Modeling and Experimental Investigation of a Periodically Excited Hybrid Energy-Harvesting Generator

Introduction – State of the Art

Hybrid energy-harvesting concepts are moving progressively into the focus of research, as these concepts provide an increase of performance and scope of energy-harvesting applications. One hybrid energy-harvesting concept for rotating applications utilizing piezoelectric and electromagnetic effects is specified in recent publications by Wurz, Kleyman, and Twiefel (2013). This article is focused on the modeling approach of this system. Introductorily an overview is given about the current situation of piezoelectric, electromagnetic and hybrid energy-harvesting concepts.

For the piezoelectric and also for the electromagnetic effect the excitation may be either translational or rotational. It depends on the application environment which form of kinetic energy is available. While piezoelectric and electromagnetic bending beam concepts are more common for vibrational excitation, electromagnetic rotor/stator combinations are used for rotating applications. Furthermore, there are a number of energy-harvesting systems that convert rotational motion into translational excitation forces and vice versa. Those serve as an inspiration for the hybrid concept presented in the following section.

Typical vibration-based energy-harvesting systems operate at their resonance frequency. Roundy and Wright (2004) present a piezoelectric generator based on a two-layer piezoelectric bending element with a tip mass (Figure 1–1). During vibration, the piezoelectric layers are compressed and elongated alternately. Thus an electric charge is getting induced. In resonance the relative motion between tip and base and consequently the electric power output is at the maximum. Beeby et al. (2007) show an energy-harvesting concept with a passive bending element and a solenoid attached at the tip of the element. During oscillation the solenoid is passing a permanent magnet, in this way a voltage is induced (Figure 1–2). Xing et al. (2009) propose a slightly different concept for an electromagnetic harvester. The bending element is made of high-permeability material and is mounted along the axial direction inside a solenoid. Two magnets with opposed magnetization are aligned outside of the solenoid. When the element vibrates its tip passes the magnets and the magnetization in the beam is reversed, thereby a voltage is induced inside the solenoid (Figure 1–3). Challa et al. (2008) developed a tunable piezoelectric bending type energy harvester. A pair of permanent magnets is placed up- and downside at the tip of a piezoelectric bending element. A second pair of magnets is attached to the inertial frame. Between the tip- and frame-magnets there are repulsive and attractive forces. By changing the distance between those magnet pairs the resonance frequency can be tuned (Figure 1–4). Challa, Prasad, and Fisher (2009) published a hybrid energy-harvesting system which combines principles of Roundy and Wright (2004) and Beeby, Tudor, and White (2006). A piezoelectric bending element has a permanent magnet at the tip. When the beam is vibrating the magnet dips inside a solenoid (Figure 1–5). Thus one energy source is used by two energy converters at once. A non-resonant power converter for translational excitations was presented by Spreemann et al. (2006). The energy harvester consists of a rotor and a stator. The rotor is a pendulum and rotates excited by vibrations. Two anti-parallel poled magnets are attached to the pendulum. The alternating magnetic field induces a voltage in an array of solenoids, which are arranged in a circle at the stator (Figure 1–6). Jung et al. (2013) designed a system that also converts vibration energy into rotation. Similar to Beeby, Tudor, and White (2006), a permanent magnet is attached at the tip of a cantilever beam. The base of the beam is connected to a gearing. When the beam is moving its tip passes a solenoid and the gearing revolves. A simple dynamo is linked to the gearing, thus a voltage is induced not only inside the solenoid but also by the dynamo (Figure 1–7).

Figure 1:

Functional principles of representative energy-harvesting systems.

Figure 2:

Classification of different types of energy-harvesting devices.

Figure 3:

Concept CAD model and photography of the energy-harvesting device and testing rig.

Figure 4:

Layout of the experimental setup.

Figure 5:

Boundary values of a bimorph.

Figure 6:

Linearized kinematics for the Euler–Bernoulli beam.

Figure 7:

Comparison of the measured and modeled system behavior.

There are also energy-harvesting concepts that convert energy directly from a rotational source. Wang, Shen, and Chen (2012) show a wideband electromagnetic harvester for tire pressure monitoring systems. That harvesting device is very similar to Spreemann et al. (2006). Permanent magnets are attached to an unbalanced rotor. Gravitation causes the rotor to rotate above a stator with an array of coils (Figure 1–8). A piezoelectric energy harvester for rotating energy sources was published by Priya (2005). An array of 12 piezoelectric bending elements is arranged along the circumference of a rotor shaft. A camshaft gear mechanism generates the torque to excite the elements (Figure 1–9). In this scenario a windmill is used as rotation energy source. This concept is picked up by Bressers et al. (2011). In their device the bending elements are exited contactless by magnetic forces. A series of magnets with alternating polarity is mounted on the rotating shaft. There are also magnets mounted on the tips of the bending elements. Rotation of the shaft induces a harmonic vibration into the piezoelectric elements via an alternating attractive/repulsive force between the magnet pairs. Their new approach is minimizing frictional losses and thus the device is more suitable for low wind speeds. Karami, Farmer, and Inman (2012) propose a piezoelectric wind turbine concept which is excited by non-harmonically magnetic forces and develop a semi-analytical model for this system.

