Digital non-Foster-inspired electronics for broadband impedance matching

Narrow bandwidths are a general bottleneck for applications relying on passive, linear, subwavelength resonators. In the past decades, several efforts have been devoted to overcoming this challenge, broadening the bandwidth of small resonators by the means of analog non-Foster matching networks for radiators, antennas and metamaterials. However, most non-Foster approaches present challenges in terms of tunability, stability and power limitations. Here, by tuning a subwavelength acoustic transducer with digital non-Foster-inspired electronics, we demonstrate five-fold bandwidth enhancement compared to conventional analog non-Foster matching. Long-distance transmission over airborne acoustic channels, with approximately three orders of magnitude increase in power level, validates the performance of the proposed approach. We also demonstrate convenient reconfigurability of our non-Foster-inspired electronics. This implementation provides a viable solution to enhance the bandwidth of sub-wavelength resonance-based systems, extendable to the electromagnetic domain, and enables the practical implementation of airborne and underwater acoustic radiators.

The radiated acoustic power can be calculated by analysis of the analogous circuit to yield the transducer bandwidth 1 .The analogous circuit for a common electroacoustic transducer 2 is shown in Fig. 2a of main text.
For the electrical part, E R is the DC voice-coil resistance.E ( ) L  is the voice-coil inductance while E ( ) R   is the voice-coil resistance caused by eddy currents, both of which have certain FD 3 .E u is the input voltage of the transducer, and the input of the electrical-mechanical gyrator e u reflects the voltage induced when the voice coil vibrates with a speed of D v .Here, the voice-coil current E i is where Bl is the force factor, in unit of N•A -1 .

D
v can be solved by where D f is the force exerted on the diaphragm and a f is the force generated by the acoustic pressure difference D p (in unit of Pa) across the diaphragm.They can be derived as The mechanical impedance m Z in unit of kg•s −1 can be expressed by where is the mechanical mass of the diaphragm and voice-coil assembly, MS

