Iranian (Iranica) Journal of Energy & Environment

This paper introduces a novel harvester to store the electrical power, which comes from the power of external applied electrical voltage. In the last decade, most of the energy harvesters have been designed and analyzed in the form of cantilever beams. In the present article, the harvesters are analyzed as a cantilever beam with the Euler-Bernoulli beam assumptions. The beam of energy harvester consists of an active Magneto-electro-elastic (MEE) layer attached to the piezoelectric layer. Assuming that the connection of these layers is perfect, the uni-morph configuration is investigated. The magneto-electro-elastic governing coupled equations of the MEE energy harvester are derived for a harmonic external applied electrical voltage in the transversal direction based on Euler-Bernoulli theory, Gaussian law, and Faraday law. These equations are solved analytically to find out the amount of harvested power and voltage. The obtained results state that by adjusting the electromechanical parameters, up to 66% of the input power and 27% of the applied voltage can be harvested. Choosing the right geometric parameters can increase the harvested power and voltages connected to the electrodes and external coil by 120.31%, 49.05% and 60.98%, respectively. Finally, the results prove the usefulness and efficiency of the dual-usage (actuator-harvester) of the new energy harvester.

environment [1].Recently, use of wasted energy has become an option to produce electrical energy [2,3].This was led to energy harvesting technology.The electrical power produced by this promising technology is usually recyclable and contributes to the sustainability of the system's infrastructure.It is usable in cooling and heating, electricity generation, and health monitoring structures.
One of the most common energy sources for energy harvesting systems is ambient vibrations, which have high energy capacity and are suitable for small-scale devices.The kinetic energy is converted into electricity in three different ways: active materials (such as piezoelectric [4]), electromechanical coupling mechanisms (such as electrostatic [5], and electromagnetism [6]), and the use of special instruments and materials that combine the above [7].
The basis of the work of electrostatic generators is capacitors with variable capacitance.Unfortunately, these generators require an initial polarization voltage or charge.Therefore, they need a battery at the start [5].A permanent magnet and a coil are used as a converter in electromagnetic generators and work better on the macro-scale [6].The advantage of these generators is their relatively high output current.The piezoelectric effect is a characteristic of some materials (like material which is called PZT, PVDF, and MFC) that in response to mechanical stresses, the electrical potential created and vice versa.Using each of these three methods to convert the vibration oscillations into electrical power has been separately investigated.The energy harvesting system used one of these methods is called a vibrational energy harvester (VEH).
Using composite and functionally graded, and smart materials attract many researchers due to their special vibrational behaviours [8].Among them, some smart materials known as MEE composites have received significant interest in material science recently.These materials have both piezoelectric and piezo magnetic properties, and they can convert electrical, magnetic, and mechanical energy into each other.Thus, using MEE composites in energy harvesting systems are recommended [9].
To the best of the author's knowledge, no one was used the energy harvester system under both the electrical excitation and the ambient vibration.One can import an electrical excitation to the energy harvester to amplify the system's vibration and harvest more energy.The results prove that there is an opportunity to save the remaining output voltage in addition to the ambient vibration source of energy.

