An investigation on stress distribution effect on multi- piezoelectric energy harvesters

Abstract

With the fast development of piezoelectric materials and due to its green and renewable characteristics, the piezoelectric energy harvesting technology has been paid more and more attention by pavement engineers. The stress distribution will significantly affect the piezoelectric material performance. In this paper, the effects of multiple piezoelectric elements on the generation of electrical energy and output power are studied. In the case of constant external load, the number of the piezoelectric units does not necessarily produce more energy. When the same multi piezoelectric units work together, if the stress state of the piezoelectric units is different, the total output energy affected by the connection mode. For uneven stress distribution, the optimal output mode is that each of the piezoelectric units rectified before connected in parallel.

Keywords

piezoelectric transducer / uneven stress / impedance matching / optimal energy output

Introduction

As the improved performance of piezoelectric materials and its green and renewable characteristics, the pavement engineers paid more and more attention on the piezoelectric energy harvesting technology. The basic mechanism of piezoelectric material is to collect environmental mechanical energy and convert it into electrical energy base on piezoelectric effect. Transportation infrastructures such as bridge and road vibrate when vehicles pass on them. There have been many researches in the asphalt pavement performance area [1–3]. The corresponding generated mechanical energy is of great value if it can be properly collected and used.

Many researchers from various countries have made significant research in piezoelectric materials, transducer structure and interface circuit. Torres et al. found that piezoelectric energy harvesting technology has the highest power density too collect ambient vibration energy by analyzing three methods that are electromagnetic, electrostatic and piezoelectric methods [4]. Innowattach carried out a pilot project to demonstrate the application of piezoelectric energy harvesting technology in the road energy collection on the Israeli National Road. Xiong et al. developed a new renewable energy harvester and tested the mechanical performances [6]. Duarte et al. designed a piezoelectric transducer that can capture the kinetic energy of the road [7]. Zhao et al. found that circular cross-section pile transducer has the best energy output efficiency compared with the square or hexagonal cross-section by using Finite Element Analysis [8]. Kim et al. studied the open circuit voltage of two kinds of piezoelectric materials under different wheel load using Asphalt Pavement Analyser (APA). The transducer can produce up to 2.67 milliwatt of power when the vehicles speed was 45 mph [9]. Xiong et al. studied the relationship between axle load and output energy [10]. Their result shows that generated electric energy is correlated with single axle load and output electric energy is related to total axle load.

Xue et al. developed a series of array piezoelectric cantilever beam energy harvester [11]. Erturk et al. designed L-shaped cantilever to overcome the limitation of the traditional linear cantilever structure [12]. Ali et al. presented the piezoelectric energy harvester based on the vibration model, where the harvesting vibration energy powers a wireless device on the bridge [13]. Chure et al. studied effect of dimensional size on the power generation characteristics of PZT piezoelectric ceramics using drop weight impact techniques. The results showed that the PZT piezoelectric ceramic body with greater ratio of thickness and cross-sectional area ratio (t/D²) can generate higher electrical energy [14]. Xu et al. and colleagues presented a high performance and simple structure bi-stable piezoelectric energy harvester based on simply supported piezoelectric buckled beam [15]. Ferrari et al. designed a dual steady state energy converter for piezoelectric and magnetic components of micro electromechanical system and the analysis and simulation results showed that a nonlinear bistable converter provides better performance in linear systems with broadband excitation, and increases the bandwidth and output levels [16].

Up to now, most of the studies are focused on promoting the efficiency of a certain piezoelectric conversion. In the actual engineering projects, due to the fragility of piezoelectric ceramic, very large volume of piezoelectric harvester may not be suitable, and multiple piezoelectric units need to work together to get enough energy. Piezoelectric transducers used in pavement usually contain multiple piezoelectric units. Under the vehicle load, the internal stress of the piezoelectric units is different which would affect on the piezoelectric energy harvesting efficiency. In this paper, the effects of multiple piezoelectric elements on the generation of electrical energy and output power are studied. The ultimate goal is to simulate and obtain the stress state and energy harvesting state in real engineering projects.