Figure 8:

Sequence diagram: calculation of electric output from a non-harmonic excitation force for electromechanical systems.

Figure 9:

Waveform of the force measurement and the corresponding Fourier coefficients.

Figure 2 shows a classification for the presented energy-harvesting concepts addicted to their physical energy conversion effect and the type of excitation. The vibration-based energy-harvesting systems are well developed. There are many different electromagnetic concepts and some piezoelectric devices are even on the market. The concepts by Priya (2005) and Bressers et al. (2011) show how the piezoelectric effect can also be utilized for rotating applications however electromagnetic concepts are more well established. Until now hybrid concepts are only studied for translational excitation, there is no hybrid rotational concept published. Our introduced hybrid energy-harvesting system fills this gap.

In the following the energy-harvesting concept is presented (section “Concept of a Hybrid Piezoelectric and Electromagnetic Energy Harvester”). The main part of this article deals with the description of the prototype unit by an analytical modeling approach (section “Modeling”). The model considers not only the properties of the solenoid and of the piezoelectric bimorph also the non-harmonically electromagnetic excitation forces. Furthermore, the analytical results are compared to the measurements of the prototype unit and the wideband performance of the concept is estimated (section “Analysis”).

Concept of a Hybrid Piezoelectric and Electromagnetic Energy-Harvester

The following concept is based on the idea of a rotational piezoelectric energy harvester by Bressers et al. (2011). This magnetically driven piezoelectric harvester is extended by a solenoid with a ferromagnetic core on the tip.

The device is excited by four magnets attached to a rotating aluminum disc as shown in Figure 3. The attractive force between magnets and core material initiates a bimorph oscillation. Simultaneously the magnetic flux gradient induces a voltage in the solenoid. The total volume of the piezoelectric bimorph is 0.75 cm3 and the solenoid has core diameter of 1 cm. The solenoid is bonded to the tip of the bimorph whereas the other end of the bimorph is clamped as shown in Figures 3 and 4. The bimorph is a typical resonant piezoelectric bending element and therefore it reaches the maximum electric power output related to the mechanical excitation power at its resonance frequency. Whereas the power output of the solenoid is quadratic to the applied mechanical power and thus to the revolution speed. By combination of both transducers in one unit a broadband energy-harvesting device is build. At low excitation speeds and high electric loads the piezoelectric part supplies the bigger part of electric energy. With increasing frequencies the power output of the solenoid grows and rises above the piezoelectric part. Thus such an energy-harvesting system may be used in an environment with highly fluctuating rotational speeds such as miniature wind turbines.

On the experimental setup (Figure 4) the disc is powered by an electric motor. Thereby the electric output variables (voltage and current) are measured for different electric load-resistance and different revolution speeds by a digital oscilloscope. Measurements have shown, that the magnetic attraction force leads to a non-harmonic excitation. Karami, Farmer, and Inman (2012) proposed how the non-harmonic excitation of the bimorph can be simulated by discretization and numerical integration of the non-linear equation of motion. Nevertheless we want to submit an analytical modeling approach in the next section.

Modeling

In this section linear analytical models for the piezoelectric bimorph (section “Analytical Model of a Piezoelectric Parallel Bimorph for Steady-State, Harmonic Vibrations”) and for the solenoid (section “Electromagnetic Solenoid”) of the hybrid energy harvester are presented.

The model for the bimorph applies to steady-state and harmonic system oscillations. However, it is assumed that the energy harvester is excited non-harmonically by the magnetic force. In order to take even non-harmonic system behavior into account, the non-harmonic input variables are decomposed into their harmonic components. Using the model harmonic system responses are calculated from each harmonic input variable and then composed to one non-harmonic response (section “Non-harmonic Excitation Force Applied to Linear Oscillating Systems”).

Coupling Effects

The coupling effects between the two transducers can be divided into an electric coupling effect (caused by the electric connected of the transducers) and an electromechanical coupling effect (caused by Lenz law). The electric coupling effect can be avoided by preventing the charge flow between the transducers. Therefore either the transducers should be connected to separate loads or an electric wave rectifier should be installed, as shown in Figure 10. In this paper the transducers are always connected to separate loads.

Figure 10:

Wave rectifier circuits to avoid direct backward coupling effects between piezoelectric and electromagnetic generators.

The electromechanical coupling effect can be determined from measurements. Therefore the power output of one transducer is measured separately while the other is either in short-circuit or in open-loop condition. Measurements on an experimental setup, as shown in Figure 4, have led to the assumption that the electromechanical coupling effect is negligible. The differences in power output between the operation modes are so small that they can be regarded as being within the accuracy of a measurement. Thereby the electromechanical coupling is not considered in the model.