C
is the mechanical compliance of the diaphragm suspension in unit of m•N −1 and MS R is the mechanical loss in the suspension in unit of N•s•m −1 .
Eventually, the input-output relationship between E u and D V can be obtained as when the transducer is loaded with the digital non-Foster-inspired circuit, Supplementary Eq. ( 5) will be rewritten as Eq. ( 2).Over the low frequency range, the transducer is conceived to operate as a rigid piston mounted in a flat infinite baffle board 4 , which can be treated as an acoustic monopole model 2,4 .The analogous circuits are widely used to estimate the basic vibration and radiation characteristics of transducers, and rad Z on a piston in an infinite baffle can be calculated by Supplementary Eq. ( 6) as follows 2 The acoustical elements a is piston radius of the electroacoustic transducer.0  and 0 c are air density and sound velocity in air, respectively.
Eventually, the analogous circuit can be modeled, as shown in Supplementary Fig. 1a.The obtained theoretical impedance is consistent with the measured results extracted by the power analyzer (PW6001, HIOKI), as shown in Supplementary Figs.1b and c.
Based on the above analysis, the acoustic power ar P of the transducer ( in unit of W) with our proposed circuit can be calculated by Eq. ( 1).According to the half-power bandwidth definition 1 , the frequency corresponding to the maximum acoustic power m P needs to be acquired by Supplementary Note 2: Stability analysis.Although a large gain can guarantee the steady-state response of the system and small steady-state errors, attention must be paid to the transient response 5 .The system is unstable if any pole of the system transfer function moves to the right half of the complex plane.So as to analyze the stability of loading the electroacoustic transducer with the digital non-Foster-inspired electronics, the general circuit structure shown in Fig. 3a is simplified as the transfer function block diagram (Supplementary Fig. 2a) 6 .Further, according to Mason's gain formula, the block function can be derived in Supplementary Fig. 2b as where where T R and T L denote the equivalent resistance and inductance of the electroacoustic transducer, respectively.A feedforward gain f K is often added to handle the conflict between steady-state response, transient response, and stability 7 .Additionally, the self-adaptive PR controller is the software core of our proposed circuit.The transfer function of the PR controller is introduced 8 where c  represents the cutoff bandwidth around the resonant frequency 0  .Also, p K and r K are the proportional term and the integral term of the controller, respectively.
Supplementary Fig. 2c By evaluating the poles' locations of the closed-loop transfer function as p K , r K , f K , and c  vary, we can observe what parameter ranges can assure the system stability.According to the above equations, the characteristic equation ( ) 0 D s  of system can be derived.( ) 0 D s a s a s a s a s a s a s a s a where coefficients i a (i=0, 1, 2, 3, 4, 5, 6, 7) are respectively The self-adaptivity requires that the PR controller can react to different operating frequencies with the optimal response 10 .Therefore, the spectrum characteristics of the controller, determined by p K and r K , ought to vary with different operating frequencies.In our design, we set different frequency-domain characteristics by altering p K and r K in different frequency ranges, thus ensuring the optimal performance of the controller.
According to Supplementary Eqs. ( 14)- (15) and the parameters in Supplementary Table 2, the root locus can be plotted as shown in Supplementary Figs.2d and e.The poles remained at the left half of the complex plane under a wide range of control parameters.For instance, when r 7250 K  , the system's poles are all still located on the left half of the complex plane.When Kr further increases, the system poles will move to the right half plane, which means that the overall system tends to be unstable 11 .
In addition, only when fs is determined, as a linearized time-invariant system, the dynamic performance and stability of the proposed non-Foster-inspired circuit can be analyzed by the open-loop gain/phase margins and closed-loop bandwidth as shown in Supplementary Figs.2f and g.Consequently, the parametric design of PR controller for stability is easy to achieve.
The DSP (TMS320F28335, Texas Instruments) is used to execute programs here.In the main program (Supplementary Fig. 3a), initialization of peripherals and PR controller, as well as the relevant interrupt configuration is complete.Afterwards, the DSP remains in a waiting state until interrupts occur.
The purpose of the timer interrupt is to generate the PWM output pulses, and the period of the pulse is equal to that of the timer.Whenever the counter overflows, an interrupt will be generated by the timer.Then, an interrupt request will be sent to the CPU where the ISR is handled.Meanwhile, the program pointer will be automatically located to the starting address of the ISR function 12 .The primary work of the timer ISR, shown in Supplementary Fig. 3b, is divided into the following steps After discretization with ZOH 13 , the relation between ref i and s u can be expressed as 14 where coefficients A, B, C and D are respectively R and T X are the resistance and reactance of the electroacoustic transducer, respectively.D R and D X manifest the equivalent negative R and C/L of the digital non-Foster-inspired circuit.
(3) Subsequently, the error between ref i and the output current o i will be calculated, which is the input of the PR controller.
(4) Tustin transform 15 Supplementary Eq. ( 19) is employed to discretize the PR controller By solving it, the PR controller can be discretized as Eventually, it can be can be further simplified as where the coefficients A i and B i (i=0, 1, 2) are (5) Closed-loop feedback regulation.After comparing the output of PR controller with the triangular carrier wave 16 , the pulse sequences outputted from the ePWM module will be adjusted to the action of switching devices towards the direction of the error reduction 17 .Since the timer ISR will be responded whenever the counter overflows, the PR controller can continuously adjust the error to realize the zero steady-state error of the sinusoidal reference current signal by a large gain over a where represents the wave number.
First, when r varies from zero to 1 r as shown in Supplementary Fig. 4, 1 P can be solved by Supplementary Eq. ( 25) illustrates the calculation method of the infinitesimal Next, when r varies 1 r from to 2 r as shown in Supplementary Fig. 4, 2 P can be solved as well by Similarly, Supplementary Eq. ( 27) illustrates the calculation method of the infinitesimal Finally, the SPL in front of the transducer at the axial position (z, 0) can be obtained by Supplementary Eq. ( 28).Here, the reference sound pressure ref P is 2×10 than the switching frequency fsw so as to filter the higher harmonics out 20 .In this work, when fsw=40kHz, the cutoff frequency of the LC filter is suggested to be below 4 kHz.
In addition, it is apparent that the proposed self-adaptive PR controller must have an infinite gain at the operating frequency fs to realize the desired negative impedance tracking with zero sinusoidal steady-state error when fs is equal to the resonance frequency of PR controller f0.
The proposed adaptive PR controller was discretized by Tustin method as Supplementary Eq. ( 19), which leads a deviation on Δf0 between f0 (equal to fs) , the resonance frequency of PR controller in s-domain, and fd0, the resonance frequency of PR controller in z-domain.It can be seen from Supplementary Fig. 5a that Δf0 with respect to the expected fs becomes greater as the sampling time (1/fsw) and fs increase.
At an operating frequency of 1400 Hz, Δf0 will reach 20 Hz at fsw=40 kHz.It means a significant gain loss as shown in Supplementary Fig. 5b.Although the controller gain can be improved by parameter adjustment, the adjustment process is also limited by stability and transient performance.When s f is far beyond the resonant point ( s 100Hz f  ), the steady-state errors as a whole rise progressively due to the gain loss.When s f is equal to 1400 Hz, the error of the negative resistance is up to 13.72% while that of the negative inductance is 11.81%.
In order to quantify the control precision, the equivalent negative R and L/C with or without FD are measured through the power analyzer.The errors between the expected values and actual values have been displayed.According to Supplementary Figs.5b and c, the closer the frequency is to the resonant point (40~100 Hz), the larger the steady-state errors between the desired and obtained equivalent impedance.This is because there is appreciable distortion in the actual voltage and current waveforms near the resonant frequency of transducer 21 .This causes the imprecise measurement of the transducer impedance and the inaccurate sensing of ref i for the controller.
Supplementary Note 6: Power and loss calculation of the digital non-Foster-inspired electronics.The usage of switch-mode electronics breaks the power limitation of conventional analog op-amp-based non-Foster circuits.The cancellation of excess stored energy through the digital non-Foster-inspired electronics makes sure that the negative reactance LD(ω) is equal to the eddy current reactance of the high power transducer LE(ω).Simultaneously, the negative resistance RE(ω) of the digital non-Foster-inspired electronics ensures more energy radiation from the transducer over a wide bandwidth.With the equivalent negative R and L with FD ( D ) and the output current o i (its root mean square is o I ), the apparent power D S can be derived as According to the measurement results