The theory of piezoelectric energy and power output

Piezoelectric energy harvested is designed based on the positive piezoelectric effect, which is the electric charge generated along the polarization direction of the piezoelectric material under the action of the stress, where the mechanical energy is converted into electric energy. Uchino gives the piezoelectric effect of the piezoelectric definition [17], as follows Eq. (1):

(1){\mathit{D}}_m={\mathit{d}}_{mi}{\mathit{T}}_i+{\mathit{\epsilon }}_{mk}^{\mathbf{T}}{\mathit{E}}_k,

where, i,j=\mathrm{1},\mathrm{2}\dots \dots \mathrm{6}; m, k=1,\mathrm{ 2},\mathrm{ 3}; D is electric displacement tensor; d is the piezoelectric strain constant tensor; T is the stress tensor; {\mathit{\epsilon }}^{\mathbf{T}} is the dielectric constant tensor under constant stress condition; E is tensor for external electric field.

Under direction stress {\mathit{T}}_i,the amount of charge generated by the piezoelectric element is Q:

(2)Q=DS=dTS,

where, S is the polarization area. The piezoelectric element itself is a capacitive element,internal equivalent capacitance is c_p. According to the capacitance energy storage formula, the formula of open circuit voltage and the electric energy formula of the piezoelectric element can be obtained.

(3)U_{oc}={\displaystyle \frac{Q}{c_p}}={\displaystyle \frac{dTS}{c_p}},

(4)E_{oc}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{c}_p{U}_{oc}^{\mathrm{2}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\displaystyle \frac{Q^{\mathrm{2}}}{c_p}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\displaystyle \frac{d^{\mathrm{2}}{T}^{\mathrm{2}}{S}^{\mathrm{2}}}{c_p}}\mathrm{.}

In Eqs. (3) (4), the subscript “oc” meaning “open circuit”.

Platt et al. studies the relationship between open circuit voltage and output power and load in the case of a single piezoelectric element under load [18]. Researchers also found that the interfacial performance has a significant impact on the composite materials [19,23]. Because the piezoelectric element is a capacitive element, there is an inverse relationship between the internal impedance and the equivalent capacitance and the load frequency in an alternating electric field. The piezoelectric element equivalent resistance R_p can be calculated as follows:

(5)R_p={\displaystyle \frac{\mathrm{1}}{\omega c_p}},

where \omega is the angular frequency, c_p is the equivalent capacitance of the piezoelectric element. Set the load impedance is R_L, and under sinusoidal excitation T=T_{\mathrm{0}}\sin (\omega t), output voltage U_{out}(t) and output power P_{out}(t) of the piezoelectric element are represented by the following formulas:

(6)U_{out}(t)={\displaystyle \frac{U_{oc}{R}_L}{R_p+R_L}}={\displaystyle \frac{d{T}_{\mathrm{0}}S}{c_p}}{\displaystyle \frac{R_L}{R_p+R_L}}\sin (\omega t),

(7)P_{out}(t)={\displaystyle \frac{U_{out}^{\mathrm{2}}}{R_L}}={\left({\displaystyle \frac{d{T}_{\mathrm{0}}S}{c_p}}\right)}^{\mathrm{2}}{\displaystyle \frac{R_L}{{(R_p+R_L)}^{\mathrm{2}}}}{\sin }^{\mathrm{2}}(\omega t)\mathrm{.}

In which T_{\mathrm{0}} is the stress amplitude.

The peak value of output voltage is:

(8)U_{out\_\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}}={\displaystyle \frac{d{T}_{\mathrm{0}}S}{c_p}}{\displaystyle \frac{R_L}{R_p+R_L}}\mathrm{.}

Based the Eq. (8), the output voltage peak value increases with the load and the growth rate gradually decreases. If the load tends to infinity, the output voltage is equal to the open circuit voltage. Eq. (7) in a period of integral can get the output energy, as follows:

(9)E_{out\_\mathrm{T}}={\displaystyle \frac{\pi }{\epsilon }}{\left({\displaystyle \frac{d{T}_{\mathrm{0}}S}{c_p}}\right)}^{\mathrm{2}}{\displaystyle \frac{R_L}{{(R_p+R_L)}^{\mathrm{2}}}}\mathrm{.}

Then the average output power of the piezoelectric unit is obtained:

(10)P_{out\_\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{g}\mathrm{e}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\left({\displaystyle \frac{d{T}_{\mathrm{0}}S}{c_p}}\right)}^{\mathrm{2}}{\displaystyle \frac{R_L}{{(R_p+R_L)}^{\mathrm{2}}}}\mathrm{.}

The parameters of R_L as independent variables of type (10) derivative, to get the maximum output power corresponding to the load R_L=R_p. The maximum average output power is:

