Vacuum packaging for MEMS piezoelectric vibration energy harvesters

Analysis of the resonant and anti resonant behavior of piezoelectric vibration energy harvestersM. Renaud, R. Elfrink and R. van SchaijkIMEC / Holst Centre, Eindhoven, The NetherlandsEmail address: Michael.Renaud@imec-nl.nl

Abstract: Piezoelectric vibration energy harvesters (PVEH) are a relevant alternative to electrochemical batteries for the electrical powering of the nodes in wireless sensor networks. Usually, PVEH based on a resonant principle are designed in such a way that their resonance frequency matches the dominant harmonic of the input vibration. In some conditions it is possible to adopt a different optimization rule making profit of the frequency dependant electrical impedance of the PVEH and of its resonant/anti-resonant behavior. A theoretical description of this effect is developed and simple closed form formulas characterizing it are established. The model is compared successfully to experimental measurements on thin film AlN and ceramic PZT based devices. As a practical use of this effect, it is shown that, for a similar generated power, it is possible to tune the voltage or current delivered by the device, which is of importance for the design of rectifying electronics.

Key words: energy harvester, piezoelectric, vibration, Aluminum Nitride, PZT.

1. Introduction

Recently, wireless sensor networks opened the way for a large panel of applications such as industrial monitoring [1] or body area networks [2]. Harvesting energy from environmental mechanical vibrations, ubiquitous in many environments, is a promising alternative to electrochemical batteries for supplying electrical power to the nodes of the networks. Vibration energy harvesters act by converting a part of the mechanical energy contained in ambient vibrations into electrical energy, typically by means of electromagnetic, electrostatic or piezoelectric transduction [3]. Devices based on the latter type of electromechanical conversion are investigated in the present article.

When mechanical vibrations have a large frequency (above tens of Hz), the simplest design of piezoelectric vibration energy harvesters (PVEH) consists in a mechanical resonator supporting a piezoelectric capacitor whose electrodes are connected to a load circuit in which energy is stored or dissipated. For optimizing the power transferred to the load circuit, it is often assumed that the frequency of the input vibration should match the fundamental resonance frequency 0R of the harvester, for which the impedance (absolute value) between the electrical ports of the device is close from its minimum. However, if some particular conditions are met, high responses are also observed for frequencies ranging from 0R to 0A, the latter corresponding to the anti resonance frequency of the PVEH, for which the electrical impedance of the device is close from its maximum. The difference between the minimum and maximum impedance of PVEH can be large. On the other hand, assuming realistic piezoelectric benders, it is shown in this paper that the amount of power delivered to a load circuit with optimum impedance remains in the same range of magnitude for input vibration frequencies ranging from 0R to 0A. Therefore, by adjusting the frequency of the input in between resonance and anti resonance of the PVEH, it is possible to tune the voltage or current at which the power is delivered. This is an important consideration for the design of conditioning electronics implying for example rectifier diodes.

This effect has already been observed and discussed in the literature [4-6] in the case of ceramic or thin film PZT based PVEH. Simple analytical descriptions of this behavior were however not developed and the conditions on the design of the device leading to this effect were not clearly established. They are therefore proposed in the present article.

The paper is organized as follow: the derivation of a model used for predicting the behavior of the PVEH is first concisely described. The analytical expression of the power dissipated into a resistive load circuit is then established and analyzed in terms of resonance and anti resonance characteristics. Simple formulas describing this effect are derived. The theoretical predictions are finally compared with experimental measurements performed on ceramic PZT and thin film AlN based devices.

2. Theoretical model of the PVEH

2.1-Derivation of the lumped model The goal of the developed model consists in deriving simple, yet reasonably accurate, closed form expressions of the behavior of the power produced by PVEH at resonance and anti resonance. To this aim, an equivalent circuit model based on a lumped representation is elaborated. This type of representation is commonly used in the field of energy harvesting, particularly by designers of rectifying electronics [7, 8]. It may however be difficult for researcher with an educational background focused on electronics to get an idea of the physical assumptions involved in lumped models of PVEH. Detailed derivations of such theoretical representations are available in the literature [9, 10]. However, they involve long developments based on theoretical mechanics, which may rebuke researchers not familiar with this field. Therefore, we find useful to provide below a brief, but comprehensive description of all the major steps involved in the derivation of the lumped model used in the rest of the paper.

