comparison between electromagnetic and piezoelectric...

Comparison between electromagnetic and piezoelectric seismic vibration energy harvesters

L. Dal Bo1, P. Gardonio

1

1Università degli Studi di Udine, DPIA Via delle Scienze,208-33100 Udine, Italy

e-mail: [email protected]

Abstract This paper contrasts the principal characteristics of seismic harvesters using either electromagnetic or

piezoelectric transducers. The electromagnetic seismic harvester is formed by a cylindrical magnetic

element with an inner gap where a coil is housed. The two components are connected via soft springs and

the coil is fixed to the case of the harvester. The piezoelectric seismic harvester is formed by a cantilever

beam with a small block mass at its tip. The beam is fixed to the harvester case and is equipped with

piezoelectric patches, which are bonded on its top and bottom surfaces. The two systems are modelled

with consistent electro-mechanical lumped parameter models. This allows the derivation of a unified

formulation for the energy harvesting and thus a direct comparison of the electro-mechanical response and

energy harvesting properties of the two harvesters. The study presented in this paper is limited to the case

of energy harvesting from tonal ambient vibrations.

1 Introduction

The process of energy extraction from ambient vibrations is generally known as energy/power

harvesting/scavenging. This approach has recently attracted a great deal of interest by researchers and by

industries, as a potential local source of energy to run low-power consumption devices. As summarised in

the books and the articles listed respectively in Refs. [1-4] and [5-14] for example, a rather large number

of vibration harvesting architectures and applications have been studied during the past two decades.

Nevertheless, in most cases, the energy harvesters are formed by seismic transducers that effectively

convert vibration energy into electrical energy. Normally they are formed by a proof mass suspended on

an elastic system with in parallel an electro-mechanical transducer [15-19]. The transducer is connected to

an electrical circuit, which is conceived to accomplish three tasks: first, to maximise the energy absorption

via the electro-mechanical seismic transducer, second, to store the absorbed energy into supercapacitors or

small batteries and third to use the stored energy to activate a low power electrical apparatus. Typically,

the apparatus is activated for short periods of time and runs very low power consumption electrical circuits

that, for example, implement control actions, condition monitoring functions, communication tasks, etc. In

general, seismic harvesters can be used to effectively absorb energy from tonal or broadband ambient

vibrations. In the former case, the response of the mechanical spring–mass system is designed to have a

sharp resonance peak at the targeted frequency, while in the second case it is designed to have a broad and

smooth resonance peak in the frequency band of interest. Often, non-linear effects are used to enhance the

stroke of the harvesting transducer and operation frequency-band of the harvester [20-24].

Energy harvesting systems provide tiny quantities of power that go from the order of milli-watts, for

devices smaller than 1 kg, to the order of watts, for devices larger than 1 kg, which are normally used in

civil constructions [11,19,20,25,26]. Several applications have been explored over the years [11,27]. For

example, energy harvesting devices have been embedded on the human body to power medical devices.

Clothes have been equipped with smart piezoelectric films to power low-energy electronic apparatuses.

Energy harvesting systems have been mounted on transportation vehicles and industrial machinery to run

condition monitoring devices. Buildings and transport infrastructure have been equipped with energy

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harvesters to power condition monitoring devices and low power consumption electronic equipment.

Smart grass energy harvesting systems have been developed to collect energy from the flow of air in fields

or the flow of water in rivers [28].

Two principal configurations of seismic transducers have emerged from the vast bulk of works on

vibration energy harvesting. The first, uses a coil-magnet or voice-coil transducer [2,29-32], while the

second uses piezoelectric patch transducers [3,31,32]. Elliott and Zilletti [20] have presented a seminal

paper with an overview of the principal characteristics of the coil-magnet seismic transducer and a

detailed analysis on the efficiency of vibration energy harvesting with reference to the scale of the

transducer.

The work presented in this paper investigates and contrasts the principal properties of seismic transducers

for vibration energy harvesting using either coil-magnet or piezoelectric patch transducers. Consistent

lumped parameter models and impedance formulations are introduced for the two types of seismic

transducers, so that a unified energy formulation is derived to analyse and contrast the harvesting

properties of the two systems in a consistent framework.