3
UDF and o i in Fig. 4b of the main text, when s f is 600 Hz or 1200 Hz, D S is 107.54VA and 229.99 VA, respectively.For the conventional analog non-Foster circuit 4 , the maximum apparent power is approximately 0.44 VA based on the manifested impedance and the op-amp supply voltage.By contrast, the power level of the digital non-Foster circuit is upgraded by about three orders of magnitude.
However ， the power loss generated by switching devices can be further divided into on-state loss, off-state loss and switching loss 6 .In general, the off-state leakage current is very small so that the off-state loss is negligible.For the switching devices S1~S4 used in our designed circuit, S1 and S2 operate at on T and off

T
have been defined as switching-on and switching-off duration, respectively 22 .According to blue, black, and white are represented by 200 Hz tone, 300 Hz tone, 600 Hz tone, 1000 Hz tone, and 1200 Hz tone, respectively.The duration of ten cycles is used for each tone.Each line is transmitted followed by 500 Hz tone of sixteen cycles as the line break signal 28 .
(3) After converted to analog audio by the soundcard of a PC, FSK modulated waveforms are fed into the transducer loaded by different impedance matching approaches.
(4) To transmit information, the receiving process occurs simultaneously on the receiver node.FSK transmission waveforms are first picked up by the acoustic sensor (ISV1610, Hangzhou Aihua Instruments Co., Ltd).According to Supplementary Fig. 9a, the FSK demodulation process is divided into the following five parts.