As described by Figure 1, the devices considered in this article consist in the most common form of PVEH: they are made of an elastic cantilever supporting one or several piezoelectric layer sandwiched between metallic electrodes. A large mass is attached to the tip of the cantilever. The base of the device is excited by mechanical vibrations, so that the beam deflects and the strain developed in the piezoelectric layer generates an electrical polarization. The electrodes of the piezoelectric capacitor are connected to an electrical load circuit in which electrical energy is stored or dissipated.

Figure 1. Principle of the studied PVEH.

The derivation of the lumped model starts from the linear constitutive equations of piezoelectricity [11], which, for an infinitesimal element of piezoelectric material, relates the tensors of the stress T, strain S, electrical field E and electrical displacement D through various material properties tensors. Cross coupled and high order terms of the four physical variables are neglected in these equations. Furthermore, thermodynamically reversible and adiabatic or isothermal behavior is assumed. The crystalline symmetry of the piezoelectric material allows reducing the number of independent components in the material properties tensors.

A first simplification of the constitutive equations is obtained from mechanical considerations. The PVEH considered in this article are cantilevers with a rectangular cross section supporting a distributed mass attached to their tip. They are mechanically excited by applying a vibration on their clamped end. In a frame of reference attached to the clamped end of the PVEH, the effect of the mechanical excitation can be represented by inertial forces acting on the volume of the cantilever and attached mass. It is assumed that the direction of the input vibration is normal to the (xy) plane of Figure 1, so that the resulting inertial forces contain solely z-components. This assumption allows simplifying the linear equations of piezoelectricity by neglecting the shear stresses and strain components in the (xy) and (yz) planes. Because the thickness of the considered structures is very small compared to the other dimensions, the tensile stress and strain developed along this direction can also be reasonably ignored. Also, the shear stresses and strains in the (xz) plane are neglected as they are known to have a negligible influence on the behavior of low frequency bending resonators [12]. Finally, plane stress (tensile stress along y neglected) or plane strain (tensile strain along y neglected) is also assumed. Plane stress approximation is appropriated for slender beams while plane strain gives better results for short and wide cantilevers. Anyway, the choice of one or the other of these approximations does not modify the general form of the lumped model. All the approximations that have been discussed in this paragraph correspond to the so called Euler-Bernoulli beam theory.

At this point of the derivation, the number of non zero stress and strain components in the constitutive equations of piezoelectricity has been dramatically reduced. Similar simplifications can be done with the components of the electrical displacement and field. Because of the particular configuration of the metallic electrodes, which constitute equipotential surfaces, E and D are limited to their z components (neglecting the fringing effects that may occur on the edges of the device). According to all the previous considerations, the linear constitutive equations of piezoelectricity can be simplified in the studied case to the scalar forms given in (1) and (2). The s11E and 33T terms represent the relevant components of the elastic compliance under constant electrical field and of the permittivity under constant stress. d31 is the sole piezoelectric charge constant involved in the problem. The given form of the equations is valid for plane stress approximation. In plane strain condition, the involved material properties should be replaced by effective values [13].

(1)

(2)

The next step in the derivation of the lumped model consists in relating the mesoscopic variables Txx, Sxx, Ez and Dz to macroscopic variables relevant for the analysis of PVEH. The latter variables were chosen in our case to be the inertial force F(x) resulting from the input vibration, the vertical displacement x of the beam, the voltage V and the electrical charges q across/on the electrodes. The stresses are related to the external force by applying the fundamental law of mechanical equilibrium. Perfect adhesion between the different layers is assumed. Kinematic relations allow linking Sxx and x. The kinematic equations are developed in our case following the infinitesimal strain theory and by assuming small deflection angle. Assuming a quasi electrostatic approximation, the electrical field is related to V by using the definition of the latter (i.e. Ez=-dV/dz) and the electrical displacement to q by simple integration of Dz over the surface of the electrodes. These manipulations result in a set of two coupled partial differential equations describing the behavior of the PVEH in terms of F(x), x, V and q.

