Optimized Resonators for Piezoelectric Power Conversion

The performance of inductors at high frequencies and small sizes is one of the largest limiting factors in the continued miniaturization of dc-dc converters. Piezoelectric resonators can have a very high quality factor and provide an inductive impedance between their series and parallel resonant frequencies, making them a promising technology for further miniaturizing dc-dc converters. In this paper we analyze the impact of resonator parameters on the performance of the piezoelectric resonator based dc-dc converter, derive the optimal load impedance and efficiency limits, and analyze the impacts of varying conversion ratio and load impedance. This work is accompanied by a prototype dc-dc converter using a piezoelectric resonator fabricated from lithium niobate. The piezoelectric resonator has a quality factor of 4178 and a coupling coefficient, <inline-formula><tex-math notation="LaTeX">$k_t^2$</tex-math></inline-formula>, of 29%. The converter is able to achieve high efficiency zero voltage switching and a continuously variable conversion ratio without the use of any discrete inductors. It achieves a maximum power output of 30.9 W at an efficiency of 95.2% with a power density of 6.76 <inline-formula><tex-math notation="LaTeX">$\frac{\text {W}}{\text {cm}^3}$</tex-math></inline-formula>.


I. INTRODUCTION
The inductor presents one of the biggest obstacles in miniaturizing power converters. Almost all switch-mode converter topologies require inductive circuit elements but inductors are significantly more lossy than capacitors, the component dual. Advances in wide band-gap semiconductors have provided significant jumps in the performance of switching devices but no parallel advancements have happened for inductors. These improvements in semiconductor technology have allowed for an increase in switching frequency and a corresponding reduction in the required energy storage of inductors and capacitors. However, unlike capacitors, inductor performance scales sublinearly with increases in frequency and becomes worse with a reduction in volume [1], [2]. These factors have lead to only modest gains in converter power density for relatively large increases in switching frequency.
One possibility to eliminate the need for inductors and avoid the associated problems with frequency and power density scaling is the piezoelectric resonator. Piezoelectric materials generate a mechanical strain from an applied voltage and generate electrical charge from an applied mechanical stress, allowing for a low loss coupling between the mechanical and electrical domain. The piezoelectric effect enables the circuit to electrically couple to mechanical resonators which can have a quality factor orders of magnitude beyond what can be realized by discrete capacitors and inductors [3].
When operated around resonance, piezoelectric resonators can provide a net-inductive impedance with very low loss, replacing the need for magnetics. Piezoelectric resonators also have integration advantages as they can be fabricated with CMOS compatible processes and do not produce fringing magnetic fields which can interfere with sensitive circuits [4]. Taking into account both power dissipation and intrinsic material limits, the upper bound on power density for piezoelectric resonators exceeds what is achievable with discrete components [5]. Piezoelectric resonators can provide a pathway to bypass the scaling challenges of inductors and a continued miniaturization of dc-dc converters.
Although the piezoelectric resonator can exhibit a netinductive resonance it is not a direct replacement for the inductor. A commonly used circuit model for the piezoelectric resonator is the Van Dyke model, shown in Fig. 1. The mechanical resonance, coupled to the electric circuit via the piezoelectric effect, is modeled as a RLC series branch (L m , C m , R m ). The static input capacitance of the piezoelectric material is modeled as a parallel capacitor, C o [6]. In this model there is no dc path through the device but there is a region of inductive impedance between the series resonance of the motional branch and the parallel resonance between the motional branch and the static input capacitance. Due to the dc block characteristics of the piezoelectric resonator, standard topologies such as the buck or boost converter are not applicable. Instead, suitable circuit topologies are similar to those of resonant switched capacitor converters, which can achieve zero voltage switching (ZVS) and a variable conversion ratio using piezoelectric resonators in place of the conventional LC resonant circuit [7]- [10].
Previous researchers have demonstrated piezoelectric resonator based dc-dc converters operating at high efficiency [10]- [12] and the optimal switching patterns have been enumerated by [10]. However, prior work utilizes piezoelectric resonators made of PZT ceramics, a higher loss material compared to materials such as lithium niobate, which limits the maximum achievable power density. To demonstrate the high power density potential of piezoelectric resonator based converters, this paper explores lithium niobate as an alternative to PZT and expands on our previous conference work [13]. The contributions of this paper are to simplify the analysis of the piezoelectric resonator based dc-dc converter by decomposition into a core switching cell, analysis of the charge transfer in the converter for non-zero switching device capacitance, and calculation of power loss and efficiency bounds based on the standardized figures of merit for piezoelectric resonators. This analysis is followed by the design of an optimized piezoelectric resonator fabricated from lithium niobate and a prototype dc-dc converter using the fabricated resonator.
In Section II we present an analysis of the operation of the converter and in Section III we analyze the performance of the converter as a function of the properties of the piezoelectric resonator. In Section IV we detail the fabricated lithium niobate piezoelectric resonator, in Section V we provide details on the prototype dc-dc converter, in Section VI we provide test results, and in Section VII we conclude the paper.

