Design and Analysis of a Hybrid Solar and Vibration Energy Harvester

Hybrid Cantilever Beam

The HEH consists of a bimorph Lead Zirconate Titanate (PZT) cantilever beam having a 6061-T6 Aluminum (Al) substrate and a cylindrical NdFeB magnet as a tip mass. The cylindrical magnet is surrounded by an induction coil of 1200 turns which is coaxially configured with the magnet on a frame. The volume of the Al substrate is 100 mm × 10 mm × 0.75 mm while each of the PZT layers have a volume of 90 mm × 8 mm × 0.40 mm. The PZT layers operate in d₃₁ mode (i. e. with transverse piezoelectric strain coefficient) as the beam vibrates in transverse direction. Two sets of copper electrodes, in the shape of a hair comb, are attached on two sides of the Al substructure by keeping insulations in between. When the beam undergoes harmonic excitation, they act as the variable capacitance electrodes (C_var electrodes). The PZT layers and the two sets of C_var electrodes are assumed to be perfectly bonded to the Al substrate. A simplified two-dimensional view of the configuration is shown in Figure 1 while the actual three-dimensional configuration is demonstrated in Figures 2 and 3.

Figure 1:

Schematic diagram of the hybrid cantilever beam showing top and front views (figure not drawn to scale).

Figure 2:

(a) Configurations of the 3-D hybrid energy harvester without the fixed electrodes, (b) exploded 3-D view of the hybrid structure showing the sets of fixed electrodes on two sides of the beam.

Figure 3:

Top view of the hybrid energy harvester showing the details of fixed and C_var electrodes.

Each set of comb-shaped C_var electrodes contains 70 electrodes, each having the dimension of 9 mm × 0.75 mm × 0.3 mm. The gap between two successive C_var electrodes in the comb is 1.1 mm (Rahman and Chakravarty 2018). Two sets of similar combs of electrodes (fixed or, stationary capacitors) are attached with the base on two sides of the beam where each electrode has the dimension of 10 mm × 1 mm × 0.2 mm. This is illustrated in Figure 2(b) where the fixed electrodes are complementing the C_var electrodes by fitting into the gaps. When the fixed electrodes and C_var electrodes overlap, the gap between the electrodes is 0.45 mm as shown in Figure 3. Since the beam oscillates in transverse direction, this gap is assumed to be constant throughout the electrostatic power generation. Among the three fundamental electrostatic configurations – in-plane overlap varying, in-plane gap closing, and out-of-plane gap closing, the third one is selected for this design to avoid contacts between the moving and stationary electrodes. The relative motion between the precharged moving capacitors and stationary capacitors governs the electrostatic transduction.

The geometric and physical properties of the HEH considered for the analysis are shown in Table 1. The hybrid structure of the device is scalable and tunable. The tip mass and the length of the comb electrodes can be adjusted to operate at the resonance frequency (Rahman and Chakravarty 2018; Challa, Prasad, and Fisher 2011; Chen, Yang, and Deng 2009). Since the performance of the EH is sensitive to the smallest amount of change of its geometry (Yi, Shih, and Shih 2002), attention must be paid to the effective scaling of the design.

Photovoltaic Panel

The PV panel includes an array of 10 high-efficiency monocrystalline solar cells as shown in Figure 4. The size of the PV unit is 55 mm × 55 mm × 3 mm weighing 36 grams. The item number of PV panel is KS-M5555 which is selected from the China Solar LTD (http://www.solars-china.com/solar-panel/5v-80mA-oem-solar-panel.html). The data sheet information is available in the manufacturer’s website (http://www.solars-china.com/solar-panel/5v-80mA-oem-solar-panel.html). The output from the PV panel is DC while the PE, EM, and ES outputs are AC outputs. Therefore, the AC outputs need to be rectified to DC outputs in the combined circuit. The HEH is designed in such a way that it allows freedom of customization of the PV panel based on the power requirement (e. g. using different scale or type) because the PV panel is not an integrated part of the hybrid cantilever beam. However, the power (bias input) supplied to the C_var electrodes form the PV panel must be adequate to excite the electrodes in order to generate electrostatic power output.

