OPTIMIZATION OF THE ELECTRICAL AND MECHANICAL PARAMETERS OF A VIBRATION ENERGY HARVESTER

Transcription

1 Blucher Mechanical Engineering Proceedings May 04, vol., num. OPTIMIZATION OF THE EECTRICA AND MECHANICA PARAMETERS OF A VIBRATION ENERGY HARVESTER M. A. Clementino, M. J. Brennan, S. da Silva Universidade Estadual do Oeste do Paraná - UNIOESTE, Centro de Engenharias e Ciências Exatas, Camus de Foz do Iguaçu, PR, Brasil (marcelclementino@uol.com.r) UNESP - Univ Estadual Paulista, Camus de Ilha Solteira, Deartamento de Engenharia Mecânica, Ilha Solteira, SP, Brasil Astract. The use of iezoelectric energy harvesting devices to ower sensor networks and electronic equiment has ecome oular research toic. For ractical use of the energy converted y these transducers conversion of the alternating current (AC) roduced to direct current (DC) is generally required. This is normally done y using rectifier circuits. One imortant issue to overcome is to e ale to design a device ale to roduce the highest ower ossile. To otimize the couled mechanical-electrical system a couled model is required, including the eam structure, iezoelectric element and rectifier circuit, using the same comutational software. Thus, for a secific level of mechanical viration, the maximum ower harvested can e otained y numerical otimization of the eam and iezoelectric geometry and arameters of resistive load, caacitive filter and diodes. All simulations are carried out using Matla and the otimization is carried out y a sequential quadratic rogram (SQP) method. The results show which characteristics are imortant to consider and to modify in order to imrove the erformance of the ower harvesting device. Keywords: energy harvesting, ower electronics, unified aroach, ower otimization.. INTRODUCTION In the last twenty years or so, many researchers have een concerned with the develoment of devices that could increase the lifesan of atteries, as well as reducing the attery relacement rate, y using a rocess to transform viration into electrical energy, called energy harvesting. The reduction of energy requirements of some low-owered electronic devices has rought this closer to real alications, esecially using iezoelectric materials in energy harvesters, causing a sustantial increase in the numer of ulished aers in this area, for examle [9,,5,7,8]. These energy harvesting devices are, in essence, a ase structure to which iezoelectric materials (usually PZT ceramics) are attached, for the uroses of generating electrical energy. An electrical circuit then converts the current from alternating current (AC)

2 to direct current (DC). Several models have een used to redict the amount of ower that these energy harvesters can generate and how much of it can e extracted/stored. Models for the mechanical ehavior range from lumed arameter single-degree-of-freedom (SDOF) models [], to distriuted arameters models with Rayleigh-Ritz [6] and finite element aroximations []. Such aers generally aim to redict the harvester characteristics that lead to the maximum ower generation while the energy roduced is dissiated y a resistor, which reresents the electrical load. However, for ractical use of the energy generated y these transducers, the AC roduced is generally converted to DC using rectifier circuits. allart and Guyomar [0], Guan and iao [5] and Ottman et al. [] are a few authors that have modeled advanced harvester circuits (e.g. Synchronized Switch Harvesting on Inductor (SSHI) circuits and rectifier ridges connected to DC/DC converters), and used suercaacitors or rechargeale atteries as storage devices. Recently, researchers have concentrated their efforts on couled models i.e., models in which the interaction etween oth the mechanical structure and electronic circuit is considered. However, many of these models usually simlify either the mechanical domain, considering a SDOF sring-mass-damer models and equivalent circuit techniques [,,0], or the electrical domain, using a single resistor to reresent the external load. Such models may not corresond to reality. To achieve maximum ower generation, several authors have sought the otimal values of structural arameters [4], as well as those related to the electrical circuits [9]. The mechanical arts and the electrical arts of the energy harvesting device are usually modeled in different software ackages that may not e comatile. Consequently, the otimization rocess, which is a very imortant stage in the design rocess, can e comromised. In this aer, the mechanical and electrical arts of the system are modeled using the same software ackage, which enales the system to e otimized so that the maximum ower can e harvested. All the simulations are carried out using Matla and the otimization rocess is carried out using a sequential quadratic rogram (SQP). The aer is organized as follows. Section resents the iezoelectric harvester device and harvesting circuit models. Section descries in detail the otimization rolem. The results are resented in Section 4, which is followed y some conclusions in Section 5.. MODE OF THE PIEZOEECTRIC HARVESTING DEVICE COUPED WITH THE ENERGY EXTRACTION CIRCUIT The electro-mechanical generating device consists of a cantilever eam to which two iezoelectric elements are attached. It is shown in Fig. (), where and are the length of the eam and Piezoelectric elements, resectively; t and t are their resective thicknesses; w is the width of oth elements; x is the distance etween the iezoelectric elements and the clamed end; the suscrits and stand for the uer and lower iezoelectric elements resectively. The model is ased on that roosed y Sodano et al. [6], which has een used frequently due to its comlete formulation that accounts for multile modes. This aer highlights only the main oints of their work, since a detailed descrition of the model can e

