Modeling and Experimental Evaluation of 1-3 Stacked Piezoelectric Transducers for Energy Harvesting

Abstract

Piezoelectric energy harvesting in roadways can power distributed sensors and electronics by capturing underutilized mechanical energy from traffic. In this research, 1-3 stacked piezocomposites were developed and evaluated to determine optimal designs for multiple applications. The design of these transducers aimed at operating in a multitude of scenarios, under compressive loads (1–10 kN) at low-frequency (10 Hz) applications, intended to simulate vehicular forces. Power comparison was utilized between numerous transducers to determine the most efficient configuration for electromechanical energy conversion. Design guidelines were based on mechanical integrity, output power, active piezoelectric volume percentage, aspect ratio, and geometric factors. The forces applied in this study were reliant on the average vehicle weight. An intermediate PZT volume fraction and moderate pillar aspect ratios were found to yield the highest power output, with the stacked 1-3 composite significantly outperforming a monolithic PZT of a similar size.

1. Introduction

As interest in sustainable and autonomous energy systems continues to grow, mechanical energy harvesting, particularly through piezoelectricity, has emerged as a promising approach for powering distributed electronics. Roadways and traffic environments offer abundant and underutilized mechanical energy that, if effectively captured, could support a range of low-power applications such as structural health monitoring, smart signage, and embedded sensor networks.

While piezoelectric energy harvesting has seen increasing exploration, especially in laboratory settings, its use in kinetic environments like transportation infrastructure remains relatively nascent. Several examples underscore both the feasibility and current limitations of existing bulk piezoelectric technologies. For example, Hill et al. demonstrated between 80 and 140 mW of power generation from a single compression event using piezoelectric transducers [1], while Kim et al. reported 2.67 mW output in a controlled laboratory setting [2]. Xiong field-tested nine piezoelectric harvesters in active roadway pavement, achieving an average output of 3.1 mW per unit [3].

Traditional piezoelectric energy harvesters typically employ monolithic ceramics such as lead zirconate titanate (PZT). Although such materials can generate substantial electrical output under ideal mechanical loading, they are inherently brittle, exhibit limited mechanical compliance, and often suffer from poor acoustic and mechanical impedance matching when embedded in real-world environments like pavements. These limitations reduce their efficiency and durability under varying force conditions. Another challenge is the intrinsic ferroelectric hysteresis of PZT, which leads to non-linear, path-dependent electromechanical responses and energy dissipation under cyclic loading [4]. Advanced control techniques can compensate for hysteresis in precision actuator applications, but in passive energy harvesters this effect manifests as minor efficiency losses rather than a functional limitation.

To address these challenges, 1-3 piezoelectric composites have garnered interest as a viable alternative [5]. Originally developed for medical ultrasound and sonar transducers [6], 1-3 composites consist of vertically aligned piezoelectric rods embedded within a polymer or porous matrix, offering improved mechanical flexibility and tunable electromechanical properties [7]. Their reduced acoustic impedance, lower mechanical quality factor, and increased electromechanical coupling coefficients make them well-suited for energy harvesting applications, especially where mechanical loading is dynamic and irregular. Recent efforts have explored additional strategies to enhance piezoelectric energy harvesting. For instance, Dong et al. (2025) examined multi-configuration arrays of piezoelectric beams to maximize vibration energy extraction [8], and Song et al. (2023) demonstrated that three-dimensional printed porous piezoelectric structures can significantly improve output performance for roadway harvesters [9].

Whereas traditional transducer applications aim to maximize voltage output (e.g., for signal detection or imaging), energy harvesting systems prioritize current output and energy conversion efficiency. In this context, 1-3 composites offer several advantages: minimized cross-coupling between piezoelectric pillars, better strain transfers due to compliant matrices, and improved performance across a range of operating conditions. Gamboa et al. [10] demonstrated that mechanically stacked 1-3 transducers with parallel electrical connections outperform bulk ceramic harvesters in terms of power output and robustness. The commonly accepted figure of merit (FOM) for piezoelectric energy harvesters is the piezoelectric coefficient (d) squared divided by the dielectric permittivity (ε). Stacked configurations further enhance the FOM [11]. In addition to improved performance, studies such as those by Pole et al. [12] suggest that stacked and composite piezoelectric devices can maintain their performance over prolonged operational cycles, an essential consideration for roadway integration.