(11)P_{average\_out\_\max }={\displaystyle \frac{\mathrm{1}}{\mathrm{8}}}{\left({\displaystyle \frac{d{T}_{\mathrm{0}}S}{c_p}}\right)}^{\mathrm{2}}{\displaystyle \frac{\mathrm{1}}{R_p}}\mathrm{.}

The effect of stress distribution of the piezoelectric element on the generation of electrical energy

As shown in Fig. 1, it is assumed that there are n piezoelectric units in one piezoelectric transducer, where the electrode area of each piezoelectric element is S_{\mathrm{0}}, the equivalent capacitance of each piezoelectric element is c_p. Loaded on a piezoelectric transducer on the external force is F, each piezoelectric element stress is F_i, then F={\displaystyle {\sum }_{i=\mathrm{1}}^n{F}_i}. The stress applied to each the piezoelectric element is T_{i\mathrm{33}}={\displaystyle \frac{F_i}{S_{\mathrm{0}}}}. Base on Eq. (2), each piezoelectric element produced charge Q_i=F_i\cdot d_{\mathrm{33}}. If all piezoelectric units are connected in parallel, the total charge and the total capacitance value is

(12)Q={\displaystyle {\sum }_{i=\mathrm{1}}^n{Q}_i}={\displaystyle {\sum }_{i=\mathrm{1}}^n(F_i\cdot d_{\mathrm{33}})}=F\cdot d_{\mathrm{33}},

(13)c=\mathit{n}\cdot c_p\mathrm{.}

Based on Eq. (4), the total energy is calculated:

(14)E_{Paral}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\displaystyle \frac{Q^{\mathrm{2}}}{c}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\displaystyle \frac{d_{\mathrm{33}}^{\mathrm{2}}{F}^{\mathrm{2}}}{n{c}_p}\mathrm{.}}

Based on Eq. (12), if all piezoelectric units connected in parallel, the energy produced is not affected by the stress distribution when the external force is constant. If all the piezoelectric units are isolated from each other, the electric energy generated by each of the piezoelectric elements is

(15)E_i={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\cdot {\displaystyle \frac{Q_i^{\mathrm{2}}}{c_p}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\cdot {\displaystyle \frac{d_{\mathrm{33}}^{\mathrm{2}}\cdot F_i^{\mathrm{2}}}{c_p}}\mathrm{.}

The total electrical energy produced by all the piezoelectric elements is:

(16)E_{Isola}={\displaystyle {\sum }_{i=\mathrm{1}}^n{E}_i={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\displaystyle \frac{d_{\mathrm{33}}^{\mathrm{2}}}{c_p}}{\displaystyle {\sum }_{i=\mathrm{1}}^n{F}_i^{\mathrm{2}}}}\mathrm{.}

By Mean inequality theorem, arithmetic mean is no more than quadratic mean we can get that:

(17){\displaystyle \frac{\mathrm{1}}{n}}\cdot {({\sum }_{i=\mathrm{1}}^n{F}_i)}^{\mathrm{2}}\le {\displaystyle {\sum }_{i=\mathrm{1}}^n{F}_i^{\mathrm{2}}}

That is

(18)E_{Paral\hspace{0.17em}}\le E_{Isola}\mathrm{.}

If and only if the stress on each piezoelectric element is the same, the equality holds. In addition, if each of the piezoelectric elements is isolated from each other, the more uniform the stress distribution is, and the more the total electric energy generated.

Effect of nonuniform stress on power output

Assume there are two piezoelectric elements A and B with same material and same size, A and B have equivalent capacitance c_{\mathrm{A}}=c_{\mathrm{B}}=c_p, and the same piezoelectric coefficient d_{\mathrm{A}}=d_{\mathrm{B}}=d. They have the same placement direction but different sizes of stress, and T_{\mathrm{A}}>T_{\mathrm{B}}. Based on Eqs. (2) and (4), the amount of charge on the electric pole and electric energy production of A and B,can be calculated as follows:

(19)Q_{\mathrm{A}}=T_{\mathrm{A}}dS,

(20)Q_{\mathrm{B}}=T_{\mathrm{B}}dS,

(21)E_{\mathrm{A}}=\frac{1}{2}\cdot \frac{T_{\mathrm{A}}^2{d}^2{S}^2}{c_p},

(22)E_{\mathrm{B}}=\frac{1}{2}\cdot \frac{T_{\mathrm{B}}^2{d}^2{S}^2}{c_p}\mathrm{.}