3 found in [6]. The model is develoed using Hamilton's variational rincile with the iezoelectric linear roerties, while some additional assumtions are made. First, Euler-Bernoulli eam theory is considered, which means the model is limited to a system with small dislacements and strain, and linear ehavior. Second, it is assumed that the iezoelectric elements are erfectly onded to the surface of the eam, and the electric voltage v( t ) roduced is constant over the thickness of the iezoelectric elements. uz 5.09 z.8 x.54 t y 0. t z x Piezoelectric element Figure. Schematic of the eam of the energy harvesting device with two iezoelectric atches attached. The iezoelectric ehavior is descried y [8]: T D.54 Beam 0.97 w c e E = S e E ε S. () Where T is the stress tensor [N/m ], S is the strain tensor [m/m], E is the electric E field [N/C], D is the electric dislacement [C/m ], c is the modulus of elasticity at a constant electric field [N/m ], ε is the dielectric constants matrix at constant strain and e is the S iezoelectric couling coefficient, that relates the stress to the alied electric field. Generally, the iezoelectric constant d = is used in the modeling, since this roerty is usually e E c rovided y manufacturers. Once the iezoelectric roerties are known and their effects are considered using Hamilton's rincile, the next ste involves solving the resulting equation. However, as the eam is a continuous system, finding a solution to its governing equation requires the use of an aroximate method. A common aroach to this is to use the Rayleigh-Ritz method, which states that the dislacement of the system can e reresented y the sum of a finite numer of modes, n, each with mode-shae Φ n( x ). Hence, the equation of motion for the eam with the iezoelectric elements attached is given y [6,4,6]: ( s ) ( s ) M + M ɺɺ r( t) + Crɺ ( t) + K + K r( t) + Θv( t) = F( t) () T Θ rɺ ( t) + C vɺ ( t) = qɺ ( t). () where M and K are the mass and stiffness matrices, resectively; C is the daming matrix, assumed to e roortional to mass and stiffness y the constants α and β, resectively; T Θ = Θ is the electromechanical couling matrix; C is the iezoelectric caacitance; the t

4 suscrits s and reresent the structural and iezoelectric elements, resectively; r ( t) is the dislacement; q( t ) is the electric charge. The dot over the variales v( t ), q( t ) and r ( t) stand for the time derivatives, where qɺ ( t) is the electric current i( t ). As shown in Fig. (), the eam is sujected to an inut dislacement at the ase of the eam. This can e aroximately reresented y a force at the ti of the eam, given y F t ρaω ωt dv F ωt (4) ( ) = sin( ) = o sin( ) V = is the force amlitude, ω is the excitation frequency, A is the amli- where Fo ρ V Aω dv tude of the dislacement at the ase of the eam, V is the volume and ρ is the density of the eam. The force F ( t ) is calculated assuming only the mass of the eam, i.e. neglecting the mass of the iezoelectric elements. Equations () and () can only reresent a real Power Harvesting system, if the connection etween the iezoelectric eam and the rectifier circuit is taken into account. Figure () illustrates this connection, where a full wave diode ridge rectifier circuit is considered, with a caacitive filter C and a resistive load R that reresents the external load. i PZT = i circuit + D D + v = v PZT circuit C R - D D4 v - PZT element Figure. Schematic diagram of the couling etween the iezoelectric harvester and the full wave diode ridge rectifiers. According to Ru et al. [4], these two systems may e couled assuming a continuity condition etween the current/voltage of the mechanical and electrical domains, which means that vcircuit = vpzt and icircuit = ipzt. In addition, in eq. () qɺ ( t) reresents the current assing through the rectifier circuit and, consequently, feeds ack the iezoelectric element. The diodes used in the circuit can e descried y an equivalent resistance R D and a forward voltage dro v, since a iecewise linear model is used. Thus, due to the caacitive element, if v( t) < v + v( t), no current flows and v ( t ) is the voltage across the load. In this case, i( t ) = 0 and v ( t ) decreases while the caacitor sulies the resistive load according the equation: v( t) Cvɺ ( t) + = 0. (5) R When v( t) v + v( t), the current i( t ) that flows through the diodes D and D 4 ecomes: Beam