While large-scale energy generation (e.g., wind farms or solar power) remains essential for feeding the electrical grid, small-scale energy harvesting from ambient sources is particularly well-suited to powering autonomous systems. In such contexts, 1-3 composites have the potential to provide a practical, cost-effective energy source. When embedded in roadways, these transducers could supplement power for battery banks, environmental sensors, or microcontrollers, without the need for external power infrastructure.

Unlike prior roadway energy harvesting studies which used monolithic PZT or single-layer composites, this work is among the first to implement a stacked 1-3 composite architecture and to systematically optimize its geometry for high power output under vehicle-like loads. Key performance metrics, such as energy density, active PZT volume fraction, aspect ratio, and mechanical integrity, are evaluated. Particular attention is paid to optimizing the geometry of the composite arrays to reduce the contribution of the d₃₁ mode (which is negative in PZT) and to enhance the contribution of the d₃₃ mode (positive and dominant in vertical compression), thus improving overall energy harvesting efficiency.

2. Numerical Analysis, Material Parameters, and Experimental Testing

2.1. Finite Element Analysis

Finite Element Analysis (FEA) was used to evaluate various 1-3 composite stack designs. The physics solvers used to model the system are Solid Mechanics, Electrostatics, and Electrical Circuit. The constituent equations for Solid Mechanics are

$$-\mathbf{\rho }{\mathbf{\omega }}^2\mathbf{u}=\mathbf{\nabla }·\mathit{S}+{\mathit{F}}_{\mathit{V}}{\mathit{e}}^{\mathit{i}\mathit{\phi }}$$

(1)

$$-\mathit{i}{\mathit{k}}_{\mathit{Z}}=\mathit{\lambda }$$

(2)

where $\mathit{\rho }$ is the density of the material, $\mathit{\omega }$ is the angular frequency, $\mathit{u}$ is the mechanical displacement vector, $\mathit{S}$ is the mechanical strain, ${\mathit{F}}_{\mathit{V}}$ is the volume force vector, $\mathit{\phi }$ is the phase, ${\mathit{k}}_{\mathit{Z}}$ is the propagation direction, and $\mathit{\lambda }$ is the wavelength.

The governing equations for the piezoelectric material, in addition to (1) and (2), are

$$\mathbf{\nabla }·\mathit{D}={\mathit{\rho }}_{\mathit{V}}$$

(3)

$$\mathit{D}={\mathit{\epsilon }}_{\mathbf{0}}{\mathit{\epsilon }}_{\mathit{r}}^{\mathit{S}}\mathit{E}-\mathit{P}$$

(4)

$$\mathit{S}={\mathit{s}}^{\mathit{E}}\mathit{T}+\mathit{d}\mathit{E}$$

(5)

$$\mathit{D}=\mathit{d}\mathit{T}+{\mathit{\epsilon }}_{\mathbf{0}}{\mathit{\epsilon }}_{\mathit{r}}^{\mathit{T}}\mathit{E}$$

(6)

where $\mathit{D}$ is the electric displacement, ${\mathit{\rho }}_{\mathit{V}}$ is the electric charge volume concentration, ${\mathit{\epsilon }}_{\mathbf{0}}$ is the permittivity in free space, ${\mathit{\epsilon }}_{\mathit{r}}^{\mathit{S}}$ and ${\mathit{\epsilon }}_{\mathit{r}}^{\mathit{T}}$ are the relative permittivity of the material at constant strain and constant stress, respectively, $\mathit{E}$ is the electric field intensity, $\mathit{P}$ is the polarization of the piezoelectric, $\mathit{S}$ is the mechanical strain, ${\mathit{s}}^{\mathit{E}}$ is the material compliance at constant electric field, T is the mechanical stress, and $\mathit{d}$ is the piezoelectric charge coefficient.

