact resonant acoustic devices.

Abstract

The quality factor of a passive, linear, small acoustic radiator is fundamentally limited by its volume normalized to the emitted wavelength, imposing severe constraints on the bandwidth and efficiency of compact acoustic sources and of metamaterials composed of arrangements of small acoustic resonators. We demonstrate that these bounds can be overcome by loading a piezoelectric transducer with a non-Foster active circuit, showing that its radiation bandwidth and efficiency can be largely extended beyond what is possible in passive radiators, fundamentally limited only by stability considerations. Based on these principles, we experimentally observe a threefold bandwidth enhancement compared to its passive counterpart, paving the way toward non-Foster acoustic radiation and more broadly active metamaterials that overcome the bandwidth constraints hindering passive systems.

Techniques to widen the bandwidth over which a system resonates are technologically important in various settings and technologies given the ubiquity of resonances in wave-based systems. Radiation sources, such as electromagnetic antennas and acoustic loudspeakers, typically operate near a resonance to ensure efficient radiation toward a distant receiver. Resonances are also crucially important in the broad field of metamaterials, structures with exotic bulk responses arising from subwavelength elements whose resonance ensures strong wave-matter interactions in small volumes. Acoustic metamaterials supporting novel acoustic phenomena, such as negative refraction and cloaking, have been recently demonstrated, but their unusual features are inherently limited in frequency because of the narrowband resonances of the underlying unit elements (1–3).

In electromagnetics, the well-established tradeoff between bandwidth and electrical volume of a small resonant antenna (4) is known as the Chu limit, and it provides an important benchmark for antenna design, setting a lower bound on the fractional bandwidth of the antenna given its size (5–8). Extending the bandwidth beyond this bound is possible only by breaking one or more of its underlying assumptions, notably linearity, passivity, and time invariance. Indeed, temporal modulations of the antenna element or the connected circuitry can overcome this limit, as demonstrated in direct antenna modulation approaches (9, 10) and in switched (11–13) or parametrically modulated (14) systems. Mechanically moving antennas have also been shown to radiate with bandwidths beyond the Chu limit (15, 16). Another approach to build broadband electrically small antennas consists in loading them with active circuitry, embedding powered components such as transistors or operational amplifiers (op-amps) to break the passivity condition. This is commonly done by synthesizing a circuit that violates Foster’s reactance theorem (i.e., with a reactance that decreases with frequency, e.g., in a negative impedance converter [NIC] circuit topology) (17–23). Such non-Foster circuits have also been recently employed to realize parity-time symmetric systems (24, 25). While, in principle, the bandwidth of non-Foster antennas can be infinitely extended (26), the main issue that limits the performance of these devices is their stability (27–30).

In the following, we first extend the Chu limit to acoustic radiators, showcasing relevant differences between the electromagnetic and acoustic wave domains. We then explore ways to overcome this limit and broaden the bandwidth of small acoustic sources using non-Foster impedance matching. In this context, time modulation has been recently explored to improve the low-frequency response of a sonar projector (31, 32). Here, instead, we explore the integration of non-Foster circuit loads in acoustic piezoelectric transducers, commonly used in airborne and waterborne sound transmission, showing a viable path to enhance bandwidth and efficiency of compact sound emitters.

Acoustic Chu Limit

The quality factor $Q=\omega W/P$ of a resonant sound source is defined as the dimensionless ratio of the time-averaged stored energy, W (units of joules), at angular frequency ω to the power P (units of watts) lost in radiation or dissipation, and it is inversely related to its fractional bandwidth. Following pioneering works in electromagnetics (4, 7, 8), here we derive a lower bound for Q assuming that the radiator is linear, passive, and time invariant. As sketched in Fig. 1, we consider a sphere of radius a that encloses in its entirety the sound source. In evaluating W, we ignore the additional energy stored inside this spherical region, which depends on the specific implementation of the radiating element, and calculate the stored energy in the acoustic near fields outside the sphere. This will provide a lower bound to the radiative Q, which can be approached minimizing the stored energy within the source. We also ignore dissipation losses possibly arising within the sphere. They can certainly lower the overall Q, but only at the expenses of lowering the overall radiation efficiency, which is typically not a viable solution to broaden the bandwidth.