The paper is structured in four sections. Section two presents the electromagnetic and piezoelectric

seismic harvesters considered in this study and introduces the lumped parameter models and impedance

formulations used to derive the dynamic response of the two seismic transducers. Also, it presents a

detailed analysis of the impedance and transduction frequency response functions (FRF) that characterise

the constitutive equations of the two seismic transducers. Section three introduces the energy formulation,

which is then used to study the conditions to maximise the vibration energy harvesting. Finally, section

four summarises the main results of this study.

2 Seismic Transducer

Figure 1 shows a) the pictures, b) the schematics and c) the lumped parameter models of (I) the coil-

magnet and (II) the piezoelectric patch seismic transducers for vibration energy harvesting considered in

this study. As shown in picture (Ia), the coil-magnet seismic transducer is composed by a magnetic

cylindrical element, which is characterised by an inner cylindrical cut where it is housed a coil rigidly

connected to the case of the actuator. The magnetic element is also connected to the case via soft axial

springs, which allow a relative motion between the coil and the magnetic element, though preserving the

static connection. As shown in picture (Ib), the, magnetic element is composed by an outer ferromagnetic

thick walled cylinder and an inner cylindrical magnet, which are connected by a ferromagnetic disc cap.

This construction produces in the inner air gap a uniform magnetic field in radial direction. As a result, the

relative motion between the magnetic element and the coil generates a voltage at the terminals of the coil

and, vice versa, a current flow in the coil generates back reactive forces on the magnetic element and the

coil [29-32]. As shown in picture (IIa), the piezoelectric seismic transducer is composed by a thin

cantilever beam clamped to the case of the harvester. The beam is equipped with a tip block mass and two

piezoelectric transducers bonded on the top and bottom faces. In this case the bending strain of the

cantilever beam produces an electric displacement in the electrodes of the piezoelectric patches, and thus a

current flow through the terminals when the patches are connected to the harvesting electrical circuit.

Alternatively, the voltage generated at the terminals of the transducer produce a back bending strain effect

on the cantilever beam.

2.1 Lumped Parameter models

The dynamic response of the two inertial transducers is derived with respect to two equivalent lumped

parameter models, which, as shown in Ic and IIc of Figure 1, are characterised by a mechanical and an

electrical part. The mechanical part is composed by a base mass and a proof mass, which are connected

via a spring and a damper element, with in parallel an idealised reactive force generator, whose strength is

linked to the current flowing in the coil via a complex FRF. The electrical part is composed by an

idealised voltage generator, whose strength is proportional to the stroke velocity via a complex FRF, with

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in series an electrical component, which, for the coil-magnet transducer is formed by a resistor and

inductor in series and for the piezoelectric transducer is formed by a capacitor.

Figure 1: Electro-magnetic (I) and piezoelectric (II) proof mass vibration energy harvesters. (a) pictures,

(b) schematics, (c) lumped models.

The mechanical and electrical parameters in the lumped element model for the coil–magnet inertial

transducer are derived straightforwardly from inspection of the system shown in Ib of Figure 1. Instead,

the mechanical and electrical parameters in the lumped element model for the piezoelectric transducer

require a more complex analysis of the electro-mechanical flexural response of the cantilever beam with

the piezoelectric patches and the tip mass. In brief, the flexural response of the smart beam is derived

considering only the contribution of the first flexural natural mode of the beam with the piezoelectric

patches and tip mass, assuming the beam is clamped at its base. Thus, the lumped parameter model

considers the physical mass of the base element and the modal mass, modal stiffness and modal damping

of the first flexural mode of the clamped smart beam. These modal parameters are specifically normalised

to allow the construction of the lumped parameter model shown in IIc of Figure 1, which considers the

physical base mass and the modal proof mass connected via the modal stiffness and modal damping. This

implies the use of a virtual displacement of the modal proof mass mw , which does not correspond to the

effective displacement of the tip mass tw .

Tables 1 and 2 summarise the principal lumped parameters of the electro-magnetic and piezoelectric

seismic harvesters considered in this study.