5 Hence, the voltage across the load is now descried y v( t) v( t) v i( t) =. (6) R D v( t) v Cvɺ ( t) + + v( t) =. (7) R RD RD ikewise in this case, when v( t) v + v( t), for the negative half cycle, the current flowing through the diodes D and D is: Hence, the voltage v( t ) across the load is descried y v( t) + v ( t) + v i( t) = (8) R D v( t) + v Cvɺ ( t) + + v( t) =. (9) R RD RD The eqs. (5) to (9) reresent the full wave diode ridge rectifier circuit in terms of the variales v( t ), i( t ) and v ( t ), where the latter can e used to otain the ower harvested. In this aer, the root mean square of the ower P rms is used and is calculated y P rms vrms = (0) R where vrms = v ( t) t f and t f is the time over which the calculation is made. A convenient way to descrie the comlete electro-mechanical model of the energy harvesting device is to use a state-sace model, which is given y xɺ = Ax + B u + B u () y = Cox + Du + D u () where x = { r( t) rɺ ( t) v( t)} T is the state vector and u and u are the inuts of the system. The dynamic matrix A, the inut matrixb, the outut matrix C o and the direct transmission matrix D are given y: 0n n In n 0 n 0n 0 n A =, M K M C M Θ B = M F, B = 0n T n C 0 0 C 0 Θ 0 n 0 n Co =, = 0, = 0 n 0 D D Φ 0 ()

6 where Φ = { Φ( x) Φ( x) Φn( x)} is the vector of mode-shaes used to aroximate the transversal dislacement u ( x, t), 0 is the zero matrix and I is the identity matrix. The outut matrix C o gives the outut of the electromechanical system y = { v( t) uɺ ( x, t) u( x, t)} T. The inut vector is defined y u = sin( ωt) and u = ɺ q( t), where u must e multilied y F 0 to roduce the alied external force. The feedack current u is negative ecause of the energy extraction effect, which is why the variale on the right-hand side of eq. () has a negative sign. With the system reresented as a sace-state model, it is ossile to erform an otimization of the comlete system. All simulations are carried out within Matla, where the statesace and the rectifier circuit models are imlemented using the Simulink toolox.. OPTIMIZATION THE ENERGY HARVESTER The goal of this aer is to maximize the ower P rms generated y the iezoelectric harvester using the Sequential Quadratic Programming (SQP) otimization method. To achieve this, the ojective function which was the inverse of P rms is minimized. The otimization arameters considered are the length of the eam ( ) and iezoelectric elements ( ), the resistive load ( R ), the caacitor ( C ) and the diodes arameters R D and v. As the caacitor included in the rectifier circuit is required to store the energy harvested, this energy must e maximized, and is another requirement that must e taken into account. Thus, the energy stored in the caacitor given y E = C v (4) C must also e considered as an ojective function. Hence, the constrained minimization rolem can e written as: where the vector of variales to e otimized is suject to the constraints: Minimize f ( x) = and h( v, C) P = E (5) rms { } T R C RD v x = (6) C g ( x) : g ( x) : min max min max g ( x) : R R R g ( x) : C C C min max min max 4 g ( x) : R R R g ( x) : v v v min max min max 5 D D D 6 (7) The otimization arameters are thus each restricted to a range of feasile values.

7 4. RESUTS The otimization rocedure was alied to the system discussed under harmonic ase excitation. Tale () gives the dimensions and roerties of the model electromechanical system. The simulations were carried out considering four mode shaes ( N = 4 ) of the eam, -4 with f = ω / ( π ) = 70 Hz as the excitation frequency and A =.5 x0 m as the dislacement amlitude. Such arameters (ω and A) were chosen to reresent real oeration conditions, since the iezoelectric harvester are usually laced in locations containing these characteristics. The otimization results are shown in Fig. () and Ta. (), where the initial conditions and the constraints used in the rocedure are also resented. Tale. Parameters of the iezoelectric harvester model. Domain Parameters Values Thickness ( t ) 0.9 mm Width ( w) mm Structural element Density ( ρ ) 700 kg/m³ Modulus of elasticity ( E ) 70 Ga Material Aluminum Thickness ( t ) 0.9 mm Density ( ) ρ 7750 kg/m³ Piezoelectric element Modulus of elasticity ( E ) Ga Model PZT-5H Dielectric constant ( K ) 400 Piezoelectric constant ( d ) 74 x 0 - m/v Scaled arameters are shown in Fig. () ecause their magnitudes are very different. 5 The arameters were scaled according to: C = 0 F, R = 0 Ω, = = 0 m, R D = 5 Ω, v = 0 V. The results in Ta. () show the otimized results as well as the ranges considered in the search for the otimum arameters Tale. Results of the otimization rocedure. Configuration Parameters ower Bound Uer Bound Initial Otimal C [F] R [ Ω ] [m] [m] R [ D Ω ] v [V] Power RMS [W]

