Piezoelectric resonators for power conversion

Abstract

Described are structures and techniques for providing high-efficiency, high-power-density piezoelectric resonators (PRs) for use in power converters. In some embodiments, a power converter can include a PR for energy transfer, where the PR substantially satisfies geometry conditions disclosed herein for achieving high-efficiency and high-power-density. The geometry conditions can be defined in terms of the converter's specified (e.g., rated) voltage and power level.

Description

FIELD

This disclosure relates generally to power electronics and, more particularly, to converters based upon piezoelectric resonators.

BACKGROUND

Magnetic energy storage elements such as inductors and transformers pose fundamental limits to miniaturization for power electronics; as magnetics scale to smaller sizes, their power density capabilities fundamentally decrease. This motivates exploration of power conversion based on other energy storage technologies. Piezoelectrics, which store energy in the mechanical compliance and inertia of a piezoelectric material, have very high power density and efficiency capabilities with improved scaling properties to small sizes compared to magnetics. Piezoelectrics also offer the advantages of planar form factors, ease of batch fabrication, and the potential to use the energy storage element itself for electrical isolation.

The promise of power conversion based on only piezo-electric energy storage is evident in magnetic-less converter designs having single-port piezoelectric resonators (PRs) as well as in designs having multi-port piezoelectric transformers (PTs). The use of a high-frequency lithium niobate PR to achieve high power density has been explored as have converter implementations catered to high step-down ratios.

SUMMARY

It is recognized herein that criteria for selecting piezoelectric materials and/or designing PRs and PTs themselves remain murky in the context of power conversion, and metrics developed for other applications serve different interests.

With proper selection of materials and PR operating modes, performance characteristics which are much improved relative to convention performance characteristics can be achieved. For example, a shape (e.g., dimensions) and mode may be selected to increase, and ideally maximize, both power density and efficiency.

Described herein are techniques for deriving figures of merit (FOMs) for piezoelectric materials and vibration modes specifically for use as energy storage components in power electronics. In particular, disclosed are achievable efficiency and power handling density FOMs for PRs, which can be derived to be functions of only material properties for each vibration mode in a converter control sequence. Also described is the use of these FOMs to compare commercially-available materials and vibration modes, which are validated both numerically and experimentally. In addition to material selection, these FOM derivations aid PR geometry design and elucidate the fundamental power handling scaling properties for a PR in a converter implementation. Also described herein are techniques and means for selecting piezoelectric materials and vibration mode for use in power conversion, as well as for techniques and means to design/size PRs and PTs using these selected materials and modes to achieve both high power density and efficiency. Also described herein are power conversion systems, circuits, devices, and other structures using piezoelectric materials and modes selected to achieve such power density and efficiency characteristics.

While certain aspects and embodiments of the present disclosure are described in terms of PRs, the techniques and structures sought to be protected herein may also be applied to PTs (e.g., designs with multiple electrical ports) through the use of additional electrodes.

According to one aspect of the disclosure, a power converter having a specified voltage V_in and a specified power level P_out can comprise: a piezoelectric resonator (PR) configured for energy transfer between an input and an output of the converter. The PR can have an area A, a distance 2l between first and second electrodes of the PR, and a dimensions value G defined as

$G=G_f\frac{A}{l}=\frac{B}{B_o}$ where G_ƒ is a geometry-dependent factor of a frequency at which acoustic waves travel through the PR, B is proportional to an operating frequency of the PR times a capacitance of the PR, B_o is a geometry-normalized representation of B. The PR may satisfy the following conditions for the specified V_in and P_out:

• • 1. G is substantially equal to

$\hat{G}=\frac{P_{\mathrm{out}}}{V_{\mathrm{in}}^2{B}_o},$ and

• • 2. l is substantially equal to

$\hat{l}=2\pi \frac{B_o{V}_{\mathrm{in}}}{I_{\mathrm{Lmaxo}}},$ where I_Lmaxo is an amplitude of resonance of the PR.

In some embodiments, wherein G may be equal to Ĝ within 10%. In some embodiments, G may be equal to Ĝ within 20%. In some embodiments, l may be equal to {circumflex over (l)} within 10%. In some embodiments, l may be equal to {circumflex over (l)} within 20%.

In some embodiments, P_out is either a rated power output or a typical power output of the power converter for a given input voltage. In some embodiments, I_Lmaxo may correspond to a geometry-normalized maximum of I_Lmax, and I_Lmax is an amplitude of resonance of the PR at P_out for a given input voltage. In some embodiments, G_ƒ can correspond to a frequency-setting dimension as defined in the following:

$f=\frac{\kappa }{2\pi }{v}_a=G_f\frac{{\kappa }_o}{2\pi }{v}_a,$ in which in which ƒ is a frequency at which acoustic waves travel through the PR, κ is a wave number of the PR, and v_a is an acoustic velocity of the PR.

In some embodiments, the PR is configured to operate in at least one of: a perpendicular vibration mode; a parallel vibration mode; a shear vibration mode; or a length extensional mode. In some embodiments, the perpendicular vibration mode includes at least one of: a length extensional mode; a thickness shear mode; a contour extensional mode; or a radial mode. In some embodiments, the parallel vibration mode includes at least one of: a length extensional mode; a thickness extensional mode; and a thickness shear. In some embodiments, the PR is one of a plurality of PRs comprised within the power converter, wherein at least two of the plurality of PRs are configured to operate in a length extensional mode in parallel.

In some embodiments, the power converter can further comprise: a plurality of switches configured to operate in accordance with one or more switching sequences to transfer energy between the input and output of the converter via the PR; and a controller for controlling the switching sequences. In some embodiments, ones of the one or more switching sequences include a sequence of voltages to be applied to the PR, wherein ones of the voltages can include V_in, −V_in, V_in−V_out, V_out−V_in, V_out, −V_out, or Zero, where V_in and V_out correspond to the input and output voltage of the converter in that order or in reverse order, and Zero corresponds to the PR being short-circuited. In some embodiments, the switching sequences include at least one of: V_in−V_out, Zero, V_out; V_in, V_in−V_out, V_out; or V_in, V_in−V_out, Zero, V_out. In some embodiments, the switching sequences include at least one of: V_in Zero, V_out−V_in; in V_out−V_in, V_out; or V_in Zero V_out−V_in, V_out.

BRIEF DESCRIPTION OF THE DRAWINGS

The manner and process of making and using the disclosed embodiments may be appreciated by reference to the figures of the accompanying drawings. It should be appreciated that the components and structures illustrated in the figures are not necessarily to scale, emphasis instead being placed upon illustrating the principals of the concepts described herein. Like reference numerals designate corresponding parts throughout the different views. Furthermore, embodiments are illustrated by way of example and not limitation in the figures.

FIG. 1 is a block diagram of a system including a dc-dc converter based on a piezoelectric resonator (PR), according to some embodiments.

FIG. 1A is a block diagram of another system including a PR-based converter, according to some embodiments.

FIGS. 2A-2G shows examples of PR vibration modes that can be used within a PR-based converter, according to some embodiments.

FIG. 3 is a plot illustrating PR impedance magnitude and phase in the proximity of a vibration mode.

FIG. 4 is a circuit diagram showing a Butterworth-Van Dyke circuit model for PRs.

FIG. 5 is a plot illustrating a sinusoidal approximation of i_L and resulting resonance amplitude based on the charge transfer required of each of several switching stages.

FIG. 6A is a plot showing PR loss ratio minimum.

FIG. 6B is a plot showing PR volumetric energy handling density (or areal power handling density) maximum.

FIG. 7 is a plot showing a comparison of hard lead zirconate titanate (PZT) materials for power conversion.

FIG. 8 is a photograph of various PRs tested for experimental FOM validation, according to the present disclosure.

FIG. 9 is a photograph of an illustrative circuit board for PR-based power conversion, according to the present disclosure.

FIG. 10 is a circuit diagram showing an illustrative circuit for PR-based power conversion, according to the present disclosure.

FIG. 11 is a plot showing minimum loss ratio that can be exhibited by disclosed embodiments of PR-based converters.

FIG. 12A is a plot showing PR vibration modes in which the applied and induced electric fields are parallel.

FIG. 12B is a plot showing PR vibration modes in which the applied and induced electric fields are perpendicular.

DETAILED DESCRIPTION

Referring to FIG. 1, an illustrative system 100 can include an input voltage 102, an output voltage 104, a dc-dc converter 106 disposed between the input and output voltages, and a switching controller 108, according to some embodiments. Output voltage 104 may correspond to a voltage source load. Converter may be provided as a step-up or step-down converter.

Converter 106 can include one or more piezoelectric resonators (PRs) and one or more switches arranged in given topology to selectively couple the input and output voltages 102, 104 to the PR electrodes. The one or more PRs may comprise all, or substantially all, of the energy transfer components of converter 106. For example, converter 106 may not include any capacitors, magnetics, or other energy storage components other than the one or more PRs. Thus, converter 106 may be referred to as a “PR-based” converter. Converter 106 can have one of the various PR-based converter topologies known in the art, including but not limited to any of the topologies shown and described in PCT Pat. App. No. PCT/US2020/037418, filed on Jun. 12, 2020, the entire contents of which are incorporated herein by reference. In some embodiments, the negative terminal of input voltage 102 may be coupled to the negative terminal of output voltage 104 (i.e., system 100 may be a common-negative system).

Switching controller 108 can include hardware and/or software configured to control switches within converter 106 according to one or more switching sequences. A switching sequence can be selected to provide low-loss soft charging of the PR capacitance.

In some embodiments, converter 106 can include a plurality of switches (such as illustrated in FIG. 10 and embodiments of PCT Pat. App. No. PCT/US2020/037418) configured to operate in accordance with one or more switching sequences to transfer energy between the input and output of the converter 106 via the PR. Controller 108 can control the switching sequences. In some embodiments, controller 108 can be provided as an application specific integrated circuit (ASIC).

A particular switching sequence can include a sequence of voltages to be applied to the PR, wherein ones of the voltages can include V_in, −V_in, V_in−V_out, V_out−V_in, V_out, −V_out, Zero, or Open, where V_in and V_out correspond to the input and output voltage of the converter in that order or in reverse order, Zero corresponds to the PR being short-circuited, and Open corresponds to the PR being open-circuited.

In some embodiments, the switching sequences include at least one of: V_in−V_out, Zero, V_out; V_in, V_in−V_out, V_out; or V_in, V_in−V_out, Zero, V_out. Such sequences may be efficient for use in a step-down converter (i.e., where V_in>V_out).

In some embodiments, the switching sequences include at least one of: V_in, Zero, V_out−V_in; V_in, V_out−V_in, V_out; or V_in, Zero V_out−V_in, V_out. Such sequences may be efficient for use in a step-up converter (i.e., where V_in<V_out).

Various other examples of switching sequences that can be implemented within controller 108 are described in PCT Pat. App. No. PCT/US2020/037418.

In some embodiments, PR-based converter 106 can include a PR having a perpendicular vibration mode, a parallel vibration mode, a shear vibration mode, and/or a length extensional mode. Perpendicular vibration modes can provide high volumetric energy handling densities and extremely planar shapes. Perpendicular vibration modes can include length extensional (with side electrodes), thickness shear (with end electrodes), contour extensional, and radial modes. Parallel vibration modes can provide high areal power handling densities and less-extreme planar shapes. Parallel vibration modes can include length extensional (with end electrodes), thickness extensional, and thickness shear (with side electrodes). Shear vibration modes (with side or end electrodes) can provide high mechanical efficiency and high areal power density. Length extensional mode (with end electrodes), also referred to as k₃₃ mode, can provide high mechanical efficiency and high areal power density for very low power applications. In some embodiments, multiple PRs of this mode may be utilized in parallel to multiply power capability.

In terms of materials, lithium niobate can provide highly planar shapes operating at higher switching frequencies (e.g., compared to PZT) for the same volumetric power density. Lithium niobate shear modes combine the advantages of lithium niobate and shear modes, providing high mechanical efficiency, high areal power density, extremely planar shapes, operating at higher switching frequencies for a given volumetric power density. The shear mode can have either side or end electrodes, creating a parallel or perpendicular shear vibration mode. Lithium niobate shear-mode cuts may include 41-degree X-cut, 163-degree rotated Y-cut, and 10-degree rotated Y-cut. PZT can provide less-extreme planar shapes operating at lower switching frequencies (e.g., compared to lithium niobate) for the same volumetric power density. PZT shear modes combine the advantages of PZT and shear modes, providing high mechanical efficiency, high areal power density, less-extreme planar shapes, operating at lower switching frequencies for a given volumetric power density. The shear mode can have either side or end electrodes, creating a parallel or perpendicular shear vibration mode.

In some embodiments, PR-based converter 106 can include a PR having a geometric configuration that provides improved, and ideally maximum, efficiency at a nominal operating point (i.e., voltage and load) by satisfying G in equation (17) herein, with mode dependent parameters and values as defined in Table 5 and elsewhere herein. In other words, the actual value G for the PR may be substantially equal to the ideal value Ĝ (for maximum efficiency) calculated using equation (17) for the input/output voltage V_in and power level P_out specified for the converter 106. In some cases, P_out may be either a rated power output or a typical power output of the converter 106 for a given input voltage, V_in.

In some embodiments, PR-based converter 106 can include a PR having a geometric configuration that provides improved, and ideally maximum, power density at a nominal operating point by satisfying l in equation (25) herein, with mode dependent parameters and values as defined in Tables 4 and 5, and elsewhere herein. In other words, the actual value l for the PR may be substantially the same as the ideal value {circumflex over (l)} calculated using equation (25) for the specified voltage V_in and power level P_out. The power density may be limited by maximum loss density

$\left(\frac{P_{\mathrm{loss},\max }}{A_s}\right),$ maximum electric field (E_max), maximum mechanical stress (T_max), or maximum mechanical strain (S_max).

In some embodiments, PR-based converter 106 may be configured to “substantially” satisfy the geometry conditions of equations (17) and/or (25). For example, PR-based converter 106 may be configured to satisfy the condition of equation (17) and/or of equation (25) within 1%, 2%, 5%, 10%, 15%, 20%, 25%, etc.