A. CONVERTER TOPOLOGY
Possible configurations and switching schemes for a dc-dc converter utilizing a single piezoelectric resonator are enumerated through by [10]. Most of these configurations, including the configurations with the highest efficiency and utilization of the piezoelectric resonator, are derivative of a single core switching cell, shown in Fig. 2. This switching cell consists of two bridge legs, one connected to the input and one connected to the output, with the piezoelectric resonator connected between the switching nodes of the two bridge legs. C sw models the switching device capacitance. This core switching cell, through different switching patterns, can achieve arbitrary conversion ratios with bidirectional power transfer for positive input and output voltages. Only an AC return path is required between the two ports, so additional configurations can be synthesized by stacking the input and output ports of the switching cell. This allows the switching cell to process only differential power and achieve a higher efficiency over a narrower voltage conversion range.
There are multiple switching sequences that can achieve zero voltage switching but differ in the charge transfer characteristics of the piezoelectric resonator, and thus efficiency. For optimal utilization of the piezoelectric resonator there are separate switching sequences for the V in > V out case and V in < V out case. However, these sequences are time reversal symmetric and the converter is bidirectional, so the analysis of the converter can be further simplified to only analyzing only the V in > V out case.

B. CONVERTER OPERATION
Due to the high quality factor of the piezoelectric resonator, the current in the motional branch, labeled I m in Fig. 2 Fig. 3 we provide an example converter waveform with the states annotated.
The converter operation and losses can be analyzed based on the charge that is transferred in each of the 7 converter states. During the four open states, [A, C, E , G], one switching node is connected to a supply rail and the other is commutated between the supply rails by the charge transferred from the motional branch. The charge to commutate the resonator input capacitance, C o , remains internal to the resonator while the charge to commutate the switch capacitances impacts the total charge transferred per cycle between the converter input and output. Previous analyses have ignored the impact of the switching device capacitance on the charge transfer between the input and output ports of the converter, which can lead to errors when the capacitances are of similar magnitude to C o . Here we analyze the converter in terms of the capacitance, C sw1 , of the two input side switches, S 1 and S 2 , and the capacitance, C sw2 , of the two output side switches, S 3 and S 4 . The total charge supplied by the resonator motional branch during any commutation period is only dependent on the converter voltages and capacitances and can be expressed as: During the three connected states, [B, D, F ], the charge transferred is dependent on the desired output power and conversion ratio. During state B charge is transferred from the input of the converter to the output through the resonator. During state D the input side of the resonator is grounded and the resonator transfers charge to the output. During state F both sides of the resonator are connected to ground and no power is transferred. The existence of state F and the charge transferred during the other states is constrained by the requirements for conservation of energy and charge during periodic steady state operation. The charge transferred in states B and D, where charge is delivered to the converter output port, and state F , where the resonator is shorted, is related by the constraints of conservation of energy and charge in the resonator. If the system is constrained for unidirectional power flow, which minimizes circulating power and is achieved by the constraint I m = 0 at the start of state A, the system of equations for the connected states can be expressed as: In Table 2 we list the charge transferred from the resonator, from the input port, and to the output port, during each switching state. The sum of the output charge for the entire 7 state sequence multiplied by the converter operating frequency provides the output current of the converter. The desired charge transfer and operating frequency are constrained by the equivalent circuit values of the piezoelectric resonator. A method to calculate the unique solution for the exact switch timing and frequency of the switching sequence, F sw , is given by [10] and involves solving a large system of equations. While exact switch timings are needed for operation of the converter, the switching frequency of the converter remains close to the series resonant frequency of the piezoelectric resonator. Assuming that the switching frequency is equal to the resonant frequency of the converter greatly simplifies analysis of the converter without introducing appreciable error: From (5) it can be seen that when I out = 2F s C sw1 V in both q B and q D are zero. This represents a minimum output current for which unidirectional power flow can be achieved and is due to the impact of the switching device capacitance on the system.