For all simulations PZT-5H is an active material, epoxy is an inactive material, and copper is an electrode material. PZT-5H has a piezoelectric coefficient d₃₃ of 440 pC/N and a set of relative permittivity of ε₁₁ = ε₂₂ = 3130 and ε₃₃ = 3400 at room temperature (measured in authors’ lab using a d₃₃-Meter and an LCR meter at 1 kHz, respectively). Figure 1 illustrates the 1-3 composite geometric design where Figure 1a is the piezoelectric pillar array, Figure 1b is the electrode array, and Figure 1c is the epoxy material. Figure 1d–f show the corresponding 1-3 composite stacked in three layers. The bottom of the stack was held in a fixed position while on the top surface a compressive load was applied ranging from 1 to 10 kN at a frequency of 10 Hz. These parameters were chosen based on typical axle loads and vehicle speeds, utilizing data collected by the Texas Department of Transportation (TxDOT) (see, e.g., [13]).

We focus on a three-layer design as an optimal and representative configuration for this study. This design aligns with our fabricated prototype (Sample B, see

Table 1. Illustration of three prototype stack PZT transducer configurations.

UTSA-0	Sample-A	Sample-B
stack of PZT layers (device of 21 layers shown)	commercial stack PZT	3-stack of 1-3 composites

), allowing for direct validation of the simulation model. The three-layer stack forms a practical two-terminal device with alternating poling directions and electrode interfaces, resembling a multilayer capacitor structure. While additional layers could increase output power, they would also introduce fabrication challenges, interfacial reliability concerns, or diminishing returns due to stress non-uniformity. Limiting the simulation to three layers reduces computational overhead while capturing the essential electromechanical behavior typical of multilayer harvesters. For configurations exceeding three layers, output power can be estimated by linear scaling of the three-layer results under quasi-static conditions. However, we note that under dynamic or high-frequency loading, interlayer stress propagation and mechanical damping effects may result in sublinear performance gains, requiring time-dependent simulation for accurate assessment.

All stacks are mechanically assembled in series but connected electrically in parallel to a load resistance (implemented via the Electrical Circuit module in COMSOL version 5.4), which was tuned to maximize power output for each configuration. Power output measurement was carried out systematically for each composite sample after matching resistance to achieve the maximum power output. The COMSOL analysis was harmonic with data reported per cycle at 10 Hz. PZT pillar length, PZT pillar width, PZT pillar height, overall composite length, overall composite width, and overall composite height were all swept to determine optimal geometric preferences. Overall length, width, and height are the accumulation of rod length, width, and height as well as the spacing between each rod, respectively. A few assumptions were made for the evaluation and development of the transducers: (1) the length and width of the PZT pillars are equal, (2) the length and width of the overall composite are equal, (3) there is equal spacing between pillars, and (4) there is an equal number of PZT pillars in the x-direction and y-direction. The equations below are used to determine the volume percentage of PZT in the composites:

$$Vol\mathrm{ \%}={\displaystyle \frac{{Vol}_{PZT}}{{Vol}_T}}$$

(7)

where Vol % is the percentage of PZT volume in the composite, Vol_PZT is the volume of PZT, ${Vol}_T$ is the volume of the entire composite. The overall length of the composite is determined as below:

$$l_T=a\sqrt{n_p}+b(\sqrt{n_p}+1)$$

(8)

where $l_T$ is the overall length of the composite, a is the length and width of the PZT pillar, b is the length of the spacing between pillars, and n_p is the number of PZT pillars per layer. The dependence of the number of PZT pillars on the overall composite length, the length of the PZT, and the spacing between pillars is derived as below:

$$\sqrt{n_p}={\displaystyle \frac{l_T-b}{a+b}}$$

(9)

