Effect of mean load, load range, and frequency on the power generated by piezoelectric crystals

S.M. Allameh, O. Akogwu, W.O. Soboyejo

Abstract

This paper presents the results of a preliminary study of the effect of loading parameters on the performance of piezo crystals. The output power of the crystals was observed to increase with parameters such as frequency and the dynamic load range but slightly decreased with mean load. The efficiency of the crystal was calculated based on the mechanical energy applied to the piezo crystal. The ratio of the electrical output to mechanical energy input was taken ad the efficiency of the crystal. This ratio was seen to increase with the cycling frequency and with the dynamic load range. However, increasing mean load caused the efficiency to drop significantly. The implications of this study in the design and applications of piezoelectric power generators are discussed.

I. Introduction

Conversion of kinetic energy to electrical energy has been investigated by a number of researchers for electrical power generation [1-6].Piezoelectric materials have been investigated as essential elements for high-power pulse generation [1-6]. Electrical energy is generated when an applied force overcomes internal inductance or capacitance of a material. For a piezoelectric material this internal fields are stored the ohmic lattice of the material. Mechanical deformation of this material by applying load in static or dynamic cases generates the output voltage required to power the piezoelectric pulse generator. Experiments have been performed by applying quasi-static or dynamic stress [7]. They reported that dynamic and quasi-static loading produced equal magnitudes of output voltage. They also reported that the dynamic loading produced a unidirectional voltage while the quasi-static case generated a bi-directional voltage. However, Engel at al [5] have reported different results for similar experiments performed on the piezoelectric material for both stresses. They discovered that the dynamic loading yielded a much higher output voltage (up to 10 times more) than the quasi-static case. They compared their experimental measurements on power generation with predictions of simulations. The theory finds that the material thickness to cross sectional area ratio (TAR=h_piezo/A) can be used to maximize output power. A higher TAR results in a higher output voltage but a lower output current. In an effort to maximize output power, the voltage and current have to be maximized. The overall effect of thickness to area ration (TAR) will be dominated by the larger contribution of voltage to the product of voltage and current. In other words, piezoelectric power output increases with increasing TAR.

To achieve this, two models have been proposed. The mechanical model that provides information on the displacement (deformation) of the material due to applied force and thus the generated voltage. The electrical model is used to find circuitous conditions needed to generate maximum current.

The purpose of this study is to investigate and present the output power generated by the piezoelectric power generator based on the proposed models. In this effort, we set up experiments comparable to earlier works and dynamically loaded the system at different frequencies, load ranges and applied different loads (resistor values) at the output. The results obtained are also discussed.

II. Experimental Procedure

Piezoelectric crystals were obtained from Staveley Sensors Inc., East Hartford, CT. Various characteristics of the crystal is tabulated in Table. They had a diameter of 25 and a thickness of 6.3 mm. The crystals were sandwiched between two Al blocks to acquire or impart electric signal to them. One Al block were then placed in load train insulated from other components by a ZrO₂ block. The ZrO₂, was housed in an Al block connected to load cell which was mounted on an x-y stage. The other Al block covering the piezo crystal was connected to a piezoelectric actuator made by Polytec PI, Inc. (MA, USA). The latter was powered by a wave function generator that produced sinusoidal signals, with different frequencies, different amplitudes, and different offsets. The signal from the wave function generator was then sent to a PI amplifier and then fed to the PI piezoelectric actuator.

A range of frequencies from 0.1 to 20 Hz were chosen for the first set of tests conducted at a cyclic load amplitude of 52 N and a constant mean load of 30 N. One set of experiments was conducted with a varying load range of 27 to 54 N at a constant mean load 36 N. Another set of tests was performed at a constant load range of 53 N but at a varying mean load of 3.7 N to 4.4 N. The set up is shown in (Figure 1).

III. Results and Discussion

(a) Effect of Loading

The results of mechanical testing of the piezo crystal are summarized in (Figure 2)(Figure 3)(Figure 4). The variation of the output voltage with load is shown in (Figure 5). The output voltage increases with the load range in a near-linear manner. It clearly shows that the dynamic range of the applied load is directly responsible for the output power.