Figure 4:

Configuration of the 5 V 80 mA solar panel (Item no: KS-M5555) (http://www.solars-china.com/solar-panel/5v-80mA-oem-solar-panel.html).

The operating voltage and current of the PV panel are 5 V and 80 mA, respectively. The other important specifications are shown in Table 2.

Photovoltaic Equations

Considering a PV array with N_S number of cells in series and N_P number of strings in parallel. the output current I and output voltage V can be obtained from eqs. (1)–(4) (Rahman and Chakravarty 2018; Sood and Bhalla 2013).

where IPH is the light-generated current or photocurrent, IS is the cell saturation of dark current, q = 1.6 × 10^–19 C is the electron charge, k = 1.38 × 10−23 J/K is Boltzmann’s constant, T_c is the cell’s working temperature, A is diode’s ideality factor (typically between 1 and 2), R_SH is the shunt resistance, and R_S is the series resistance. The photocurrent IPH can be calculated as follows:

where I_SC is the short-circuit current of a cell at reference temperature of 25 °C and reference irradiance of 1000 W/m², K₁ = 0.032 is the cell’s short-circuit current temperature coefficient (Sood and Bhalla 2013), T_ref is the cell’s reference temperature, and λ is the solar insolation (irradiance) in W/m². The cell’s saturation current varies with the change in cell temperature, which is expressed as

where I_RS is the cell’s reverse saturation current at a reference temperature and irradiance, and E_G is the band energy of the semiconductor used in the cell. The reverse saturation current IRS at the reference temperature can approximately be obtained as

Several parameters including the open-circuit voltage, V_OC (at I = 0), short-circuit current, I_SC (at V = 0), and the fill factor (FF) determine the performance of a solar cell, which are related as

where I_mp and V_mp are the current and voltage operating points for the maximum power Ppv, respectively. The output power is calculated by taking the product of either I and V, or, I² and R_L (Rahman and Chakravarty 2018). However, in terms of the FF, the maximum PV power output is determined by

Piezoelectric Equations

Assuming the Euler-Bernoulli beam, the coupled electromechanical equation for a bimorph PZT cantilever beam of length L with a tip mass M_t and mass per unit length m_L can be represented as follows (Rahman 2016; Erturk and Inman 2008; Rahman and Chakravarty 2018):

Here, CsIc is the internal (Kelvin-Voigt) damping and Ca is the viscous air damping. The sum of the harnonic base displacement wb=Y0ejωt and relative displacement wrel gives the total transverse displacement w at any time t and location x of the beam as follows:

The electromechanical coupling term ϑ for series connection of bimorph PZT is given by

where e31 is the piezoelectric strain constant. The vibration response relative to the base can be represented as a convergent series of the Eigen functions by following modal damping (Erturk and Inman 2008, 2009; Rahman and Chakravarty 2018) as follows:

where ∅n and ηn are the mass normalized Eigen function and the modal coordinate of the clamped-free beam for the n^th mode, respectively. Now, the governing equation indicates that, four boundary conditions are required in terms of x and two initial conditions are needed in terms of t for the solution (Rahman and Chakravarty 2018). At the fixed end (i. e. x = 0),

At the free end (i. e. x = L), moment M is zero, but shear force is equal to the applied force Fn at the tip, i. e.

where meff represents the effective mass of the hybrid beam and g is the gravitational acceleration. The modal mechanical response ηn can be obtained from the mass normalized reduced partial differential equation as follows:

Here, fn is the mass normalized tip force on the beam. The backward modal electromechanical coupling term χn can be given by

Applying the Kirchhoff laws to the circuit depicted in Figure 6, the capacitance CP and piezoelectric current ipt for the series connection of the PZT layers (Rahman and Chakravarty 2018) can be obtained by

Figure 6:

(a) Series connection of the two PZT layers and, (b) electrical circuit corresponding to the series connection (Erturk and Inman 2008, 2009).

where the capacitance and the piezoelectric current for each layer of PZT can be defined as,

In eq. (17), ε33s is the dielectric permittivity of the PZT layer for the series connection.