In some embodiments, PR-based converter 106 can include a PR having a geometric configuration that provides both improved/maximum efficiency and power density at a nominal operating point by simultaneously satisfying G and l in equations (17) and (25) based on the geometry- and material-dependent parameters as described herein. Table 7 of this disclosure provides illustrative design selections and anticipated performance for different modes and materials based on available material parameters.

In some embodiments, switching controller 108 can be configured to operate/control the PR-based converter 106 according to techniques described hereinbelow. For example, in some embodiments, PR-based converter 106 can be operated with a switching frequency similar to that described by equation (20) below to achieve improved, and ideally maximum, efficiency.

Referring to FIG. 1A, according to another embodiment, a system 120 can include a PR-based converter 106 coupled between an input voltage 102 and a resistive load 122. That is, embodiments of the PR-based converters disclosed herein can be used within systems having voltage source loads, as in FIG. 1, and within systems having resistive loads, as in FIG. 1A.

Before describing various embodiments of the disclosure, various material state definitions are introduced in Table 1, and various material property definitions are introduced in Table 2.

TABLE 1
Material State Definitions
u Mechanical Displacement (m)
T Mechanical Stress (N/m2)
S Mechanical Strain
E Electric Field Strength (N/m)
D Electric Flux Density (C/m2)

TABLE 2
Material Property Definitions
Qm   Mechanical Quality Factor
k    Electromechanical Coupling Factor
va   Acoustic Velocity (m/s)
ρ    Mass Density (kg/m3)
ε    Dielectric Constant (F/m)
s    Compliance Constant (m2/N)
d    Piezoelectric Charge Constant (C/N)
c    Elastic Modulus (N/m2)
e    Piezoelectric Strain Modulus (C/m2)
σ    Poisson's Ratio
Tmax Maximum Mechanical Stress (N/m2)
Smax Maximum Mechanical Strain
Emax Max. Electric Field Strength (N/m)

2. Piezoelectric Resonator Model

Turning to FIGS. 2A-2G, PRs can be produced in a variety of shapes and electrode patterns, and each configuration has a unique set of compatible vibration modes depending on the PR's polarization direction, electrode placement, and boundary conditions. FIGS. 2A-2G shows various different PR vibration modes, with electrodes denoted by shaded areas, displacement direction(s) marked with arrows, and nodes/nodal planes marked with dashed lines. The polarization direction of the PR is denoted with “P”, and each electrode is assumed to have area A with distance 2l between electrodes. All surfaces are assumed to have no externally-applied stress (i.e., all surfaces are traction-free), and the origin is assumed to be at the PR's center for analysis in Appendix A.

To derive power conversion metrics for piezoelectric materials, focus can be directed to the fundamental frequencies of the vibration modes displayed in FIGS. 2A-2G. These vibration modes can be grouped into two categories that permeate throughout this disclosure: modes for which the applied and induced electric fields are parallel (II), and modes for which these fields are perpendicular (+).

FIG. 2A shows a length extensional mode 202 with side electrodes and perpendicular (+) fields. FIG. 2B shows a length extensional mode 204 with end electrodes and parallel (II) fields. FIG. 2C shows a thickness shear mode 206 with side electrodes and parallel (II) fields. FIG. 2D shows a thickness shear mode 308 with end electrodes and perpendicular (+) fields. FIG. 2E shows a thickness extensional (plate) mode 210 having parallel (II) fields. FIG. 2F shows a contour extensional (square plate) mode 212 having perpendicular (+) fields. FIG. 2G shows a radial (circuit plate) mode 214 having perpendicular (+) fields.

Deriving power conversion metrics for piezoelectric materials first requires appropriate models for a PR's mechanical and electrical behaviors. Piezoelectric materials are governed by coupled constitutive relations between mechanical strain (S), mechanical stress (T), electric field strength (E), and electric flux density (D) due to the direct and converse piezoelectric effects:

$\begin{array}{cc}{S}_{\mathrm{ij}}=\frac{1}{2}\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)=s_{\mathrm{ijmn}}^E{T}_{\mathrm{mn}}+d_{\mathrm{mij}}{E}_m& \left(1\right)\end{array}$ $\begin{array}{cc}{D}_i=d_{\mathrm{imn}}{T}_{\mathrm{mn}}+{\epsilon }_{\mathrm{im}}^T{E}_m& \left(2\right)\end{array}$

States and parameters for these equations are defined in Tables 1 and 2, wherein superscripts indicate constant conditions for measurement, and indices designate tensor components. Combined with the Newtonian equation of motion, (1) and (2) reduce to an acoustic wave equation that dictates sinusoidal time- and space-dependent solutions for u, S, T, and E; this is described with more detail in Appendix 10. With all surfaces in FIGS. 2A-2G assumed to be traction-free, the maximum amplitudes of S, T, and E each occur at the center of the PR.

TABLE 3
Piezoelectric Resonator Model Parameters
Parameter	Parallel (∥)	Perpendicular (+)
Gf	$\frac{1}{l}$	$\frac{1}{a}$
Yo	{square root over (π2 − 8k2)}	$\sqrt{{\pi }^2+8\frac{k^2}{1-k^2}}$
Cp	${\epsilon }^T\left(1-k^2\right)\frac{A}{2l}$	${\epsilon }^T\left(1-k^2\right)\frac{A}{2l}$
C	$\frac{8k^2}{{\pi }^2-8k^2}{C}_{p}$	$\frac{8k^2}{{\pi }^2\left(1-k^2\right)}{C}_{p+}$
L	$\frac{1}{2G_f^2{k}^2{v}_a^2{C}_{p}}$	$\frac{1-k^2}{2G_f^2{k}^2{v}_a^2{C}_{p+}}$
R	$\frac{1}{Q_m}\sqrt{\frac{L_{}}{C_{}}}$	$\frac{1}{Q_m}\sqrt{\frac{L_{+}}{C_{+}}}$

Additional radial mode model parameters are shown in Appendix 10(c).

The frequency ƒ of the acoustic wave propagating through the PR is expressed as:

$\begin{array}{cc}f=\frac{\kappa }{2\pi }{v}_a=G_f\frac{{\kappa }_o}{2\pi }{v}_a& \left(3\right)\end{array}$ in which κ is the wave number (in rad/m) and v_a is the acoustic velocity (in m/s) of the PR material. Herein, κ_o is defined to be the geometry-normalized wave number (in rad), from which the geometry-dependent factor G_ƒ is extracted. G_ƒ is different for parallel and perpendicular vibration modes and is displayed for each in Table 3; as observed in equation (3), it is proportional to the resonant frequency of the PR.

In some embodiments, mounting and/or contacts can be placed at nodes and/or at nodal planes as illustrated in FIGS. 2A-2G to mitigate loss and damping. Electrode patterning (e.g., using a mesh-patterned electrode, a backbone-and-rib pattern, etc.) can be used to provide required electrical distribution while mitigating mechanical loss for the desired operating mode of a design.

Turning to FIG. 3, the PR's electrical impedance can be likewise derived from the acoustic wave solution as a function of ƒ. FIG. 3 is a plot 300 showing PR impedance magnitude 302 and phase 304 in the proximity of a vibration mode, where ƒ_r is the resonant frequency and ƒ_ar is the anti-resonant frequency. The PR exhibits inductive behavior in the region 306 between the resonant (ƒ_r) and anti-resonant (ƒ_ar) frequencies; this region is of most interest to power conversion since it enables zero voltage switching and other high-efficiency behaviors. The inductive region 306 spans the following ranges of κ_o for parallel (∥) and perpendicular (+) vibration modes, respectively:

$\begin{array}{cc}\frac{{\gamma }_{o}}{2}<{\kappa }_{o}<\frac{\pi }{2}& \left(4\right)\end{array}$ $\begin{array}{cc}\frac{\pi }{2}<{\kappa }_{o+}<\frac{{\gamma }_{o+}}{2}& \left(5\right)\end{array}$ for which factor γ_o is displayed in Table 3.

Turning to FIG. 4, the PR impedance characteristic shown in FIG. 3 can be modeled by an equivalent electrical circuit (i.e., the Butterworth-Van Dyke model) as illustrated in FIG. 4. Illustrative circuit model 400 includes a first capacitor 402 (C_p) to model the PR's physical electrical capacitance and an LCR branch including an inductor 404 (L), a second capacitor 406 (C), and a resistor 408 (R) to model the PR's electromechanical resonance properties.

For excitation of a PR's fundamental frequency, the circuit model 400 can have the parameters shown in Table 3, which are generalized to parallel and perpendicular vibration modes. A is the electrode area, 2l is the distance between the electrodes, and the material parameters of interest are shown for each vibration mode in Appendix A. This electrical model serves as a basis for conceptualizing the PR's behavior: ƒ_r corresponds to series resonance between L and C, and ƒ_ar occurs at parallel resonance between C_p and the series combination of L and C.

The model's full derivation for all considered modes can be found in J. Erhart, P. Pulpan, and M. Pustka, “Piezoelectric Ceramic Resonators,” Springer, 2017; and J. D. Boles, P. L. Acosta, Y. K. Ramadass, J. H. Lang, and D. J. Perreault, “Evaluating piezoelectric materials for power conversion.” in “Proc. IEEE Workshop on Control and Modeling for Power Electronics,” Aalborg, Denmark, November 2020, pp. 1-8 (“Perreault et al.”) provides this derivation for the length extensional mode. The vibration mode analyzed in Perreault et al. is referred to as the thickness extensional mode, though it assumes the k₃₃ coupling coefficient. Thus, the analysis in Perreault et al. corresponds to this disclosure's length extensional mode with end electrodes (e.g., mode 204 illustrated in FIG. 2B) analysis, except with one electrode solidly mounted.

3. Amplitude of Resonance Model

The circuit model described in Section 2 can be employed to analyze the PR's behavior in a power converter (e.g., within converter 106 of FIG. 1). Because PRs tend to have very high quality factors in the proximity of their resonant frequencies, i_L can be assumed to be sinusoidal. The amplitude of i_L (I_L) then provides insight into the amplitude of the PR's mechanical resonance, which dictates its mechanical energy storage and loss.

A. Model Introduction

Turning to FIG. 5, the PR's ideal amplitude of resonance can be calculated from the total magnitude of charge it transfers via i_L during each resonant cycle, as shown, which depends on the converter's specific switching sequence and operating point. FIG. 5 is a plot 500 showing a sinusoidal approximation 502 of i_L (and resulting amplitude of resonance I_L) based on the charge transfer q_n required of each switching stage n (e.g., within the six stages labeled 504a-504f). See J. D. Boles, J. P. Piel, and D. J. Perreault., “Enumeration and analysis of dc-dc converter implementations based on piezoelectric resonators,” in “IEEE Transactions on Power Electronics,” vol. 36, no. 1. pp. 129-145, 2021 (“Boles et al.”).

To provide an accurate representation of the PR's behavior as used in a converter, this charge transfer calculation includes both the “connected/zero stages” (i.e., stages when the PR is connected to the source-load system) and “open stages” (stages when the PR is open-circuited and C_p charges/discharges through resonance) of the switching sequence.

To evaluate piezoelectric materials, either of the highest-efficiency step-down switching sequences analyzed in Boles et al. can be assumed, which have the following I_L:

$\begin{array}{cc}{I}_L=\frac{\pi }{2}f{Q}_{\mathrm{total}}=\pi \left(\frac{P_{\mathrm{out}}}{V_{\mathrm{in}}}+{\mathrm{fC}}_p{V}_{\mathrm{in}}\right)& \left(6\right)\end{array}$

In accordance with the switching sequences proposed in Boles et al., this I_L equation assumes resonant “soft” charging/discharging of the PR's C_p, zero voltage switching of all switches, and all-positive instantaneous power transfer for high efficiency. Thus, equation (6) models the PR's amplitude of resonance for quintessential converter operation, which is assumed throughout this disclosure.

B. Material Limits

The maximum permissible I_L for a given PR may be set by its material's limits for strain (S_max), stress (T_max), and electric field (E_max). Such potential limits include yield stress and strain, as well as depolarization field (often referred to as “coercive” field). The relationship between I_L and the PR's physical states can be derived using the constitutive relations of equations (1)-(2) and the equation of motion. It should be noted that V_in in equation (22) is the converter's input voltage, and that the voltage exposed to the PR is not purely sinusoidal. Thus, V_in cannot be directly affixed to physical limits like T_max and E_max. This results in the following relationship between I_L and the amplitude of mechanical displacement (A) as derived in Appendix B:

$\begin{array}{cc}{I}_L=\kappa A{G}_f\frac{v_{a}d}{s^E}\Delta \sin \left({\kappa }_o\right)& \left(7\right)\end{array}$

Then, S, T and E can each be related to I_L through Δ. This strategy can be utilized to determine the amplitudes of resonance that correspond to maximum strain (I_LmaxS), maximum stress (I_LmaxT), and maximum E field (I_LmaxE) as demonstrated in Appendix B. It is shown that I_LmaxS, I_LmaxT, and I_LmaxE have identical geometry terms, so their geometry-normalized limits (referred to as I_LmaxSo, I_LmaxTo and I_LmaxEo, respectively) are summarized for each considered vibration mode in Table 4.

