Mechanical effect of combined piezoelectric

and electromagnetic energy harvesting

Mickael LALLART∗and Daniel J. INMANDepartment of Mechanical Engineering, Center for Intelligent Material Systems and Structures

Virginia Tech, Blacksburg, Virginia, 24061, USA

ABSTRACTThe recent progress in microelectronics, combined with the increasing demand from numerous industrial fields interms of autonomous sensor networks, has placed a particular attention on ways to power up such devices. In orderto bypass the drawbacks of batteries, harvesting power from ambient sources has been proposed. Among all theavailable sources, particular attention has been placed on scavenging energy from vibrations, which are commonlyavailable in many environments. In this case, converting mechanical energy into electrical energy is typically doneusing either piezoelectric or electromagnetic transduction. Recently, the combination of these two conversion mech-anisms has been proposed, allowing benifits from the advantages of the two techniques and leading to the conceptof hybrid energy harvesting. This paper proposes an investigation of the combination of the two conversion effectson the mechanical behavior of the host structure, both in terms of damping and stiffness changes. Particularly, forhighly coupled, weakly damped systems, it is shown that combining the piezoelectric and electromagnetic effectsdoes not lead to a power increase, but allows enhancing both the bandwidth and the load independency of theharvester.

1 INTRODUCTION

The increasing demand in terms of consumer electronics has raised the issue of powering up devices using ambientharvested energy in order to replace batteries that raise maintenance and environmental issues. Hence, the futureof sensing should be self-powered, allowing devices to operate using the surrounding energy sources for supply-ing electrical circuits ([1, 2, 3]). Such a trend is encouraged by progresses in micropower electronics, as well as anincreasing demand from various industrial fields (for instance aeronautic, civil and biomedical engineering, andhome automation) in terms of “place and forget” sensors and sensor networks.

Among all the available energy for small-scale devices (solar, magnetic or thermal), a particular attention has beenplaced on vibration energy harvesting, as mechanical energy from vibrating parts is one of the most commonlyavailable sources in many environments ([4]). When dealing with electromechanical conversion, several physicaleffects can be considered, but the two most common ones rely on piezoelectricity ([5, 6, 7, 8, 9, 10, 11]) and electro-magnetism ([12, 13, 14]). While a significant number of studies on energy harvesting has been devoted to only oneor the other conversion effect, a recent trend consisted in combining piezoelectric and magnetic energy harvestinginto a single device, leading to the concept of hybrid energy harvesting ([15]).

The purpose of this paper is to investigate the mechanical effects that arise when using such a harvester, demon-strating both the frequency shift and damping effects generated by the harvesting processes. The paper is orga-nized as follows. Sections 2 and 3 review the basic principles of energy harvesting using piezoelectric and magnetictransducers, and expose the derivation of the expected output power and the effect of harvesting on mechanicalvibrations when using both transduction mechanisms. Section 4 aims at validating the proposed model throughexperimental measurements, and Section 5 finally concludes the paper.

2 HYBRID HARVESTING PRINCIPLES

When designing an energy harvester that aims at supplying an electrical circuit from vibrations, the basic operationconsists of rectifying the voltage of the transducer. This is typically done using AC/DC converters. However, from

∗mickael.lallart@insa-lyon.fr

Proceedings of the IMAC-XXVIIIFebruary 1–4, 2010, Jacksonville, Florida USA

©2010 Society for Experimental Mechanics Inc.

the transducer’s point of view and assuming that only the first harmonic of the vibrations has a significant effect,the converter can be seen as a single linear load represented by its internal impedance (Figure 1)1.

Under these assumptions, the energy harvesting processes can be considered separately in an electrical point ofview (but not in terms of mechanical effect). As well, the dual nature of the piezoelectric and electromagneticeffects leads to the following differences ([16]):

• Mechanical aspects and positioning:

– Piezomaterials are sensitive to the absolute strain within the structure, and have to be placed near aclamped edge for simple structures such as cantilever beams.

– Electromagnetic elements are sensitive to the relative speed of the structure with respect to the secondpart of the transducer (e.g., coil), and have to be placed near a free end for simple structures such ascantilever beams (large displacement).

• Electrical aspects:

– Piezomaterials act as voltage sources, these latter being proportional to the strain or stress (or, in amacroscopic point of view, to the displacement).

– Magnetic elements act as current sources, these latter being proportional to the relative speed.