Because T_{\mathrm{A}}>T_{\mathrm{B}}, Q_{\mathrm{A}}>Q_{\mathrm{B}}, E_{\mathrm{A}}>E_{\mathrm{B}}. The A and B are electrically connected in parallel, the total charge and electric energy are:

(23)Q_{\mathrm{A||B}}=(T_{\mathrm{A}}+T_{\mathrm{B}})dS,

(24)E_{\mathrm{A||B}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\cdot {\displaystyle \frac{{(T_{\mathrm{A}}+T_{\mathrm{B}})}^{\mathrm{2}}{d}^{\mathrm{2}}{S}^{\mathrm{2}}}{\mathrm{2}{c}_p}}\mathrm{.}

It is obvious that Q_{\mathrm{A+B}}>Q_{\mathrm{A}}>Q_{\mathrm{B}}, but the relationship between E_{\mathrm{A+B}} and E_{\mathrm{A}} is not determined, and is related to the stress state of the A and B.

Set\hspace{0.17em}\alpha =\frac{T_{\mathrm{A}}}{T_{\mathrm{B}}}(T_{\mathrm{A}}\ge T_{\mathrm{B}}),\beta =\frac{E_{\mathrm{A}}}{E_{\mathrm{A}\left|\right|\mathrm{B}}}=\frac{2T_{\mathrm{A}}^2}{{(T_{\mathrm{A}}+T_{\mathrm{B}})}^2}=\frac{2{\alpha }^2}{{(\alpha +1)}^2},

If , then , , which means the electric energy generated by A and B parallel is more. If \alpha =\sqrt{\mathrm{2}}+\mathrm{1}, then \beta =\mathrm{1}, E_{\mathrm{A}}=E_{\mathrm{A||B}}, which means The electrical energy produced by A and B in parallel can be as much as that produced by A alone. If \alpha >\sqrt{\mathrm{2}}+\mathrm{1}, then \beta >\mathrm{1}, E_{\mathrm{A}}>E_{\mathrm{A||B}}, which means the electric energy generated by A and B parallel is less than electric energy generated by A alone.

Effect of stress distribution on power output

Assume that there are two piezoelectric elements A and B, they have the same the material parameters and geometric dimensions. They were subjected to sinusoidal load, the amplitude of the load are T_{\mathrm{A}} and T_{\mathrm{B}}, at the same frequency and phase and different amplitude (T_{\mathrm{A}}>T_{\mathrm{B}}). Based on Eq. (5), internal equivalent capacitance and resistance of A and B are same, that is c_{p\_\mathrm{A}}=c_{p\_\mathrm{B}}=c_p, R_{p\_\mathrm{A}}=R_{p\_\mathrm{B}}={\displaystyle \frac{\mathrm{1}}{\omega c_p}}. Load is R. Based on Eq. (8) and (10), we can get the output peak voltage and the average power of the piezoelectric elements A and B respectively.

(25)U_{out\_\mathrm{A}}={\displaystyle \frac{d{T}_{\mathrm{A}}S}{c_p}}{\displaystyle \frac{R_L}{R_p+R_L}},

(26)U_{out\_\mathrm{B}}={\displaystyle \frac{d{T}_{\mathrm{B}}S}{c_p}}{\displaystyle \frac{R_L}{R_p+R_L}},

(27)P_{out\_\mathrm{A}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\left({\displaystyle \frac{d{T}_{\mathrm{A}}S}{c_p}}\right)}^{\mathrm{2}}{\displaystyle \frac{R_L}{{(R_p+R_L)}^{\mathrm{2}}}},

(28)P_{out\_\mathrm{B}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\left({\displaystyle \frac{d{T}_{\mathrm{A}}S}{c_p}}\right)}^{\mathrm{2}}{\displaystyle \frac{R_L}{{(R_p+R_L)}^{\mathrm{2}}}}\mathrm{.}

BecauseT_{\mathrm{A}}>T_{\mathrm{B}}, then U_{out\_\mathrm{A}}>U_{out\_\mathrm{B}}, P_{out\_\mathrm{A}}>P_{out\_\mathrm{B}}.

As shown in Fig. 2 there are 4 power supply methods. P-1 and P-2 in Fig. 2 represent the piezoelectric element A and B respectively. The first power supply mode is A and B respectively after rectification and then parallel supply power. The second is the A and B connected in parallel before rectifier power supply. The third power supply mode is A rectifier power supply alone. The fourth power supply mode is B rectifier power supply alone.