The derived PZT volume percentage and total composite volume are shown below:

$${Vol}_{PZT}=nh{n}_p{a}^2$$

(10)

$${Vol}_T={{(l}_T)}^2\left(nh+nt+t\right)$$

(11)

where n is the total number of layers, h is the height of the PZT pillar, and t is the electrode thickness. Substituting the equations above gives the final PZT volume percentage with respect to particular factors important to the designer of such composites, as shown below:

$$Vol\mathrm{ \%}={\displaystyle \frac{nh{n}_p{a}^2}{l_T^2(nh+nt+t)}}={\displaystyle \frac{nh{(\frac{l_T-b}{a+b})}^2{a}^2}{l_T^2(nh+nt+t)}}={\displaystyle \frac{nh{n}_p{a}^2}{{[a\sqrt{n_p}+b(\sqrt{n_p}+1)]}^2(nh+nt+t)}}$$

(12)

Figure 2 illustrates the designated variables described above. The left image is a side view of the 1-3 stacked composite, at n = 3, and the right image is a top–down view of the composite.

2.2. Experimental Evaluation and Validation

The piezoelectric power output of composite transducers was experimentally evaluated using principles underlying 1-3 biphasic composites and stacked PZT transducer architectures [11]. Mechanical excitation was applied via an MTS Acumen III electrodynamic test system (MTS Systems Corporation, Eden Prairie, MN, USA), while the generated electrical output was continuously recorded using a PC-interfaced digital multimeter (Metrahit, Messtechnik GmbH sourced from San Antonio, TX, USA).

All stacked transducers were mechanically assembled in series and electrically connected in parallel, thereby increasing output current. Power output was evaluated after resistive matching to ensure optimal energy transfer for each composite sample. Comparative testing between stacked 1-3 composite transducers and their bulk counterparts was conducted to assess performance gains. Geometric parameters including PZT pillar length, width, and height, as well as overall composite dimensions (length, width, and height), were systematically varied to identify optimal configurations. Overall composite dimensions reflect the accumulation of rod dimensions and the spacing between adjacent PZT pillars, providing a direct link between microstructural design and macroscopic transducer behavior.

For benchmarking purposes, three prototype transducer configurations were tested: (1) a UTSA-fabricated multilayer PZT stack (UTSA-0), (2) a commercial PZT stack (Sample-A), and (3) a three-layer stacked PZT 1-3 composite (Sample-B), as depicted in Table 1. The UTSA-0 design comprises a variable number (1 to 21) of active PZT-5H plates assembled using indium, copper mesh, insulating epoxy, and silver conductive epoxy, each plate measuring 20 × 20 × 2 mm³. Sample-A is a commercially sourced stack featuring over 50 thin PZT-5H layers with total dimensions of 14 × 14 × 20 mm³. Sample-B, the 1-3 composite stack, includes three layers, each containing 25 PZT-5H rods (2.5 × 2.5 × 11 mm³ each) arranged within a 19 × 19 mm² cross-sectional area. The 1-3 composite layers for Sample-B were fabricated using a dicing-and-filling technique—PZT-5H ceramic blocks were diced into 2.5 mm × 2.5 mm rod elements, arranged in a 5 × 5 grid, and the voids were filled with a two-part epoxy resin which was then cured to form a solid composite plate. Three such plates were stacked with conductive interlayers (copper electrodes and epoxy bonds) to create the final three-layer composite transducer.

When compared with theoretical predictions, UTSA-0 demonstrated good agreement, with an average deviation of less than 5.9% within a load range of 1–3 kN. These discrepancies are largely attributed to manual fabrication variability. Accurate theoretical comparisons for Sample-A and Sample-B were hindered by the limited availability of detailed material properties (e.g., piezoelectric, dielectric, and mechanical constants) for active and inactive components. These limitations underscore the necessity of combining robust experimental validation with finite element modeling (FEA) to explore a broader design space and guide transducer optimization. A combined experimental and computational study was conducted to examine structure–property relationships across different composite designs [14]. Key electromechanical parameters—including impedance, piezoelectric coefficients, and dielectric permittivity—were measured to facilitate the accurate interpretation of performance variations across transducer configurations. Power output was characterized as a function of both compressive force and applied pressure, with power density calculated relative to both the PZT active volume and the total transducer volume.