• Representation:

– Piezomaterials are equivalent in the electrical point of view to a current source in parallel with a capaci-tor, and in a mechanical point of view to a force in parallel with a spring.

– Magnetic elements are equivalent in the electrical point of view to a voltage source in series with aninductor, and in a mechanical point of view to a force in series with a mass.

3 THEORETICAL DEVELOPMENT

Based on the previous statements, this section proposes to derive the power expression obtained when using ahybrid device, as well as to investigate the mechanical effect of harvesting, in terms of resonance frequency shiftand vibration damping, induced by the combination of piezoelectric and magnetic energy harvesting.

3.1 Modeling

In the followings, a simple single degree of freedom (SDOF) model of the mechanical structure that relates quitewell the mechanical behavior of the device near one of its resonance frequencies ([17, 18]) is considered. It can beshown that the governing electromechanical equations of the entire device are given by ([16]):

Figure 1: Hybrid energy harvesting schematic

1which can nevertheless depends on the load and frequency

Mu + Cu + KEu = −µ1Ma − αVp − AIm

Ip = αu − C0Vp

Vm = Au − L0Im − rLIm

, (1)

where the parameters are those defined in Table 1. The duality of the transducers can also be demonstrated fromthese equations (e.g., piezovoltage vs. magnetic current, piezocapacitance vs. magnetic inductance...).

3.2 Power derivation

When connecting each transducer to its corresponding converter that presents a given input impedance, the elec-trical equation of Eq. (1) becomes:

Vp

Rp

= αu − C0Vp

RmIm = Au − L0Im − rLIm

, (2)

where Rp and Rm represent the piezoelectric and electromagnetic equivalent loads, respectively. Hence, expressingthese relationships in the frequency domain yields:

Vp(ω) =jωRpα

1 + jRpC0ωu(ω)

Im(ω) =jωA

(Rm + rL) + jLωu(ω)

, (3)

and therefore the transfer function giving the displacement as a function of the applied acceleration is given by:

u(ω)

a(ω)=

−µ1M

−Mω2 + jCω + KE +jωRpα

2

1 + jRpC0ω+

jωA2

(Rm + rL) + jLω

. (4)

Hence, for weakly damped systems, the resonance frequency ω0 can be found by cancelling the real part of thedenominator, leading to:

[

(Rm + rL)2

+ (Lω0)2

]

+ C0 (RP αω0)2

[

(Rm + rL)2

+ (Lω0)2

]

+ L (Aω0)2

[

1 + (RpC0ω0)2

]

= 0, (5)

and the corresponding vibration magnitude is given by:

Parameter Definition

MechanicalM Dynamic massC Structural damping coefficientKE Natural stiffnessµ1 Correction factor ([18])a Base acceleration

ElectricalVp Piezoelectric voltageIp Piezoelectric currentC0 Piezoelectric transducer clamped capacitanceVm Electromagnetic voltageIm Electromagnetic currentL0 Electromagnetic transducer inductancerL Electromagnetic transducer leakage resistance

Electromechanicalα Piezoelectric force factorA Magnetic force factor

Table 1: Model parameter definition

um|res =µ1MaM

Cω0 +ω0Rpα

2

1 + (RpC0ω0)2

+ω0A

2 (Rm + rL)

(Rm + rL)2

+ (Lω0)2

. (6)

Consequenly, these last two equations demonstrate that the harvesting process induces both a resonance frequencyshift of the structure (Eq. (5)) and a damping of the vibrations at the resonance (Eq. (6)), depending on both thepiezoelectric and magnetic energy harvesting processes.

The power harvested only by the piezoelectric element can be expressed as:

Pp =VpVp

∗

2Rp

=1

2

Rpω02α2

1 + (RpC0ω0)2uM

2? (7)

with uM the vibration magnitude at the considered frequency (obtained using Eq. (4)), while the power harvestedonly by the magnetic element is given by:

Pm =1

2RmImIm

∗ =1

2

Rmω02A2

(Rm + rL)2

+ (Lω0)2uM

2. (8)

Hence, when combining both transduction mechanisms, the total output power yields:

Ptotal =

[

Rpα2

1 + (RpC0ω0)2

+RmA2

(Rm + rL)2

+ (L)2

]

×(ω0uM )

2

2. (9)

3.3 Theoretical comparison

Here it is proposed to compare and evaluate the performance and mechanical effects of harvesting energy by com-bining both the piezoelectric and electromagnetic conversion principles, with regard to systems using only one ofthem.