C. SYSTEM LOSSES AND EFFICIENCY
In this section we analyze the losses in the converter in order to lay the framework for optimizing the sizing of the resonator. There are three primary loss mechanisms in the piezoelectric resonator based dc-dc converter: the mechanical loss in the resonator, modeled as an I 2 R loss in the motional branch of the circuit model, the I 2 R loss in the switching devices, and the dielectric loss in the static input capacitance of the piezoelectric resonator. All switching transitions achieve ZVS and three of the four switching transitions achieve zero current switching, so in most cases the switching losses are negligible.
The I 2 R loss in the motional branch and the dielectric loss in the input capacitance are the most direct losses to calculate. The I 2 R loss in the motional branch is dependent on the RMS current in the resonator and can be computed based on the sum the charge transferred per cycle, the operating frequency, and the resonator motional resistance, R m . Based on (3)(4)(5) and the values provided in Table 2 the peak amplitude of the sinusoidal current in the motional branch, I m and the loss in the motional branch, P m can be expressed as: Unlike the resonator current, the voltage across the static input capacitance is not sinusoidal. Analysis is simplified if the dissipation factor of the dielectric remains constant with frequency, which is an accurate approximation for PZT and lithium niobate piezoelectric materials at typical power conversion frequencies and relevant harmonics harmonics [14], [15]. If this approximation holds, the dielectric loss can be calculated in terms of the peak to peak voltage swing across the resonator without having to decompose the switching waveform into its harmonics through the definition of Q = ω energy stored power loss . The dielectric loss in the resonator, P d , can be expressed in terms of the switching frequency, the input capacitance of the piezoelectric resonator, C o , and the quality factor of the dielectric of the piezoelectric resonator, Q E = 1/tan(δ): Calculating the I 2 R losses in the switching devices is more difficult as the current in the switching devices is nonsinusoidal. During open states, the current in the switches of the commutating bridge leg is zero, while the opposite side switches have a reduced current as they only carry the current charging C sw and not C o . The total losses in all the switches can be calculated by subtracting out the I 2 integral of these periods from the I 2 integral for a sinusoidal current. In the interest of providing understandable and compact closed form expressions, the switch resistance can be incorporated as part of the resonator motional resistance with minimal error in most operating conditions. This approximation will over-estimate the switch conduction loss and have minimized error on the system efficiency when the charge transferred during the connected periods is much larger than the charge transferred during the open periods. Using this approximation the total switch loss, P sw , and the system efficiency can be approximated in terms of the input side switch resistance, R sw1 , and output side switch resistance, R sw2 , as:

A. RESONATOR EQUIVALENT CIRCUIT VALUES
The Van Dyke equivalent circuit model for the piezoelectric resonator consists of 4 components, the static input capacitance, C o , and the motional branch values, R m , L m , and C m . C o represents the static capacitance of the piezoelectric material. This capacitance is determined by the geometry of the piezoelectric resonator and the dielectric constant of the piezoelectric material; for the thickness mode resonators used in this paper it can be modeled as a parallel plate capacitor.
The circuit values of the motional branch are related to the value of C o by a mechanical quality factor, Q m , a piezoelectric coupling coefficient, 1 k 2 t , and the series resonant frequency, F s . Q m represents the quality factor of the mechanical resonance of the piezoelectric resonator and determines the loss in the motional branch. k 2 t represents how well the mechanical resonance is electrically coupled to through the piezoelectric effect and determines the impedance of the motional branch relative to C o .
Typically, a resonator can be geometrically scaled in a way that changes C o but does not impact the other parameters, which allows C o to be used as a scaling factor. Expressions for the motional branch circuit values in terms of these values vary in literature, we use the notation detailed in [16]: Both Q m and k 2 t are determined by the properties of the piezoelectric material and resonant mode. Upper bounds on these values can be computed from material properties. Expressing system loss and efficiency in terms of these values allows for different types of piezoelectric resonators to be more easily compared.