The experimental results indicated that Sample-B exhibited the highest power density (output power per unit active PZT volume) among the prototypes, under equivalent pressure [11]. Figure 3 presents a comparison between the experimentally measured and FEA-simulated output power of the 1-3 composite stack (Sample-B) across a range of applied compressive loads. The fabricated device has an estimated active PZT volume fraction of ~41.4%, which corresponds closely to simulated cases within comparable volume fractions (37.8% and 41.0%). The experimental data follows a power-law trend with increasing load, and a fitted curve is included to highlight this behavior. While the measurements were conducted up to a moderate load range, the curve captures the expected non-linear increase in output. For reference, an additional projection is shown for an optimized design with ~21.8% PZT volume fraction—identified in the simulation (to be discussed in Section 3) as near the optimal range for maximizing power density, demonstrating the potential performance gain achievable through geometric tuning. The close agreement between simulation and experiment supports the validity of the FEA model for design optimization purposes.

Figure 4 presents a comparative electrical impedance spectroscopy (EIS) analysis of the 3-layer stacked 1-3 composite transducer (Sample B) and the 21-layer monolithic PZT stack (UTSA-0). The stacked 1-3 composite, despite its smaller overall size, displays a markedly lower resonant frequency (~56 kHz) than UTSA-0 (~95 kHz), along with an overall lower admittance magnitude throughout the frequency range. This comparison reinforces the distinction between the electromechanical properties of monolithic and composite designs and supports the design advantages of the 1-3 stacked composite for robust energy harvesting under dynamic mechanical loading. While the piezo stacks are operated non-resonantly, well below the stack’s resonance frequency, these data are presented to show how the device’s effective electrical admittance varies and to provide additional insight into the dynamic behavior of the transducers.

3. Results and Discussion

The models generated were evaluated based on the power generation of the composites as well as mechanical integrity. Stress analysis of 1-3 composite stacks indicates that the harder material (PZT compared to the given epoxy) is responsible for absorbing approximately 80 percent of the mechanical load; illustrated in Figure 5. The literature indicates that, for soft PZT (PZT-5H), to avoid depolarization or mechanical failure (cracks or grain boundary damage), the stress level was not to exceed 50 MPa [15]. Piezoelectric energy harvesters must be designed to withstand the various mechanical loads they will be exposed to in roadways. As a result, optimized electromechanical energy conversion is also dependent on long-term mechanical stability.

The height of individual PZT pillars varied from 4 mm to 3.83 mm, 3.5 mm, 3 mm, 2 mm, and 1 mm. The length and width of the PZT pillars ranged from 0.47 mm to 19.1 mm. The PZT volume percentage of the composite energy harvesters ranges from 96.64% (bulk) to 4.91%. The spacing between the PZT pillars ranges from 0 mm (bulk) to 1.97 mm. The total number of PZT rods per layer ranges from 1 to 484 rods. These dimensions are chosen to give designers a wide range of composites with variations in force at low-frequency applications. Assuming an equal distribution of pressure on the composite, the compression for safe operation is approximately between 10,000 and 45,000 Newtons (maintaining a pressure less than 50 MPa).

3.1. Effect of PZT Volume Fraction on Output Power Vs. Force for Various Designs

Figure 6a–d presents the composite output power as a function of applied compressive force for various PZT volume fractions, for four different device footprints: 15 × 15 mm², 19 × 19 mm², 25 × 25 mm², and 31 × 31 mm² (shown in panels a, b, c, and d, respectively). The PZT volume percentages are calculated using Equation (12), which incorporates the variables pillar length a, spacing between pillars b, number of PZT rods per layer n_p, and pillar height h, with the number of stacked layers (n) held constant at three layers for all configurations. The thickness (t) of the electrode layer is assumed to be 0.1 mm in the total volume evaluation for all samples evaluated.