Fig. 1.

An acoustic source connected to a non-Foster element to overcome the Chu bound. The emitted sound field consists of inductive energy stored in the near fields and resistive energy lost in radiation to the far field. Adding a capacitive element can create a resonance with a Q constrained by the size of the radiator. However, with the use of a non-Foster element such as a negative inductor, the constraint can be overcome.

The total stored energy in the near fields of an acoustic source is given by the sum of kinetic and potential energy contributions. In order to induce a resonance so that the source can efficiently radiate the input energy, an additional reactive element must be added to impedance match the structure. This tuning reactive element must store sufficient energy to make up for the difference between kinetic and potential energies around the source. After impedance matching, the total stored energy is twice the value of either the kinetic or potential energy in the near fields, depending on which is larger: $W=2\text{max}\left\{W_k,W_p\right\}$. Assuming no losses other than those associated with radiation, the quality factor at resonance is therefore

$$Q=\frac{2\omega \text{max}\left\{W_k,W_p\right\}}{P_r}.$$ [1]

Following the procedure described in Materials and Methods, we derive the quality factor in terms of each outward propagating spherical harmonics with index l, with Q_l being the quality factor when the radiation consists solely of harmonic l. As shown in Fig. 2, the values of Q_l increase with index l. As the total Q is a weighted sum of these individual values, as a function of the different contributions of each harmonic to the overall radiation, the minimum achievable Q is Q₀. Therefore, the minimum quality factor for a small acoustic radiator scales with the radius of the sphere enclosing the source as 1/ka, where k is the wavenumber and a the radius, and it can be approached for a purely monopolar source. In electromagnetics, this limit instead scales as 1/(ka)³, attained for electric or magnetic dipolar radiation; hence, our derivation shows that acoustic monopolar sources offer broader bandwidths than electromagnetic antennas in the subwavelength limit. In order to approach this lower bound, a passive acoustic source must store as little energy as possible in the region r < a. As the size of a passive radiator shrinks, the fields at resonance grow close to the origin, contributing more to the near-field stored energy than to the radiated energy. This fundamentally limits the overall achievable bandwidth.

Fig. 2.

Quality factors for multipolar orders 0 to 3. (A) Expressions for acoustic and electromagnetic multipoles. The Chu bound (final row) is the lowest possible Q in the limit $ka\ll 1$. (B) Plot of the acoustic Q showing the divergence for subwavelength radiators. The lowest curve, corresponding to Q₀, is the ultimate bound for acoustic radiators.

Broadband Impedance Matching Using Non-Foster Circuits

At resonance, the cancellation of excess kinetic or potential stored energy through an impedance matching element makes sure that the input reactance of the system goes to zero. While a system largely detuned by its natural resonance has a very large reactance, corresponding to the large stored energy in its near fields, we induce a subwavelength resonance by loading it with a reactive lumped element that compensates for it. However, this form of impedance matching can be achieved only at a single frequency when considering passive loads, due to the fact that Foster’s reactance theorem requires both the input and load reactance to grow with frequency. On the contrary, an active non-Foster matching element, not bound by this passivity constraint, can compensate the input reactance over a broad frequency range.

We consider, as a common acoustic source, an electroacoustic transducer, which radiates an acoustic signal proportional to the input voltage signal applied at its terminal and finds widespread use in research and commercial applications. A common way to couple the electrical and acoustic domains is the piezoelectric effect, based on which an applied electric field creates mechanical strain in the material. When the structure is small relative to the radiation wavelength, a lumped-parameter circuit model can well describe the piezoelectric response, as shown in Fig. 3A. An input voltage is applied to the clamped electrical capacitance C_e and to the mechanical portion of the circuit through a transformer with coupling constant N. The mechanical portion of the circuit is a resonant mass spring system moving at velocity, v, and radiating into the surrounding medium with radiation impedance, Z_rad.