When considering that the system is excited at its resonance frequency (Eq. (5)), the corresponding expected outputpower and displacementt magnitude are given in Figure 2 when each transducer has the same coupling coefficientdefined as:

kp =√

α2/ (C0KE + α2) for the piezo device

km =√

A2/ (L0KE + A2) for the magnetic device

, (10)

and the resistive losses of the magnetic transducer are null, Figure 3 when each transducer has the same couplingcoefficient but the magnetic transducer presents resistive losses, and Figure 4 when the coupling coefficients ofthe two transducers are different and taking into account resistive losses of the magnetic device. These charts arenormalized along the x and y-axis according to the optimal loads given by:

(Rp)opt=

1

C0ω0

for the piezo

device

(Rm)opt =

√

(L0ω0)2

+ rL2 for the magnetic

device

, (11)

where the magnetic transducer is considered as perfect (i.e. rL = 0), and along the z-axis according to the maximalharvested power ([7]):

Pmax =(µ1MaM )

2

8C. (12)

These figures are indexed with repsect to the figure of merits given by the product of the mechanical quality factorQM of the structure (giving the amount of energy that is available) with the squared coupling coefficient k2 (reflect-ing the part of the available energy that can actually be harvested).

(a) (b)

(c) (d)

Figure 2: Evolution of the normalized displacement magnitude and normalized harvested power as a function ofnormalized loads for several values of k2QM and same coupling coefficients between the piezoelectric and magnetictransducers, with a perfect electromagnetic transducer (rL = 0)

(a) (b)

(c) (d)

Figure 3: Evolution of the normalized displacement magnitude and normalized harvested power as a function ofnormalized loads for several values of k2QM and same coupling coefficients between the piezoelectric and magnetictransducers, with an imperfect electromagnetic transducer (angle of losses: ϕL = 84 ˚ )

(a) (b)

(c) (d)

Figure 4: Evolution of the normalized displacement magnitude and normalized harvested power as a functionof normalized loads for several values of k2QM and different coupling coefficients between the piezoelectric andmagnetic transducers, with an imperfect electromagnetic transducer (angle of losses: ϕL = 84 ˚ )

These figures demonstrate that combining both piezoelectric and magnetic devices leads to an increase of the har-vested energy for low coupled and/or strongly damped structures, but as the coupling coefficient and/or the me-chanical quality factor increase, the damping effect cannot be neglected any longer, leading to a harvested powerlimit and a split in the optimal loads. In this latter case, compared to classical energy harvesting techniques thatconsider only one conversion effect, the combination of the piezoelectric and magnetic devices allows decreasingthe sensitivity to a load shift (e.g. a shift in the piezoelectric load would be compensated by an increase of theharvested energy by the magnetic transducer, and conversely), however without allowing a global gain in termsof harvested energy as the power limit is reached. In addition, when taking into account the internal losses in theelectromagnetic transducer, the expected power output significantly decreases for low magnetic load values. Thisis explained by the fact that the same energy is extracted from the device (so that the mechanical effect is the same),but a great part of this energy is dissipated in the loss resistance. In a mechanical point of view, it can be seen thatharvesting energy leads to vibration damping at the resonance, the damping effect being greater as the harvestedenergy increases, whether by a better load matching or by an increase of the figure of merit k2QM .

The frequency behavior of the hybrid harvester, as well as the comparison with single conversion harvesters, isdepicted in Figure 5. As previous, the power is normalized with respect of the maximal harvested power Pmax,while the frequency axis is normalized such that 0 is the resonance frequency and ±0.5 the −3 dB cut-off frequencies.