The figures also reveal the influence of design parameters on output power. The five top-performing power converters in terms of composite footprint, 15 mm × 15 mm, featured 25–49 PZT rods per layer, pillar spacing ranging from 1 mm to 1.667 mm, pillar heights between 3.5 and 4 mm, and a constant pillar length of 1 mm. The 19 mm × 19 mm composites showed optimal performance with 64–100 PZT rods per layer, pillar spacing between 0.818 and 1.22 mm, pillar heights of 3.5–4 mm, and a pillar length of 1 mm. In the 25 mm × 25 mm composites, the top five performers had 81–196 PZT rods per layer, pillar spacing of 0.85–1.5 mm, pillar heights from 3.5 to 3.833 mm, and pillar lengths ranging between 0.8 and 1.1 mm. The 31 mm × 31 mm composites performed best with 121–225 PZT rods per layer, pillar spacing between 1 mm and 1.667 mm, pillar heights of 3.5–4 mm, and a fixed pillar length of 1 mm.

Each curve corresponds to a group of configurations binned by PZT volume fraction. The shaded region around each curve indicates the variation within ±3% of the nominal PZT volume percentage for that bin, capturing variability due to geometric variables within each subset. The data reflects the top-performing designs ranked by power output under matching resistance load determined experimentally for a given transducer geometry.

As seen in Figure 6a–d, output power increases quadratically with compressive force. A trade-off is observed between the PZT volume fraction and the transferred mechanical force. As the number of pillars increases, corresponding to a higher volume fraction, the output power initially increases due to enhanced overall stress transfer and energy conversion. This output reaches a peak around the intermediate volume fraction of PZT (~20%). Beyond these optimal points, however, output power begins to decline. This decline occurs because, under a given applied force, the stress distributed to each PZT pillar becomes lower to generate charges, even though the total PZT content continues to rise. This highlights the importance of balancing pillar density and stress localization to achieve optimal power output.

3.2. Effect of PZT Volume Fraction on Active Power Density vs. Force for Various Designs

Figure 7a–d illustrates the active power density (power output per unit PZT volume) as a function of applied force for composites with varying PZT volume percentages. Panels (a), (b), (c), and (d) of Figure 7 correspond to device footprints of 15 × 15 mm², 19 × 19 mm², 25 × 25 mm², and 31 × 31 mm², respectively. The PZT volume percentages are calculated using Equation (12), which incorporates the variables pillar length a, spacing between pillars b, number of PZT rods per layer n_p, and pillar height h, with the number of stacked layers (n) held constant at three for all configurations. The thickness (t) of the electrode layer is assumed to be 0.1 mm in the total volume evaluation for all samples evaluated.

For the 15 mm × 15 mm composites, the five top-performing configurations (in terms of power density) typically contained 25–36 PZT rods per layer, except for one outlier with 225 rods. Pillar spacing ranged from 1 mm to 1.667 mm, pillar heights were between 2 and 4 mm, and pillar length was consistently 1 mm. In the 19 mm × 19 mm composites, the best five designs featured 49–144 rods per layer, spacing of 0.78–1.5 mm between pillars, pillar heights of 3–4 mm, and lengths ranging from 0.74 to 1 mm. For the 25 mm × 25 mm composites, the top designs included 81–196 rods per layer, spacing between 0.92 and 1.5 mm, pillar heights from 3 to 3.5 mm, and pillar lengths from 0.8 to 1.17 mm. Finally, the 31 mm × 31 mm composites showed top performance with 121–225 rods per layer, spacing between 1 and 1.667 mm, pillar heights from 1 to 4 mm, and a consistent pillar length of 1 mm.

For each footprint, the configuration that maximized total power (Section 3.1) was not always the one that maximized power per unit PZT. In fact, designs with slightly fewer or thinner PZT pillars sometimes yielded higher active-material efficiency. Generally, the highest PZT-normalized power density still occurred at intermediate volume fractions—confirming that adding too much PZT can be counterproductive even when evaluating output per material volume. Beyond a certain packing of PZT rods, the stress per rod reduces, limiting per-volume effectiveness, and the active power density tends to flatten or decline at high volume fractions. It is thus noted that stiffness matching and stress concentration should be optimized for each footprint. This analysis is relevant for cost-sensitive designs—achieving more power per unit of piezoelectric material means a more efficient use of the active material.