1)
We have to translate analog FSK tones into digital data by a soundcard with a sampling frequency a f of 44.1 kHz at 16 bits per sample.
2) Digital processing of the demodulated signal can significantly improve the picture quality especially when the environment is noisy 28 .Inspired by slow-scan TV transmission, we adopt an approach as shown in Supplementary Fig. 9b to process signals for restoring image quality.The received digital FSK tones y(k) goes through the finite impulse response filters 29 to obtain each modulation frequency separately, and then consistency of each modulation frequency is guaranteed by normalization.Finally, all the modulated signals are added together to generate the processed audio signal yy(k).
3) By picking up the signal peaks, the locations of the line-break frequency are retrieved.Therefore, we decouple the demodulation process of each line to avoid mutual influence.
4) Discrete Fourier Transformation (DFT) is used to identify the modulation frequency of each pixel signal 30 .The pointer m points to the line being demodulated and the pointer k points to the position of the pixel being identified.B represents the signal length for DFT, which is equal to 222 here to ensure that all six modulation frequencies can be accurately identified.To uniformly compare the image receiving effect under different impedance matching approaches, the threshold values (TH) of the same color are set to be identical.The THs of yellow, red, blue, and black are respectively 0.02, 0.07, 0.19, 0.1.If the actual value after processing is greater than TH, the pixel is received; otherwise, it is shown in white.requirements but also the stability of the system.c  is the cutoff bandwidth.T R and T X represent the resistance and reactance of electroacoustic transducer, respectively.D R and D X represent the manifested negative resistance and inductance with FD of the proposed digital non-Foster-inspired circuit, respectively.
Operating frequency s D S is the area of the diaphragm.D V represents the volume velocity emitted by the diaphragm in unit of m 3 •s −1 and rad Z is the acoustic radiation impedance in unit of Pa•s•m −3 .

2 P
, cannot be solved directly without knowing D R and D L , much less D ( ) R  and D ( ) L  with FD.Thus, the values of l f and h f at the half-power points m cannot further be mathematically derived.For this reason, the exhaustion method is applied to sweep over D R and D L Fig. 2b of the main text, where the maximum bandwidth point 2 p ( RD, LD) and 4 p (0, LD) are marked.E ( ) Z   with FD is the main component of the transducer impedance at frequencies far from the resonance frequency.It severely limits the operation bandwidth.The system bandwidths by impedance matching configurations 1 p ( D ( ) R  , D ( ) L  ) and 3 p (0, D ( ) L  ) are further derived and also marked in Fig. 2b of the main text.

pK
ensures good transient performance and stability, while r K can eliminate the amplitude and phase steady-state errors 5 .
illustrates the corresponding open-loop transfer function block diagram.By simplifying the open-loop control block diagram, and the transfer function from Iref to Io can be derived as

( 1 )( 2 )
The DSP generates triggers for starting A/D (AD7656, Analog Devices) conversion to collect and restore the actual electrical signals ( s u and o i ) at the moment.Generation of the reference signal ref i .The reference current iref is generated according to s i needs to be discussed respectively in relation to the magnitude of the reactance reference Xref and the transducer total reactance XT.Under those two cases, ref i can be expressed as S4 are operated at s f .Consequently, the losses of S1 and S2 are mainly the switching losses while those of S3 and S4 are the on-state losses.The switching energy sw E consists of turn-on switching energy on E and turn-off switching energy off E , which can be solved through the integral of product of drain-source voltage ds ( ) V t and the drain current d ( ) I t during the switching transient.Therefore, sw E , on E and off E

5 )
Recover color information from the FSK-modulated signals and generate the received image.Eventually, an exclusive program using the described algorithms is designed and tested.The resultant images are shown in Fig. 5c of the main text.function block diagram of the digital non-Foster-inspired electronics.d.Root locus diagram of (s) 0 D  with the parameter r K .e. Enlarged root locus diagram of (g).f.The open-loop bode plot of the digital non-Foster-inspired electronics at an operating frequency s f of 900 Hz. co  indicates the cutoff frequency at which the open-loop gain is 0 dB.g.The closed-loop bode plot of the digital non-Foster-inspired electronics at an operating frequency s f of 900 Hz.BW  indicates the closed-loop bandwidth, which is the frequency range of -3 dB down from the peak gain.