TABLE 4
Maximum Amplitudes of Resonance Based on Material Limits
Vibration Mode	ILmaxSo	ILmaxTo	ILmaxEo	ILmaxLo
Length Extensional (s)	$v_a\frac{d_{31}}{s_{11}^E}{S}_{\max }\sin \left({\kappa }_o\right)$	$v_a{d}_{31}{T}_{\max }\cot \left(\frac{{\kappa }_o}{2}\right)$	vak312ε33TEmaxtan(κo)	$\sqrt{\frac{4Q_m{k}_{31}^2{\epsilon }_{33}^T{v}_a}{\pi }H}$
Length Extensional (e)	$v_a\frac{d_{33}}{s_{33}^E}{S}_{\max }\sin \left({\kappa }_o\right)$	$v_a{d}_{33}\left(1-k_{33}^2\right)T_{\max }\cot \left(\frac{{\kappa }_o}{2}\right)$	$v_a{k}_{33}^2{\epsilon }_{33}^S{E}_{\max }\frac{\sin \left({\kappa }_o\right)}{\cos \left({\kappa }_o\right)-k_{33}^2}$	$\sqrt{\frac{4Q_m{k}_{33}^2{\epsilon }_{33}^S{v}_a}{{\gamma }_o}H}$
Thickness Shear (side)	$v_a\frac{d_{15}}{s_{55}^E}{S}_{\max }\sin \left({\kappa }_o\right)$	$v_a{d}_{15}\left(1-k_{15}^2\right)T_{\max }\cot \left(\frac{{\kappa }_o}{2}\right)$	$v_a{k}_{15}^2{\epsilon }_{11}^S{E}_{\max }\frac{\sin \left({\kappa }_o\right)}{\cos \left({\kappa }_o\right)-k_{15}^2}$	$\sqrt{\frac{4Q_m{k}_{15}^2{\epsilon }_{11}^S{v}_a}{{\gamma }_o}H}$
Thickness Shear (end)	$v_a\frac{d_{15}}{s_{55}^E}{S}_{\max }\sin \left({\kappa }_o\right)$	$v_a{d}_{15}{T}_{\max }\cot \left(\frac{{\kappa }_o}{2}\right)$	vak152ε11TEmaxtan(κo)	$\sqrt{\frac{4Q_m{k}_{15}^2{\epsilon }_{11}^T{v}_a}{\pi }H}$
Thickness Extensional	vae33Smaxsin(κo)	$v_a\frac{e_{33}\left(1-k_t^2\right)}{c_{33}^E}{T}_{\max }\cot \left(\frac{{\kappa }_o}{2}\right)$	$v_{\alpha }{k}_t^2{\epsilon }_{33}^S{E}_{\max }\frac{\sin \left({\kappa }_o\right)}{\cos \left({\kappa }_o\right)-k_t^2}$	$\sqrt{\frac{4Q_m{k}_t^2{\epsilon }_{33}^S{v}_a}{{\gamma }_o}H}$
Contour Extensional	$\frac{2d_{31}{v}_a}{s_{11}^E\left(1-\sigma \right)}{S}_{\max }\sin \left({\kappa }_o\right)$	$2v_a{d}_{31}{T}_{\max }\cot \left(\frac{{\kappa }_o}{2}\right)$	vakp2ε33TEmaxtan(κo)	$\sqrt{\frac{4Q_m{k}_p^2{\epsilon }_{33}^S{v}_a}{\pi }H}$
Radial	$\frac{4d_{31}{v}_a}{s_{11}^E\left(1-\sigma \right)}{S}_{\max }{J}_1\left({\kappa }_o\right)$	$\frac{2v_a{d}_{31}\left(1+\sigma \right)}{\frac{1}{2}\left(1+\sigma \right)-\Psi }{T}_{\max }{J}_1\left({\kappa }_o\right)$	$v_a{k}_p^2{\epsilon }_{33}^T\left(1+\sigma \right)E_{\max }\frac{J_1\left({\kappa }_o\right)}{\Psi }$	$\sqrt{\frac{2Q_m{k}_p^2{\epsilon }_{33}^T{v}_a}{\frac{{\kappa }_{o,r}}{1+\sigma }-\frac{1-\sigma }{{\kappa }_{o,r}}}H}$

In Table 4,

$H={\left(\frac{P_{loss}}{A_s}\right)}_{\max }$ is substituted for I_LmaxLo. For radial mode, Appendix 10(c) contains the definition for WY and series expansions for Bessel functions.

The lowest-magnitude limit can be considered the geometry-normalized maximum for I_L:

$\begin{array}{cc}{I}_{\mathrm{Lmaxo}}=\frac{I_{\mathrm{Lmax}}}{A{G}_f}=\min \left(I_{\mathrm{LmaxSo}},I_{\mathrm{Lmax}\mathrm{To}},I_{\mathrm{LmaxEo}}\right)& \left(8\right)\end{array}$

Thus, the geometry-normalized limit I_Lmaxo (and which physical limit constrains it) can be determined based on only material parameters and limits. I_Lmaxo is employed to determine maximum power handling density in Section 5.

C. Loss Density Limit

Thermal management limitations may confine a PR's amplitude of resonance to lower bounds than its material's physical limits, so an I_L limit can be derived based on loss density. For this limit, assume that most PR heat extraction occurs through a surface with area A_s, and that a thermal design can safely accommodate a certain quantity of PR loss per A_s. This areal loss density relates to the PR's operation as follows, considering only mechanical loss as defined in equation (12):

$\begin{array}{cc}\frac{P_{loss}}{A_s}=\frac{1}{2}\frac{I_L^{2}R}{A_s}=\frac{1}{2}{I}_{Lo}^2{R}_o& \left(9\right)\end{array}$ in which geometry parameters from I_L and R have been extracted, assuming A_s=A for parallel (∥) modes and A_s=G_ƒAl for perpendicular (+) modes as defined in Table 6, and R_o as defined in Table 5.

For contour extensional and radial modes (i.e., modes for which both non-l dimensions are equal),

$\frac{P_{loss}}{A_s}=\frac{4}{\hat{G}}\frac{P_{loss}}{A}$ may be used to scale a representative areal loss density based on A (assuming a PR design will adhere to the Ĝ condition for maximum efficiency as described in Section 4(b)). Since Ĝ varies by operating point, it should be noted that the resulting I_LmaxLo would also depend on operating point information.

For these assumed surfaces, all geometry terms cancel and equation (9) shows a direct relationship between areal loss density and I_Lo. This equation can be rearranged to define a loss-limited maximum for I_Lo:

$\begin{array}{cc}{I}_{L\max \mathrm{Lo}}=\sqrt{\frac{2}{R_o}{\left(\frac{P_{loss}}{A_s}\right)}_{\max }}& \left(10\right)\end{array}$

I_LmaxLo is displayed in Table 4 for each mode and can be directly compared to I_LmaxSo, I_LmaxTo and I_LmaxEo in (8) to determine I_Lmaxo It can also be utilized in Section 5 to calculate loss-limited energy and power handling densities.

4. Mechanical Efficiency Figure of Merit

To quantitatively compare piezoelectric materials and vibration modes for power conversion, focus can first be given to achievable PR efficiency; this has implications for both operating cost and thermal management. Efficiency can be expressed as

$\begin{array}{cc}\eta =\frac{P_{out}}{P_{out}+P_{loss}}=\frac{1}{1+\frac{P_{loss}}{P_{out}}}& \left(11\right)\end{array}$

Thus, the impact of piezoelectric material properties on efficiency can be examined through loss ratio P_loss/P_out, which is desired to be as low as possible. Material-dependent losses in the PR include mechanical loss and dielectric loss; since mechanical loss typically dominates close to resonance, this disclosure focuses on only mechanical loss for this FOM.

A. Minimum Mechanical Loss Ratio

To derive the minimum mechanical loss ratio for a piezoelectric material, mechanical loss can be estimated using the PR's amplitude of resonance as in equation (6) and resistance as follows:

$\begin{array}{cc}{P}_{loss}=\frac{1}{2}{I}_L^{2}R& \left(12\right)\end{array}$

P_loss can then be divided by P_out to produce the following mechanical loss ratio equation:

$\begin{array}{cc}\frac{P_{loss}}{P_{out}}=\frac{\frac{1}{2}{I}_L^{2}R}{P_{out}}=\frac{{\pi }^2}{2}\left(\frac{P_{out}}{V_{\mathrm{in}}^2}R+2BR+\frac{V_{\mathrm{in}}^2{B}^2}{P_{out}}R\right)& \left(13\right)\end{array}$ in which the following substitution is made:B=ƒC_p  (14)

This loss ratio equation has only two operating point parameters (V_in and P_out) and two PR-dependent parameters (B and R). The PR's material and geometry properties can be explicitly separated by extracting all geometry parameters from B and R, which have the same lumped geometry term G (as reciprocals), leaving only the material-dependent B_o and R_o, respectively. Thus, equation (13) can be rewritten as

$\begin{array}{cc}\frac{P_{loss}}{P_{\mathrm{out}}}=\frac{{\pi }^2}{2}\left(\frac{P_{\mathrm{out}}}{V_{\mathrm{in}}^2}\frac{R_o}{G}+2B_o{R}_o+\frac{V_{\mathrm{in}}^2}{P_{out}}G{B}_o^2{R}_o\right)& \left(15\right)\end{array}$

• • where

$\begin{array}{cc}G=G_f\frac{A}{l}=\frac{B}{B_o}=\frac{R_o}{R}& \left(16\right)\end{array}$

These parameters are displayed for each vibration mode in Table 5. It should be noted that Q_m is treated as a material property provided by the manufacturer for the purposes of this study, though Q_m may vary from this value based on PR shape, vibration mode, and mounting structure.

TABLE 5
Mechanical Efficiency Figures of Merit and Relevant Parameters
Vibration Mode	G	Bo	Ro	FOMM	γo
Length Extensional (s)	$\frac{4b}{l}$	${\epsilon }_{33}^T\left(1-k_{31}^2\right)\frac{{\kappa }_o{v}_a}{4\pi }$	$\frac{\pi }{2Q_m{k}_{31}^2{\epsilon }_{33}^T{v}_a}$	$4Q_m\frac{k_{31}^2}{1-k_{31}^2}\frac{1}{{\pi }^2{\stackrel{\_}{\kappa }}_o}$	$\sqrt{{\pi }^2+8\frac{k_{31}^2}{1-k_{31}^2}}$
Length Extensional (e)	$\frac{A}{l^2}$	${\epsilon }_{33}^T\left(1-k_{33}^2\right)\frac{{\kappa }_o{v}_a}{4\pi }$	$\frac{{\gamma }_o}{2Q_m{k}_{33}^2{\epsilon }_{33}^T\left(1-k_{33}^2\right)v_a}$	$4Q_m{k}_{33}^2\frac{1}{\pi {\gamma }_o{\stackrel{\_}{\kappa }}_o}$	{square root over (π2 − 8k332)}
Thickness Shear (side)	$\frac{A}{l^2}$	${\epsilon }_{11}^S\frac{{\kappa }_o{v}_a}{4\pi }$	$\frac{{\gamma }_o}{2Q_m{k}_{15}^2{\epsilon }_{11}^S{v}_a}$	$4Q_m{k}_{15}^2\frac{1}{\pi {\gamma }_o{\stackrel{\_}{\kappa }}_o}$	{square root over (π2 − 8k152)}
Thickness Shear (end)	$\frac{4b}{l}$	${\epsilon }_{11}^S\frac{{\kappa }_o{v}_a}{4\pi }$	$\frac{\pi }{2Q_m{k}_{15}^2{\epsilon }_{11}^T{v}_a}$	$4Q_m\frac{k_{15}^2}{1-k_{15}^2}\frac{1}{{\pi }^2{\stackrel{\_}{\kappa }}_o}$	$\sqrt{{\pi }^2+8\frac{k_{15}^2}{1-k_{15}^2}}$
Thickness Extensional	$\frac{A}{l^2}$	${\epsilon }_{11}^S\frac{{\kappa }_o{v}_a}{4\pi }$	$\frac{{\gamma }_o}{2Q_m{k}_t^2{\epsilon }_{33}^S{v}_a}$	$4Q_m{k}_t^2\frac{1}{\pi {\gamma }_o{\stackrel{\_}{\kappa }}_o}$	{square root over (π2 − 8kt2)}
Contour Extensional	$\frac{4a}{l}$	${\epsilon }_{33}^T\left(1-k_p^2\right)\frac{{\kappa }_o{v}_a}{4\pi }$	$\frac{\pi }{2Q_m{k}_p^2{\epsilon }_{33}^T{v}_a}$	$4Q_m\frac{k_p^2}{1-k_p^2}\frac{1}{{\pi }^2{\stackrel{\_}{\kappa }}_o}$	$\sqrt{{\pi }^2+8\frac{k_p^2}{1-k_p^2}}$
Radial	$\frac{\pi a}{l}$	${\epsilon }_{33}^T\left(1-k_p^2\right)\frac{{\kappa }_o{v}_a}{4\pi }$	$\frac{{\kappa }_{o,r}^2-\left(1-{\sigma }^2\right)}{Q_m{\kappa }_{o,r}\left(1+\sigma \right)k_p^2{\epsilon }_{33}^T{v}_a}$	$Q_m\frac{k_p^2}{1-k_p^2}\frac{2}{\pi {\stackrel{\_}{\kappa }}_o}\frac{{\kappa }_{o,r}\left(1+\sigma \right)}{{\kappa }_{o,r}^2-\left(1-{\sigma }^2\right)}$	Appendix 10(c)

Referring to Table 5, from (21),

${\overline{\kappa }}_o=\frac{\pi {\gamma }_o}{\pi +{\gamma }_o}$ (see Appendix 10(c) for radial mode).

From here, it can be assumed that the designer has the flexibility to choose the PR's geometric dimensions.

FIG. 6A is a plot 600 showing PR loss ratio minimum 602 and FIG. 6B is another plot 640 showing PR volumetric energy handling density 642 (or areal power handling density) maximum. Both plots 600, 640 are shown with logarithmic axes. The loss ratio equation reached in equation (15) is a second-order equation with respect to G as illustrated in FIG. 6A. Minimizing equation (15) with respect to G reveals the following G condition (denoted by Ĝ) and corresponding minimum mechanical loss ratio:

$\begin{array}{cc}\hat{G}=\frac{P_{out}}{V_{\mathrm{in}}^2{B}_o}& \left(17\right)\end{array}$ $\begin{array}{cc}\Rightarrow {\left(\frac{P_{loss}}{P_{out}}\right)}_{\min }=2{\pi }^2{B}_o{R}_o& \left(18\right)\end{array}$

The condition in equation (17) cancels the operating point and PR geometry parameters in the loss ratio equation, so the minimum achievable mechanical loss ratio for a PR depends on only its material properties (i.e., quantities assumed in Table 2). Taking the inverse of equation (18), the following unitless factor can therefore be considered a mechanical efficiency FOM (FOM_M) for PR materials and vibration modes:

$\begin{array}{cc}{\mathrm{FOM}}_M=\frac{1}{2{\pi }^2{B}_o{R}_o}& \left(19\right)\end{array}$

FOM_M is desired to be as large as possible, and it is summarized in Table 5 for the considered vibration modes.