Without consideration of the loss of the rectifier bridge, according to the Eqs. (2), (3), (6), (7), (10), we can obtain the output voltage peak value and the average output power of each mode, when the load is R, as follows:

(29)

(30)U_{\mathrm{I}\mathrm{I}}=\frac{dS(T_{\mathrm{A}}+T_{\mathrm{B}})}{c_p}\cdot \frac{R}{R_p+2R},

(31)U_{\mathrm{I}\mathrm{I}\mathrm{I}}=\frac{d{T}_{\mathrm{A}}S}{c_p}\cdot \frac{R}{R_p+R},

(32)U_{\mathrm{I}\mathrm{V}}=\frac{d{T}_{\mathrm{B}}S}{c_p}\cdot \frac{R}{R_p+R},

(33)

(34)P_{\mathrm{I}\mathrm{I}}=\frac{1}{2}\frac{d^2{S}^2{(T_{\mathrm{A}}+T_{\mathrm{B}})}^2}{c_p^2}\cdot \frac{R}{{(R_p+2R)}^2},

(35)P_{\mathrm{I}\mathrm{I}\mathrm{I}}=\frac{1}{2}{\left(\frac{d{T}_{\mathrm{A}}S}{c_p}\right)}^2\cdot \frac{R}{{(R_p+R)}^2},

(36)P_{\mathrm{I}\mathrm{V}}=\frac{1}{2}{\left(\frac{d{T}_{\mathrm{B}}S}{c_p}\right)}^2\cdot \frac{R}{{(R_p+R)}^2}\mathrm{.}

By comparison, the following conclusions can be drawn:

If R=\frac{T_{\mathrm{B}}}{T_{\mathrm{A}}-T_{\mathrm{B}}}\cdot R_p, then P_{\mathrm{I}}=P_{\mathrm{I}\mathrm{I}}=P_{\mathrm{I}\mathrm{I}\mathrm{I}}>P_{\mathrm{V}\mathrm{I}}. When the load is equal to a certain value, the output power of the parallel output power is equal to that of A. If ,then P_{\mathrm{I}}=P_{\mathrm{I}\mathrm{I}}>P_{\mathrm{I}\mathrm{I}\mathrm{I}}>P_{\mathrm{V}\mathrm{I}} When the load is lower than a certain value, the output power of the parallel output power is greater than that of any single voltage power supply. If R>\frac{T_{\mathrm{B}}}{T_{\mathrm{A}}-T_{\mathrm{B}}}\cdot R_p,then . When the load is higher than this value, the output power of the parallel output power is smaller than that of the single output power.

This particular value is closely related to the stress distribution. The greater the stress difference, the smaller the certain value, the more unfavorable for the parallel output. The best solution is that each of the piezoelectric units alone is rectified, and then output in parallel.

According to the mothed II, the optimal load is R=\mathrm{0.5}{R}_p, and the maximum output power is

(37)P_{\mathrm{I}\mathrm{I}\_\mathrm{M}\mathrm{A}\mathrm{X}}=\frac{1}{16}\frac{d^2{S}^2{(T_{\mathrm{A}}+T_{\mathrm{B}})}^2}{c_p^2{R}_p}\mathrm{.}

According to the mothed III, the optimal load is R=R_p, and the maximum output power is

(38)P_{\mathrm{I}\mathrm{I}\mathrm{I}\_\mathrm{M}\mathrm{A}\mathrm{X}}=\frac{1}{8}\frac{d^2{S}^2{T}_{\mathrm{A}}^2}{c_p^2{R}_p}\mathrm{.}

Set \gamma =\frac{P_{\mathrm{I}\mathrm{I}\mathrm{I}\_\mathrm{M}\mathrm{A}\mathrm{X}}}{P_{\mathrm{I}\mathrm{I}\_\mathrm{M}\mathrm{A}\mathrm{X}}}=\frac{2T_{\mathrm{A}}^2}{{(T_{\mathrm{A}}+T_{\mathrm{B}})}^2}, as known \alpha =\frac{T_{\mathrm{A}}}{T_{\mathrm{B}}}(T_{\mathrm{A}}\ge T_{\mathrm{B}}), then \gamma ={\displaystyle \frac{\mathrm{2}{\alpha }^{\mathrm{2}}}{{(\alpha +\mathrm{1})}^{\mathrm{2}}}}.