−5 0 5 10 15 20 25

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency

Pow

er

Piezo only (k2QM

=0.18)

Magnetic only (k2QM

=0.18)

Hybrid (k2QM

=0.36)

Piezo only (k2QM

=1.9)

Magnetic only (k2QM

=1.9)

Hybrid (k2QM

=3.8)

Piezo only (k2QM

=5.1)

Magnetic only (k2QM

=5.1)

Hybrid (k2QM

=10)

(a)

−5 0 5 10 15 20 25

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency

Pow

er

Piezo only (k2QM

=0.18)

Magnetic only (k2QM

=0.18)

Hybrid (k2QM

=0.36)

Piezo only (k2QM

=1.9)

Magnetic only (k2QM

=1.9)

Hybrid (k2QM

=3.8)

Piezo only (k2QM

=5.1)

Magnetic only (k2QM

=5.1)

Hybrid (k2QM

=10)

(b)

−5 0 5 10 15 20 25

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency

Pow

er

Piezo only (k2QM

=0.18)

Magnetic only (k2QM

=0.045)

Hybrid (k2QM

=0.22)

Piezo only (k2QM

=1.9)

Magnetic only (k2QM

=0.49)

Hybrid (k2QM

=2.4)

Piezo only (k2QM

=5.1)

Magnetic only (k2QM

=1.4)

Hybrid (k2QM

=6.5)

(c)

−5 0 5 10 15 20 25

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency

Pow

er

Piezo only (k2QM

=0.18)

Magnetic only (k2QM

=0.045)

Hybrid (k2QM

=0.22)

Piezo only (k2QM

=1.9)

Magnetic only (k2QM

=0.49)

Hybrid (k2QM

=2.4)

Piezo only (k2QM

=5.1)

Magnetic only (k2QM

=1.4)

Hybrid (k2QM

=6.5)

(d)

Figure 5: Evolution of the normalized maximal harvested power as a function of the normalized frequency forseveral values of k2QM , same ((a);(b)) and different ((c);(d)) coupling coefficients between the piezoelectric andmagnetic transducers, with a perfect ((a);(c)) and imperfect ((b);(d)) electromagnetic transducer (angle of losses:ϕL = 84 ˚ )

Parameter Value

MechanicalDynamic mass M 40 gStructural damping coefficient C 0.14 N.s.m−1

Natural stiffness KE 450 N.m−1

Correction factor µ1 12

ElectricalPiezoelectric clamped capacitance C0 54 nFMagnetic transducer inductance L0 3 mHMagnetic transducer leakage resistance rL 4.3 Ω

ElectromechanicalPiezoelectric force factor α 0.82 mN.V−1

Electromagnetic force factor A 1.2 N.A−1

Table 2: Experimental model parameter identification

This chart clearly shows that the use of the hybrid configuration allows a great enhancement of the harvester, asthe resonance frequency range is extended, starting from the case where the piezoelectric element is short-circuitedand the electromagnetic element left in open circuit, with the corresponding resonance frequency given by:

(ω0)low =√

K/M, (13)

to the configuration where the piezoelectric element is in open circuit condition and the electromagnetic elementshort-circuited, where the resonance frequency is equal to:

(ω0)high =√

(K + α2/C0 + A2/L0) /M. (14)

Hence the combined effect of piezoelectric and magnetic actuation permits to siginificantly extend the effectiveenergy harvesting bandwidth.

4 EXPERIMENTAL VALIDATION

4.1 Experimental set-up

This section aims at validating the previously exposed results giving the theoretical output power and displacementmagnitude. The experimental setup, depicted in Figure 6, consists of a cantilever beam bonded with piezoelectricinserts on each side, hence shapping a bimorph piezoelectric transducer. One of the bimorphs is used for vibrationsensing ; hence only one piezoelectric material is connected to the harvesting circuit. The beam also features atip mass made of rare earth permanent magnets that go into a coil, therefore constituing the magnetic transducer.The system is vibrated at approximately at 0.1 g peak using a shaker, driven by a function generator through apower amplifier. The piezoelectric and magnetic transducers are each connected to a pure resistor, simulating theinput impedance of the AC/DC converter, in a independent fashion, and the signal waveforms are monitoredusing a digital oscilloscope. Preliminary measurements have also been performed in order to identify the modelparameters, which are listed in Table 2.

Figure 6: Hybrid energy harvesting experimental setup

4.2 Experimental results and discussion

The first set of measurements consisted of exciting the structure at its resonance frequency3 for several piezoelectricand electromagnetic load values. Results showing the harvested power and displacement magnitude as a functioinof the connected load are depicted in Figure 7(a) and in Figure 7(b), respectively. Theoretical predictions are alsoshown in these chart for comparison.