Before continuing the description of the derivation of the lumped model, it is important to discuss the representation of the parasitic damping mechanisms occurring in PVEH. These dissipative phenomena can be classified in two categories [14]: those resulting from the interaction of the device with its environment (squeeze film, Couette damping) and those being caused by internal mechanisms (thermoelastic, anchor losses, internal friction, dielectric damping). If some conditions are respected, it is possible to represent the first type of damping by a viscous term (i.e. proportional to velocity) in the mechanical equilibrium equation. Out of this particular situation, general solutions for the flow of a fluid around a vibrating structure require numerical analysis. Internal dissipations result from complex effects such as for example the mechanical friction between the grains in a polycrystalline material, the propagation of elastic waves through the anchor of the structure or the delay between the application of an electrical field and the orientation of the electrical dipoles in the material along this field. For problems involving a frequency domain approach, the use of complex material properties is generally admitted as an acceptable phenomenological way of representing these mechanisms [15]. Following this method in the case of piezoelectric materials, all the internal parasitic effects related to mechanical phenomena can be represented by a complex compliance in (1) while dielectric losses can be included through a complex electrical permittivity in (2). Holland [15] suggests even that, for having a complete phenomenological representation of the internal losses in a piezoelectric material, a complex value of d31 should be used in (1) and (2).

For the studied PVEH, experimental measurements described later show that the losses are dominated by internal mechanical losses. Therefore, the representation of the parasitic effects occurring in our devices is limited to the use of a complex compliance in the way expressed by (3), in which Qmp is the mechanical quality factor of the piezoelectric material and j is the complex unity. As the considered structures are composite beams made of different material layers, each of them may exhibit a different material quality factor. The mechanical quality factor Qm of the full structure depends on all the different material quality factors involved in the beam. A short observation should be done at this point: the linear constitutive equations of piezoelectricity, which are the basis of the proposed model, imply reversible and adiabatic or isothermal thermodynamic processes. On the other hand, introducing parasitic damping in the model contradicts these assumptions. The developed equations can then only produce reasonable theoretical predictions in case of small damping amplitude.

(3)

Parasitic dissipations can be included in the partial differential equations describing the behavior of the PVEH by making use of the complex compliance described above. Analytical solutions, which are one of the goals of the developed model, are not known for the complete form of this equation. To avoid this problem, the terms related to the rotary inertia of the bending structure are neglected. This assumption is justified for low frequency bending resonators [12]. Analytical solutions of the problem can now be found, but they are still too tedious for being the base of a lumped model.

To further simplify the problem, the mass of the cantilever is considered negligible compared to the one of the attached mass, so that the distributed inertial forces F(x) resulting from the motion of the clamped end can be reduced to a concentrated force F acting on the center of gravity of the proof mass. Also, while solutions can be found for the deflection (x) for any x along the length of the beam and proof mass, it is convenient to limit the analysis of the deflection to the point where F is applied, i.e. at middle length of the proof mass. The deflection at this location is labeled in the following as A last simplification is implemented: the duration required for an acoustic wave to travel from the clamped end to the free end of the beam is considered to be much smaller than the period of the input vibration, i.e. mechanical quasi staticity is assumed.

From all the reasoning steps described above, the lumped equations describing the electromechanical dynamics of the PVEH are finally obtained as given in (4) and (5), in which a dot superscript indicates a time derivative of the corresponding variable. In these equations, k represents the lumped stiffness of the structure at the centre of gravity of the proof mass, is a macroscopic equivalent of the e31 piezoelectric constant and Cp is the clamped (i.e. motion restrained) capacitance of the device. These parameters can be expressed in terms of dimensions and material properties as given in [10]. Qm is the mechanical quality factor of the structure and m corresponds to the proof mass attached to the tip of the beam. Note that it is possible to circumvent the previously described assumption of negligible mass of the beam by making use of the notion of effective mass [12]. Finally, A0 and represent the amplitude and angular frequency of the input acceleration (assumed sinusoidal) on the clamped end.

(4)

(5)

Analogies between the differential equations governing electrical and mechanical phenomena allow representing (4) and (5) in the form of the electrical network of Figure 2. Network representations constitute an excellent tool for reasoning purposes. Also, they can be easily integrated into electrical simulation software which allows coupling the harvester to non linear power management electronics. The mass, stiffness and parasitic damping are represented in the electrical domain by respectively an inductance m, a capacitance 1/k and a resistor k/(Qm. These elements are associated in series. The deflection is represented by the charges flowing in the mechanical side. The electrical part of the system consists in the clamped capacitance Cp of the piezoelectric layer(s), in parallel with the impedance ZL of the load circuit. The coupling between the electrical and mechanical side is realized through an ideal transformer of ratio 1/. The effect of the input vibration is represented by the voltage mA0sin(t) on the mechanical ports of the network.

