Acoustically driven electromagnetic radiating elements

The low propagation loss of electromagnetic radiation below 1 MHz offers significant opportunities for low power, long range communication systems to meet growing demand for Internet of Things applications. However, the fundamental reduction in efficiency as antenna size decreases below a wavelength (30 m at 1 MHz) has made portable communication systems in the very low frequency (VLF: 3–30 kHz) and low frequency (30–300 kHz) ranges impractical for decades. A paradigm shift to piezoelectric antennas utilizing strain-driven currents at resonant wavelengths up to five orders of magnitude smaller than electrical antennas offers the promise for orders of magnitude efficiency improvement over the electrical state-of-the-art. This work demonstrates a lead zirconate titanate transmitter > 6000 times more efficient than a comparably sized electrical antenna and capable of bit rates up to 60 bit/s. Detailed analysis of design parameters offers a roadmap for significant future improvement in both radiation efficiency and data rate.

is the antenna complex impedance and is the source complex impedance.
Matched total antenna efficiency is a measure of both how efficiently power can be delivered from a source to the antenna and how well the antenna radiates the power delivered to it: Therefore the matched total efficiency of the antenna, , is defined as the product of the mismatch efficiency, Γ , and the radiation efficiency, , where , , are the power available from the source, power delivered to the antenna and power radiated by the antenna respectively.
The radiated power from an antenna is defined as: where the circuit elements are shown in Fig. S1. Maximum radiated power occurs for the case of a lossless antenna ( = 0) with a conjugate match between the source and antenna impedances ( = * ) such that all the power available from the source (50%) is delivered to the antenna and 50% of the power is dissipated in the source resistance. This maximum theoretical limit is the same for both ADMIRE and ESA antennas using the same source.
The ADMIRE antenna is matched so that = = 0 and the antenna losses are represented by the motional resistance ( = ). Equation (2) can be modified for the ADMIRE as follows: On the other hand, infinitesimal dipole antennas require matching to cancel out the large reactive part that occurs in non-resonant ESAs. In that case = − and = + ℎ where represents the ESA conduction/dielectric losses and ℎ is the loss due to the finite quality factor of the matching element. Equation (2) can be modified for ESAs as follows: ESA is used here for comparison. Using all previously stated assumptions and dividing (3) by (4), the matched total antenna efficiency ratio, which is the ratio of radiated power from ADMIRE to radiated power from ESA using the same source, is expressed as: The matched efficiency boost of the ADMIRE antenna is calculated using equation (5) in conjunction with equations (6)-(13).
Where , 2 are the mechanical damping coefficient and the electromechanical coupling coefficient of the ADMIRE antenna respectively, = 2 × 33 kHz is the resonant frequency for both antennas and is the free space wavelength of the electromagnetic radiation. The piezoelectric coefficient, , and stiffness constant, , are given generally in (7) since different, or even multiple, piezoelectric coefficients and stiffness constants can be used to generate a magnetic field. For the ADMIRE PZT disc demonstrated here operating in the dilation mode, analytical calculations indicate, and FEM simulations confirm, that 31 and 11 are the dominant contributors to radiation. The ESA is assumed to be made from copper with bulk conductivity and matched with an inductor with a of 200. Using (6)-(13) in conjunction with Supplementary  Table 1 and =50 Ω, yields an improvement in matched antenna efficiency of more than 6400x.
The previous discussion assumes the same source for both ADMIRE and ESA with the same source resistance and same available power for radiation where the matched efficiency is defined as the ratio of power radiated to the maximum power available for radiation. An alternative way to define the matched efficiency ratio would be to consider defining the matched total efficiency as the radiated power to input power ratio assuming different power sources are available that can match each type of antenna separately. In that case equation (5) can be modified as follows: which results in 56x matched antenna efficiency improvement if compared with ESA, where = + + ℎ and = + .
The ADMIRE antenna parameters are either characterized by the manufacturer (*) or measured post-fabrication.

PZT material properties
The PZT discs (PIC-181) were bought commercially from PI ceramics (www.piceramic.com). Relevant material properties for the ADMIRE design are given below. The demonstrated PZT ADMIRE utilizes for proof of concept and is far from the conceivable limit for acoustically driven antennas. As seen from (7), incorporating different provides the potential for a further order of magnitude improvement in radiation efficiency.

Spectrum Measurements and Mechanical Response
The below figure shows the magnetic field frequency response at 1m (indoors) that is radiated by the PZT disk measured with a loop antenna connected to a spectrum analyzer while at the same time the disk edge velocity is measured using an optical interferometer. The figure shows a complete correlation between the two frequency responses. The used drive voltage = 25 V peakto-peak and the frequency sweep range is limited by the noise floor of the spectrum analyzer Fig. S4. Magnetic field and velocity frequency responses. The magnetic field frequency response at 1m (indoors) measured with a loop antenna connected to a spectrum analyzer. At the same time the disk edge velocity is measured using an optical interferometer (drive voltage = 25 V pk-pk). :   Fig. S5. Axis-symmetric simulation region. Air around PZT is represented as a half circle. An arbitrary voltage amplitude is applied to the top electrode while the bottom electrode is grounded and all other surfaces are floating. A free mechanical boundary condition is assigned to the whole PZT disk to reduce any anchor damping.

Further Analysis on Additional Current Elements:
There are additional significant currents that need to be explained to show that the intended radiating element is the only element that is being measured. There will be large components of radial currents but these currents will cancel each other as they have the same magnitude and opposite direction. This is clarified in the following figure that show the surface voltage and radial current distribution on the cross-section of the Disk: The current flowing in the leads is opposite in direction and ~20% in magnitude of the current in the exposed region. For the simulated case, the terminal current from the source equals the applied voltage divided by the motional resistance ~ 16 mA (confirmed by simulation), the polarization current in the exposed region is ~ 82 mA (confirmed with supplied equations in the manuscript and simulated using the voltage or electric field within the exposed region) as shown in the previous figure where the surface voltage amplified by the quality factor reaches an average of ±50 V on the top/bottom exposed surfaces. This results in a 20% reduction in measured magnetic. The previous discussion assumes that the current in the leads contributes as a dipole not as a loop because in our measurement setup we reduce the loop area and put it in an orientation that generates a magnetic field that is orthogonal to the receiver loop used to measure the magnetic field. This can be confirmed by the magnetic field measurement done that has 1/R magnetic field decay, which can never be due to the loop as theoretically loops have 1/R3 magnetic field decay in the near-field region.
With all previous considerations in mind, we assume that the intended radiating element is the only element that is being measured.

Magnetic Field Calculation from Electrostatic Simulations:
Electric and magnetic field components for infinitesimal electric dipole radiation can be written as follows: = 4 sin (1 + 1 ) − The wave impedance in air can be calculated and plotted as follows: Using the simulated near-field electric field and calculated wave impedance in air we can calculate the corresponding magnetic field. Note that, the main assumptions here are: 1. Wave impedance is a material property and doesn't change if the material is fixed. The surrounding material is always air and the relative permittivity of the transmitter is the one that changes. So, by simulating the near-field in the air region and knowing the wave impedance the corresponding magnetic field can be calculated. 2. The far-field component is only dependent on the current so changing the permittivity of the transmitter does not change the far-field as long as the current is fixed. The far-field is the first term in the brackets of (14) and (15).