B. OPTIMAL PIEZOELECTRIC RESONATOR SIZING
In this section we derive the optimal resonator size for maximum efficiency. Here we only consider the I 2 R loss in the motional branch of the resonator, ignoring the dielectric loss (which is typically small) and assuming ideal switches with no on resistance or junction capacitance. Substituting (13) into (7) allows for efficiency to be calculated for a given load impedance in terms of C o , k 2 t and Q m , with C o as the independent variable. This optimization represents the trade-off between the circulating charge due to C o and resistive losses due to R m , which are proportional and inversely proportional to size of the piezoelectric resonator, respectively. The function is convex so the optimal sizing of C o and the maximum efficiency at that point can be computed by finding the zero of the derivative: From (15) a figure of merit for piezoelectric resonators used in power conversion is apparent, Q m k 2 t . This value as a figure of merit is largely to be expected, as from (13) it can be seen that Q m k 2 t ∝ 1/R m and Q m k 2 t is the standard figure of merit for piezoelectric resonators in other applications [3], [17]. In Fig. 4 we present a plot of the maximum achievable efficiency for optimal resonator sizing across the conversion voltage range for three values of k 2 t Q m representative of values that are achievable with different designs of piezoelectric resonators. From (14) we can also calculate a load resistance for maximum efficiency, R optimal , given a fixed sizing of C o . In Fig. 5 we present a plot of the maximum achievable efficiency over the load range, normalized to this optimal load resistance, for a nominal conversion ratio of 1:1:  From these plots we can see the advantages of high k 2 t Q m resonators. Beyond a high maximum efficiency, they maintain a significant efficiency advantage over variations in both conversion ratio and load impedance. Although our optimizations do not take into account the dielectric loss due to the complexity it adds to the analytical solutions, we can look at the impact of dielectric loss. From (7) and (8) a ratio of the power lost in the resonator dielectric to the power lost in the motional branch can be calculated for operation with an optimally sized resonator and a 1:1 conversion ratio: In Section IV we will discuss what range of k 2 t Q m and Q E values are realizable, but in the context of evaluating the above equation, the lithium niobate resonators produced for this paper have a k 2 t Q m value of above 1000. Q E is largely a material property and lithium niobate has a Q E in excess of 1000 [14]. This equates to P d P m ≈ 0.2, suggesting that dielectric loss will not unduly impact the nominal efficiency of the converter. However, as dielectric loss is constant for given input and output voltages and independent of the load impedance, dielectric loss may become significant at load impedances significantly above the optimal load impedance or for materials with a lower Q E relative to Q m k 2 t , such as PZT ceramics.

C. OPERATING FREQUENCY
For the analysis of the converter the operating frequency was assumed to be equal to the series resonant frequency of the piezoelectric resonator in order to simplify the provided equations. For the system without this assumption the resonator motional branch current, I m , resonator motional branch impedance, Z m , and fundamental component of the resonator voltage, V m,0 , are related by the switching frequency: However, there is no simple closed form expression of V m,0 . The peak to peak amplitude of V m will always be V in + V out but the harmonic content will change as a function of load impedance.
As load impedance decreases and the commutation time becomes negligible the waveform of V m will approximate a stepped rectangle with voltages and timings determined by the conversion ratio dependent connected state charge transfer. The operating frequency will asymptotically approach the series resonant frequency, F s . As load impedance increases the commutation time will become dominant and V m will approximate a sinusoid. The operating frequency will asymptotically approach the parallel resonant frequency of the resonator motional branch and the static input capacitance, F p . This can be expressed in terms F s and k 2 t : In lieu of an analytical solution, in Fig. 6 we present a plot of the normalized operating frequency range as a function of load impedance as calculated from sampling the closed form operating equations for the converter. From this plot it can be seen that the converter operation will span most of the inductive load region with modest changes in load impedance, but variation in the conversion ratio has little impact.
The modeled converter operates under the constraint of unidirectional power flow, which optimizes efficiency by minimizing circulating current. If the unidirectional power flow constraint is ignored it is possible to operate the converter at a frequency closer to the series resonant frequency by circulating additional power. Therefore, Fig. 6 provides an upper bound on the operating frequency.