Figure 2. Electrical equivalent circuit of the lumped equations representing the behavior of PVEH.

Based on the electrical circuit of Figure 2, the power dissipated in a load resistor RL (i.e. ZL=RL) is analyzed in the next section.

2.2-Analysis of the harvested power In order to determine an expression of the power dissipated into the load resistor, the voltage or current across it should first be determined. Applying Kirchhoffs law to the two branches of the circuit in Figure 2 in the frequency domain and for ZL=RL leads to (6) and (7), in which the overscore indicates the complex transform of the corresponding variable.

(6)

(7)

From these two equations, the expression of the voltage can be derived.

(8)

In order to simplify further analysis, the non dimensional parameters given in Table 1 are introduced in the model. 0s represents the fundamental mechanical angular resonance frequency (short circuit) of the PVEH. corresponds to the ratio of the frequency of the input vibration over 0s. is an electrical load parameter and it corresponds to the ratio of 0s to the cut-off angular frequency 1/RLCp of the electrical circuit. Finally, K is the generalized (or effective) electromechanical coupling factor (GEMC) of the piezoelectric bender [16]. The GEMC is the equivalent for a piezoelectric bending structure of the material property k312/(1- k312). The GEMC correspond to the efficiency of the energy conversion between the mechanical and electrical domain during a quasi static thermodynamic cycle defined in [11]. It is shown in [17] that the GEMC depends only on k312 and on the relative thicknesses and compliances of the piezoelectric and elastic layer(s) in a PVEH. Commercial ceramic PZT benders have typically K values of about 0.3 while the GEMC is generally smaller for thin film based PVEH, around 0.05 for the MEMS devices studied in this article.

Table 1. Definitions of the dimensionless parameters used in the representation of the PVEH.Short circuit mechanical angular frequency

Normalized mechanical frequency

Ratio of the mechanical and electrical angular frequency

Generalized electromechanical coupling factor (GEMC)

Using the definitions of Table 1, the amplitude of the voltage can be written as:

(9)

The average power P dissipated in the load resistor is equal to the square of the voltage amplitude over twice the load resistor. Its expression is given in (10). For simplifying further discussions, the normalized power Pn is defined as P normalized to mA02/(20R).

Note that the expression of the parasitic damping term in (12) slightly differs from the one that was proposed in a paper of the same author [18]. This is explained by the fact that the parasitic damping is represented by a complex stiffness in the present case, while it consists of a viscous damper in [18] (furthermore, a typographical error is present in the expression of the harvested power given in [18]: the exponent of in the numerator should be 6 instead of 2).

(10)

(11)

with

(12)

A first observation can be done on (11): two characteristic frequencies expressed by the terms 2-1 and 2-1-K2 are clearly identifiable. =1=0s and =(1+K2)1/2=0o correspond respectively to the short and open circuit mechanical oscillation frequencies of the PVEH. When a frequency sweep of the input vibration is done, 0s and 0o correspond to the frequencies at which the maximum short circuit current and open circuit voltage are obtained respectively. The shift between the short and open circuit characteristic frequencies is pronounced if the generalized electromechanical coupling factor is large. When parasitic dissipations are not included in the model (i.e. =0), a non physical result is obtained from (10) when the bender is short circuited and excited at 0s or when the bender is open circuited and excited at 0o, i.e. P. This non-physical result from the fact that the deflection of the bender is also mathematically infinite in these cases, so that in a practical situation, even if the parasitic dissipations are extremely small, the output power is limited by physical constraints such as for example the yield limit of the structure or the limited space available in a package.

In practice, parasitics are always present and they should be considered when performing the optimization of the generated power. The expression of Pn given in (11) depends on one hand on intrinsic parameters of the PVEH (K2 and Qm) and on the other hand on extrinsic parameters related to the characteristics of the input vibration and load circuit (and). As illustrated by Figure 3, numerical analysis reveals two distinct behaviors of Pn in terms of the extrinsic parameters: when the product K2Qm is above a certain value, two local maxima and a saddle point of the power are found. The frequencies at which the maximums are found are denoted as resonance 0R and anti resonance 0A frequencies. The corresponding load parameters are labeled asoptR and optA. The parameters 0 andoptare attributed to the saddle point. In the case of low K2Qm, a single local maximum exists. Note that no numerical scales are present in Figure 3 as the graphics are only proposed for a qualitative description of the behavior. a) b)

Figure 3. The two distinct behaviors of the harvested power. a) K2Qm is large and two peaks of power and a saddle points exist. b) K2Qm is small and a single maxima of power is found. On both graphics, Pn and are represented along arbitrary linear scales while is plotted along an arbitrary logarithmic scale.

