0885-8993 (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2018.2877699, IEEE Transactions on Power Electronics IEEE TRANSACTIONS ON POWER ELECTRONICS 1 Piezoelectric Disc Transformer Modeling Utilizing Extended Hamilton’s Principle Oliver M Barham, Mona Mirzaeimoghri, and Don L DeVoe, Member, IEEE Abstract—Piezoelectric transformers (PTs) are resonant elec- tromechanical voltage conversion devices used in a variety of applications including voltage boosting and galvanic isolation. They offer advantages over electromagnetic voltage transformers (EMTs), including high energy density as well as the potential for monolithic fabrication, making them well suited to small-scale mi- croelectromechanical (MEMS) applications. Among various PT topologies, circular discs lend themselves to microfabrication, and are considered here. In order to gain fundamental understanding of disc PT dynamics, this work applies the Extended Hamilton’s Principle of variational calculus to the piezoelectric electric enthalpy, using cylindrical coordinates, in order to derive electro- mechanical constitutive equations for bulk disc transformers. The use of the Hamilton approach in this work supports the integration of mechanical tethers that physically support the disc, allowing the model to be applied to device designs that are compatible with monolithic microfabrication from sheets of bulk piezoelectric material. Using the integrated model, voltage gain (output voltage / input voltage) is predicted as a function of multiple variables including electrode area ratio, device size, tether stiffness, internal material damping and output load impedance, and is compared against finite element numerical and experimental prototype results for verification. Prototype 4mm diameter tethered disc PTs on the order of .002cm 3 , two orders smaller than the bulk PT literature, were fabricated to validate the proposed model, and had peak voltage gains over 2. I. I NTRODUCTION Voltage transformation is employed in many electronic or electromechanical systems that require source voltage to either be increased (boosted) or decreased (bucked) to meet system requirements. The two most prevalent technologies for trans- forming voltage are the piezoelectric transformer (PT) [1]– [15] and electromagnetic transformer (EMT) [16]–[22]. EMTs transform input electrical energy into an electromagnetic field, and back to output electrical energy, while PTs transform input electrical energy into mechanical vibrations, and back to output electrical energy. PTs are a newer technology, with practical devices first patented in the 1950s [23], and on centimeter size scales they have been commercialized in electronics powering laptop liquid crystal displays, but on the millimeter and micrometer scales, they are still relegated to academic research. Both devices are operated at or near resonant normal modes for best performance, and can achieve efficiencies over 90% and voltage transformation ratios (gain) higher than 10:1 (output/input), for various device topologies [16], [18], [24], [25]. The present work is concerned with enabling microrobotic and other small-scale systems, and so The authors are with the University of Maryland MEMS and Microflu- idics Lab, Department of Mechanical Engineering (correspondence email: Oliver@OliverMBarham.com; mirzaei.mona@gmail.com, ddev@umd.edu) is not only interested in device performance metrics such as voltage transformation, but also performance as a function of volume. In order to determine whether EMT or PT devices would be better suited for small-volume systems, prototype devices from the literature were compared on the basis of power and energy density. These metrics are appropriate because EMTs temporarily store energy in electromagnetic fields, and PTs temporarily store energy in mechanical vibrations during normal operation. Energy storage is not their primary function, but they are designed to quickly and efficiently transform and transfer energy from their input to their output, so higher energy and power density performance would indicate that as designs are scaled down, they could transform high voltage levels, at higher output power, for a given device volume. PT and EMT prototype devices reported in the literature are compared in Fig. 1 using a Ragone chart [26], which directly compares devices on the basis of power and energy density. The results are from different researchers, with varying goals, and were not all designed to maximize their power and/or energy density, so these results are not an absolute measure of the relative worth of different prototypes, but rather form a useful general comparison between the two competing tech- nologies. On average, for a given power density, PT devices trend towards higher energy density compared to EMTs, and therefore would be more advantageous to microsystem designers concerned with minimizing overall system volume. Data labels on the chart correspond to reference numbers in the bibliography, and capacitor device ranges are from [27]; included for comparing PT and EMT performance to other typical electronic system components. Given that PTs, in general, are advantageous to microsystem designers, which specific topologies should be considered? To answer that question, we consider two topologies from the literature: “Rosen bar” and circular disc. The most common topology, which has been commercialized in consumer lap- top liquid crystal display electronics, is the Rosen bar [23], [28], manufactured by mechanically joining two rectangular sections of piezoelectric material, each one of which has separately been electrically poled in different directions (in order to align electric dipoles within the material). The need to pole two different material sections, with electrodes deposited both on top/bottom faces and sidewalls, and mechanically join them to create a working device is incompatible with tradi- tional microfabrication techniques such as photolithography and micromachining, and uniform topologies fabricated from a single piezoelectric section with metal deposited only on top and bottom faces are preferable. Circular disc PTs can