Fig. 3.

Non-Foster cancellation of input susceptance in a piezoelectric radiator. (A) An acoustic piezoelectric circuit model with tuning element, electrical capacitance, electromechanical turns ratio, mechanical resistor-inductor-capacitor (RLC) components, and radiation impedance. (B and C) The total susceptance of the circuit (dashed) is the sum of the transducer (green) and tuning (blue) susceptances. (B) The total susceptance is small only around one frequency when the transducer is tuned with an inductor. (C) The susceptance is cancelled over a broad frequency range when tuned with a non-Foster negative capacitance.

These transducers are commonly operated in the frequency range around their mechanical resonance, for which the reactance in the mechanical branch is small, while the electrical capacitance dominates the reactive response of the system. A common technique to improve the transducer performance around this frequency is to use a tuning inductor to compensate for this capacitance (33). The overall quality factor of the dissipation-free source is therefore typically larger than the Chu limit due to the additional presence of stored energy in the reactive elements C_e, L_m, and C_m, in addition to the energy in the tuning inductor. At resonance, the wavelength of the elastic wave is generally on the same order of magnitude as the size of the structure. The wave speed of commonly used piezoelectric ceramics is about ten times higher than the one in air and about twice than in water, so the source typically operates at a moderate value of ka, with k being the wavenumber in the surrounding medium. The Chu limit is not particularly constraining at these frequencies since ka is not small.

Here, we are interested in the source operation in the deeply subwavelength regime. In this frequency range the transducer response is capacitive, as seen in Fig. 3, where we show the input susceptance, which linearly increases at low frequencies. A resonance can be induced in this range by using a tuning inductor with a larger inductance than the typical tuning inductor. However, the susceptance of a passive inductor, negative at all frequencies, can cancel the input capacitance only at a single frequency in this low-frequency regime (Fig. 3B). This is due to Foster’s reactance theorem, which requires the reactance (or susceptance) of any passive, lossless circuit to always increase with frequency. In the context of Fig. 3B, the slope of any possible blue curve can only be positive and therefore cancel the low-frequency positive slope of the transducer (green curve) at one or more isolated frequencies.

By breaking the assumption of passivity, the slope of the matching reactance can be made negative using a non-Foster component, and the transducer’s low-frequency susceptance can be cancelled over a broader frequency range. The susceptance of a negative capacitor is the mirror reflection of the transducer’s positive capacitance, and it can compensate the capacitive response over a broad frequency range (Fig. 3C). A source connected to this transducer with cancelled total susceptance can then radiate with high efficiency over the entire low-frequency range.

When deriving the lower bound on Q in the previous section, we did not make any assumption on the specifics of the radiator and, in particular, on the additional energy stored in the piezoelectric element, which largely contributes to the total reactance, resulting in a Q factor larger than the lower bound. A non-Foster matching element, powered by its own energy source, can partially compensate for the stored energies in the transducer and acoustic near fields, offering an effective way to lower the overall Q factor and extend the resonance bandwidth well beyond what achievable with a passive tuning inductor.

Non-Foster Piezoelectric Radiator

We employ an NIC to realize our non-Foster matching load. An NIC circuit inverts the sign of the impedance of a circuit element by flipping the sign of its input voltage or current. For example, a current inversion NIC can transform the impedance of a component from Z to −Z by supplying current to a source instead of drawing current from it in response to an applied voltage. The flow can be reversed because the NIC is an active circuit with its own power source. These circuits have historically found use in the design of active resistor-capacitor (RC) filters and, more recently, in the study of non-Hermitian and parity-time symmetric systems. NIC circuits can be synthesized using op-amps or transistors. Floating (ungrounded) NICs require the use of multiple op-amps (34). We choose instead to use a grounded NIC topology, which can be realized with a single op-amp. This choice necessitates the use of a current source or a voltage source with a large internal resistance, R_s. The goal is to combine this grounded NIC negative capacitor in parallel with a piezoelectric transducer to create a broad low-frequency resonance (Fig. 4).