The effect of mean load is illustrated in detail in (Figure 6). The tests were performed at a constant load range to exclude its effect on the output voltage. The frequency was also maintained at 10 Hz. The effect of mean load on the output voltage is seen to be small. The results show an inverse dependence of the output voltage on the mean load. This implies that for a given dynamic load range, the highest output power is achieved at lowest possible mean load. Mean load here may be thought as isostatic pressure in piezo crystal that affects the mobility of electrons in the crystal lattice.

The effect of frequency on the output voltage is shown in (Figure 7). The change in frequency is seen to change the out put voltage significantly. However, this change in output voltage on the loading frequency is linear as seen in (Figure 3). The linear dependence of the output voltage on frequency can be understood given the fact that more mechanical work is performed when the piezo crystal is loaded at a higher frequency. In fact the mechanical work that is performed on the piezo crystal is proportional to the area under the load-displacement curve. However, this area does not significantly change with the strain rate for this crystal, (e.g. same amount of work is performed at a higher strain rate), then more cycles will mean more energy input into the crystal. This higher energy input is seen to have yielded a higher energy output (manifested by a higher output voltage). Doubling the frequency to 20 Hz translates into a near-doubling in the output voltage as seen in (Figure 3).

All these tests were performed with a resistor attached to the crystal. This resistor was chosen to be 1.1 MW. In this manner, the output voltage was representative of the power output of the crystal through the relationship: P=V²/R. The dependence of the power output on frequency and dynamic load will then be through this power law, meaning, doubling the frequency will quadruple the output power. An interesting outcome of this experiment is that the power output increases with square of the values for frequency and dynamic load range. This means that the efficiency of the crystal is a function of the frequency and load range, increasing with both of these factors.

Two important conclusions can be made based on the outcome of these experiments. The first one points to the application of piezoelectric crystals in higher frequency applications where higher efficiencies are achieved. The second one points to the higher dynamic load range applications where efficiency maximizes. In both of these applications the mean load should be minimized in order for the efficiency to be optimum.

The effect of dynamic load range is shown in (Figure 8). Increasing the dynamic load range at constant mean load at a frequency of 10 Hz is seen to increase the output voltage of the piezo crystal. The increase in the output voltage with the load range is summarized in (Figure 8). There is a significant change in the output voltage with an increase in the dynamic load range. The effect is seen to be much greater at higher load ranges. The magnified effect of load range will greatly boost the efficiency at higher load ranges.

The output voltage of the crystal varied linearly with the value of the resistor used across the two leads of the multimeter. This variation is depicted in (Figure 4). The output voltage increases linearly with the resistance attached to the crystal. Since power is a square function of voltage, and increase in the resistance will greatly increase the power extracted from the piezo crystal.

The effect of cyclic loading on the output voltage can be compared to the results of Engel et al.[1]. The output voltage predicted from the formula

(1)

where V_piezo is the voltage across the piezo crystal, d33 is the force sensitivity of the crystal, C_piezo is the capacitance of the crystal F is the applied force. Equation (1) predicts a voltage of about 10 V at a load level of 50 N. Without the external resistor, the cyclic voltage, measured in this study was bi-directional with a total range (minimum to maximum) of ~ 5 V.

The power output of the piezo crystal as determined by the voltage across a resistor placed parallel to the piezo varies with the resistor. This variation in illustrated in (Figure 9). This log-log plot clearly shows an increase in the consumable power extracted from the piezo crystal with the value of the external resistance. It should be noted that power consumed in the external resistance is inversely proportional to the value resistance.

(2)

However, the output voltage increases with increasing resistance. Since the effect of voltage on the power is greater than the effect of resistance, the net effect of increasing resistance on the amount of extracted power is positive.

(b) Efficiency of Piezo Crystal

For piezo electric crystal with a capacitor of 2.8 nF under cyclic loading, the stored energy per cycle time t_cycle can be calculated as:

(3)

With no external resistance, the output power of the piezo will be ~ 3.75 x 10^-7 W. As expected, this value is greater than the energy consumed by the external 2 MW resistance (e.g. 1.75 x 10^-7 W). To calculate the efficiency of the crystal, under different loading conditions, we need to determine the amount of mechanical work performed on the crystal.