B. Geometry Condition

A PR can be designed to achieve its material's minimum mechanical loss ratio at a nominal operating point by satisfying G=Ĝ with its geometric dimensions. The order of magnitude of B_o ranges from 10⁻⁸ to 10⁻⁶ for most piezoelectric materials, which requires Ĝ>>1 in (17) for most power conversion applications. Thus, Ĝ often dictates l to be the shortest geometric dimension, resulting in primarily planar PR designs. Parallel vibration modes can be particularly advantageous for satisfying large Ĝ values in that both the numerator (A) and denominator (l²) have squared length dimensions. Mode configurations for which l is not the shortest dimension (length extensional and thickness shear, each with end electrodes) have limited practicality; their relative dimension assumptions can only be maintained for very high

$\frac{V_{\mathrm{in}}^2}{P_{out}}.$ C. Operating Frequency

Though B_o and R_o have no geometry dependence in the derivation of Section 4(a), B_o depends on κ_o, which spans the ranges of equations (4) and (5) for the inductive region 306 shown in FIG. 3. Although the assumed switching sequence described in Section 3(a) naturally spans this inductive region, frequency is not an independent control variable. To maintain the high-efficiency behaviors described in Section 3, the exact operating frequency (and therefore the exact κ_o) is dictated by the operating point.

Thus, frequency and κ_o that correspond to the minimum loss ratio in equations (17)-(18) are determined for use in the FOMs of Sections 4 and 5 (though this is also useful for broader converter design). Inserting Ĝ into the amplitude of resonance model of equation (6) reveals both addends to be equal for this condition. Based on the amplitude of resonance model's origin, this implies that the PR's charge transfer is split evenly between connected/zero stages (resonating at ƒ_r) and open stages (resonating at ƒ_ar). Thus, one can approximate the highest-efficiency operating frequency to be the harmonic mean of ƒ_r and ƒ_ar (i.e., the reciprocal of the average of

$\frac{1}{f_r}$ and

$\frac{1}{f_{\mathrm{ar}}}$

$\begin{array}{cc}\stackrel{\_}{f}=\frac{2f_r{f}_{\mathrm{ar}}}{f_r+f_{\mathrm{ar}}}& \left(20\right)\end{array}$ which has the following geometry-normalized wave number (also derived for radial mode in Appendix A(c)):

$\begin{array}{cc}{\stackrel{\_}{\kappa }}_o=\frac{\pi {\gamma }_o}{\pi +{\gamma }_o}& \left(21\right)\end{array}$

Thus, assuming the switching sequence constraints listed in Section 3 are met, this is the operating frequency for which equation (17) is satisfied and the minimum loss ratio occurs. This is the operating frequency assumed for the remainder of this disclosure.

5. Power Density Figures of Merit

A second point of comparison for piezoelectric materials and vibration modes is achievable power handling density, which poses a boundary for converter miniaturization. Useful power density metrics must consider how the PR is to be utilized in a converter (i.e., not just energy storage capability), so the amplitude of resonance model in Section 3 can again be assumed. To model the PR's power handling capability as a function of operating constraints, equation (6) can be rearranged such that power delivered is the following function of I_L:

$\begin{array}{cc}{P}_{\mathrm{out}}=\frac{1}{\pi }{V}_{\mathrm{in}}{I}_L-C_p{\mathrm{fV}}_{\mathrm{in}}^2=\frac{1}{\pi }{V}_{\mathrm{in}}{I}_L-{\mathrm{BV}}_{\mathrm{in}}^2& \left(22\right)\end{array}$

With equation (22), the maximum energy and power handling densities can be derived considering the physical and loss-density limits for I_L presented in Sections 3(b) and 3(c), respectively.

A. Volumetric Energy Handling Density

Volumetric power density is a common metric for converter power handling capability with respect to size. The PR's volumetric power density can be derived by first dividing (22) by volume, where vol=2Al:

$\begin{array}{cc}\frac{P_{\mathrm{out}}}{\mathrm{vol}}=\frac{P_{\mathrm{out}}}{2\mathrm{Al}}=G_f\left(\frac{1}{2\pi l}{V}_{\mathrm{in}}{I}_{\mathrm{Lmaxo}}-\frac{1}{2l^2}{B}_o{V}_{\mathrm{in}}^2\right)& \left(23\right)\end{array}$ in which I_L has been set equal to I_Lmax and geometry terms have been separated from material properties. It is evident that a PR's volumetric power density is directly proportional to G_ƒ, and therefore ƒ. The operating frequency of a converter determines its driving and control requirements, so it is elected to normalize equation (23) to ƒ for even comparison between PR vibration modes with respect to converter capability. Such normalization results in the following expression for volumetric energy handling density (i.e., the quantity of energy the PR can deliver to the load in one resonant cycle, divided by volume):

$\begin{array}{cc}\frac{E_{\mathrm{out}}}{\mathrm{vol}}=\frac{P_{\mathrm{out}}}{\mathrm{vol}·f}=\frac{1}{{\kappa }_o{v}_a}\left(\frac{1}{l}{V}_{\mathrm{in}}{I}_{\mathrm{Lmaxo}}-\frac{\pi }{l^2}{B}_o{V}_{\mathrm{in}}^2\right)& \left(24\right)\end{array}$

Of note, energy handling density should not be confused with energy storage capability; not all of the PR's stored energy can be delivered to the load in one resonant cycle since the energy is stored mechanically.

This expression can be maximized by assuming the designer has the flexibility to choose l. Maximizing equation (24) with respect to l as illustrated in FIG. 6B results in the following l condition (denoted {circumflex over (l)}) and maximum volumetric energy handling density:

$\begin{array}{cc}\hat{l}=2\pi \frac{B_o{V}_{\mathrm{in}}}{I_{\mathrm{Lmaxo}}}& \left(25\right)\end{array}$ $\begin{array}{cc}{\left(\frac{E_{\mathrm{out}}}{\mathrm{vol}}\right)}_{\max }=\frac{I_{\mathrm{Lmaxo}}^2\epsilon }{16{\pi }^2{B}_o^2}& \left(26\right)\end{array}$

Thus, all operating point and geometry terms cancel in equation (26), and the maximum volumetric energy handling density of a PR depends on only its material properties (defined in Table 2). The following can therefore be considered a volumetric energy handling density FOM for direct comparison between PR materials and vibration modes:

$\begin{array}{cc}{\mathrm{FOM}}_{\mathrm{VED}}=\frac{I_{\mathrm{Lmaxo}}^2\epsilon }{16{\pi }^2{B}_o^2}& \left(27\right)\end{array}$ which has units J/m³. FOM_VED is summarized for various PR vibration modes in Table 6; the minimum-loss wave number κ_o as detailed in Section 4(c) may be used. The corresponding volumetric power density can be calculated by multiplying FOM_VED by the operating frequency, though frequency depends on {circumflex over (l)} for parallel modes.

TABLE 6
Power Density Figures of Merit
Vibration Mode	FOMVED	FOMAPD	As
Length Extensional (s)	$\frac{I_{\mathrm{Lmaxo}}^2}{{\stackrel{\_}{\kappa }}_o^2{v}_a^2{\epsilon }_{33}^T\left(1-k_{31}^2\right)}$	$\frac{I_{\mathrm{Lmaxo}}^2}{\pi {\stackrel{\_}{\kappa }}_o{\epsilon }_{33}^T\left(1-k_{31}^2\right)v_a}$	4bl
Length Extensional (e)	$\frac{I_{\mathrm{Lmaxo}}^2}{{\stackrel{\_}{\kappa }}_o^2{v}_a^2{\epsilon }_{33}^T\left(1-k_{33}^2\right)}$	$\frac{I_{\mathrm{Lmaxo}}^2}{\pi {\stackrel{\_}{\kappa }}_o{\epsilon }_{33}^T\left(1-k_{33}^2\right)v_a}$	A
Thickness Shear (side)	$\frac{I_{\mathrm{Lmaxo}}^2}{{\stackrel{\_}{\kappa }}_o^2{v}_a^2{\epsilon }_{11}^S}$	$\frac{I_{\mathrm{Lmaxo}}^2}{\pi {\stackrel{\_}{\kappa }}_o{\epsilon }_{11}^S{v}_a}$	A
Thickness Shear (end)	$\frac{I_{\mathrm{Lmaxo}}^2}{{\stackrel{\_}{\kappa }}_o^2{v}_a^2{\epsilon }_{11}^S}$	$\frac{I_{\mathrm{Lmaxo}}^2}{\pi {\stackrel{\_}{\kappa }}_o{\epsilon }_{11}^S{v}_a}$	4bl
Thickness Extensional	$\frac{I_{\mathrm{Lmaxo}}^2}{{\stackrel{\_}{\kappa }}_o^2{v}_a^2{\epsilon }_{33}^S}$	$\frac{I_{\mathrm{Lmaxo}}^2}{\pi {\stackrel{\_}{\kappa }}_o{\epsilon }_{33}^S{v}_a}$	A
Contour Extensional	$\frac{I_{\mathrm{Lmaxo}}^2}{{\stackrel{\_}{\kappa }}_o^2{v}_a^2{\epsilon }_{33}^T\left(1-k_p^2\right)}$	$\frac{I_{\mathrm{Lmaxo}}^2}{\pi {\stackrel{\_}{\kappa }}_o{\epsilon }_{33}^T\left(1-k_p^2\right)v_a}$	4al
Radial	$\frac{I_{\mathrm{Lmaxo}}^2}{{\stackrel{\_}{\kappa }}_o^2{v}_a^2{\epsilon }_{33}^T\left(1-k_p^2\right)}$	$\frac{I_{\mathrm{Lmaxo}}^2}{\pi {\stackrel{\_}{\kappa }}_o{\epsilon }_{33}^T\left(1-k_p^2\right)v_a}$	πal

Referring to Table 6, from equation (21),

${\stackrel{\_}{\kappa }}_o=\frac{\pi {\gamma }_o}{\pi +{\gamma }_o}$ (see Appendix 10(c) for radial mode).

For parallel modes, maximum volumetric power density occurs for

$l=\frac{3}{4}\hat{l},$ though with diminishing returns with respect to frequency for l<{circumflex over (l)}. The volumetric power density at

$l=\frac{3}{4}\hat{l}$ is approximately 18.5% greater than that at l={circumflex over (l)}. For perpendicular modes, frequency depends on a non-l dimension and can therefore be manipulated independently.B. Areal Power Handling Density

In some applications, power handling capability per footprint area may be more useful to the designer than volumetric energy handling density. Areal power density also becomes more relevant for highly-planar PR designs, as often dictated by Ĝ in (17) for maximum efficiency. Like Section 3(c), it is assumed that the area of interest A_s=A for parallel modes and A_s=G_ƒAl for perpendicular modes as displayed in Table 6. Similar to equation (23), the areal power density can be written as follows with I_L=I_Lmax and geometry terms extracted:

$\begin{array}{cc}\frac{P_{\mathrm{out}}}{A_s}=\frac{1}{\pi l}{V}_{\mathrm{in}}{I}_{\mathrm{Lmaxo}}-\frac{1}{l^2}{B}_o{V}_{\mathrm{in}}^2& \left(28\right)\end{array}$

This expression can be likewise maximized with respect to l, resulting in the same {circumflex over (l)} condition and the following maximum areal power density:

$\begin{array}{cc}\hat{l}=\frac{2\pi B_o{V}_{\mathrm{in}}}{I_{\mathrm{Lmaxo}}}& \left(29\right)\end{array}$ $\begin{array}{cc}{\left(\frac{P_{\mathrm{out}}}{A_s}\right)}_{\max }=\frac{I_{\mathrm{Lmaxo}}^2}{4{\pi }^2{B}_o}& \left(30\right)\end{array}$

Like the maximum volumetric energy handling density in equation (26), the maximum areal power density for a PR depends on only its material properties. Thus, equation (30) serves as an areal power handling density FOM for PR material and vibration modes:

$\begin{array}{cc}{\mathrm{FOM}}_{\mathrm{APD}}=\frac{I_{\mathrm{Lmaxo}}^2}{4{\pi }^2{B}_o}& \left(31\right)\end{array}$ with units W/m². FOM_APD is likewise summarized Table 6 for each considered vibration mode. It should be noted that this FOM_APD derivation assumes the area A_s corresponds to the PR's relevant footprint, though the validity of this assumption depends on the PR's shape, vibration mode, and mounting structure. For contour and radial modes (i.e., modes for which both non-i dimensions are equal), FOM_APD may be scaled by

$\frac{4}{\hat{G}}$ for a representative areal power density based on A (assuming a PR design will adhere to the Ĝ condition for maximum efficiency). Since Ĝ varies by operating point, this scaled FOM_APD for such configurations also depends on operating point information unless FOM_APD is limited by loss density (which cancels the Ĝ dependence).

6. Material and Vibration Mode Comparison

Equipped with FOMs for mechanical efficiency and power density, it is now demonstrated how to use these FOMs to evaluate piezoelectric materials and vibration modes for power conversion. First relative capabilities of commercially-available variants of hard PZT are compared. PZT is the most widely-utilized piezoelectric material for sensing, actuation, transduction, and energy harvesting applications. This is followed by a comparison of the seven vibration modes discussed herein for PZT and lithium niobate (LiNbO₃), a second piezoelectric material of emerging interest for power conversion. It should be noted that the results of these studies are meant to be more demonstrative of the FOMs than prescriptive of the materials; their results are sensitive to variation between manufacturers in terms of measurement procedure and/or reporting of material properties (particularly for E_max and Q_m).

A. Comparing Materials

Commercial PZT materials can be compared based on efficiency and power density for the thickness extensional and thickness shear vibration modes (with side electrodes). The following properties for 30 hard PZT materials from eight different manufacturers can be collected and/or calculated: Q_m, ε₃₃^S, ε₁₁^S, k_t, k₁₅, and v_a for each mode (

$v_a=\frac{\pi N}{{\kappa }_{o,r}},$ where N is the manufacturer-provided frequency constant). Sinusoidal amplitude E_max can be conservatively estimated to be 500 V/mm for all materials, and power density capability is assumed to be limited by this E_max. These properties can be used to calculate FOM_M and FOM_APD for both modes with each material.