The forward modal coupling term φn can be expressed as,

The cross-section of the hybrid beam through the C_var electrodes in y-z plane is shown in Figure 7.

Figure 7:

Cross-section of the beam through the C_var electrodes.

The equivalent bending stiffness EcIc and the mass of the beam m without the tip mass can be given by,

Therefore, the fundamental natural frequency ωn of the hybrid beam can be found by,

where the effective stiffness and effective mass of the beam are keff=3EI/L3 and meff=0.236m+Mt, respectively (Rahman and Chakravarty 2018). The maximum output power is obtained at the resonance condition, i. e. at ω = ωn. The expression of power output PPEω for the PE generator at resonance can be given by eq. (23) (Rahman and Chakravarty 2018; Cottone 2011).

where RL is the load resistance and ζn is the damping ratio of the bimorph PZT cantilever beam. At resonance, Fn is equal to resonance-equivalent static load (Challa, Prasad, and Fisher 2009) at the tip, i. e. Fn = Ftip = m+Mtg/2ζn.

Electromagnetic Equations

The expression for the maximum EM power output PEMω at the resonance frequency (Rahman and Chakravarty 2018; Cottone 2011; Kim et al. 2010) is given by the following equation:

where the electrical coupling force factor α, characteristic cut-off frequency ωc, electromechanical conversion factor δc, and coil self-inductance Li are expressed as follows (Rahman and Chakravarty 2018; Cottone 2011; Kim et al. 2010; Miller et al. 2016):

where B is the magnetic flux density, l is the length of wire in the coil, N is the number of turns in the coil, μ0 is the permeability of the coil material, rc is the coil radius, and hc is the coil height.

Electrostatic Equations

The maximum power output expression for the ES mechanism PESω at the resonance frequency (Cottone 2011) is given by,

In eq. (29), the mechanical damping dm and the electrical damping de can be expressed as,

where the electric force Fe can be given by,

where ε0 is the permittivity at free space, A_c is the overlap area of the comb electrodes, and d_c is the gap between two electrodes.

Combined Power Output and Efficiency

The combined maximum power output Pcombined is not simply the sum of all the stand-alone outputs because the damping of the system causes power loss Ploss. Therefore, the total power output combining all the generators is determined by

where the total power output Pout is given by

The power loss due to the mechanical and electrical damping (Xu et al. 2017) of the hybrid system can be calculated by

In eq. (35), the average power losses PD, PCoil, and PElectrodes due to the mechanical damping dm, internal resistance of the coil RCoil, and internal resistance of variable capacitance electrodes RElectrodes, respectively, are expressed as

The bias input Pin,ES for the ES mechanism, which is supplied by either the battery or the PV panel to the C_var electrodes, should be considered as an input while calculating the efficiency of the HEH. If the input power Pin is a sum of the Pin,ES, solar power input, and vibratory power input, then the overall efficiency η of the HEH (Miller et al. 2016) becomes,

The magnitude of the vibrational power input P_vib is calculated as follows (Rahman 2016; Rahman and Chakravarty 2018), i. e.

The solar power input Psolar is given by,

where λ is the irradiance and APV is the effective photovoltaic area.

Fundamental Natural Frequency

The fundamental natural frequency ωn of the proposed HEH is found 37.16 Hz by eq. (22). Since the power output of the HEH is sensitive to the natural frequency ωn, finite element (FE) modeling is conducted in ANSYS Workbench 15.0 to verify the analytical result for ωn. The FE analysis result for ωn of the HEH is found 35.867 Hz with a deviation of 3.48 % as compared to the analytical result. Figure 8 shows the first resonant mode shape of the HEH obtained by the FE analysis which indicates a good agreement with the first normalized mode shape of a typical cantilever beam with a tip mass (Rahman 2016; Rahman and Chakravarty 2018).