Fig. 4.

A non-Foster piezoelectric radiator. (A) An NIC is used to induce a low-frequency resonance in a piezoelectric transducer. (B) The real part (resistance) and imaginary part (reactance) of the input impedance to the NIC circuit, showing experimental (solid) and theoretical (dashed) results. The NIC is composed of a single op-amp whose feedback gain is controlled by two resistors and turns capacitor C_i into a negative capacitor with a non-Foster, negatively sloped reactance.

The NIC negative capacitor topology, shown in the inset of Fig. 4B, is a noninverting amplifier with the capacitor placed between its input and output. The complex impedance of the NIC is a function of the closed-loop gain G,

$$Z_{\text{NIC}}=\frac{1}{s{C}_i\left(1-G\right)},$$ [2]

where s is the complex frequency. When the closed-loop gain is 2, the NIC impedance is the one of a negative capacitor, −C_i. However, since the gain is a dispersive complex function, there is an accompanying real part of Z_NIC, and the effective negative capacitance is bandwidth-limited. This limitation is captured by modeling the open loop gain of the op-amp with a single pole model, $A_0/\left(1+s/{\omega }_p\right)$, where A₀ is the zero-frequency open loop gain and ω_p is the 3 dB frequency. The closed-loop gain is then

$$G=\frac{A_0}{1+\frac{s}{{\omega }_p}+\frac{A_0}{G_0}}.$$ [3]

Resistors R_f and R_g form a voltage divider where G₀ is the reciprocal of the voltage division ratio: $G_0=1+R_f/R_g$. The amount of closed-loop feedback is controlled by adjusting the ratio of R_f and R_g. The desired closed-loop feedback gain is much less than the zero-frequency open loop gain of a typical op-amp. This feedback gain is achieved by keeping G₀ small such that $G_0\ll A_0$. As seen from Eq. 3, the feedback gain G is then given by G₀ at low frequencies.

We physically implemented this NIC circuit and measured its input impedance to compare with our circuit model predictions (Fig. 4B), finding an overall good agreement. We observe the desired non-Foster negative slope of the negative capacitor in the frequency range of interest with some accompanying negative resistance. The synthesized non-Foster element can be used to induce a broadband low-frequency resonance in our piezoelectric transducer.

Experimental Realization of a Non-Foster Acoustic Source

The non-Foster acoustic source is completed by placing the fabricated NIC circuit (Fig. 5A) in parallel with a cylindrical piezoelectric transducer (Fig. 5B) and a waveform generator. A swept sine input is used for input, and the velocity of the transducer is measured from 0 to 45 kHz with a laser Doppler vibrometer (Fig. 5C). We see that the amount of negative capacitance provided by the NIC is enough to induce a resonance near 20 kHz, significantly lower than the transducer’s 120 kHz natural resonance. As a point of comparison, the same measurement is performed when an inductor is used as the tuning element. The inductor is subject to Foster’s reactance theorem, and the bandwidth is limited. With the Foster-violating NIC circuit, we see that the bandwidth shows a 320% improvement. The peak level of the NIC response is also modestly higher, resulting in a combined gain bandwidth improvement of 450%.

Fig. 5.

Bandwidth of a non-Foster acoustic source. (A) Implementation of the NIC circuit. External connections are to the cylindrical piezoelectric transducer, shown in B, to the power source for the op-amp, and to the function generator. (C and D) Experimental and modeled velocity spectra of the piezoelectric transducer when connected to the Foster (blue) and non-Foster (red) tuning elements. The non-Foster tuning extends the measured low-frequency resonant bandwidth while ensuring stability.