MODELLING OF AN MEE ENERGY HARVESTER UNDER ELECTRICAL EXCITATION
An electrically excited MEE energy harvester is shown in Figure 1.In this figure, Vin, VE, and VM represent input voltage, generated voltages on the electrodes attached to the MEE layer and the voltage induced in the N-turn coil, respectively.
A Cartesian coordinate system (x,y,z)=(x1,x2,x3) is used to derive the governing constitutive equations.The coupled constitutive equations for an anisotropic and three-dimensional linear MEE energy harvester beam may be expressed as follows [10]: where σi, Di and Bi are the stress, electric displacement, and magnetic induction (i.e., magnetic flux); εi, Ei and Hi are the strain, electric and magnetic fields; Cij, hij, and μij are the elastic, dielectric, and magnetic permeability coefficients; eij, fij and gij are the piezoelectric, piezomagnetic and magnetoelectric coefficients, respectively.The standard contraction indices have been used for the elastic variables (i.e., ε4 = ε23, etc.).
Because the thickness of the beam is significantly less than its radius of curvature under different loads, the inplane electric and magnetic fields can be ignored (i.e., E1 = E2 = 0 and H1 = H2 = 0).So, the coupled constitutive Equations (1) reduced into Equations (2).and the analytical equations are derived using Euler-Bernoulli beam theory [10].
The constitutive equation of the piezoelectric layer is as follows [10]: Assuming the electric field is uniform throughout the constant piezoelectric layer thickness hp, then the electric field  3  , exist in terms of the input voltage Vin (connected to the resistive loads Rin) across the thickness of the  , in terms of the voltage VE can be expressed as follows [8]: With the help of Faraday's law, the magnetic field H3 (t) can be obtained in terms of the current iM (t) and the number of turns of the coil N as expressed in the following relation [11].
The basic equations listed will be used to derive the coupled magneto-electro-elastic equations from investigating the performance of a MEE energy harvester.
The relative deflection of the neutral axis, w(x,t), is caused by electrical excitation, Vin.The following equation is the governing equation of vibration of the beam under base excitation [12]: where I is the moment of the cross section of the beam, cm viscous air damping coefficient, ckI equivalent composite structural damping coefficient, and m is the mass per unit length of the beam.Also, M(x,t) is the internal bending moment of the beam which is calculated for Uni-morph configuration of the MEE energy harvester by Equation (7) .
where b is the width of the beam, ha, hb, hc, hM, and hp, are defined in Figure 2, and ̅ is the location of the neutral axis.Substituting Equations (2) and Equation (3) into Equations ( 7), the internal bending moment for each configuration is obtained as follows [12]: where   ,   and   are the electro-and magnetomechancial coupled terms and defined as follows: where ρm and ρp are the density of MEE and piezoelectric layers, respectively.By substituting Equations (8-10) into Equation ( 6), the governing differential equation of the harvester beam for each configuration is achieved.Using the separation of variables method, one can write the relative deflection as follows [13]: where Tr(t) and Wr(x) are the time response and the r-th normal mode shape of the cantilever beam, respectively.Then, the motion equation in modal space can be written as Equation (12).
where FVin,r(t) is the electrical excitation induced force and expressed as follows: where; where hMc is the distance between the neutral and mid axes of the active MEE and sgn(z) is the sign function.
Next, integrating the electric displacement in Equation ( 14) over the surface of the electrodes, using Gauss's law, the electrical charge QE (t) appearing on the electrode attached to the MEE layer can be expressed as follows [12]: By imposing Equation (15) into Equation ( 16), for each configuration, and calculating the integral, an expression for the charge is obtained as a function of time.
CM denotes the internal capacitance of the MEE layer, and ϑ r EM and χ ME are the coefficients of modal and magnetic to electric coupling terms in the circuit connected to the electrodes, respectively.That are obtained in the following forms: Faraday's law states that the voltage V is induced proportional to the time derivative of magnetic flux linkage, ϕ, [14].As a result, the voltage induced in the Nturn coil VM may be calculated as follows: Generally, ϕ, is calculated as follows: According to substitution of Equation ( 5) into Equation (2c) and using By solving the Equation ( 29) and substituting the results in Equation ( 27): Substituting Equation (31) into Equation ( 12), the displacement of each point on the MEE energy harvester under electrical excitation is obtained.The electric current frequency response functions are obtained by dividing the resulting voltages by the load resistance of the electrical circuits.
In order to compute the MEE energy harvester efficiency under electrical excitations, the input and output power must be compared.Thus, using Ohm's law ( = ), and  = 0.5, the total power made by energy harvester , Ptotal(t), are expressed as follows: The MEE energy harvester efficiencies under electric excitation, ΛE, ΛM and ΛME, are introduced as the ratio of harvested to the input parameters and expressed as follows:

RESULTS AND DISCUSSION
In this section, the results obtained from the analytical solutions in the previous section are presented.Numerical results will be carried out for BaTiO3-CoFe2O4 composite as the MEE layer with a volume fraction  = 0.5.The geometry and material properties used in the following case studies are listed in Table 1.
The performance of the MEE energy harvesting system under external electrical excitation is considered.The effect of the parameters extracted from the previous section, RM, RE, Rin, hp and hM on the MEE energy harvester efficiencies, ΛE, ΛM and ΛME has been investigated.
The effects of the non-dimensional electrical excitation frequency, η, on the power efficiency, ΛME, for four different values of the electrical load resistances connected to the applied circuit voltage (Rin = 10 6 Ω,  31  (C m -2 ) Piezoelectric coefficient of the Piezoelectric layer -6.5  concluded that in order to harvest the maximum power, the range 10 5  ≤   ≤ 10 5.5  should have a higher priority in the choice of RE.It is obvious from Figure 5 that for RE = 10 6 Ω, 26.87% of the applied voltage can be stored in the circuit connected to the electrodes, which is a significant amount to reduce the consumption of electronic circuits.In order to store higher voltage values, the range 10 5.5  ≤   ≤ 10 6  must be selected.Also, the maximum efficiency ΛM occur in RE = 10 5 Ω to the numerical value of ΛM= 0.0014 V/V that is insignificant compared to the ΛE.Finally by considering that the thickness of harvesters is the limiting parameter of the design, harvester efficiencies ΛME , ΛE and ΛM were checked for different thickness ratios in Figures 10 to 12, respectively.The obtained results show that the equal thickness of the piezoelectric and MEE layers has a better effect on the power and voltage efficiencies of the discussed harvester.

CONCLUSION
In this paper the performance of the energy harvesters made of active MEE layers and under electrical excitation was analyzed.The MEE beam consisted of the piezoelectric material as the substrate layer and MEE composite as the active layer.The unimorph and bimorph configurations of the MEE layers (including serial, parallel, and single-layer connections) were studied, assuming that the connections between these layers are perfect.The coupled magneto-electro-elastic governing equations were obtained using Euler-Bernoulli theory, Faraday laws and Gauss, and then, these equations were solved analytically to obtain the generated electrical power and voltage.The effects of RM, RE, and Rin, on the MEE energy harvester efficiencies, have been investigated.The results show that a significant amount of electrical excitation voltage can be stored from the circuits connected to the electrodes.Also, the load resistance RM connected to the external coil can have a decisive role on the power efficiency.For energy harvesting purposes, one can use the electrical excitation as the second source added to the ambient vibration.

Figure 1 .
Figure 1.The MEE energy harvester under the electrical excitation

Figure 2 .
Figure 2. The cross-section area of the Unimorph configuration MEE energy harvester

μ33 (N s 2 C - 2 )Figure 3 .
Figure 3. Frequency response curves of the harvested power ratio ΛME , of the unimorph MEE beam around the first natural frequency for different values of Rin

Figure 4 .
Figure 4. Frequency response curves of the harvested power ratio ΛME, of the unimorph MEE beam around the first natural frequency for different values of RE

Figures 7 to 9 Figure 5 .Figure 6 .
Figure 5. Frequency response curves of the generated voltage over electrodes ratio ΛE, of the unimorph MEE beam around the first natural frequency for different values of RE

Figure 7 .Figure 8 .
Figure 7. Frequency response curves of the harvested power ratio ΛME, of the unimorph MEE beam around the first natural frequency for different values of RM

Figure 9 .Figure 10 .Figure 11 .
Figure 9. Frequency response curves of the generated voltage over external coil ratio ΛM, of the unimorph MEE beam around the first natural frequency for different values of RM

Figure 12 .
Figure 12.Frequency response curves of the generated voltage over external coil ratio ΛM, of the unimorph MEE beam around the first natural frequency for different thickness ratios Now, the differential equation of the electrical circuit of the two ends of the electrodes used around the MEE layers is extracted.Initially, the two ends of the electrodes and external coils were connected to external electrical resistors RE and RM to use the voltages generated by VE and VM.

Table 1 .
Material properties and geometric dimensions of the layers of the harvester