If , then , , the piezoelectric units are connected in parallel before rectification can output more energy than the single unit output. If \alpha =\sqrt{\mathrm{2}}+\mathrm{1}, then \gamma =\mathrm{1}, E_{\mathrm{A}}=E_{\mathrm{A}\left|\right|\mathrm{B}}, the piezoelectric units are connected in parallel before rectification can output the same energy with the single unit A output. If \alpha >\sqrt{\mathrm{2}}+\mathrm{1}, then \gamma >\mathrm{1},E_{\mathrm{A}}>E_{\mathrm{A}\left|\right|\mathrm{B}}, the piezoelectric units are connected in parallel before rectification output less energy than the single unit A output. Therefore, the stress no uniformity between the piezoelectric elements will not only affect the energy generation but also have the same effect on the power output.

Experimental verification

We have developed a piezoelectric cantilever beam with four piezoelectric elements. The piezoelectric material is PZT-5H, and the piezoelectric coefficient d₃₁ is 186 pc/N, the length, width and thickness of the piezoelectric units is respectively 6.5 cm, 4 cm, and 0.3 cm, where its structure as shown in Fig. 3. In the process of vibration, A and B were subjected to the stress with the same direction and phase and different amplitude. The capacitance value of A or B is same 275 nF.

As shown in Fig. 4, under sinusoidal vibration excitation, the open circuit voltage curves of A and B are close to the sine curve, respectively. The open circuit peak-peak voltages are 36.2 V and 13.28 V respectively. The open circuit voltage peak-peak of A & B in parallel is 25.4 V. The open circuit peak voltage is half of the peak-peak voltage. The average stress of A and B can be calculated by bringing the open circuit peak voltage of A and B into the Eq. (3).

T_{\mathrm{A}}=10.03\mathrm{M}\mathrm{P}\mathrm{a},T_{\mathrm{B}}=3.78\mathrm{M}\mathrm{P}\mathrm{a};\alpha =\frac{T_{\mathrm{A}}}{T_{\mathrm{B}}}=2.73>\sqrt{2}+1

Based on Eq. (4) we can also calculate the energy generated by A and B respectively. The value is 0.045 mJ and 0.006 mJ. The electric energy value of A & B in parallel is 0.044 mJ. Conform to the previous conclusion that if \alpha >\sqrt{\mathrm{2}}+\mathrm{1}, then .

Therefore, when the piezoelectric units are under non-uniform stress, If \alpha >\sqrt{\mathrm{2}}+\mathrm{1}, piezoelectric the generated energy in connection mode II is less than that in connection mode I.

According to the power supply mode of Fig. 2, change the load resistance from 10 kW to 100 kW each increment 10 kW. The average output power of the four power supply modes was measured.

As shown in Fig. 5, the curves of mode I/II/III intersect at one point. When the load is lower than that of the point, the output power of the mode I and mode II is higher than mode III, and the power of the mode I and mode II are similar. When the load is higher than the value, the output power of the mode I and mode III is higher than mode II, and the power of the mode I and mode III are similar. From Fig. 5 when \alpha >\sqrt{\mathrm{2}}+\mathrm{1}, the output power of mode II is less than that of mode III and verify the above conclusion.

Conclusions

In this paper, the stress distribution effects on multi- piezoelectric energy harvesters are studied. The following conclusions can be drawn based on the current research:

(1) when the external load is consistent, if the piezoelectric transducer is connected in parallel, the stress distribution has no effect on the piezoelectric capacity. If each piezoelectric unit is isolated from each other, the sum of the electrical energy generated by the piezoelectric elements is larger than that of the parallel connection mode, and the worse the stress uniformity is, the more energy generated.

(2) When Several piezoelectric units are connected in parallel, the electric energy generated by all the piezoelectric units maybe not much more than the electric energy generated by the single work of the piezoelectric units which are under the stress of the maximum stress. With two piezoelectric units as an example, stress ratio for \sqrt{\mathrm{2}}+\mathrm{1} as the dividing line, When the ratio is less than the value, the parallel mode produces more power, and when the ratio is larger than the value, the parallel connection can generate less energy than single unit which is under the larger stress. That is, the worse the uniformity of stress, the less suitable for parallel output mode. The same conclusion can apply to the piezoelectric energy output.

(3) When multi-units working at the same time, the power supply of each piezoelectric unit is rectified and output in parallel, which is favorable for the output of the power supply.

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