The theoretical model of the non-Foster acoustic source is completed by retrieving the circuit model parameters for the piezoelectric transducer as described in Materials and Methods. With a fully specified circuit model, we can examine the predicted velocity of the transducer per input volt. The transfer function between the velocity output and voltage input is a function of the complex variable s. To see how our circuit model compares to our experimental results, we plot the magnitude of the transfer function as a function of the real frequency ω by letting s = iω (Fig. 5D). We find that it captures the salient features of the measured response. An even better fit could be obtained by using a more complex circuit model to describe the transducer response. The level of circuit complexity we chose to use here ensures that the model is able to make accurate predictions (for example, for the stability analysis in the following) while maintaining enough simplicity for the results to be easily interpretable.

We further our results by exploring the system response as the parameters of the NIC are varied. Experimentally, we observe that, by increasing the amplifier feedback gain G₀, we can adjust the bandwidth and peak amplitude of the radiator with great flexibility (Fig. 6A). In numerical simulations, we employ our circuit model to explore combinations of G₀ and C_i over a wide range. A suitable performance metric is the gain bandwidth product (peak amplitude multiplied by bandwidth) of the non-Foster system normalized to the one of a passive matching network. The numerical results reveal a band in parameter space in which the performance is maximized, reaching gain bandwidth improvements of about eight (Fig. 6B).

Fig. 6.

Stability bound of non-Foster acoustic source. (A) The velocity frequency response as the amplifier’s feedback gain, G₀, is increased from 2 to 2.26. (B) Numerical examination of the non-Foster gain bandwidth (GBW) normalized to the Foster gain bandwidth, GBW₀, as G₀ and C_i are varied in the NIC. The red curve is the boundary of stability predicted by the model. (C) To validate the numerical stability results, we compare the theoretical instability curve (red, dashed) to measured points of instability onset (black, solid).

However, the magnitude of the transfer function only contains information about the steady-state response of the transducer. Attention must be paid to the transient response, as it may grow without bound in systems with active components and become oscillatory or unstable. A system is unstable if any pole of the transfer function (i.e., any frequency s at which it diverges) lies in the right half of the complex plane. By evaluating the location of the transfer function poles as G₀ and C_i are varied, we can see what combinations assure stable operation (Fig. 6B, red curve). We see that the band of highest gain bandwidth follows the region of instability, indicating a tradeoff between maintaining a margin of stability and achieving the best possible performance. We also have experimentally validated the instability contour by measuring the onset of instability at a few locations in parameter space (Fig. 6C), verifying that it indeed follows our theoretically derived contour. This result makes clear that, within the region of high gain bandwidth, there is increased sensitivity to the system parameters, which is helpful in understanding why there is not more agreement with the experimental results in Fig. 5. A more complex model of the underlying circuits may be able to capture better the dispersion in this region, but at the price of a less intuitive physical picture. Anyhow, both the experimental demonstration and the analysis of our model confirm that the proposed technique can practically achieve broadly enhanced gain bandwidth products for acoustic radiation from a subwavelength transducer, well beyond the limitations of passive radiating elements.

Discussion

Our results outline the exciting opportunities offered by non-Foster circuits in the context of active acoustic sources, offering bandwidth enhancements well beyond the limits of passive systems. Our derived quality factor limit provides a general lower bound on the bandwidth of any passive acoustic resonator, given its size. To approach this limit, a transducer needs to minimize its internal stored energy and loss. For a piezoelectric transducer, this requires low moving mass, high compliance, high electromechanical coupling, and low electrical and mechanical losses. A typical piezoelectric transducer operates far from this limit, and our approach to active circuit loading can compensate for this excess stored energy, extending significantly the available bandwidth for low-frequency resonances.

Stability considerations determine the ultimate limits of performance in these scenarios. In our work, we have presented an analytical model that captures the main phenomena describing active radiating elements, which can accurately predict the stability boundaries of this approach. Our simple non-Foster op-amp circuit loading a piezoelectric transducer already shows a significant bandwidth enhancement, and further improvements can be expected by considering more complex circuit layouts and dispersion engineering to extend the stability region. Further optimization of the dispersion of the active tuning elements paves the way toward even larger gain bandwidths from subwavelength acoustic sources of interest in a wide range of applications.