DYNAMICS OF ENERGY HARVESTERS 683

Parameter Value

Base mass kg 10115 3bm

Proof mass kg 10185 3mm

Spring stiffness N/m 2921mk

Damping coefficient Ns/m 274.7 mc

Natural frequency Hz 20nf

Damping ratio 6.0 16.0

Transduction

coefficient N/A 5.22Bl

Resistance 22eR

Inductance mH 35.4eL

Table 1: Lumped parameters for the

electromagnetic seismic harvester

Parameter Value

Base mass kg 10115 3bm

Proof mass kg 10161 3mm

Spring stiffness N/m 2562mk

Damping coefficient Ns/m 4.245.6 mc

Natural frequency Hz 20nf

Damping ratio 6.016.0

Transduction

coefficient N/V 0077.0

Capacitance F 10845.2 7pC

Equivalent

coordinate m 1086.1 3x

Table 2: Lumped parameters for the

piezoelectric seismic harvester

2.2 Constitutive equations

Considering time-harmonic vibrations described with phasors given in the form )exp()()( tjwtw ,

where )(w is the complex amplitude, is the circular frequency and 1j , the electromechanical

response of the two lumped parameter models can be expressed in the frequency domain with the

following two equations

hfibmib iTwZf , (1)

heibewh iZwTe , (2)

where bf ,

bw are the complex amplitudes of the force and velocity at the base of the transducer and he ,

hi are the complex amplitudes of the voltage and current across the terminals of the transducer. The

mechanical and electrical impedance FRFs and the two electromechanical transduction FRFs are given by:

a

mmm

b

ib

b

miZ

ZZZ

w

fZ

h

0

, (3)

a

mm

fi

wa

b

fiZ

Z

i

fT

b

0

, (4)

a

ewfi

e

wa

a

eiZ

Zi

eZ

b

0

, (5)

a

mmew

ib

aew

Z

Z

w

eT

h

0

, (6)

where

mmm mjZ , (7)


bb mjZ . (8)

Also, for the coil magnet transducer

eee LjRZ , (9)

m

mm c

j

kZ

, (10)

mmm

mmma mjcj

kZZZ

, (11)

while for the piezoelectric transducer

p

eCj

Z1

, (12)

e

ewfi

mm

mZ

cj

kZ

, (13)

m

e

ewfi

mm

mmma mjZ

cj

kZZZ

. (14)

Finally, for the coil magnet transducer

Blfi , (15)

Blew , (16)

where � is the magnetic field in the gap where the coil is housed and � is the length of the winding, and for

the piezoelectric transducer,

efi Z , (17)

eew Z , (18)

with

Lx

dx

xd

x

)(

)(

1 1

1

, (19)

where )(1 x is the first flexural mode of the clamped beam, x is the position of the virtual displacement

mw of the modal tip mass. Also, )(31 sp hhbe represents the piezoelectric coupling coefficient for

flexural vibrations of the beam and two piezoelectric patches. Here 31e is the stress/charge constant for the

piezoelectric material, b is the width of the beam and piezoelectric patches and sh ,

ph are the thicknesses

of the beam and piezoelectric patches respectively. The terms fiT and ewT appearing in Eqs. (1) and (2)

represent the electromechanical coupling coefficients for both seismic transducers, which give the base

force effect produced by the transducer per unit current in the blocked seismic transducer, i.e.

0

bwabfi ifT

, and the electric voltage at the terminals of the transducer per unit velocity at the base of

the open circuit seismic transducer, i.e. 0

hi

baew weT . Also, 0

bwaaei ieZ

is the electrical impedance

of the blocked seismic transducer while 0

hi

bbmi wfZ is the mechanical impedance of the open circuit

seismic transducer.


2.3 Impedance and transduction frequency response functions

Figures 2 and 3 show the 1 Hz – 2 kHz spectra of the mechanical and electrical impedances and the two

electromechanical transduction FRFs that form the constitutive Eqs. (1) and (2) for the electromagnetic

and the piezoelectric seismic transducers shown in Figure 1. Considering first the FRFs for the

electromagnetic seismic transducer, the following considerations are made. The mechanical impedance

miZ given in Eq. (3) is characterized by mass behaviours at low and high frequencies, which are

proportional respectively to the total mass of the transducer mb mm and to the mass of the magnetic

elementmm . These two asymptotic behaviours are connected via a peak at about the fundamental natural

frequency of the transducer, i.e. 20 Hz, and an antiresonance low at about 35 Hz. The transition between

the two mass effects is effectively smoothened when the coil – magnet assembly is characterized by high

viscous damping produced by the air flow in the narrow cut of the magnetic element where the coil is

housed. As can be deduced from Eq. (5), the electrical impedance eiZ is given by the superposition of the

electrical impedance of the coil eZ and the electro-mechanical impedance aewfi Z . The electrical

impedance of the coil eZ is characterized by a low frequencies resistive effect and, above the cut off

frequency at about 800 Hz, a higher frequencies inductive effect. The electro-mechanical impedance

aewfi Z introduces a peak, at about the fundamental natural frequency of the seismic transducer, i.e.