In order to establish the circumstances leading to one or the other of the behaviors, a first approach consists in determining the conditions leading to the existence of several real solutions to the couple of equations dPn/d=0 and dPn/d=0. This approach leads however to excessively complex computations and does not give insights into the physical origin of this phenomenon. A more fruitful approach involves transforming the circuit of Figure 2 into its Thevenin equivalent given in Figure 4. The Thevenin equivalent impedance Zth corresponds in fact to the impedance that would be measured from the electrical ports of the circuit of Figure 2 when no mechanical input is applied, i.e. when the mechanical ports of Figure 2 are short circuited.

Figure 4. Thevenin representation of the PVEH.

(13)

From circuit theories, it is known that, for the circuit of Figure 4, maximum power is transferred to the load when the impedance of the later matches the complex conjugate of Zth. In mathematical terms, this translates to Re(Zp)=RL and Im(Zp)=0. Making use of (13) and remembering that RL=/Cp0s, these equations can be written as:

(14)

Analysis of (14) with the symbolical computations software Mathematica reveals that the couple of equations admits four solutions. From those four solutions, two are real and positive only if the condition K2Qm>2 is respected. From numerical analysis of (11), it is also found that the resonance and anti resonance power maxima are only observed if this condition is observed. It can then be concluded that the occurrence of the resonance and anti resonance power peaks is related to the discussed impedance matching considerations. Two peaks of power, corresponding to the maximum power transfer points of the system, are observed only if the characteristics of the bender are such that it is possible to match its impedance with the one of the load resistor, i.e. when K2Qm>2. Impedance matching requires furthermore particular couples of values for and , which were previously labeled as (0R,optR) and (0A,optA). The exact expressions of these parameters and of the corresponding power from the solutions of (14) are tedious, but reasonable approximations, leading to an error of a few percent for K2Qm>3, are given in Table 2.

We have now explained the physical meaning of the two power peaks when the condition K2Qm>2 is respected. As described in Figure 3, a saddle point corresponding to (0opt) is also observed when full impedance matching is possible. To interpret the characteristics corresponding to this point, it is necessary to introduce the notion of efficiency of the energy conversion into the problem. is defined as the ratio of the power dissipated into the load over the sum of the later with the power dissipated by the parasitic dissipations (resistor k/Qm of Figure 2). By analysis of the equivalent circuit of Figure 2, it can be expressed as (15). It can easily be shown that (15) is maximized when =1/. Assuming that this condition is respected, numerical analysis shows that the saddle point corresponds approximately to the maximum power that can be achieved while the condition of maximum efficiency is simultaneously respected. Approximated expressions corresponding to the load, the frequency and the power corresponding to this particular condition are given in Table 2.

(15)

Table 2. Simplified expressions of the resonance, anti resonance and maximum efficiency parameters.

ResonanceMaximum efficiencyAntiresonance

Optimum normalized frequency

Optimum load parameter

Optimum power (W)

Efficiency

It can be seen from these expressions that the resonance power peak is found in the neighborhood of the short circuit frequency for a small value of the load resistor. At the opposite, the optimum load is large at the anti resonance peak, found close from the open circuit frequency. As for the short and open circuit characteristic frequencies, the shift between resonance and anti resonance is pronounced if the GEMC is large. The saddle point is observed in an intermediate region, both in terms of load parameter and frequency of excitation. The efficiency at resonance and anti resonance is equal to 50 % while it can reach values close from 100 % at 0if K2Qm is largeNote that, as long as the dynamics of the vibration source can be assumed unchanged under the action of the PVEH, the efficiency of the energy conversion is not a critical parameter for the type of harvesters discussed in this article. It is more important for devices based on shock or impact energy harvesting [19]. In terms of generated power, the performances are similar for the three characteristic points assuming realistic PVEH. It is then possible to obtain a similar quantity of power for a large range of load impedance by adjusting the frequency of the input vibration between 0R and 0A. The current or voltage at which the power is generated can then be tuned to a desired value.