Figure 8:

First mode shape of the hybrid cantilever beam with tip mass at ωn = 35.866 Hz.

Based on the fundamental natural frequency, a mesh independence or, convergence study is conducted using ANSYS where the value of ωn converges to 35.867 Hz with the increase of degrees of freedom. Figure 9(a) depicts the default converged 3-D mesh of the hybrid beam where 38,402 elements are connected with 121,087 nodes. The convergence of the value of ωn with the number of nodes is shown in Figure 9(b).

Figure 9:

(a) Computational mesh of the hybrid beam and, (b) Convergence of ωn value with the increase of DOF.

Stress and Fatigue Analysis

Stress and fatigue analyses of the HEH are also done in ANSYS Workbench 15.0 to study the stress and failure criteria of the hybrid structure. The maximum bending stress σ of the beam is found 2.939 MPa analytically following the equation σ=Mc/Ic where c = h_p + h_s/2 (Rahman 2016). The FE analysis result for the bending stress is found 2.868 MPa which substantiated the accuracy of the analytical calculation by yielding 2.12 % deviation. A fatigue analysis is also done in ANSYS for the substrate material, i. e. Aluminum, since it is clamped to the base, carries the PZT layers (which are not clamped to the base, starts from 0.1 mm on the substrate), Cu electrodes, and tip magnet, and undergoes the maximum stress at the root. Using a fully reversed cyclic load of F=meff.g = 0.10455 N at the tip, a fatigue life of nearly 10⁷ cycles is found following the Goodman stress-life approach. Figure 10 shows the distribution of bending stress and its maximum value at the fixed end of the hybrid cantilever beam which is obtained from the FE analysis in ANSYS Workbench.

Figure 10:

Distribution of stress on the hybrid beam due to tip loading.

Power and Optimum Load Resistance at the Resonance

The results for the output power of the HEH are obtained in MATLAB R2015a following the mathematical model. The parameters considered for the analytical calculation and numerical simulation are listed in Table 3.

The PV power output along with the I-V characteristics is obtained by developing a MATLAB algorithm based on the maximum power point tracking (MPPT) technique at an irradiance level of 1000 W/m² (AM 1.5) at 25 °C temperature (Rahman, Sarker, and Chakravarty 2019). Figure 11 shows the maximum power output from the PV panel along with the I-V characteristics as obtained from the MPPT algorithm.

Figure 11:

Power output from the PV panel.

The calculations for PE, EM, and ES power outputs are done considering 1 g harmonic base excitation which shown in Figure 12.

Figure 12:

Harmonic base excitation with 1 g (9.81 m/s²) amplitude.

The peak of the output power for the PE, EM, and ES generators occurred at resonance, i. e. the fundamental natural frequency ωn which is found 37.16 Hz. The power outputs around the resonance frequency bandwidth for the PE, EM, and ES methods are shown in Figure 13.

Figure 13:

Power outputs at the optimum load resistance from the three vibration-based standalone mechanisms: (a) PE, (b) EM, and (c) ES.

The optimal load resistance is considered as 180 kΩ for the maximum PE output at 37.16 Hz. This is shown in Figure 14 where the resistance at the peak power is the optimal load resistance.

Figure 14:

Optimal load resistance R_L at resonance.

In general, energy harvesters are designed to operate in the fundamental natural frequency, i. e. the first mode of vibration which typically provides the maximum deflection and therefore gives the maximum electrical energy (Rahman and Chakravarty 2018). Erturk and Inman (2008, 2009) mentioned that the sign of the mode shapes of a cantilever beam during vibration is analogous to the sign of axial strain distribution along the length and any change in sign reduces the output voltage (Rahman and Chakravarty 2018). As the first mode shape does not change its sign (i. e. stays at one side of the x axis), the corresponding axial strain curve also remains unchanged in its sign. This ensures the maximum voltage generation by the PZT layers. Therefore, strong emphasis is given on determining the fundamental natural frequency and the first mode shape of the structure before calculating the power output and efficiency.