These results show a good agreement between the previously proposed model and the reality. Hence, for low mag-netic load values, the parasitic resistance of the electromagnetic transducer absorbs the major part of the extractedenergy, leading to a low harvested energy and a significant damping effect. As the electromagetic load increases,both displacement and harvested power increase, almost independently to the value of the piezoelectric load, untilthe electromagnetic load reaches its optimal value. Around this point, the converse piezoelectric effect also affectsthe mechanical behavior of the structure, and the displacement is strongly related to the value of the piezoelectricload. Once the magnetic load becomes far greater than its optimal value, only the piezoelectric effect is active interms of energy harvesting (the magnetic transducer only influencing the resonance frequency of the system, notthe damping ratio).

It can also be noted that there is no split in the optimal loads, as the figure of merit k2QM is below its critical valueand the leakage resistance of the magnetic transducer is relatively large. Another interesting point, also shown bythe theoretical predictions, is that the optimal load pair giving the maximum harvested power is not exactly equal tothe couple given by the optimal load of the piezoelectric element alone and the optimal load of the electromagneticelement alone.

The behavior of the system in terms of harvested power as a function of the frequency is also depicted in Figure 8,along with theoretical predictions (considering either hybrid, piezoelectric or eletromagnetic harvesting) for com-parison. Again, the experimental results closely match the theoretical predictions. As the figure of merit k2QM ofeach transducer is below its critical value and as the leakage resistor rL is important, the combination of the piezo-electric and the electromagnetic transducers allows a gain in terms of harvested energy. This gain is equal to 10%compared to the piezoelectric element alone and 30% compared to the electromagnetic element alone. The band-width is also slightly enhanced, going from 0.9 Hz and 0.83 Hz in the purely electromagnetic and purely piezolectric

104

105

106

0

0.5

11.5

Rp (Ω)

P (

mW

)

Rm

=1Ω

104

105

106

0

0.5

11.5

Rp (Ω)

P (

mW

)

Rm

=5.6Ω

104

105

106

0

0.5

11.5

Rp (Ω)

P (

mW

)

Rm

=10Ω

104

105

106

0

0.5

11.5

Rp (Ω)

P (

mW

)

Rm

=22Ω

104

105

106

0

0.5

1

1.5

Rp (Ω)

P (

mW

)

Rm

=82Ω

Experimental results

Theoretical predictions

(a)

104

105

106

0

1

2

Rp (Ω)

u (m

m)

Rm

=1Ω

104

105

106

0

1

2

Rp (Ω)

u (m

m)

Rm

=5.6Ω

104

105

106

0

1

2

Rp (Ω)

u (m

m)

Rm

=10Ω

104

105

106

0

1

2

Rp (Ω)

u (m

m)

Rm

=22Ω

104

105

106

0

1

2

Rp (Ω)

u (m

m)

Rm

=82Ω

Experimental results

Theoretical predictions

(b)

Figure 7: Experimental and theoretical results at the resonance frequency, as a function a the piezoelectric andelectromagnetic transducers’ load: (a) harvested power; (b) displacement magnitude

3which varies with the connected load according to Eq. (5)

15 15.5 16 16.5 17 17.5 18 18.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Frequency (Hz)

Pow

er (

mW

)

Experimental resultsTheoretical predictionsPiezo only (theoretical)Magnetic only (theoretical)

Figure 8: Theoretical and experimental maximum harvested power at a function of the frequency

case respectively, to 1.1 Hz in the hybrid case. This bandwitdh magnification would be even more important in thecase of strongly coupled systems, as shown in Section 3.3.

5 CONCLUSION

This paper investigated the effect of combined piezoelectric and electromagnetic energy harvesting from vibra-tions. Based on a simple SDOF model that approximates quite well the behavior of such a harvester near one ofits resonance frequencies, it has been demonstrated that both conversion effects lead to damping and a shift in theresonance frequency. The study also showed that, although the use of a hybrid energy harvesting can increase theenergy level for low coupled, highly damped structures, the harvested energy has the same limit as devices usingonly one conversion principle due to damping effect when the electromechanical coupling is high and the structureweakly damped. However, in this case, it has been also demonstrated that the use of both piezoelectric and mag-netic energy harvesting leads to a greater robustness when facing load drifts, as well as an enhanced bandwitdh interms of harvested power and vibration magnitude.

Acknowledgements

The authors would gratefully acknowledge the support of the U.S. Department of Commerce, National Instituteof Standards and Technology, Technology Innovation Program, Cooperative Agreement Number 70NANB9H9007,and the Air Force Office of Scientific Research MURI Grant No. F955-06-1-0326.

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