Devices in which the full impedance matching condition can be achieved are the focus of this paper. However, it is necessary to shortly discuss the other case for which a single maximum of power is found (Figure 3b). This situation result from low values of either the generalized electromechanical coupling factor or the mechanical quality factor. Simplified formulas describing each case can be derived but are not presented here. In respectively the former and latter situation, the maximum of the power is found close from the short and open circuit resonance frequency. The power generated is in both situations smaller than the values predicted by the formulas of Table 2.

In the next section, the theoretical predictions are compared to experimental measurements performed on thin film AlN and ceramic PZT based PVEH.

3. Experimental characterization The two types of devices that are investigated are illustrated in Figure 5. The first type corresponds to MEMS fabricated PVEH whose characteristics and manufacturing process were already presented in the literature [20]. As illustrated by Figure 5a, they consist of a silicon cantilever supporting a piezoelectric capacitor and attached to a large proof mass. AlN is used as piezoelectric material. The total length (beam and proof mass) of the structure is 4.4 mm and its width 3 mm. The thickness of the silicon beam and piezoelectric capacitor are 50 m and 1.2 m, respectively. The device is encapsulated in vacuum conditions (< 0.1 mbar) by a top and bottom glass layer. The proof mass is 14 mg. The second studied device consists in a bimorph made of a brass shim and of two metalized ceramic PZT layers electrically connected in a parallel arrangement. The device is a BM 120/36/350 ordered from Piezomechanik [21]. As indicated in Figure 5b, its free length is 36 mm and its width 10 mm. The thicknesses of the shim and of a single PZT layer are about 0.15 and 0.3 mm. One end of the structure is glued to a PCB board and a steel mass (6.4 g) is attached to the other.

a) b) Figure 5. Schematic and picture of the tested samples. MEMS AlN (a) and ceramic PZT (b) PVEH.

In the following, the measurements of the relevant parameters for the two devices of Figure 5 are presented.

3.1-Measurements of the network parametersWhile a large amount of methods can be used to measure the network parameters of PVEH [10], the experiments presented here are limited to the simplest possible. The sole experiments required for determining K2 and Qm consists in measuring the frequency dependence of the short circuit current and open circuit voltage of the PVEH when they are excited by a mechanical vibration. The input vibration is produced by an electrodynamic shaker. Note that, for clarity, all the experiments are presented in terms of dimensional parameters (frequency f, load resistor RL and power P) rather than in terms of the non dimensional ones (normalized frequency , load parameter and normalized power Pn) used in the theoretical part of the article. The results of the two discussed measurements are presented in Figure 6. As for all the experimental results presented in this paper, the amplitude of the input acceleration is 0.32 G for the MEMS AlN device and 0.01 G for the ceramic PZT harvester.

a) b) Figure 6. Short circuit current (solid line) and open circuit voltage (dashed line) for the MEMS AlN device (a) and the ceramic PZT harvester (b).

The maximum of the short circuit current (open circuit voltage) is obtained at the short circuit frequency f0s (open circuit frequency f0o). Note that PVEH have generally high output electrical impedance so that measurement setups with high input impedance should be used for this measurement. As implied by formulas which have already been described, the GEMC can be related to f0s and f0o by (16). The quality factor Qm of the device can be measured by applying the half bandwidth method on the frequency response of the short circuit current. It was mentioned in the theoretical part of the article that, for the two studied devices, parasitic damping is dominated by internal mechanical dissipations. In the case of the MEMS AlN harvester, this element is justified in [20], where the contributions to the parasitic damping in the discussed devices are investigated in terms of the pressure in the cavity containing the harvester. It is shown that below 0.1 mbar, the parasitic damping is indeed dominated by internal mechanisms. For justifying this assumption in the case of the ceramic PZT device, we measured the short circuit current in both atmospheric and vacuum (0.01 mbar) conditions. For the same input acceleration, there was no dependence of Qm on pressure. We then conclude that the parasitic damping mechanisms in this PVEH are also dominated by internal contributions. Note however that, solely in the case of the PZT device, we observed a dependence of the quality factor on the input acceleration. The parasitic damping amplitude appears then in this case to depend on the amplitude of the cantilever deflection. An exercised eye can observe that the frequency responses of the ceramic PZT bender in Figure 6b present some degree of non linearity (i.e. asymmetric curves), which may be attributed to this effect.