Calculation of Total Power Output and Efficiency

The efficiency of the HEH is determined by eq. (39). The magnitude of P_vib is found 2196 µW for 1 g base excitation. On the other hand, considering 17 % efficiency and an irradiance level of 1000 W/m² at AM 1.5, the solar energy input P_solar yielded a value of 2.35 W. 10 % of the PV power output is utilized for biasing the ES mechanism. The alternate way to provide the bias input for the ES generator is to use the low voltage battery of the circuit. In that case, the rated amount of PV output, i. e. 0.4 W is harvested which contributes to its maximum efficiency. The total power loss due to damping of the system is calculated by using eqs. (36)–(38). The power inputs and the total amount of power loss are shown in Figure 15.

Figure 15:

Amounts of input power and power loss for the hybrid energy harvester.

The total power output of the HEH is determined by adding all the stand-alone power outputs and deducting the power losses from that amount. While calculating the efficiency, the total input power is considered as the summation of the solar input, base excitation input, and the initial bias input to the ES generator. First, it is assumed that the bias input for the ES generator is supplied by the battery through the microcontroller. The total power output of the HEH in that case is found 772.972 mW with an overall efficiency of 32.3 %, which is greater than the stand-alone power outputs of the PV, PE, EM, and ES generators. The results are shown in Figure 16.

Figure 16:

Results for power outputs and efficiency (%) with biasing by the battery.

Instead of the battery, if the PV panel provides 40,000 µW solar power for biasing the ES generator, the total power output of the HEH drops slightly due to the lack of PV power output. In that case, efficiency of the PV panel decreases to 15.3 % and the total power output becomes 732.973 mW with an overall efficiency of 30.6 %. This is illustrated in Figure 17.

Figure 17:

Results for power outputs and efficiency (%) with biasing by the solar panel.

Results displayed in Figures 16 and 17 indicate that the PV and EM generators are the major contributors in the total power output. Results also indicate that, if the battery or the PV panel fails to supply the bias input, the EM generator will still be able to provide the bias input to the ES generator in order to maintain a continuous power transduction.

Effect of Change in Scale of the Beam

Since the fundamental natural frequency ωn plays an important role in the performance of the vibration energy harvesters, the effects of change in scale or size of the harvester are studied considering the value of ωn. A parametric study is conducted to observe the effects of change in length and width on the natural frequency of the beam and the total power output from the PE, EM, and ES generators. As the expression for the ωn suggests, the increases in length L and mass meff result in the decrease in the value of ωn for a constant width and thickness. Figures 18 shows the effect of changing length on the fundamental natural frequency of the beam.

Figure 18:

Effect of change in length on the fundamental natural frequency of the beam.

On the other hand, at a given length and thickness, the fundamental natural frequency increases with the increase of width of the beam. This effect is illustrated in Figure 19 where the additional width increases the effective mass of the beam but contributes to the increase of the fundamental natural frequency value by increasing the area moment of inertia.

Figure 19:

Effect of change in width on the fundamental natural frequency of the beam.

Knowing the variation of natural frequency with the structural parameters is also critical for tuning the vibration energy harvesters. For a given input excitation, any change in natural frequency as a result of adjusting the tip mass or changing the scale, can change the power output of the energy harvester (Rahman 2016). Therefore, it is necessary to optimize the length and width of the beam depending on the value of the fundamental natural frequency. The variations of natural frequency and power output with the change in length and effective width at a given thickness of the vibration-based generators (i. e. PE, EM, and ES generators) are shown in Table 4.

As mentioned earlier, the size of the PV panel (also the number of arrays) can be adjusted based on the power requirement. The design of the HEH is very flexible in incorporating any small-size PV panel as it is externally connected to the vibration-based system.