FIG. 7 is a plot 700 displaying these results with one FOM on each axis, and the considered materials exhibit wide variation in terms of capability. In particular, FOM_M is represented by horizontal axis 702x and FOM_APD is represented by vertical axis 702y. FIG. 7 shows a comparison of hard PZT materials for power conversion based on FOM_M and FOM_APD, overlaid with areal-loss-density contour lines (e.g., contour line 704). Filled and unfilled markers (e.g., filled marker 706 and unfilled marker 708) indicate the thickness shear with side electrodes (e.g., mode 206 of FIG. 2C) and thickness extensional vibration modes (e.g., mode 210 of FIG. 2E), respectively. The two considered operating modes are generally clustered, with thickness extensional mode (denoted by unfilled markers) demonstrating higher power density capability given the same E_max and thickness shear mode (denoted by filled markers) showing higher efficiency capability.

FIG. 7 is further overlaid with loss-density contour lines (e.g., contour line 704) to help identify thermal management limits, which vary based on design and application constraints. For a given areal loss density limit, all points above the corresponding contour line can be projected downward (in the −y direction, keeping constant FOM_M) onto the allowable heat transfer line itself; this new point conveys their maximum power densities given the assumed limit. It can be inferred that many materials would in fact never reach the assumed E_max limit without aggressive thermal management; for a given thermal capability, higher power density is instead enabled by a higher FOM_M. In the context of this disclosure, this implies that the thickness shear mode can enable higher power densities than the thickness extensional mode under practical loss density constraints (≤1 W/cm²).

B. Comparing Vibration Modes

Next, the seven vibration modes analyzed herein for hard PZT and lithium niobate are compared. Using material properties respectively from “Material coefficients PIC181,” PI Ceramic and “Lithium niobate,” Boston Piezo Optics Inc. (available online), FOM_M, FOM_VED, and FOM_APD are calculated each vibration mode and display in Table 7. For lithium niobate, material properties stated in the later reference are assumed as-is, though different cuts can be similarly compared.

TABLE 7
Vibration Mode Comparison for Hard PZT and LiNbO3
	Hard PZT (PIC181)	LiNbO3
		FOMVED	FOMAPD		Theoretical		FOMVED	FOMAPD		Theoretical
Mode	FOMM	(J/m3)	(W/cm2)	Ĝ*	Design*	FOMM	(J/m3)	(W/cm2)	Ĝ*	Design*
Length	63.4	15100	63.4	264	l = 240 μm,	1.26	4.14	1.26	5190	l = 390 μm,
Ext. (s)					b = 16 mm					b = 510 mm
Length	345	1930	345	395	N/A	74.0	224	74.0	4910	N/A
Ext. (e)
Thickness Shear (side)	303	2740	302	553	l = 77 μm, $\sqrt{\frac{A}{4}}=910\mu m$	1260	7490	1520	5020	l = 11 μm, $\sqrt{\frac{A}{4}}=410\mu m$
Thickness	338	3210	337	581	l = 71 μm,	1388	8630	1680	5240	l = 11 μm,
Shear (end)					b = 10 mm					b = 14 mm
Thickness Extensional	138	631	138	330	l = 150 μm, $\sqrt{\frac{A}{4}}=1.3\mathrm{mm}$	72.0	198	72.0	4450	l = 56 μm, $\sqrt{\frac{A}{4}}=1.9\mathrm{mm}$
Contour	302	13200	33.0*	302	l = 36 μm,	3.14	12900*	3.14	5090	l = 7 μm,
Extensional					a = 2.8 mm					a = 8.9 mm
Radial	230	15600	67.3*	230	l = 34 μm,	2.71	8040*	3.12	4030	l = 9 μm,
					a = 2.5 mm					a = 11 mm

(*) Assuming an example operating point of V_in=100V and P_out=10 W. PZT power density is E-field-limited for contour extensional and radial modes at this assumed operating point. All other modes (including for LiNbO₃) are loss-limited.

In Table 7, the same Q_m and E_max are assumed for all modes of each material; Q_m is assumed to be 2,200 for PIC181 and 10,000 for LiNbO₃ (as reported in the datasheets), and E_max is assumed to be 633 V/mm for PIC181 and 7000 V/mm for LiNbO₃ (these values are ⅓ the coercive field of each material). A loss density limit of 1 W/cm² is also assumed, so the displayed FOMs are based on the lower of these two limits as described in Section 3(b). Contour and radial mode FOMs that have been scaled by Ĝ are marked with asterisks in Table 7; these FOMs assume V_in=100V and P_out=10 W. For this operating point, Ĝ and a corresponding theoretical design for both maximum efficiency and maximum power density (based on Ĝ and {circumflex over (l)}) are displayed for each mode. Modes for which this theoretical design violates the relative dimension assumptions in FIGS. 2A-2G are marked with “N/A”.

Table 7 shows the modes of each material to have significantly varying capabilities with respect to mechanical efficiency, volumetric energy handling density, and areal power density. Higher FOM_M is associated with higher k² (since the same Q_m is assumed for all modes of the same material), and particularly high FOM_Ms are shown for LiNbO₃ shear modes. Perpendicular vibration modes tend to have higher FOM_VED and lower FOM_APD compared to parallel modes, which implies that perpendicular modes are capable of higher volumetric power densities for a given frequency but tend to be more planar in nature, requiring more footprint area. It is also noted that some modes would have to operate at more than an order of magnitude higher frequency than other modes to achieve the same volumetric power density.

Differences in geometry requirements are further highlighted by Ĝ and the displayed theoretical designs. LiNbO₃ generally requires an order of magnitude higher Ĝ than PZT, resulting in more planar-shaped geometries that often require more footprint area or higher operating frequencies for similar density. Perpendicular modes likewise require more extreme planar shapes for the same Ĝ compared to parallel modes (as described in Section 4(b), parallel modes have the advantage that the length dimensions comprising Ĝ are squared). This is less pronounced at lower G values but more exaggerated for higher Ĝ, making parallel modes more advantageous for high power applications. As Ĝ approaches infinite, all modes besides length extensional (end electrodes) are capable of designs that meet the relative dimension assumptions in FIGS. 2A-2G. As Ĝ approaches zero, the set of compatible modes reduces to just the two length extensional modes and thickness shear (end electrodes) mode (out of the modes considered).

Of note, length extensional with side electrodes mode and thickness shear with end electrode modes are theoretically capable of meeting the relative dimension assumptions for any Ĝ since α is unconstrained, though the resulting shape may be heavily distorted.

Thus, the most appropriate piezoelectric materials and vibration modes depend heavily on the target application space. In addition to PZT and LiNbO₃, there are numerous other piezoelectric materials that may be similarly evaluated for power conversion using these FOMs.

7. Figure of Merit Validation

To evaluate their utility, the FOMs derived herein are now validated with a numerically-obtained periodic steady state solution (PSSS) of converter behavior and experimental results.

A. Periodic Steady State Solution

As is known, a PSSS details the PR's state trajectories throughout a converter switching sequence based on its circuit model parameters and the desired operating point (V_in, V_out, and P_out); it has no dependence on the amplitude of resonance model or other derivations herein. With an “exact” PSSS (i.e., considering resistor 408 (R) in the circuit model FIG. 4, which requires a numerical solution), useful information can be extracted from the PR's state trajectories like loss and output power for validating the derived FOMs.

First, validation is done of the minimum loss ratio and maximum power densities with geometry and material data for 572 APC International discrete PR parts listed on “Ready to deliver piezoelectric ceramics,” APC International (available online); these parts consist of round and rectangular PRs of varying dimensions, spanning nine total materials. For each part, the following can be performed:

• • 1. Calculate its circuit model parameters (as shown in Table 3) for a given vibration mode.
  • 2. Based on its dimensions, determine V_in and P_out corresponding to its minimum loss ratio and maximum energy handling density using equations (17) and (25), assuming I_Lmax to be limited by a loss density of 1 W/cm².
  • 3. For this operating point, solve for the exact PSSS of the PR's states as detailed in Boles et al., assuming the converter switching sequence and high-efficiency behaviors corresponding to equation (6) and V_out=0.55V_in (I_L is predicted to be constant with respect to V_out for

$V_{in}>V_{\mathrm{our}}>\frac{1}{2}{V}_{in}$

• • 4. Extract output power and loss information from the PR's PSSS state trajectories.

Ultimately, all discrete parts of the same material yield the same minimum loss ratio, maximum energy handling density, and maximum areal power density for a given vibration mode. In Table 8, these PSSS-calculated results for the k₃₃ mode are compared with their derived estimates and demonstrate very small error for all low-loss-ratio materials (though the error is higher for lossier materials since equation (6) models an ideal PR).

TABLE 8
PSSS Figure of Merit Validation
Material	Qm	k33	$\frac{P_{\mathrm{loss}}}{P_{\mathrm{out}}}\mathrm{vs}.\left(18\right)$	κo vs. (21) (radians)	$\frac{E_{\mathrm{out}}}{\mathrm{vol}}\mathrm{vs}.\left(26\right)$ (J/m3)	$\frac{P_{\mathrm{out}}}{A}\mathrm{vs}.\left(30\right)$ (W/cm2)	$\frac{P_{\mathrm{loss}}}{A}\mathrm{vs}.1.$ (W/cm2)
840	500	0.72	.0102 | .0102	1.371 | 1.364	429.6 | 432.3	97.97 | 98.07	1.002
841	1400	0.68	.0041 | .0041	1.389 | 1.385	1073 | 1079	241.0 | 241.9	0.9912
842	600	0.71*	.0086 | .0086	1.373 | 1.366	499.3 | 502.0	116.3 | 116.3	1.001
844	1500	0.65*	.0043 | .0044	1.404 | 1.405	1009 | 1009	228.6 | 228.6	.9940
840-844 Average Error:	0.37%	0.31%	0.45%	0.13%	0.43%
880	1000	0.62	.0083 | .0083	1.433 | 1.436	523.2 | 523.7	119.5 | 120.0	.9939
881	1000	0.73*	.0047 | .0047	1.360 | 1.350	906.4 | 910.4	213.6 | 212.9	1.002
880-881 Average Error:	0.20%	0.50%	0.26%	0.33%	0.39%

(*) Calculated using k₃₃²=Y₃₃^Ed₃₃²/ε₃₃^T. Not shown in Table 8: Soft PZT materials 850, 851, and 855 have loss ratios up to 0.069, with an average error (compared to (18)) of up to 6%. These materials have significantly more loss than hard PZT and are therefore less conducive to power conversion.

Thus, the PSSS validates the following:

• • The independence of

${\overline{\kappa }}_o,{\left(\frac{P_{loss}}{P_{out}}\right)}_{m\mathrm{in}},{\left(\frac{E_{out}}{\mathrm{vol}}\right)}_{max},{\left(\frac{P_{out}}{A_s}\right)}_{m\mathrm{ax}},\mathrm{and}\frac{P_{loss}}{A_s}$ from PR geometry and operating point information; these quantities depend on only material properties.

• • The minimum loss ratio value of equation (18), geometry condition of equation (17), and operating point of equation (21).
  • The maximum energy density and power density values of equations (26) and (30), respectively, and geometry condition of equation (25).
  • The dependence of energy, power, and loss densities of equations (26), (30), and (10), respectively, on normalized amplitude of resonance I_Lo.B. Experimental Results

FIG. 8 illustrative various PRs 802a-802f tested for experimental FOM validation. The illustrated PRs 802a-802f may be identified by part #'s 1553, 2040, 1105, 790, 1268, and 1817, respectively. The wire attachment locations for the various PRs 802a-802f are marked with respective dots 804a-804f.

FIG. 9 shows an experimental prototype board 900 that can be used for testing the PRs 802a-802f of FIG. 8. The various PRs can be mounted proximate to PR pads 904a, 904b. In more detail, part #790 (e.g., PR 802d in FIG. 8) and part #1817 (e.g., PR 802f in FIG. 8) can be mounted in an upright position on the board, with two wires mechanically supporting the PR. All other PRs can be positioned horizontally with the bottom electrode making contact with the copper ribbon and the other attached with a non-rigid wire. The board 900 of FIG. 9 also includes switches 906a-906d (S1-S4). One or more of the switches 906a-906d may be provided as field-effect transistors (FETs), such as EPC 2019 GaN FETs. In some embodiments, third switch 906c (S3) and fourth switch 906d (S4) may be provided as diodes, such as ON Semiconductor NRVSTA4100 Schottky diodes.

FIG. 10 shows a prototype topology 1000 that may be used within the board of FIG. 9. Like elements of FIG. 9 are show using like reference designators in FIG. 10. In addition to switches 906a-906d, the topology 1000 includes a PR model 1002, an input voltage 1004, a resistive load 1006, and a capacitor 1008, arranged as shown. PR model 1002 may be the same as or similar to circuit model 400 of FIG. 4. As illustrated, switches 906c, 906d (S3, S4) can be implemented as active switches 908c, 908d or diodes 910c, 910d (i.e., topology 1000 may include either the active switches 908c, 908d shown or the diodes 910c, 910d shown).

The mechanical efficiency FOM is further validated experimentally using six of the PR parts considered in Section 7. These PRs consist of APC International's highest FOM_M materials (841, 844, 880, and 881) in different shapes and sizes as pictured in FIG. 8. These parts are selected to be low in frequency (<600 kHz) to minimize frequency-dependent loss and potential damping effects due to mounting when validating the FOMs. With each PR, the following can be performed:

• • 1. Plot the PR's impedance characteristic (e.g., generate a plot similar to plot 300 of FIG. 3) for a given vibration mode using a network analyzer.
  • 2. Estimate Q_m and k based on the impedance characteristic, as detailed in Appendix D(a).
  • 3. Calculate the minimum loss ratio for the estimated Q_m and k using equation (18), as well as the minimum-loss-ratio P_out dictated by equation (17), assuming V_in=100 V (detailed in Appendix D(b)).
  • 4. Test the PR in a prototype converter, such as shown in FIG. 9 (which has the topology in FIG. 10) with the V_in−V_out, Zero, V_out switching sequence and high-efficiency behaviors assumed in Section 3. V_in=100 V and V_out=49 V. To identify the minimum loss ratio, sweep through multiple power levels surrounding the calculated P_out. More information about the converter prototype and switching sequence can be found in Boles et al.
  • 5. Reduce whole-converter power measurements to an approximate PR loss ratio by extracting other known loss (i.e., the voltage drop incurred by the Schottky diode rectifier). This is likewise detailed in Appendix D(c).