A. MATERIAL AND RESONANT MODE
The characteristics of a piezoelectric resonator is determined by the material and geometry. The choice of the piezoelectric material determines an upper bound on Q m due to intrinsic material loss and an upper bound on the coupling coefficient. The resonator geometry determines the resonant mode, and, by scaling, the resonant frequency and impedance of the equivalent circuit. The possible combinations of resonator material and resonant mode are too extensive to enumerate in this paper, but in this section we highlight some variants that have particular interest for power conversion and in Table 4 we highlight specific resonator designs representative of the state of current research in the field.
Three piezoelectric materials hold significant promise for power conversion: Lead Zirconate Titanate (PZT) ceramics, Lithium Niobate, and Aluminum Nitride [4], [29]- [31]. In Table 3 we compare the properties of these three materials. 2 PZT ceramics have been commonly used in previous research on piezoelectric based power conversion [32]- [34] and are widely used for commercial piezoelectric actuators and sensors. These ceramics have the advantage of a high dielectric constant and piezoelectric coupling coefficient, which allows for low impedance resonators at lower frequencies, 2 Here we compare materials by the definition of coupling coefficient k 33 = d 33 / T 33 s E 33 [22]. This represents an upper bound for k t under the optimal boundary conditions. providing a better match to low voltage applications. PZT ceramics are fabricated from pressed and sintered ceramic powders, which allows for complex structures to be fabricated. However, PZT ceramics have a relatively high mechanical loss and dielectric loss compared to other materials, limiting efficiency, and a relatively low curie temperature, limiting high power operation. Lithium niobate has very low loss, both electrically and mechanically, as well as a high k 2 t and a high curie temperature. It is a crystalline material and resonators must be fabricated via subtractive processes, posing some fabrication challenges. It has been used for both bulk and MEMS scale resonators [26], [27], [31], [35]. The lower dielectric constant of the material can be offset by operation at higher frequencies, which is now more practical due to advancements in wide-bandgap semiconductors. Very low loss and high coupling coefficient make lithium niobate a promising material for power conversion with excellent power density [5].
Aluminum Nitride has a low mechanical loss but a low k 2 t , its main advantage is that it can be deposited as part of a CMOS compatible process flow, allowing for a high degree of integration with IC processes [4], [29].
Common piezoelectric resonant modes for bulk devices include radial mode, shear mode, lateral mode, and thickness resonant modes [27], [30], [33], [34]. Most of these resonant modes can also be applied in MEMS scale devices, along with other resonant modes such as lamb wave [29], [31], [35], [36]. Of these resonant modes, radial mode resonators operate at a low frequency for a given size, providing a high impedance and limited power density. Lateral mode and lamb wave resonators have limited electrode coverage which also makes achieving a low impedance difficult. Shear mode and thickness mode resonators have advantages in that they provide good utilization of the piezoelectric material and have an easy to fabricate planar structure.
For our design we chose to fabricate thickness mode bulk acoustic wave resonator using lithium niobate. The thickness mode resonator, shown in Fig. 7, consists of a slab of material with parallel top and bottom faces, each covered in an electrode. The acoustic impedance mismatch with air, or an acoustic reflector, at the top and bottom of the resonator confines a standing wave between the two faces, with the thickness equal to half of the wavelength of the acoustic wave. Laterally, the acoustic energy is confined by the electrodes. Areas without electrodes are inactive and can be used for mounting the resonator. The thickness mode resonator has only one critical dimension, the thickness, t, which along with the acoustic velocity of the material, v determines the series resonant frequency of the fundamental mode, F s = v/2t. The thickness also determines the impedance per unit area, which scales with frequency as 1/F 2 s .