Non-Foster loads also offer an approach to improve the bandwidth of acoustic metamaterials, which have been shown to enable exotic sound-matter interactions but typically only over a narrow frequency range. This limitation is tied to the frequency response of the individual subwavelength resonators that constitute the material. By using active non-Foster loads, the bandwidth of each individual element can be extended beyond the passivity limitations while maintaining a subwavelength footprint that ensures the possibility of homogenizing the array of elements as a bulk material response. In SI Appendix, Supplementary Information Text, we study a metamaterial made of an array of subwavelength non-Foster elements like the one discussed in this paper, showing that it can support exotic bulk material properties over large bandwidths, beyond the limitations of passive acoustic metamaterials. Our findings therefore open interesting opportunities for metamaterial technology, not necessarily limited only to the acoustic domain but extendable also to mechanics, electromagnetics, and nano-optics.

Materials and Methods

Derivation of the Acoustic Chu Limit.

The total kinetic and potential energies in the waves generated by an acoustic source are composed of both stored and radiated energy and they are infinite as the traveling portion extends to infinity. The sum of kinetic and potential stored energies can be calculated by subtracting from the total energy the portion traveling outward:

$$W_k+W_p=\underset{a}{\overset{\infty }{{\displaystyle \int }}}\underset{0}{\overset{2\pi }{{\displaystyle \int }}}\underset{0}{\overset{\pi }{{\displaystyle \int }}}\left(\frac{p{p}^{\ast }}{4{\rho }_0{c}_0^2}+\frac{{\rho }_0\overrightarrow{v}\cdot {\overrightarrow{v}}^{\ast }}{4}-\frac{1}{2c_0}\text{Re}\left(p{\overrightarrow{v}}^{\ast }\right)\cdot \widehat{r}\right)r^2\sin \theta d\theta d\phi dr,$$ [A1]

where * denotes the complex conjugate and $\widehat{r}$ is the normalized radial vector. The first and second terms in the integral are indeed the kinetic and potential energy densities, proportional to the squares of the local acoustic pressure and velocity. The last term is the subtracted energy density of the traveling wave. The ambient density and speed of sound are ρ₀ and c₀. The difference between the two stored energies is given by the imaginary part of the equation for the conservation of energy, describing the power flow, corresponding to the reactive acoustic intensity (35):

$$W_k-W_p=\frac{1}{4\omega }\text{Im}\underset{0}{\overset{2\pi }{{\displaystyle \int }}}\underset{0}{\overset{\pi }{{\displaystyle \int }}}p\left({\overrightarrow{v}}^{\ast }\cdot \widehat{r}\right)a^2\sin \theta d\theta d\phi .$$ [A2]

Combining Eqs. A1 and A2, we can derive the stored kinetic and potential energies, which can be used in Eq. 1. The radiated power is the flux of the power density along any surface that encloses the source:

$$P_r=\frac{1}{2}\text{Re}{\displaystyle \mathrm{\oint }}p{\overrightarrow{v}}^{\ast }\cdot d\overrightarrow{S}.$$ [A3]

By generally expressing the radiated field as a sum of outward propagating spherical harmonics centered at the source location, the quality factor can be calculated in terms of a multipole expansion. The pressure is generally given by a sum of multipoles with complex amplitudes A_lm,

$$p={\displaystyle \sum }_{l=0}^{\infty }{\displaystyle \sum }_{m=-l}^l{A}_{lm}{h}_l^{\left(2\right)}\left(kr\right)Y_l^m\left(\theta ,\phi \right),$$ [A4]

where $h_l^{\left(2\right)}\equiv j_l-i{y}_l$ is the spherical Hankel function of the second kind, defined with spherical Bessel functions of the first and second kinds, and $Y_l^m$ is the spherical harmonic function of degree l and order m. The velocity can be calculated from the pressure gradient using the momentum equation $\overrightarrow{v}=i\nabla p/\omega {\rho }_0$, with the $e^{i\omega t}$ time convention. Using these expressions, we can evaluate Q in closed form, arriving at