Of note, the switching sequence assumed in Section 3 has maximum and constant efficiency for the range

$V_{in}>V_{\mathrm{out}}>\frac{1}{2}{V}_{in},$ which requires an active switch for S3. Efficiency is continuous across the

$V_{\mathrm{out}}=\frac{1}{2}{V}_{in}$ boundary and then decreases for

$V_{\mathrm{out}}<\frac{1}{2}{V}_{in},$ though the latter region permits used of two diodes for S3 and S4 as detailed in Boles et al. In the absence of closed-loop control, testing can be done at V_out=0.49V_in with a diode rectifier; this operating point is very close to the boundary

$\mathrm{of}{V}_{\mathrm{out}}=\frac{1}{2}{V}_{in}$ and therefore still provides representative behavior of the assumed operating region as demonstrated in Boles et al.

The results of these experiments, along with material, vibration mode, and frequency information for each PR, are displayed in Table 9.

TABLE 9
Experimental Figure of Merit Validation
Part No.	Material	Vibration Mode	Qm	k	Cp (nF)	f vs. (20) (kHz)	Pout vs. (17) (W)	$\frac{P_{\mathrm{loss}}}{P_{\mathrm{out}}}\mathrm{vs}.\left(18\right)$ (PR only)	Converter Efficiency
1817	841	Length	850	.32	.879	56.7 | 56.6	0.61 | 0.78	.0449 | .0402	95.02%
		Ext. (s)
1105	841	Thickness	2500	.31	.602	601 | 596	6.4 | 6.6	.0247 | .0156	96.90%
		Ext.
2040	880	Contour	1600	.52	1.64	80.6 | 80.9	1.6 | 1.7	.0095 | .0069	98.36%
		Ext.
1553	841	Radial	1700	.58	1.50	80.4 | 80.9	1.6 | 1.6	.0070 | .0049	98.61%
790	844	Radial	1400	.55	3.53	126 | 122	2.8 | 4.9	.0068 | .0069	98.63%
1268	881	Radial	1600	.52	.643	128 | 129	1.3 | 1.5	.0094 | .0072	98.36%

All quantities in Table 9 are either measured values or calculations based on the measured values for ƒ_r, ƒ_ar, C_p, P_in, and P_out.

FIG. 11 is a plot 1100 showing experimental minimum loss ratio 1102 compared to the estimate 1104 in equation (18) for the PRs 802a-802f of FIG. 8 operated in the converter prototype 900 of FIG. 9. The estimated loss ratio tracks the trend of the experimental loss ratio as visualized in plot 1100 of FIG. 11, albeit a slight underestimation for most PRs. Sources for the observed discrepancies may include parasitic capacitances on the board such as switch capacitances (the PRs were characterized alone on the network analyzer, without the board), as well as differences between the PR's small-signal (as characterized) and large-signal (as tested) characteristics. Nevertheless, equation (18) provides a close approximation of the loss ratio to be expected by a given material and vibration mode, validating the utility of FOM_M.

8. Fundamental Scaling Properties

PRs have been previously suggested to have advantageous power density and efficiency scaling properties compared to magnetics. Using FOM derivations described herein, it can now be explored how PR capabilities scale with size for a converter implementation.

If a PR is scaled in all three dimensions by linear scaling factor α as has been postulated for magnetics, its volume scales by α³. In the case of a PR with fixed I_Lo, maximum volumetric power density scales inversely with α as does operating frequency, both due to G_ƒ. However, the minimum loss ratio, the maximum volumetric energy handling density, the maximum areal power density, and the areal loss density have no geometry dependence and therefore remain fixed regardless of α. Table 10 summarizes these characteristics.

TABLE 10
Piezoelectric Resonator Scaling Properties
	Geometry
Property	Dependence	Scaling (fixed ILo)
$\left(\frac{P_{\mathrm{out}}}{\mathrm{vol}}\right)\left(23\right)$	Gf	α−1
${\left(\frac{E_{\mathrm{out}}}{\mathrm{vol}}\right)}_{m\mathrm{ax}}\left(26\right)$	none	constant
${\left(\frac{P_{\mathrm{out}}}{A_s}\right)}_{m\mathrm{ax}}\left(30\right)$	none	constant
${\left(\frac{P_{\mathrm{loss}}}{P_{\mathrm{out}}}\right)}_{min}\left(18\right)$	none	constant
$\frac{P_{\mathrm{loss}}}{A_s}\left(9\right)$	none	constant
f (3)	Gf	α−1

Thus, as volume is scaled downward (α<1), maximum volumetric power density increases while minimum loss ratio (and therefore maximum efficiency) stays constant. These are favorable scaling properties for converter miniaturization.

9. Conclusions

Disclosed herein are techniques and structures for evaluating and selecting piezoelectric materials and vibration modes for power conversion. FOMs are established for achievable efficiency, volumetric energy handling density, and areal power density based on power converter operation. These FOMs depend on only material properties (and loss density limits if considered) and correspond to PR geometry conditions for realizing both maximum efficiency and maximum power density in PR designs.

The mechanical efficiency FOM depends on only k² and Q_m, and its corresponding geometry condition dictates the relative PR dimensions required to achieve maximum efficiency for a given operating point. Parallel modes are particularly advantageous for satisfying this condition with less-extreme planar shapes. Which vibration modes are compatible with a given operating space is likewise dictated by this geometry condition, which favors vibration modes with l as their smallest dimensions for most converter applications. The operating frequency at which maximum efficiency occurs is found to be the geometric mean of the PR's resonant and anti-resonant frequencies for the assumed converter operation.

The volumetric energy handling density and areal power density FOMs are constrained by the PR's geometry-normalized amplitude of resonance, which can be calculated based on material limits or an acceptable loss density. The volumetric energy handling density FOM is normalized to frequency, which permits direct comparison between vibration modes for a given converter switching and control requirement. The areal power density FOM provides insight into footprint requirements for planar-shaped PRs, though the utility of this FOM depends on whether its assumed area corresponds to the footprint area of a given PR configuration (this FOM is often scaled by the maximum-efficiency geometry condition when representing contour and radial modes). Both of these FOMs have the same geometry condition for l that corresponds to maximum power density for a given operating point.

The present disclosure illustrates the utility of the proposed FOMs to compare the capabilities of 30 PZT-based materials and seven PR vibration modes for PZT and lithium niobate. These materials and vibration modes vary immensely with respect to the FOMs, though FOMs tend to cluster according to vibration mode for variants of the same material. With the same Q_m assumed for each mode, the shear modes demonstrate the highest achievable efficiencies, particularly for lithium niobate. In general, materials and modes with higher efficiency figures of merit are capable of achieving higher power densities for loss density limits. Perpendicular modes are capable of higher volumetric power densities for a given frequency but tend to be more planar, requiring more footprint area than parallel modes to meet the maximum-efficiency geometry condition. Lithium niobate also necessitates more extreme planar dimensions than PZT to meet this geometry condition, requiring more footprint area and/or higher frequency for the same volumetric power density.

Appendix A—Piezoelectric Resonator Wave Solution

In this Appendix, the PR's acoustic wave solution for each vibration mode is derived. Piezoelectric materials are governed by the following constitutive relations between mechanical strain (S), mechanical stress (T), electric field strength (E), and electric flux density (D):

$\begin{array}{cc}{S}_{ij}=\frac{1}{2}\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)=s_{ijmn}^E{T}_{mn}+d_{mij}{E}_m& \left(32\right)\end{array}$ $\begin{array}{cc}{D}_i=d_{imn}{T}_{mn}+{\epsilon }_{im}^T{E}_m& \left(33\right)\end{array}$ and equation of motion:

$\begin{array}{cc}\frac{\partial T_{\mathrm{ij}}}{\partial x_i}=\rho \frac{{\partial }^2{u}_j}{\partial c^2}& \left(34\right)\end{array}$

Together, these equations determine the behavior of the PR states, which varies by vibration mode. Parameters for these equations are defined in Table 1, and i, j, m, and n specify tensor components using index notation.

A. One-Dimensional Stress/Strain Modes

Most vibration modes considered (length extensional, thickness shear, and thickness extensional) can be modeled by one-dimensional stress/strain (i.e., only one tensor component for each PR state in equations (32)-(34) may be considered). Thus, the coupled constitutive relations take the reduced form:S=S^ET+dE  (35)D=dT+ε^TE  (36)

To illustrate the PR behaviors for each vibration mode, a generalized Voigt notation can be adopted for all parameters without indices; the tensor components pertinent to each vibration mode are displayed in Table 11. A location along the l-dimension axis for parallel modes and a-dimension axis for perpendicular modes are herein referred to generally as “x”.

TABLE 11
Parameters for each Vibration Mode
Mode	va	d	sE	εT	k
Length Ext. (s)	$\sqrt{\frac{1}{\rho s_{11}^E}}$	d31	s11E	ε33T	$k_{31}=\frac{d_{31}}{\sqrt{s_{11}^E{\epsilon }_{33}^T}}$
Length Ext. (e)	$\sqrt{\frac{1}{\rho s_{33}^D}}$	d33	s33E	ε33T	$k_{33}=\frac{d_{33}}{\sqrt{s_{33}^E{\epsilon }_{33}^T}}$
Thickness shear (side)	$\sqrt{\frac{1}{\rho s_{55}^D}}$	d15	s55E	ε11T	$k_{15}=\frac{d_{15}}{\sqrt{s_{55}^E{\epsilon }_{11}^T}}$
Thickness shear (end)	$\sqrt{\frac{1}{\rho s_{55}^E}}$	d15	s55E	ε11T	$k_{15}=\frac{d_{15}}{\sqrt{s_{55}^D{\epsilon }_{11}^S}}$
Thickness Extensional	$\sqrt{\frac{c_{33}^D}{\rho }}$	**	**	ε33T	$k_t=\frac{e_{33}}{\sqrt{c_{33}^D{\epsilon }_{33}^S}}$
Contour Extensional	$\sqrt{\frac{1}{\rho s_{11}^E\left(1-{\sigma }^2\right)}}$	d31	s11E	ε33T	$k_p=\sqrt{\frac{2k_{31}^2}{1-\sigma }}$
Radial	$\sqrt{\frac{1}{\rho s_{11}^E\left(1-{\sigma }^2\right)}}$	d31	s11E	ε33T	$k_p=\sqrt{\frac{2k_{31}^2}{1-\sigma }}$

In Table 11, the following measurement condition conversions apply: ε₁₁^S=(1−k₁₅²)ε₁₁^T ε₃₃^S=(1−k₃₃²)ε₃₃^T S₁₁^D=(1−k₃₁²)S₁₁^ES₃₃^D=(1−k₃₃²)S₃₃^E S₅₅^D=(1−k₁₅²)S₅₅^E C₃₃^E=(1−k_t²)C₃₃^D. (**) Thickness extensional mode is more commonly represented by e₃₃ and c₃₃.

With the equation of motion (34), (35)-(36) create an acoustic wave equation for mechanical displacement u:

$\begin{array}{cc}\frac{{\partial }^{2}u}{\partial t^2}=v_a^2\frac{{\partial }^{2}u}{\partial x^2}& \left(37\right)\end{array}$

The PR is assumed to resonate in the proximity of its lowest-frequency vibration mode for traction-free boundaries in FIGS. 2A-2G. Thus, the acoustic wave solution is sinusoidal in form:u=Δ sin(κx)e^jωt  (38)

Inserting equation (38) into the constitutive relations and enforcing traction-free boundaries yields the solutions for S, T, and E; these solutions are displayed for each mode in Table 12.

TABLE 12
One-Dimensional Vibration Mode Derivations
	Length Ext. (s) and Thickness	Length Ext. (e) and
	Shear (e)	Thickness Shear (s)	Thickness Extensional
S	κΔcos(κx)ejωt	κΔcos(κx)ejωt	κΔcos(κx)ejωt
T	$\frac{1}{s^E}\kappa \Delta \left(\cos \left(\kappa x\right)-\cos \left({\kappa }_o\right)\right)e^{j\omega t}$	$\frac{1}{s^D}\kappa \Delta \left(\cos \left(\kappa x\right)-\cos \left({\kappa }_o\right)\right)e^{j\omega t}$	cDκΔ(cos(κx) − cos(κo))ejωt
E	$\frac{\kappa \Delta }{d}\left(\cos \left({\kappa }_o\right)\right)e^{j\omega t}$	$\frac{\kappa \Delta }{d\left(1-k^2\right)}\left(-k^2\cos \left(\kappa x\right)+\cos \left({\kappa }_o\right)\right)e^{j\omega t}$	$\frac{c^D\kappa \Delta }{e\left(1-k^2\right)}\left(-k^2\cos \left(\kappa x\right)+\cos \left({\kappa }_o\right)\right)e^{j\omega t}$
IL	$\kappa {\mathrm{AG}}_f\frac{v_{a}d}{s^E}\Delta \sin \left({\kappa }_o\right)$	$\kappa {\mathrm{AG}}_f\frac{v_{a}d}{s^E}\Delta \sin \left({\kappa }_o\right)$	κAGfvaeΔsin(κo)
ΔS	$\frac{1}{\kappa }{S}_{\max }$	$\frac{1}{\kappa }{S}_{\max }$	$\frac{1}{\kappa }{S}_{\max }$
ΔT	$\frac{s^E}{\kappa }\frac{1}{1-\cos \left({\kappa }_o\right)}{T}_{\max }$	$\frac{s^D}{\kappa }\frac{1}{1-\cos \left({\kappa }_o\right)}{T}_{\max }$	$\frac{1}{c^D\kappa }\frac{1}{1-\cos \left({\kappa }_o\right)}{T}_{\max }$
ΔE	$\frac{d}{\kappa }\frac{1}{\cos \left({\kappa }_o\right)}{E}_{\max }$	$\frac{d}{\kappa }\frac{\left(1-k^2\right)}{\left(\cos \left({\kappa }_o\right)-k^2\right)}{E}_{\max }$	$\frac{e}{c^D\kappa }\frac{1}{\left(\cos \left({\kappa }_o\right)-k^2\right)}{E}_{\max }$
ILmaxS	${\mathrm{AG}}_f\frac{v_{a}d}{s^E}{S}_{\max }\sin \left({\kappa }_o\right)$	${\mathrm{AG}}_f\frac{v_{a}d}{s^E}{S}_{\max }\sin \left({\kappa }_o\right)$	${\mathrm{AG}}_f\frac{v_{a}d}{s^E}{S}_{\max }\sin \left({\kappa }_o\right)$
ILmaxT	${\mathrm{AG}}_f{v}_a{\mathrm{dT}}_{\max }\cot \left(\frac{{\kappa }_o}{2}\right)$	${\mathrm{AG}}_f{v}_{a}d\left(1-k^2\right)T_{\max }\cot \left(\frac{{\kappa }_o}{2}\right)$	${\mathrm{AG}}_f{v}_a\frac{e}{c^D}{T}_{\max }\cot \left(\frac{{\kappa }_o}{2}\right)$
ILmaxE	AGfk2εTvaEmaxtan(κo)	${\mathrm{AG}}_f{k}^2{\epsilon }^S{v}_a{E}_{\max }\frac{\sin \left({\kappa }_o\right)}{\cos \left({\kappa }_o\right)-k^2}$	${\mathrm{AG}}_f{k}^2{\epsilon }^S{v}_a{E}_{\max }\frac{\sin \left({\kappa }_o\right)}{\cos \left({\kappa }_o\right)-k^2}$