Analysis

In this analysis the open-electrode voltage is used as a reference to compare the time domain signals and therewith the systems dynamics of the model and the prototype device. For the bimorph Figure 11 shows the simulated versus the measured voltage signals in time domain during one revolution period for three exemplary chosen revolution speeds. The simulation data is a reconstruction from the transfer function and the excitation force as described in section “Modeling”. As electric boundary condition the load impedance Zload is set to infinity, this is equal to open-electrode condition. It is evident that the simulations (Figure 11 red) can reproduce the real system behavior (Figure 11 blue) almost exactly. From the Fourier coefficients on the bottom of Figure 11 and from the signals theirself it can be seen that the rate of higher harmonic oscillations in the simulated signal is marginally overrated. Since there are four magnets on circumference of the disc the dominant Fourier coefficient index is k=4. Therefore the Fourier coefficients are normalized to this value. From this it can be followed that at higher bending modes the calculated transfer function differs slightly more from the real system. However at lower frequencies it matches the system almost exactly.

Figure 11:

Comparison between simulation data and measurements of the open-electrode voltage at the bimorphs electrodes and corresponding Fourier coefficients for different revolution speeds.

Figure 12 shows a comparison between the measurement of open-electrode induction voltage from the solenoid and a calculation at a certain rotation speed. It is evident that both waveforms are very similar, this is a proof for a correct solenoid model. The magnitude of the measured voltage signal is bigger as the real system has got a high permeability core, which increases the magnetic flux flowing through the solenoid. Comparing the RMS values of both signals a correction factor of 1,4 is calculated. This factor is almost constant for all considered revolution speeds.

Figure 12:

Comparison between simulation data and measurements of the open-electrode voltage at the solenoids electrodes. Simulation data is scaled to measurements magnitude.

For optimization purposes of an energy-harvesting system the study of the power output for different excitation speeds and electric loads is very important. The RMS power output in this analysis is calculated from the multiplication of the current and voltage across a varying load resistance (see Figure 4 for experimental setup). Figure 13 shows a comparison of the simulation data based on the models in section “Modeling” and measurements on the prototype. Both, the bimorphs model as well as the solenoids model, show a very good agreement with the measurements. From this data it is evident that the maximum power output of the two transducers is reached at different electric loads (marked by black dotted lines in Figure 13).

Figure 13:

Comparison of power-RMS values between simulation data and measurements at the piezo-electrodes (above) and solenoid (down) for different electric loads.

The optimum load-resistance for the bimorph is Zopt,piezo=100 kΩ and the optimum for the solenoid is reached at only Zopt,sol=450Ω. Figure 14 shows the comparison between RMS power output of the models and the measurements for the optimum load resistance and underline the correlation between model and measurement results. Measurements and simulation show that the solenoid provides more power at low electric loads and decreases strongly at higher loads (Figure 15). Figure 15 compares directly the simulated power output of the two transducers for different loads and excitation speeds. At higher loads and lower excitation speeds the power output of the bimorph overexceeds the power output of the solenoid.

Figure 14:

RMS power output at optimum electric load for the piezoelectric bimorph (left) and the solenoid (right).

Figure 15:

RMS power of the bimorph and solenoid model.

This power compensation is the intended benefit of the hybrid concept.

Finally, it can be stated that with our modeling approach it is possible not only to prove the performance of the prototype but also to optimize our concept for different load conditions and excitation frequencies. With a proper adjustment of the system parameters a broadband behavior can be achieved.

Conclusion

The volume of an energy-harvesting system is often limited by a defined space. In addition to this the excitation frequency and the electric load may vary with time. In this paper a modeling approach for a hybrid piezoelectric and electromagnetic energy-harvesting concept has been proposed which enables to optimize system parameter for maximum broadband energy harvesting. Therefore a brief overview of existing piezoelectric, electromagnetic and hybrid energy-harvesting concepts for rotational and vibrational energy sources was given. From this a new approach for rotating applications was developed and presented. The concept is based on a piezoelectric parallel bimorph, structural combined with a solenoid on its tip. Permanent magnets attached to a rotating object excite the bimorph and induce a voltage into the solenoid. Since the magnetic attraction force is non-harmonic, a method was submitted, which allows to determine the electric output of the bimorph by Fourier transformation and superimposition. For the bimorph an analytical model in the form of a compact transfer matrix is derived. The model is based on the Euler–Bernoulli beam theory and considers the piezoelectric coupling as well as the multilayer aspects of a bimorph. Compared to a FEM calculation, this method offers a reduced computation time, quick investigation of load influences as well as a simple way to combine several passive and active elements by matrix multiplication.

The calculated electric output signals of the bimorph and of a simple modeled solenoid have been compared to measurements of a prototype device. The comparison of these signals has shown that the simulation matches the measurements almost exactly and thus the modeling can be used to optimize the device in future. The RMS values of the power output, which depend on revolution speed and electric load, were measured and simulated to evaluate the frequency-domain performance of the device. The concept shows a broadband behavior and an equalization of the power drops over a stand-alone piezoelectric vibration harvester, which is a primary goal in energy-harvesting research. Thus the concept is ideal for application with highly fluctuating revolution speeds.