B. RESONATOR DESIGN
One of the main challenges to the use of piezoelectric resonators in power conversion are spurious resonant modes, which are minor resonant modes that are not predicted by a 1D approximation of the resonator or captured in the basic Van Dyke circuit model. These spurious modes are difficult to calculate analytically and can lead to regions of high loss within the operating frequency range of the piezoelectric resonator, limiting converter operating range and posing challenges for closed loop control. In thickness mode resonators, the largest spurious modes are typically related to the lateral dimensions of the resonator body and electrodes. In RF applications apodized electrodes have been used, where instead of rectangular electrodes a shape having no parallel edges is used. The lack of parallel electrode edges disperses lateral resonances [17]. However, apodized electrodes lead to reduced utilization of the piezoelectric material and mainly serve to spread out the spurious modes; instead of a narrow frequency range with high losses, there is a wider frequency range with somewhat reduced losses. These impacts make apodization relatively unsuitable for high-efficiency power conversion, and instead we focus on minimizing the number of spurious modes and moving them out of the frequency range of interest.
The electrode geometry was constrained to a rectangle with a side ratio of 2:1 such that the spurious modes from the length and width would be harmonically related. Based on the geometry of available wafers, the size of the lithium niobate piece is limited to 32 x 13 mm, allowing 4 resonators to be fabricated on a single 3" wafer. The electrode area was limited to a nominal 20x10 mm to keep the active area away from the resonator edges and mounting areas and reduce mechanical damping.
While the dimensions of the lithium niobate crystal are defined in the wafer dicing step, the electrode size is defined lithographically and can be adjusted during fabrication for each resonator, allowing limited control over the spurious modes. COMSOL, a finite element method modeling program, was used to aid in maximizing the spacing between the series resonant frequency and the first spurious mode by optimizing the electrode size, maximizing the operating range for high power operation. Accurately modeling the resonator requires a simulation mesh size that is a fraction of an acoustic wavelength in the material, 1mm for our resonator, and requires an individual simulation to be run at each frequency point, making full 3D simulation of the resonator difficult without access to significant compute resources. To identify spurious modes, a 2D cross section along the major axis of the resonator was simulated. This spans the 32 mm width of the lithium niobate piece and includes the FR-4 substrate on which the resonator is mounted on and the aluminum electrodes. Between simulations the electrode width was incremented.
Through this method, the frequency difference between the series resonant frequency and the first spurious mode was identified to be periodic with a 1 mm change in the length of the resonator, indicative of a spurious lateral resonance mode that is defined by the electrode boundary and propagates at a similar velocity to the thickness mode wave. This 1 mm change in electrode dimension only represents 5% of the electrode width. Although the electrode size is accurately defined with a lithographic process other discrepancies in the lithium niobate thickness or properties may cause a mismatch with simulation. To ensure that an optimized resonator was fabricated despite possible mismatches the four resonators on the wafer were designed with a 0.25 mm increment in size between each, ensuring that at least one resonator from each wafer would have a maximally distanced spurious mode.

C. RESONATOR FABRICATION AND CHARACTERIZATION
The resonators were fabricated on commercially available 0.5 mm thick Y+36 • cut lithium niobate wafers, a cut which maximizes the piezoelectric coupling coefficient for the thickness resonant mode [37]. Electrodes consisting of 300 nm of aluminium and a 10 nm titanium adhesion layer were patterned on both sides with an e-beam evaporator and a liftoff lithographic process. In Fig. 8 we show a photograph of the resonators during fabrication shortly before application of metalization to the second side. A set of four resonators from  a single wafer were mounted to carrier PCBs for characterization with a VNA. In Fig. 9 we provide a plot of the impedance close to resonance of two resonators with different electrode dimensions, showing the impact small differences in electrode size have on the spurious modes close to the series resonant frequency.
The resonator with the minimal spurious modes was the design with electrode dimensions of 20.25x10.125 mm. In Fig. 10 we provide a plot of the impedance of this resonator along with the best fit line for the Van Dyke circuit model. In Table 5 we provide the component values for the extracted Van Dyke model equivalent circuit values.
Despite attempts to minimize the spurious modes it can be seen that the resonator has severe spurious modes across its entire frequency range. These spurious modes pose one of the biggest challenges to the design of piezoelectric resonators and converters and their use in power conversion. Compared to other resonator types, thickness mode resonators often have a greater number of spurious modes. However, the planar design and high material utilization still make them a promising candidate for use in power conversion.
In addition to causing regions of high loss, spurious modes also impact the fit of the resonator impedance to the Van Dyke circuit model. In the desire to achieve a high output power the fit to the Van Dyke circuit model was optimized for the region close to the series resonant frequency where the converter operates at the highest power levels.
The power handling of the fabricated resonator is thermally limited due to the limited temperature range of the epoxy used for attachment and thermally induced frequency shift interfering with the open loop control used in the converter. However, lithium niobate resonators are capable of operating up to several hundred degrees Celsius with little change in quality factor or coupling coefficient and only modest frequency shift [38]. If the resonator was not thermally limited, power would be limited by two properties of lithium niobate, the coercive field strength, which limits the maximum resonator voltage, or the material strain limit, which limits the maximum resonator current. Within the power limits of our converter the resonator does not approach either of those limits.