3.3. Normalized Performance—Power Density per Transducer vs. Force

Figure 8a–d presents the power density (output power normalized by the total transducer volume) as a function of compressive force for four transducer sizes: 15 × 15 mm², 19 × 19 mm², 25 × 25 mm², and 31 × 31 mm² (corresponding to Figure 8a, 8b, 8c, and 8d, respectively). All stacked composites consist of three layers, and total volume is straightforward to compute from known dimensions. However, power density varies with design parameters such as the number of PZT rods, pillar spacing, pillar height, and pillar length. In all configurations, the electrode thickness was held constant at 0.1 mm.

For the 15 × 15 mm² designs, the top five power converters had 25–36 PZT rods per layer (with one outlier at 225), a pillar spacing between 1.286 and 1.667 mm, a pillar height between 3 and 4 mm, and a pillar length of 1 mm. For 19 × 19 mm², the top performers had 64–100 rods per layer, a spacing from 0.818 to 1.22 mm, a pillar height of 3.5–4 mm, and a pillar length of 1 mm. For 25 × 25 mm², they contained 49–196 rods per layer, spacings of 0.92–1.5 mm, pillar heights from 3.5 to 3.83 mm (with an outlier at 2.4 mm), and pillar lengths of 0.8–1.1 mm. For 31 × 31 mm², the range was 121–289 rods per layer, a spacing between 0.778 and 1.667 mm, and a pillar height generally of 1 mm (with an outlier at 4 mm), and a pillar length of 1 mm.

Power density per total volume as a function of compressive force reflects system-level energy density. It is noted that compact designs (15 mm and 19 mm) exhibit higher volume-normalized efficiencies, and these could be critical for applications where space and integration volume are constrained. The highest power density per transducer volume reported (>0.6 mW/mm³) is comparable to previously reported values in the literature.

3.4. Overall Comparison—Top Power Density Designs

The top five performers from each plot were then compared relative to each other in order to gain deeper insight. The figures indicate an optimal PZT volume fraction in the ~20–22% range, when considering energy density, absolute output power, and mechanical stability together. While earlier figures (Figure 6, Figure 7 and Figure 8) use applied force as the independent variable to reflect realistic vehicle loading conditions (up to 10 kN, based on TxDOT axle weight limits), Figure 9, Figure 10 and Figure 11 switch to pressure-based scaling to enable geometry-independent comparison. Using pressure allows for a normalized evaluation of power density and transducer efficiency across designs with different footprint areas and active PZT volumes.

Figure 9 compares output power vs. pressure for the top five designs, highlighting the effect of different PZT volume percentages. Figure 10 and Figure 11 show the active-material power density and total-volume power density vs. pressure, respectively, for those top designs. Each figure plots the five top-performing configurations for each size, with labels indicating PZT volume fractions.

Similarly, when comparing the top designs’ power densities normalized by PZT volume (Figure 10) and by total volume (Figure 11) as a function of pressure, the designs with ~20% PZT consistently outperform those with very high (or very low) PZT fractions. This consistency across metrics further validates ~20% as an optimal active volume fraction for these stacked composites. The insight can be gained that smaller transducers (15 × 15 mm and 19 × 19 mm) exhibit higher volume-normalized efficiencies, which is critical for applications where space and integration volume are constrained. Additional FEA simulations for a smaller 10 × 10 mm², three-layer composite (not originally in our prototype set) indicate that while the maximum output power under a given pressure is lower (e.g., ~0.15–0.2 W at a 2.0 kN load, compared to that of ~500+ mW produced by 15 × 15 mm in Figure 9), the device’s power density and optimal design parameters remain consistent with those of larger devices (≈20% PZT fraction yielding the highest efficiency).