FIGS. 12A and 12B illustrate the relative amplitudes of u, S, T, and E along the axis of G_ƒ in a PR. FIG. 12A is a plot 1200 illustrating vibration modes in which the applied and induced E fields are parallel (∥). FIG. 12B is another plot 1220 illustrating vibration modes in which the applied and induced E fields are perpendicular (+). These states retain similar spacial dependencies across x for each vibration mode, though their specific directions may be different. Moreover, the maximum amplitudes for S, T, and E each occur at the center of the PR (at x=0).

This wave solution provides means to analyze the PR's mechanical and electrical limits, which is utilized in Section 5 and Appendix B for calculating maximum power density. Also, the Butterworth-Van Dyke circuit model (FIG. 4) can be derived from E.

B. Contour Extensional Mode

For contour extensional mode, it can be assumed that the PR to be under plane stress (i.e., consider only the normal stress components along the two axes perpendicular to the applied E field). These normal stresses are denoted with T₁ and T₂, and their coordinate plane is defined by the displacement arrows in FIG. 2F with the origin at the center of the PR. The constitutive relations then have the form:

$\begin{array}{cc}{T}_1=\frac{1}{s_{11}^E\left(1-{\sigma }^2\right)}\left(\frac{\partial u_1}{\partial x_1}+\sigma \frac{\partial u_2}{\partial x_2}\right)-\frac{d_{31}}{s_{11}^E\left(1-\sigma \right)}E& \left(39\right)\end{array}$ $\begin{array}{cc}{T}_2=\frac{1}{s_{11}^E\left(1-{\sigma }^2\right)}\left(\sigma \frac{\partial u_1}{\partial x_1}+\frac{\partial u_2}{\partial x_2}\right)-\frac{d_{31}}{s_{11}^E\left(1-\sigma \right)}E& \left(40\right)\end{array}$ $\begin{array}{cc}D=\frac{d_{31}}{s_{11}^E\left(1-\sigma \right)}\left(\frac{\partial u_1}{\partial x_1}+\frac{\partial u_2}{\partial x_2}\right)+{\epsilon }_{33}^T\left(1-k_p^2\right)E& \left(41\right)\end{array}$

Considered along with equation (34), equations (39)-(41) constitute an acoustic wave equation for mechanical displacement u. Its solution can be approximated by two separate waves, which can be assumed to be identical along their respective dimensions:u₁=Δ sin(κx₁)e^jωt  (42)u₂=Δ sin(κx₂)e^jωt  (43)

Inserting equations (42) and (43) into equations (39) and (40) and enforcing the boundary conditions of T₁=0 at x₁=±a and T₂=0 at x₂=±a provides the analytical solution shown in Table 13 for the PR's S, T, and E states. This solution follows the relative amplitudes illustrated in plot 1220 of FIG. 12B.

TABLE 13
Planar Vibration Mode Derivations
	Contour Extensional	Radial
S	S1 = κΔcos(κx1)ejωt	$S_{\mathrm{rr}}=\frac{\kappa }{2}\Delta \left(J_0\left(\kappa r\right)-J_2\left(\kappa r\right)\right)e^{j\omega t}$
	S2 = κΔcos(κx2)ejωt	$S_{\mathrm{\theta \theta }}=\frac{1}{r}\Delta J_1\left(\kappa r\right)e^{j\omega t}$
T	$T_1=\frac{\kappa \Delta e^{j\omega t}}{s_{11}^E\left(1-{\sigma }^2\right)}\left(\cos \left(\kappa x_1\right)+\sigma \cos \left(\kappa x_2\right)-\left(1+\sigma \right)\cos \left({\kappa }_o\right)\right)$	$T_{\mathrm{rr}}=\frac{\Delta e^{j\omega t}}{s_{11}^E\left(1-{\sigma }^2\right)}\left(\frac{\kappa }{2}\left(J_0\left(\kappa r\right)-J_2\left(\kappa r\right)\right)+\frac{\sigma }{r}{J}_1\left(\kappa r\right)-\kappa \Psi \right)$
	$T_2=\frac{\kappa \Delta e^{j\omega t}}{s_{11}^E\left(1-{\sigma }^2\right)}\left(\mathrm{\sigma cos}\left(\kappa x_1\right)+\mathrm{\sigma cos}\left(\kappa x_2\right)-\left(1+\sigma \right)\cos \left({\kappa }_o\right)\right)$	$T_{\mathrm{\theta \theta }}=\frac{\Delta e^{j\omega t}}{s_{11}^E\left(1-{\sigma }^2\right)}\left(\frac{\mathrm{\kappa \sigma }}{2}\left(J_0\left(\kappa r\right)-J_2\left(\kappa r\right)\right)+\frac{1}{r}{J}_1\left(\kappa r\right)-\kappa \Psi \right)$
E	$\frac{\mathrm{\kappa \Delta }}{d_{31}}\left(\cos \left({\kappa }_o\right)\right)e^{j\omega t}$	$\frac{\mathrm{\kappa \Delta }}{d_{31}\left(1+\sigma \right)}\Psi e^{j\omega t}$
IL	$2\kappa {\mathrm{AG}}_f\frac{v_a{d}_{31}}{s_{11}^E\left(1-\sigma \right)}\Delta \sin \left({\kappa }_o\right)$	${\kappa }_o{v}_a\frac{2\pi d_{31}}{s_{11}^E\left(1-\sigma \right)}\Delta J_1\left({\kappa }_o\right)$
ΔS	$\frac{1}{\kappa }{S}_{\max }$	$\frac{2}{\kappa }{S}_{\max }$
ΔT	$\frac{s_{11}^E\left(1-\sigma \right)}{\kappa \left(1-\cos \left({\kappa }_o\right)\right)}{T}_{\max }$	$\frac{s_{11}^E\left(1-{\left(\sigma \right)}^2\right)}{\kappa \left(\frac{1}{2}\left(1+\sigma \right)-\Psi \right)}{T}_{\max }$
ΔE	$\frac{d_{31}}{\kappa }\frac{1}{\cos \left({\kappa }_o\right)}{E}_{\max }$	$\frac{d_{31}\left(1+\sigma \right)}{\kappa \Psi }{E}_{\max }$
ILmaxS	$2A{G}_f{v}_a\frac{d_{31}}{s_{11}^E\left(1-\sigma \right)}{S}_{\max }\sin \left({\kappa }_o\right)$	$A{G}_f\frac{4d_{31}{v}_a}{s_{11}^E\left(1-\sigma \right)}{S}_{\max }{J}_1\left({\kappa }_o\right)$
ILmaxT	$2A{G}_f{v}_a{d}_{31}{T}_{\max }\cot \left(\frac{{\kappa }_o}{2}\right)$	$A{G}_f\frac{2v_a{d}_{31}\left(1+\sigma \right)}{\frac{1}{2}\left(1+\sigma \right)-\Psi }{T}_{\max }{J}_1\left({\kappa }_o\right)$
ILmaxE	AGfvakp2ε33TEmaxtan(κo)	${\mathrm{AG}}_f{v}_a{k}_p^2{\epsilon }_{33}^T\left(1+\sigma \right)E_{\max }\frac{J_1\left({\kappa }_o\right)}{\Psi }$

C. Radial Mode

For the radial vibration mode, it can be assumed that only planar stress components parallel to the electrodes; E is again applied only in the polarization direction. This time, a cylindrical coordinate system is adopted in which r is the radial coordinate and θ is the hoop coordinate. Thus, the constitutive relations are:

$\begin{array}{cc}{T}_{\mathrm{rr}}=\frac{1}{s_{11}^E\left(1-{\sigma }^2\right)}\left(\frac{\partial u_r}{\partial r}+\sigma \frac{u_r}{r}\right)-\frac{d_{31}}{s_{11}^E\left(1-\sigma \right)}E& \left(44\right)\end{array}$ $\begin{array}{cc}{T}_{\theta \theta }=\frac{1}{s_{11}^E\left(1-{\sigma }^2\right)}\left(\sigma \frac{\partial u_r}{\partial r}+\frac{u_r}{r}\right)-\frac{d_{31}}{s_{11}^E\left(1-\sigma \right)}E& \left(45\right)\end{array}$ $\begin{array}{cc}D=\frac{d_{31}}{s_{11}^E\left(1-\sigma \right)}\left(\frac{\partial u_r}{\partial r}+\frac{u_r}{r}\right)+{\epsilon }_{33}^T\left(1-k_p^2\right)E& \left(46\right)\end{array}$

Together with the equation of motion (34), these constitute a wave equation for mechanical displacement u_r with solutionu_r=ΔJ₁(κr)e^jωt  (47)for which J_n is the Bessel function of first kind and nth order. Inserting equation (47) into equation (44) and enforcing the boundary condition T_rr=0 at r=a yields the analytical solution in Table 13 for the PR's S, T, and E states, which follows the illustrative plot 1220 of FIG. 12B for x=r. In this solution, the following substitution is made:

$\begin{array}{cc}\Psi =\frac{1}{2}{J}_0\left({\kappa }_o\right)-\frac{1}{2}{J}_2\left({\kappa }_o\right)+\frac{\sigma }{{\kappa }_o}{J}_1\left({\kappa }_o\right)& \left(48\right)\end{array}$

The geometry-normalized wave numbers corresponding to ƒ_r and ƒ_ar in radial mode are solutions to the following transcendental equations, respectively:

$\begin{array}{cc}{\kappa }_{o,r}\frac{J_0\left({\kappa }_{o,r}\right)}{J_1\left({\kappa }_{o,r}\right)}=1-\sigma & \left(49\right)\end{array}$ $\begin{array}{cc}{\kappa }_{o,\mathrm{ar}}\frac{J_0\left({\kappa }_{o,\mathrm{ar}}\right)}{J_1\left({\kappa }_{o,\mathrm{ar}}\right)}=1-\sigma -\frac{k_p^2}{1-k_p^2}\left(1+\sigma \right)& \left(50\right)\end{array}$

If needed, Bessel functions J₀(κ_o), J₁(κ_o), and J₂(κ_o) can be approximated by series expansions around κ_o,r; second-order expansions around κ_o=2 are shown in Table 14.

TABLE 14
Bessel Function Expansions
Function                         Expansion Around κo = 2
κ                      o                                        ⁢                                                                                            J                          0                                                (                                                  κ                          o                                                )                                                                                              J                          1                                                (                                                  κ                          o                                                ) 0.776 − 1.525(κo − 2) − 0.789(κo − 2)2
J0(κo)                           0.224 − 0.577(κo − 2) − 0.0322(κo − 2)2
J1(κo)                           0.577 − 0.645(κo − 2) − 0.2(κo − 2)2
J2(κo)                           0.353 − 0.224(κo − 2) − 0.0560(κo − 2)2

The geometry-normalized wave number that corresponds to the minimum-loss-ratio operating frequency is then:

$\begin{array}{cc}{\overline{\kappa }}_o=\frac{2{\kappa }_{o,r}{\kappa }_{o,\mathrm{ar}}}{{\kappa }_{o,r}+{\kappa }_{o,\mathrm{ar}}}& \left(51\right)\end{array}$

Finally, E in Table 13 can be reduced to the circuit model of FIG. 4 with the parameters shown in Table 15.