(4)

Here force (F), applied on the crystal, is measured directly by the load cell in the load train. It will include forces originating from the acceleration of the piezo mass, damping of the piezo crystal and the elastic response of the crystal. Since the sum of all forces is measured instantaneously, there is no need to calculate various force components separately. Using parameters presented in Table 1 for the piezo crystal used in this study, the efficiency is determined as the ratio of the electrical power output of the crystal to the mechanical power provided to the crystal to the consumed output using Equation (4).

The results of variation of the efficiency with the dynamic load range is presented in (Figure 10). As expected the efficiency of the piezo crystal increases with increasing load range. For all data points, the mean load has been maintained constant at 30 N. The increase in the efficiency is nearly linear. A similar trend is observed in the variation of efficiency with frequency (Figure 11). More mechanical work applied on the crystal generates more electrical power. Frequencies used in this study are small enough to allow full relaxation of the crystal between loadings. This leads to additive effect of sequential cycles that impose mechanical work on the crystal.

The most intriguing result of this study is illustrated in (Figure 12). The efficiency of the piezo crystal drastically decreases with increasing mean load. There are two contributions to this effect. The smaller contribution comes from the hydrostatic pressure that is applied to the crystal at higher mean loads. This reduces the output voltage, possibly by affecting the mobility of the electrons in the crystal lattice of the piezo material. The more important factor in the loss of efficiency with the mean load comes from the huge increase in the mechanical work expended in the actuation of the piezo crystal. Higher mean load dramatically increases the mechanical work. In conjunction with a constant load range, it leads to a significant drop in efficiency.

The results of this study shed light on a number of important parameters that affect the efficiency of piezo electric crystals. With the current renewed interest in alternative sources of energy (as a result of higher oil and gas prices), optimization of processes that produce electric energy from natural forces such as wind and waves becomes important.

Some factors such as higher TAR (thickness to area) ratio that are known to increase the power output of the piezo crystals. Our results show the effects of some other factors such as the effect of mean load that reduces the efficiency of power generation through the use of piezo crystals. This implies that for better efficiency, the dynamic load range should be chosen in a manner that minimizes the mean load.

The power extraction from the piezo is also shown to depend on the external resistance used to measure it. At a higher electrical resistance, power extraction is higher. This means lower loads on piezo electric generators will yield higher efficiencies.

IV. Summary and Conclusions

Based on the results obtained in cyclic loading experiments performed on a piezo crystal the following conclusions can be made:

(1) The output voltage of a piezo crystal across a resistor is an indicator of the power output of the crystal. The magnitude of this output voltage increases with dynamic load range, frequency and decreases with increasing mean load

(2) The power output of the crystal, as expressed by V²/R greatly increases with the dynamic load range and with frequency.

(3) The efficiency of a piezo crystal, as a function of the power consumption through a resistor depends on frequency and load range. It has been demonstrated that efficiency increases with frequency and with dynamic load range. The efficiency of the piezo drastically decreases with increasing mean load

(4) The optimum power generation is shown to commence with highest frequency, highest dynamic load range and lowest mean load.

References

1. T.G. Engel, C. Keawboonchuay, and W.C. Nunnally, "Energy conversion and high power pulse production using miniature piezoelectric compressors," IEEE Transactions on Plasma Science, 2000, 28 (5) pp. 1338-1341.

2. T.G. Engel, W.C. Nunnally, J. Becker, R. Rahman, and C. Keawboonchuay, "Research progress on compact kinetic-to-electrical energy converters," Presented in Proceedings of the 12th International Pulsed Power Conference, 1999, Monterey, CA, USA, C.K. Stallings, H., Editor, IEEE, pp. 1287-1290 vol.1282.

3. T.G. Engel, W.C. Nunnally, and N.B. VanKirk, "Compact kinetic-to-electrical energy conversion," Presented in Digest of Technical Papers 11th IEEE International Pulsed Power Conference, 1997, Baltimore, MA, USA, G.V. Cooperstein, I., Editor, IEEE, pp. 1503-1507 vol.1502.

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Table I. Characteristics of the piezo electric crystal used in this study

Properties	Symbol	Magnitude	Units
Diameter	d	15	mm
Thickness (mm)	h	3	mm
Young’s Modulus	E	111	GPa
Capacitance	C_p	2.08	nF
Load proportionality constant	d₃₃	598	C/N