(16)

For being able to determine all the dimensional parameters, it is necessary to measure the clamped capacitance Cp of the devices in addition to the previous parameters. Cp was extracted from measurements of the electrical impedance Zth of the PVEH using formulas given in [10]. A summary of the measured properties is given in Table 3.

Table 3. Summary of the measured properties.

Thin film AlN PVEHCeramic PZT PVEH

f0s (Hz)1082.458.8

f0o (Hz)1083.961.2

Cp (nF)0.3630

K20.00280.0840

Qm120042

K2Qm3.43.5

Assuming the measured values for the different parameters, the three dimensional graphic relating the predicted power to the frequency of the input vibration and to the load resistor can now be plotted, as given in Figure 7. Both benders have a value of K2Qm above 2, so that they exhibit the resonance and anti resonance behavior, including two power peaks and a saddle point. a) b)

Figure 7. Theoretical prediction of the power vs. the load resistor and the input frequency for (a) the MEMS AlN and (b) the ceramic PZT PVEH.

For confirming the theoretical predictions expressed before, the measurements presented in the next sections are not limited to the three characteristic points (resonance, maximum efficiency and anti resonance) but two complete possible paths of optimization are followed. In the first path, the optimum frequency for each value of the load resistor is determined. It corresponds to the black solid lines in Figure 7. In the second path, the load resistor leading to the optimum power for each value of the input frequency is found (white line of Figure 7).

3.2-Power at optimum frequency for each loadIn order to obtain this relation, the first step consists in measuring the output power delivered to various load resistors while the frequency of the input vibration is swept, as illustrated by Figure 8a and Figure 9a. In these figures, the values of the load resistor were chosen to correspond to RoptR, Ropth and RoptA. Almost perfect fit for AlN device Fit no so good for PZT, but still reasonable prediction of the resonance and anti resonance behaviour. Some discussion on why the fit is not so good. In both situations, it is possible to tune the voltage or current at which an almost constant level of power is generated.

a) b)

Figure 8. Experimental results on the AlN based device. a) Harvested power vs. the load resistor for three characteristic frequencies, b) Harvested power at optimum frequency and optimum frequency vs. the load resistor (the red elements correspond to the optimum frequency and the black to the power). In both graphics, the markers correspond to experimental measurements, while the solid lines describe theoretical predictions.

a) b)

Figure 9. Experimental results on the PZT based device. a) Harvested power vs. the load resistor for three characteristic frequencies, b) Harvested power at optimum frequency and optimum frequency vs. the load resistor (the red elements correspond to the optimum frequency and the black to the power). In both graphics, the markers correspond to experimental measurements, while the solid lines describe theoretical predictions.

3.3-Power at optimum load for each frequencyThe results of the measurements are presented in Figure 10 and Figure 11.Almost perfect fit for AlN device Fit no so good for PZT, but still reasonable prediction of the resonance and anti resonance behaviour. Some discussion on why the fit is not so good. In both situations, it is possible to tune the frequency at which an almost constant level of power is generated.

a) b)

Figure 10. Experimental results on the AlN based device. a) Harvested power vs. the input frequency for three characteristic load resistor, b) Harvested power at optimum load and optimum load vs. the frequency (the red elements correspond to the optimum load and the black to the power). In both graphics, the markers correspond to experimental measurements, while the solid lines describe theoretical predictions.

a) b)

Figure 11. Experimental results on the PZT based device. a) Harvested power vs. the input frequency for three characteristic load resistor, b) Harvested power at optimum load and optimum load vs. the frequency (the red elements correspond to the optimum load and the black to the power). In both graphics, the markers correspond to experimental measurements, while the solid lines describe theoretical predictions.

4-ConclusionsThe resonance and anti resonance characteristics of piezoelectric vibration energy harvesters have been established in this paper. Conditions for observing the two power peaks were established from a theoretical point of view and simple design considerations leading to this behavior were derived. It was shown that this behavior is observed when it is possible to match the impedance of the load circuit with the one of the piezoelectric bender. The theoretical predictions were confirmed by experiments on ceramic PZT and thin film AlN based piezoelectric harvesters. It was shown that, by making use of the demonstrated effect, it is possible to generate an almost constant level of power over a large range of load resistor. This is an important consideration for the design of rectifying electronics.

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