TABLE 15
Circuit Model Parameters for Radial Vibration Mode
Parameter Radial Mode Expression
Gf        1                    a
Cp        ε                        T                                            (                                              1                        -                                                  k                          p                          2                                                                    )                                        ⁢                                          A                                              2                        ⁢                        l
C         2                        ⁢                                                  (                                                      1                            +                            σ                                                    )                                                                                                                      κ                                                      o                            ,                            r                                                    2                                                -                                                  (                                                      1                            -                                                          σ                              2                                                                                )                                                                                      ⁢                                                                  k                        p                        2                                                                    1                        -                                                  k                          p                          2                                                                                      ⁢                                          C                      p
L         1                                              2                        ⁢                                                  G                          f                          2                                                ⁢                                                  v                          a                          2                                                ⁢                                                  C                          p                                                                                      ⁢                                                                  1                        -                                                  k                          p                          2                                                                                            k                        p                        2                                                              ⁢                                                                                            κ                                                      o                            ,                            r                                                    2                                                -                                                  (                                                      1                            -                                                          σ                              2                                                                                )                                                                                                                      κ                                                      o                            ,                            r                                                    2                                                (                                                  1                          +                          σ                                                )
R         1                                              Q                        m                                                              ⁢                                                                  L                        C

Appendix B—Maximum IL Derivation

In this Appendix, maximum permissible I_L is determined based on physical limits for the PR's S, T, and E states. To begin, derive the relationship between I_L and the PR's maximum displacement amplitude (Δ) using the coupled state equations in (32)-(34) and the solution in Tables 12 and 13. Illustrating with a uni-dimensional vibration mode, inserting equation (32) into equation (33) yields:

$\begin{array}{cc}D=\frac{d}{s^E}\frac{\partial u}{\partial x}+{\epsilon }^T\left(1-k^2\right)E& \left(52\right)\end{array}$

Integrating across the volume of the PR (i.e., across electrode area A and distance 2l) gives:

$\begin{array}{cc}Q=A{G}_f\frac{d}{s^E}u\left(G_f^{-1}\right)+\frac{A}{2l}{\epsilon }^T\left(1-k^2\right)v_{p,1}& \left(53\right)\end{array}$ for which v_p,1 is the first harmonic approximation of v_p. Finally, taking the time derivative yields:

$\begin{array}{cc}{i}_{in}=j\omega A{G}_f\frac{d}{s^E}u\left(G_f^{-1}\right)+C_p\frac{{\mathrm{dv}}_{p,1}}{\mathrm{dt}}& \left(54\right)\end{array}$ in which i_in can be considered the current entering the PR through its top terminal in the model of FIG. 4. This process can be repeated for the planar modes, resulting in a similar form as equation (54) for contour extensional mode and the following form for radial mode:

$\begin{array}{cc}{i}_{in}=j\omega a\frac{2\pi d_{31}}{s_{11}^E\left(1-\sigma \right)}\Delta J_1\left({\kappa }_o\right)e^{j\omega t}+C_p\frac{{\mathrm{dv}}_{p,1}}{\mathrm{dt}}& \left(55\right)\end{array}$

These expressions for i_in correspond to FIG. 4 such that I_L equals the magnitude of the first term in equation (54) or equation (55). I_L is shown for each operating mode in Table 12.

From here, S, T and E can each be related to I_L through Δ. As shown in FIGS. 12A and 12B, the maximum S, T and E each occur at x=0, so their respective equations in Tables 12 and 13 can be rearranged to focus on x=0 to reach Δ_S, Δ_T, and Δ_E as functions of S_max, T_max and E_max, respectively (S_max and T_max refer to the maximum principal strains and stresses for planar modes; this requires taking the limit of each quantity as r→0 for radial mode). Each Δ can then be directly inserted into I_L, resulting in the strain-limited (I_LmaxS), stress-limited (I_LmaxT), and electric-field-limited (I_LmaxE) maximum amplitudes of resonance displayed in Tables 12 and 13. It should be noted that I_LmaxS, I_LmaxT, and I_LmaxE all have the same geometry terms (AG_ƒ), which allows direct comparison of their geometry-normalized quantities as shown above.

Appendix C—Extended PSSS Validation Results

PSSS results validating the length extensional mode (with end electrodes) FOMs are shown in Table 13. Results for all other considered modes are displayed in Table 16.

TABLE 16
Extended Figure of Merit Validation Using APC International PZT Parts
Vibration Mode	Material	$\frac{P_{loss}}{P_{out}}\mathrm{vs}.\left(18\right)$	κo vs. (21) (radians)	$\frac{E_{out}}{vol}\mathrm{vs}.\left(26\right)$ (J/m3)	$\frac{P_{out}}{A_s}\mathrm{vs}.\left(30\right)$ (W/cm2)	$\frac{P_{loss}}{A_s}\mathrm{vs}.1.$ (W/cm2)
Length Ext.	840	.0602 | .0570	1.607 | 1.613	112.3 | 112.1	17.52 | 17.54	1.054
(s)	841	.0236 | .0232	1.603 | 1.608	248.5 | 247.8	43.12 | 43.13	1.018
	880	.0414 | .0399	1.597 | 1.601	142.0 | 142.4	24.91 | 25.04	1.032
Thickness	840	.0108 | .0107	1.379 | 1.373	803.2 | 809.4	92.80 | 93.14	.9992
Shear (s)	841	.0043 | .0044	1.395 | 1.394	1986 | 1979	230.1 | 229.1	.9984
	880	.0103 | .0103	1.455 | 1.461	840.2 | 836.3	96.68 | 96.62	1.000
Thickness	840	.0093 | .0092	1.810 | 1.795	1004 | 1009	108.7 | 108.4	1.006
Shear (e)	841	.0038 | .0038	1.776 | 1.769	2377 | 2396	260.2 | 261.3	.9918
	880	.0096 | .0096	1.683 | 1.689	938.2 | 932.8	104.3 | 104.1	1.001
Thickness	842	.0243 | .0240	1.483 | 1.490	195.1 | 193.5	41.88 | 41.70	1.019
Extensional	844	.0096 | .0096	1.483 | 1.490	485.5 | 483.6	104.2 | 104.3	.9999
	881	.0160 | .0159	1.491 | 1.497	293.8 | 293.2	62.79 | 62.94	1.005
Contour	840	.0159 | .0158	1.708 | 1.712	271.8 | 272.4	62.96 | 63.22	1.001
Extensional	841	.0054 | .0054	1.715 | 1.718	826.1 | 826.7	185.3 | 185.8	.9921
	880	.0124 | .0123	1.658 | 1.665	362.1 | 361.2	80.99 | 81.16	1.000
Radial	840	.0163 | .0161	2.221 | 2.092	269.5 | 271.0	62.3 | 62.30	1.014
	841	.0056 | .0056	2.292 | 2.276	809.5 | 811.1	180.1 | 180.1	1.008
	880	.0126 | .0125	2.147 | 2.148	355.4 | 357.0	79.45 | 79.85	1.002

Each mode is validated with the hard PZT materials for which APC International provides all relevant parameters (or the ability to calculate such). It should be noted that the energy and power density quantities are validated for all materials based on loss density constraint (9), though the practical relevance of considered area A_s in this calculation depends on the specific vibration mode and mounting structure.

Appendix D—Characterization and Calculations for Experimental Validation

This Appendix provides the characterization and loss calculations utilized for experimental validation.

A. PR Characterization

First, characterize each PR in order to calculate its FOMs based on physical properties. Q_m can be calculated based on the PR's 3 dB bandwidth at resonance as measured using a network analyzer:

$\begin{array}{cc}{Q}_m=\frac{f_r}{B{W}_{3\mathrm{dB}}}& \left(56\right)\end{array}$

Further, the effective k can be calculated based on the observed resonant and anti-resonant frequencies:

$\begin{array}{cc}{k}_{\mathrm{eff}}=\sqrt{\frac{f_{ar}^2-f_r^2}{f_{ar}^2}}& \left(57\right)\end{array}$

This enables the following mode-specific k calculations:

$\begin{array}{cc}{k}_{}^2=\frac{\pi }{2}\sqrt{1-k_{\mathrm{eff}}^2}\cot \left(\frac{\pi }{2}\sqrt{1-k_{\mathrm{eff}}^2}\right)& \left(58\right)\end{array}$ $\begin{array}{cc}{k}_{+}^2=\frac{1}{1-\frac{2}{\pi }\sqrt{1-k_{\mathrm{eff}}^2}\tan \left(\begin{array}{cc}\pi & 1\\ 2& \sqrt{1-k_{\mathrm{eff}}^2}\end{array}\right)}& \left(59\right)\end{array}$ $\begin{array}{cc}{k}_{p\left(\mathrm{radial}\right)}^2=\frac{\left(1-\sigma \right)J_1\left({\kappa }_{ar}\right)-{\kappa }_{ar}{J}_0\left({\kappa }_{ar}\right)}{2J_1\left({\kappa }_{ar}\right)-{\kappa }_{ar}{J}_0\left({\kappa }_{ar}\right)}& \left(60\right)\end{array}$ B. Minimum-Loss-Ratio Power

The output power associated with the minimum loss ratio can be calculated based on the geometry condition of equation (17). To best represent the PR's behavior in the converter, switch capacitances to C_p can be added for this calculation as follows:C_eff=C_p+2C_switch  (61)

At 50 V, the switches in the converter prototype each have charge-equivalent capacitances of approximately 250 pF. The output power corresponding to the minimum loss ratio is then:P_out=ĜB_oV_in²=C_effƒV_in²  (62)C. Experimental PR Loss Ratio

To determine the experimental loss ratio of the PR, the influence of the Schottky diode rectifier (the largest non-PR source of loss in the converter prototype) can be extracted from the measured output power. The PR loss ratio is calculated from experimental results using

$\begin{array}{cc}{\left(\frac{P_{\mathrm{loss}}}{P_{\mathrm{out}}}\right)}_{\mathrm{PR}}=\frac{P_{\mathrm{in}}-P_{\mathrm{out}}^{*}}{P_{\mathrm{out}}^{*}}& \left(63\right)\end{array}$ where P*_out is the effective output power of the PR. The effective output voltage seen by the PR (due to the rectifier) is V*_out=V_out+V_fwd, where the forward voltage drop of each Schottky diode is assumed to be V_fwd=0.35 V. Thus, the effective output power of the PR is

$\begin{array}{cc}{P}_{\mathrm{out}}^{*}=P_{\mathrm{out}}\frac{V_{\mathrm{out}}+V_{\mathrm{fwd}}}{V_{\mathrm{out}}}& \left(4\right)\end{array}$ and the loss ratio for the PR is therefore approximated as:

$\begin{array}{cc}{\left(\frac{P_{\mathrm{loss}}}{P_{\mathrm{out}}}\right)}_{\mathrm{PR}}=\frac{P_{\mathrm{in}}-P_{\mathrm{out}}\frac{V_{\mathrm{out}}+V_{\mathrm{fwd}}}{V_{\mathrm{out}}}}{P_{\mathrm{out}}\frac{V_{\mathrm{out}}+V_{\mathrm{fwd}}}{V_{\mathrm{out}}}}& \left(65\right)\end{array}$

In the foregoing detailed description, various features are grouped together in one or more individual embodiments for the purpose of streamlining the disclosure. This method of disclosure is not to be interpreted as reflecting an intention that each claim requires more features than are expressly recited therein. Rather, inventive aspects may lie in less than all features of each disclosed embodiment.

References in the disclosure to “one embodiment,” “an embodiment,” “some embodiments,” or variants of such phrases indicate that the embodiment(s) described can include a particular feature, structure, or characteristic, but every embodiment can include the particular feature, structure, or characteristic. Moreover, such phrases are not necessarily referring to the same embodiment(s). Further, when a particular feature, structure, or characteristic is described in connection knowledge of one skilled in the art to affect such feature, structure, or characteristic in connection with other embodiments whether or not explicitly described.

The disclosed subject matter is not limited in its application to the details of construction and to the arrangements of the components set forth in the following description or illustrated in the drawings. The disclosed subject matter is capable of other embodiments and of being practiced and carried out in various ways. As such, those skilled in the art will appreciate that the conception, upon which this disclosure is based, may readily be utilized as a basis for the designing of other structures, methods, and systems for carrying out the several purposes of the disclosed subject matter. Therefore, the claims should be regarded as including such equivalent constructions insofar as they do not depart from the spirit and scope of the disclosed subject matter.

Although the disclosed subject matter has been described and illustrated in the foregoing exemplary embodiments, it is understood that the present disclosure has been made only by way of example, and that numerous changes in the details of implementation of the disclosed subject matter may be made without departing from the spirit and scope of the disclosed subject matter.

All publications and references cited herein are expressly incorporated herein by reference in their entirety.

Claims

1. A power converter having a specified voltage Vin and a specified power level Pout, the power converter comprising: a piezoelectric resonator (PR) configured for energy transfer between an input and an output of the converter, the PR having an area A, a distance 2l between first and second electrodes of the PR, and a dimensions value G defined as

2. The power converter of claim 1, wherein G is equal to Ĝ within 10%.

3. The power converter of claim 1, wherein G is equal to Ĝ within 20%.

4. The power converter of claim 1, wherein l is equal to {circumflex over (l)} within 10%.

5. The power converter of claim 1, wherein l is equal to {circumflex over (l)} within 20%.

6. The converter of claim 1 wherein Pout is either a rated power output or a typical power output of the power converter for a given input voltage.

7. The power converter of claim 1, where ILmaxo is a geometry-normalized maximum of IL, and IL is an amplitude of resonance of the PR at Pout for a given input voltage.

8. The power converter of claim 1, wherein Gƒ corresponds to a frequency-setting dimension as defined in the following:

9. The power converter of claim 1, wherein the PR is configured to operate in at least one of: a perpendicular vibration mode;a parallel vibration mode;a shear vibration mode; ora length extensional mode.

10. The power converter of claim 9, wherein the perpendicular vibration mode includes at least one of: a length extensional mode;a thickness shear mode;a contour extensional mode; ora radial mode.

11. The power converter of claim 9, wherein the parallel vibration mode includes at least one of: a length extensional mode;a thickness extensional mode; anda thickness shear.

12. The power converter of claim 1 wherein the PR is one of a plurality of PRs comprised within the power converter, wherein at least two of the plurality of PRs are configured to operate in a length extensional mode in parallel.

13. The power converter of claim 1, further comprising: a plurality of switches configured to operate in accordance with one or more switching sequences to transfer energy between the input and output of the converter via the PR; anda controller for controlling the switching sequences.

14. The power converter of claim 13, wherein ones of the one or more switching sequences include a sequence of voltages to be applied to the PR, wherein ones of the voltages can include Vin, −Vin, Vin−Vout, Vout−Vin, Vout, −Vout, or Zero, where Vin and Vout correspond to the input and output voltage of the converter in that order or in reverse order, and Zero corresponds to the PR being short-circuited.

15. The power converter of claim 14, wherein the switching sequences include at least one of: Vin−Vout, Zero, Vout;Vin, Vin−Vout, Vout; orVin, Vin−Vout, Zero, Vout.

16. The power converter of claim 14, wherein the switching sequences include at least one of: Vin, Zero, Vout−Vin;Vin, Vout−Vin, Vout; orVin, Zero, Vout−Vin, Vout.