8 60 Scaled Design Parameters C R R D Iterations Figure. Evolution of the scaled otimization arameters in the otimization rocedure iterations. It can e seen from Fig. () that the design arameters reach their otimum values after aout 0 iterations. Further, it can e seen from Ta. () that the rms value of the ower harvested was increased from the initial design y aout 5.%. For some arameters, the otimal values differ little from the chosen initial condition, suggesting that the otimization algorithm may not have sought the otimal solution through the whole domain. This could e ecause the Simulink Design Otimization tool does not assure a gloal otimal solution, or it could e that the initial values were quite close to the otimal values. Comaring the otimal configuration to a similar one, with the same values ut setting R D and v to 65 Ω and 8 0 V, resectively, it could e seen that the solution is not really the gloal maximum. Figure (5) resents the way in which the values of the design arameters and the excitation frequency affect the ower harvested. Examining this figure it can e seen that all otimal values remains inside a region where the ower is maximum, i.e. around 70 Hz. Since the length of the eam greatly influences the natural frequency of the system, the maximum ower varies dramatically with the length, as can e seen in Fig. (5()). The otimal values of the electrical arameters also occur around the excitation frequency. Figures (5(c)) and (5(d)) show that C and R, have a large effect on the amount of ower that can e harvested and are thus imortant to consider in the otimization rocess. v 5. CONCUSIONS The use of iezoelectric devices to harvest viration energy has een considered in this aer. A unified model of a eam with iezoelectric elements attached connected to a full wave diode ridge rectifier circuit has een develoed. This couled electro-mechanical model has enaled the ower harvested to e maximized y otimizing oth mechanical and elec-

9 trical arameters. Desite the fact that the otimization tool does not assure a gloal maximum solution, the results showed that when the diode is reresented as a linear equivalent model, its characteristics do not cause large changes to the ower harvested. However, the mechanical arameters are imortant to ensure that the natural frequency of the eam is coincident with the excitation frequency. It was also shown that the values of caacitor and the load resistor in the electrical circuit are also vitally imortant in an otimally designed ower harvester PZT ength [m] Beam ength[m] Frequency [Hz] (a) Power as a function of iezoelectric element length and excitation frequency Frequency [Hz] (). Power as a function of eam length and excitation frequency. 4 4 x x Resistance [Ω] Caacitance [F] Frequency [Hz] (c). Power as a function of caacitance and excitation frequency Frequency [Hz] (d). Power as a function of resistive load and excitation frequency. Figure 5. Influence of,, C and R on ower harvested and the excitation frequency. Acknowledgements The first author would like to thank the CAPES for his scholarshi. The authors are also thankful the financial suort rovided y National Council for Scientific and Technological

10 Develoment (CNPq), Fundação Araucária de Desenvolvimento Científico e Tecnológico do Paraná (PR) and Fundação de Amaro à Pesquisa do Estado de São Paulo (FAPESP/SP). 6. REFERENCES [] de Marqui Jr. C., Erturk A. and Inman D. J., "An electromechanical finite element model for iezoelectric energy harvester lates". Journal of Sound and Viration, 7, 9-5, 009. [] Elvin N. G. and Elvin A. A., "A general equivalent circuit model for iezoelectric generators". Journal of Intelligent Material Systems and Structures, 0, -9, 008. [] Elvin N. G. and Elvin A. A., "A couled finite element-circuit simulation model for analyzing iezoelectric energy generator". Journal of Intelligent Materials Systems and Structures, 0, , 009. [4] Franco V. R., Mineto A. T. and Varoto P. S., "Otimization strategies for otimum ower generation in iezoelectric energy harvesting from amient virations". In st International Congress of Mechanical Engineering, (Natal, RN, Brazil), UFRN/ABCM, 0. [5] Guan M. J. and iao W. H., "On the efficiencies of iezoelectric energy harvesting circuits towards storage device voltages". Smart Materials and Structures, 6, , 007. [6] Hagood N. W., Chung W. H. and Flotow A. V., "Modeling of iezoelectric actuator dynamics for active structural control". Journal of Intelligent Materials Systems and Structures,, 7-54, 990. [7] Hu Y., Hu T. and Jiang Q., "On the interaction etween the harvesting structure and the storage circuit of a iezoelectric energy harvester". International Journal of Alied Electromagnetics and Mechanics, 7, 97-09, 008. [8] IEEE, "IEEE standard on iezoelectricity" [9] Kymissis J., Kendall C., Paradiso J. and Gershenfeld N., "Parasitic ower harvesting in shoes". Second International Symosium on Wearale Comuters, -9, 998. [0] allart M. and Guyomar D., "An otimized self-owered switching circuit for nonlinear energy harvesting with low voltage outut". Smart Materials and Structures, 7, 0500, 008. [] Ottman G. K., Hofmann H. F., Bhatt A. C. and esieutre G. A., "Adative iezoelectric energy harvesting circuit for wireless remote ower suly". IEEE Transactions on Power Electronics, 7, , 00. [] Renno J. M., Daqaq M. F. and Inman D. J., "On the otimal energy harvesting from a viration source". Journal of Sound and Viration, 0, , 009.

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