3.5. Effect of Aspect Ratio

Aspect ratios are of concern due to long-term mechanical stability. Aspect ratios for pillars are defined as height divided by length or width (length = width). Specific ratios are compared to determine if there is any dependence on output power. It is important to note that the active PZT volume percentage is maintained at approximately 21 percent for this comparison. The height of the PZT pillars was 3 mm for all comparisons made. Aspect ratios range from 1.55 to 6.38 with an applied force of 10 kN.

Table 2. Power output as a function of PZT pillar aspect ratio for each device footprint (15, 19, 25, and 31 mm).

31 × 31 mm2		25 × 25 mm2		19 × 19 mm2		15 × 15 mm2
Aspect Ratio	Power [mW]	Aspect Ratio	Power [mW]	Aspect Ratio	Power [mW]	Aspect Ratio	Power [mW]
3.09	280.62	3.85	439.93	5.00	791.07	6.38	1291.97
2.88	278.85	3.57	442.07	4.69	788.25	5.88	1300.94
2.68	277.96	3.33	436.76	4.35	784.60	5.56	1283.17
2.48	276.00	3.09	441.45	4.05	784.97	5.08	1293.07
2.27	274.10	2.80	434.88	3.70	779.23	4.62	1288.72
2.05	275.22	2.56	426.78	3.37	774.61	4.23	1272.37

indicates that power density varies with aspect ratio, peaking near an optimal height-to-width range. Excessively squat or tall PZT rods are less effective, suggesting geometry-dependent stress optimization. In our simulations (with pillar height fixed at 3 mm for each device size), the best performance occurred at aspect ratios in the order of ~3–4, with deviations on either side causing a drop in output power in the order of 2–10%. This allows composite manufacturers some flexibility when designing a composite for specific force applications. We find that aspect ratios are higher for a smaller transducer footprint to reach peak power output, which calls for a balanced design that provides compact, high-power output, and mechanical robustness.

4. Conclusions

This study comprehensively evaluated the output power of stacked 1-3 piezocomposite transducers in relation to key design factors: active PZT volume fraction, pillar aspect ratio, and mechanical loading limits. Our simulations and experiments confirm that stacked 1-3 composites provide significantly enhanced electromechanical energy conversion compared to bulk PZT, in line with previous reports, and further provide the following design insights. The power assessment of composites indicates that PZT pillar length should be approximately 25–35% of the pillar height for optimal power conversion. Pillar spacing relies on the total number of PZT pillars per layer but should approximately be within 24–40% of the pillar height. The total number of pillars per layer shows a tendency to increase as the overall length and width of the composite increases. This study has found that the active (PZT) volume percentage should lie at ~20% of total volume. Mechanical stability is of utmost importance as heavy loads will be applied continuously for the applications of these composites we are studying. Aspect ratio simulations display an increase in power (an average of 6 percent) with an increase in aspect ratio. At the same time the increase in aspect ratio makes the structure weaker and prone to failure. Lower aspect ratios with the provided volume percent should be implemented to enhance mechanical stability and optimize electromechanical power conversion. Following these guidelines will help ensure the devices not only maximize power output but also survive the heavy, repeated loads of traffic environments.

In summary, this study identifies ~20% PZT by volume and moderate pillar aspect ratios (~3–4) as an optimal design space for stacked 1-3 piezocomposite harvesters, balancing high power output with mechanical robustness. While our focus was on energy harvesting, we acknowledge that these design choices could influence other performance metrics if such composites were used as actuators or active transducers. In general, increasing compliance via polymer embedding and stacking may lead to a lower blocking force and a longer electromechanical response time compared to a monolithic PZT actuator, since the compliant matrix and interfaces can absorb some energy and delay strain transfer. The multi-layer configuration might also reduce the effective resonant frequency of the device. However, these trade-offs do not detract from the energy harvesting performance and were beyond the scope of the present work. They are mentioned here to emphasize that an optimized harvester design can be further enhanced to achieve fast actuation or high force output for various applications.