A. CIRCUIT DESIGN
The prototype converter consists of the previously analyzed switching cell with the input port stacked on top of the output port. This limits the nominal voltage conversion ratio to [0,1] but increases efficiency by having the switching cell only process differential power, analogous to the difference between the buck-boost and buck converter. The relationship between the voltages and currents of the stacked converter and the switching cell can be expressed as: I out,cell = I out,conv − I in,conv (25) In Fig. 11 we provide a simplified schematic of the prototype dc-dc converter and in Table 6 we provide schematic values. Due to the high switching frequency, GaN HEMT devices, EPC8009 (65 V, 130 m ) are used and driven with PE29102 gate drivers. Gate drive power for the floating switches is derived by a bootstrap circuit, eliminating the need for any isolated power supplies [39]. The entire converter hence requires no magnetic components. Based on the ratings of the PE29102 gate drivers, the maximum input voltage to the  converter is 60 V. The maximum output power is limited by the thermal dissipation of the switching devices and resonator.
The converter is fabricated on a 4 L 1.0 mm FR-4 PCB and the edges of the resonator are directly bonded to the PCB with ultra-violet (UV) curable epoxy. Electrical attachment between the resonator and the PCB is made with with aluminum bond wires. The bounding box for the active area of the converter, consisting of the resonator and the active circuity bounded by the white box on the silkscreen, is 32x25x2.8 mm, for a total volume of 2.25 cm 3 . The metalization process adds negligible thickness to the lithium niobate material, so the finished resonator has dimensions of 32x13x0.5 mm for a total volume of 0.21 cm 3 In Fig. 12 we provide a photograph of the prototype converter.

B. CONTROL
Due to the high quality factor of the piezoelectric resonator, the converter is very sensitive to variations or deviance in operating frequency. However, time spent in each switching state as a portion of the total period is relatively insensitive to fluctuations. The controller is designed to optimize the accuracy of the switching frequency, beyond what can be achieved on a standard microcontroller. The controller synthesises a multiple of the operating frequency with a Si5338 G clock generator chip and then uses a XC7S25 FPGA to generate drive signals for the four switches with a PWM module written in Verilog using the double data rate (DDR) output peripherals of the FPGA. This provides a frequency resolution of 62.5 Hz and a PWM step resolution of 1/112 th the period (1.3 ns at an operating frequency of 6.8 MHz). The controller operates in open loop and the switching frequency and switching pattern are pre-computed on a desktop and loaded onto the FPGA over a serial link before testing.

A. MEASURED PERFORMANCE
The efficiency of the converter was measured over a range of load points to validate its performance and determine the maximum power output. Waveforms were collected to validate that zero voltage switching was maintained. Converter testing at low power was limited by the presence of spurious modes in the piezoelectric resonator frequency response. Fig. 10 shows several spurious mode free frequency ranges, but operation during testing was limited to operating between the series resonant frequency and the first spurious mode at 6.87 MHz, which limits the converter to load impedances significantly lower than the optimal load impedance for an idealized resonator. Testing at high power was limited by the thermal limits of the switching devices and resonator.
In Fig. 13 we present plots comparing the measured converter efficiency with a 50 V input for a variable conversion ratio at a fixed power output of 20 W±10% and for variable output power with a fixed output voltage of 25 V. These plots also contain a modeled efficiency curve derived from the equations presented in Section II. In testing a maximum efficiency of 96.5% was achieved at an output voltage of 25 V and an output power of 21.0 W, and a maximum output power of 30.9 W at an output voltage of 25 V was achieved with an efficiency of 95.2%.
With an output power of 30.9 W, the converter's power density is 6.76 W cm 3 . Resonant switched capacitor converters, which are topologically similar to the piezoelectric resonator based converter but use discrete inductors, have achieved power densities of 79.9 W cm 3 [9] and 57.6 W cm 3 [8]. In comparison to those designs, this converter is not yet competitive in terms of power density with designs using conventional magnetics. Few others papers that built piezoelectric resonator based converters report a bounding box size or power density but they do report the size of the piezoelectric resonator itself, which allows for calculating the power density for just the piezoelectric resonator. The power density for just the piezoelectric resonator in our converter is 148 W cm 3 . The resonator power density for other designs using PZT based resonators can be calculated as 102 W cm 3 [10], 13.2 W cm 3 [40], and 7.09 W cm 3 [12]. This difference in resonator power densities illustrates the relative advantages of lithium niobate over PZT ceramics.
In Fig. 14 we provide waveforms of the two switching nodes during operation at the 20 W 35 V output test point, showing ZVS on the two switch nodes. Some ringing can be seen on the resonator voltage at a frequency of approximately 135 MHz. Thickness mode resonators can be excited at odd harmonics of their fundamental frequency, but the coupling coefficient of the harmonics fall as 1/N 2 , so the coupling coefficient of any resonant mode around 135 MHz would be exceedingly small [41]. The observed ringing is due to the series inductance of the connections between the piezoelectric resonator and the converter circuitry oscillating with the input During testing both the FPGA controller and the gate drive circuity was powered and metered separately. The gate drive circuity draws 0.5 W (50 mA at 10 V), which is not insignificant compared to the total power dissipation of the converter. However, this is largely due to the bias current drawn by the gate drive ICs, which are relatively over-sized for the switching devices used. The gate charge of the EPC8009 GaN FET used in the design is only 375 pC at 5 V; the power required to drive all 4 gates at the switching frequency of 6.8 MHz without other losses is only 51 mW.