$$Q=\frac{{{\displaystyle \sum }}_{l,m}{\left|A_{lm}\right|}^2{Q}_l}{{{\displaystyle \sum }}_{l,m}{\left|A_{lm}\right|}^2},$$ [A5]

where

$$Q_l=\frac{1}{2}\left[ka\left(2-l{\left|h_l^{\left(2\right)}\right|}^2\right)+{\left(ka\right)}^2\left(j_l{j}_{l+1}+y_l{y}_{l+1}-j_l{j}_l^{\text{'}}-y_l{y}_l^{\text{'}}\right)+{\left(ka\right)}^3\left(j_{l-1}{j}_{l+1}+y_{l-1}{y}_{l+1}-{\left|h_l^{\left(2\right)}\right|}^2\right)\right]$$ [A6]

is the individual quality factor when only multipole harmonic of degree l and any order m is excited. It is expressed in terms of spherical Bessel and Hankel functions with argument ka (wavenumber multiplied by radius). The form of Eq. A5 reveals that Q is a weighted average of the quality factors of degree l where |A_lm|² are the weights.

Design of the Non-Foster Tuning Element.

The NIC circuit was based on the Texas Instruments OPA549 op-amp. The capacitor and feedback gain parameters—3 nF and 2.03, respectively—were chosen as they were observed to induce a subwavelength resonance at a frequency and amplitude similar to that of the Foster tuning element, which was a 22 mH inductor (Bourns RL181S-223J-RC).

An Agilent E5071C network analyzer was used to measure the S₁₁ scattering parameter (reflection coefficient) of the NIC circuit. The input impedance was then calculated from S₁₁ using $Z=Z_0\left(1+S_{11}\right)/\left(1-S_{11}\right)$, where Z₀ is the 50 Ω line impedance.

Retrieval of Circuit Model Parameters for the Piezoelectric Transducer.

The piezoelectric transducer was a 1-mm-thick, 10-mm-long PZT-4 cylinder with an outer radius of 5 mm. The overmold was Stycast 2651 and covered the cylinder and two Macor end caps. The piezoelectric material was radially poled, and the primary mechanical resonance was the radial breathing mode.

Circuit model parameters were retrieved based on a measurement of the velocity of the piezoelectric transducer connected to the Foster tuning element. Parameters were chosen that provided the best fit around the induced and mechanical resonant frequencies. In this retrieval process, loss in the tuning inductor was modeled as a series 30.8 Ω resistance, a value estimated from the quality factor and direct current resistance provided by the manufacturer. The self-resonant frequency of the inductor was sufficiently high to ensure that any effective capacitance could be safely neglected. The best fit was found with the transducer parameters C_e = 3.4 nF, n = 0.3, C_m = 0.48 nF, L_m = 3.85 mH, and R_m = 60 Ω.

Velocity Measurements.

An Agilent 33120A function generator was connected in series with a 10-kΩ resistor to provide approximately constant current output. The velocity of the transducer was measured at a central point on the curved face of the protective overmold using a Polytec PSV-500 laser Doppler vibrometer. The piezoelectric element supports predominately a breathing radial motion in this frequency range with the highest levels of vibration seen along the central equator of the cylinder. The vibrometer captures the magnitude of this motion by measuring the outward surface velocity at a point in this region of maximum response. Extraneous low-frequency noise was removed with a 1-kHz high pass filter. The function generator supplied a 3-Vpp swept frequency input, and the op-amp was powered by an Agilent E3631A supply set to a symmetric 25 V, which supplied on average 24 mA during the measurements.

To create the instability curve (Fig. 6C), a potentiometer was used to increase the feedback gain to the point where unstable oscillations were seen in the velocity and voltage signals. After swapping in a new capacitor, C_i, the process was repeated. This continued until the full parameter space was sufficiently explored.

Data Availability

All study data are included in the article and/or SI Appendix.