B. ANALYSIS
The modeled and measured efficiency have good agreement, both in individual values and general trends. A large portion of the discrepancy can likely be attributed due to the switching devices. The current path consists of the motional resistance of the resonator, which is 190m , and the series resistance of two switches, which is nominally 300m at a temperature of 50 • C. Based on the relative size of the two resistances, the majority of the loss will occur in the switching devices instead of the resonator. GaN devices have a relatively high positive temperature coefficient, meaning that the total efficiency of the converter will be sensitive to the device temperature. For the model calculations the on state resistance value of the EPC8009 FET for a temperature of 50 • C was used, but this is only slightly above ambient and thermal camera readings show the switching devices exceeding this temperature at some load points.
In Fig. 13b, the sweep over output power for a 2:1 conversion ratio, the largest discrepancy between the modeled and measured efficiency is that at the 18 W test point measured efficiency begins decreasing while the modeled efficiency continues to increase. Based on the derivations provided in (14) and (22), and the resonator properties provided in Table 5, the maximum efficiency point should occur at a power level of 2.2 W and efficiency should monotonically increase as load power is decreased until this point is reached. However, this operating point is not achievable with the fabricated resonators due to spurious resonant modes, visible in the measured resonator impedance plot provided in Fig. 10, which cause sharp increases in the resonator loss in specific frequency ranges. The first spurious mode limited the minimum power during testing and encroaching upon it likely led to the decreased efficiency at the 18 W test point during testing at a 2:1 conversion ratio. Encroaching upon the first spurious mode is also the most likely mechanism for the decreased efficiency at higher conversion ratios during testing with a constant 20 W load, as the amount of differential power the switching cell processes is reduced at higher conversion ratios. Improved resonator design would eliminate the spurious resonances and enable high efficiency over a wider load range.

VII. CONCLUSION
In this paper we present an analysis of the piezoelectric resonator based dc-dc converter in terms of a core switching cell that other converter variants can be decomposed into. We derive formulas relating optimal load resistance and maximum achievable efficiency to the properties of the piezoelectric resonator across variations in conversion ratio and load resistance. We establish that Q m k 2 t is the relevant figure of merit for piezoelectric resonators in power conversion applications and based on this criteria we fabricate a thickness mode resonator out of lithium niobate, a low loss piezoelectric material. The fabricated lithium niobate resonator has a Q m of 4180, a k 2 t of 0.29, and a resonant frequency of 6.8 MHz.
The prototype dc-dc converter using the fabricated resonator achieves high-efficiency ZVS operation and a variable conversion ratio of [0.2, 0.8] without using any magnetic components. The converter achieves a high average efficiency, with a peak efficiency of 96.5% at a power level of 21.0 W, an input voltage of 50 V, and an output voltage of 25 V. The maximum output power achieved is 30.9 W at an efficiency of 95.2%, input voltage of 50 V, and output voltage of 25 V, for a power density of 6.76 W cm 3 . Testing at lower power points, where the predicted efficiency would be higher, was limited by spurious resonant modes in the piezoelectric resonator. This work shows the promise of piezoelectric materials for high power density converters but also illustrates the challenges that spurious modes pose in their effective use.