20 Hz.

Figure 2: mechanical impedance, electrical impedance and electromechanical transduction, FRFs for the

electromagnetic seismic transducer with damping ratio � = . 6 (solid black line) and � = .6 (dashed blue line).


The electromechanical transduction coefficients fiT and ewT in Eqs. (4) and (6) are identical. At low

frequencies the modulus rises proportionally to 2 towards a resonance peak at about the fundamental

natural frequency of the seismic transducer, i.e. 20 Hz. At higher frequencies, the modulus rapidly

decreases towards a constant value, which indicates that, at frequencies above about 50 Hz, the seismic

transducer behaves as a constant force source per unit current flowing into it and as a constant voltage

source per unit stroke velocity. The phase spectrum starts at o0 and undergoes a o180 phase lag in

correspondence to the 20 Hz fundamental natural frequency of the transducer. Thus, at frequencies higher

than the fundamental natural frequency, the constant force-current or voltage-stroke velocity transduction

effects are characterised by a negative sign with respect to the notation indicated in the lumped parameter

model Ic of Figure 1.

Figure 3: mechanical impedance, electrical impedance and electromechanical transduction, FRFs for the

piezoelectric seismic transducer with damping ratio � = . 6 (solid black line) and � = .6 (dashed blue

line).

Moving now to the FRFs for the piezoelectric seismic transducer shown in Figure 3, the following

considerations are made. The spectrum of the mechanical impedance miZ is very similar to that derived

for the electromagnetic transducer. Thus also the mechanical impedance for the piezoelectric harvester is

characterised by mass behaviours at low and high frequencies, proportional respectively to the total mass

mb mm and the suspended mass mm of the cantilever beam with piezoelectric transducers and tip mass,

which are joined up by a resonance peak at about 20 Hz followed by an antiresonance low at about 35 Hz.

When the beam is affected by high damping, the resonance peak and antiresonance low are noticeably

rounded off. Also with the piezoelectric transducer the electric impedance is characterized by the

superposition of the electrical impedance eZ that characterises the piezoelectric transducer and the

electromechanical impedance aewfi Z . As can be noted in Figure 3, the capacitive impedance of the


piezoelectric transducer dominates the spectrum and is characterized by a modulus that drops

proportionally to 1 and phase 090 . The electromechanical impedance effect aewfi Z produces a

little glimpse in the modulus and a little phase variation to about 075 in correspondence to the

fundamental natural frequency of the harvester, i.e. 20 Hz. The electromechanical transduction

coefficients fiT and ewT have in this case the same modulus and opposite phase. The spectrum of the

modulus for the two transduction coefficients fiT and ewT is differs from that found for the

electromagnetic transducer. In fact, at low frequencies, it is characterised by a rising trend proportional to 1 interrupted by a resonance peak at the fundamental natural frequency of the transducer at about 20 Hz.

At higher frequencies, the modulus decreases proportionally to 1 . The phase spectra of fiT and ewT

start respectively at 090 and 090 , and in correspondence to the resonance frequency undergo through

a 0180 phase lag. Thus, in this case, the constant force-current or voltage-stroke velocity transduction

effects are characterised by a reactive behaviour.

3 Energy analysis

Since the constitutive equations for the two seismic transducers have been derived in the same form, a

unified energy formulation is introduced in this section to study the energy harvested by the

electromagnetic and the piezoelectric seismic transducers.

For harmonic vibrations, the time averaged harvested power is given by

T

hT

h dttPT

P0

)(1

lim , (20)

where the instantaneous harvested power is given by

)()()( titetP hhh , (21)

and )(teh, )(tih

are the voltage across and current through the harvesting load. Assuming time harmonic

functions, the following impedance relation holds for the harvesting element:

hhh iZe , (22)

where hZ is the electrical impedance of the harvesting circuit. Also, considering the electric mesh of the

harvesters shown in Figure 1c and recalling that the transducers yield a voltage

beww wTe , (23)

the harvesting current can be derived straightforwardly by analysing the mesh with Kirchhoff’s voltage

law, which gives

b

hei

ewh w

ZZ

Ti

. (24)

Thus, for harmonic vibrations, the time average harvested power is derived from Eqs. (20) to (24) as

follows:

2

2

2Re

2

1e

2

1)( b

hei

ewhhhh w

ZZ

TZiZRtP

. (25)

Eq. (25) suggests that the harvested power depends on the impedance of the harvesting circuit. The

complex impedance that maximizes the harvested power at each frequency can be derived using Fermat’s theorem on the stationary points for n-dimensional functions [33], which, in this case, sets the following

two conditions:


0

Re

h

h

Z

P and

0Im

h

h

Z

P . (26,27)

Solving Eqs. (26) and (27) leads to the following condition for the optimal impedance of the harvesting

circuit:

*

C, eih ZZ , (28)

where * is the complex conjugate operator. The maximum for the harvested power hP is thus obtained

when the electric impedance of the harvesting circuit ChZ , is the complex conjugate of the electrical

impedance of the freely suspended seismic transducer eiZ given in Eq. (5). This result is in line with the

the maximum power transfer theorem [33], which states that a load collects the maximum amount of

power from a source when its resistance is equal to the internal resistance of the source and when the

imaginary part of the impedance of the load is counter phase with the imaginary part of the impedance of

the source.

Substituting the optimal condition for hZ given in Eq. (28) into Eq. (25), the maximum harvested power

results:

2

2

Re8

1b

ei

ew

h wZ

TP , (29)

which, using Eqs. (5) and (6) can be specified respectively for the electromagnetic and for the

piezoelectric seismic harvesters as follows:

2

2

2

8

1b

aefiewm

mewh w

ZRc

mP

, (30)

2

2

8

1b

m

m

h wc

mP

. (31)

Figure 4 shows the real and imaginary parts of the impedances that would maximise the harvested power

at each frequency for the electromagnetic seismic harvester (Plot a) and for the piezoelectric seismic

harvester (Plot b). Plot (a) indicates that, when the electromagnetic seismic transducer is used, the

harvester circuit should be characterised by a real positive part that peaks at frequencies around the

fundamental natural frequency of the transducer and a positive or negative imaginary part. Instead, Plot

(b) shows that, when the electromagnetic seismic transducer is used, the harvester circuit should be

characterised by a real positive part that peaks in correspondence to the fundamental natural frequency of

the transducer and then goes to zero and by a positive imaginary part that decreases proportionally to 1. It should be highlighted that this study is focussed on time-harmonic energy harvesting. Thus, the spectra

shown in Figure 4 give the complex impedance that should be implemented at each frequency to guarantee

the maximum energy harvesting with the two seismic harvesters. Thus it does not represent the FRF of the

harvesting circuit.

Figure 5 shows the 1 Hz – 2 kHz spectra of the power harvested by the electromagnetic seismic harvester

(Plot a) and by the piezoelectric seismic harvester (Plot b), with reference to a 1 g base acceleration, when

the harvesting circuits are characterised by the optimal complex impedance *

C, eih ZZ . Plot (a) shows that

when the harvesting circuit implements at each frequency the optimal impedance derived in Eq. (28) and

depicted in plot (a) of Figure 4, the electromagnetic seismic harvester is particularly effective when it is

operated at frequencies close to the fundamental natural frequency of the seismic electromagnetic

transducer, i.e. 20 Hz, where it generates about W04.0 . It is interesting to note that, according to Plot (a)

in Figure 4, at this frequency, the harvester should be characterised by a purely real impedance. Plot (b) in

Figure 5, shows that when the harvesting circuit implements at each frequency the optimal impedance

derived in Eq. (28) and depicted in plot (b) of Figure 4, a constant level of power equal to the maximum


value obtained with the electromagnetic seismic transducer, i.e. W05.0 , is harvested in the whole

frequency range. This is a remarkable result, which however should be carefully analysed. In fact, Plot (b)

of Figure 4 shows that to obtain this result, the impedance of the harvesting circuit is either characterised

by a very large imaginary part at low frequencies or by a null real part at frequencies above the

fundamental natural frequency of the piezoelectric seismic transducer. Thus it is likely that in practice, the

piezoelectric seismic harvester is also operated at frequencies close to the fundamental natural frequency

of the piezoelectric seismic harvester where, according to Plot (b) of Figure 4, the impedance of the

harvesting circuit is characterised by comparatively large real part and small imaginary part. The dashed

blue lines in the two plots of Figure 5 show that, as one would expect, when the mechanical damping of

the transducer is increased, the harvested power decreases.

Figure 4: Spectra of the Real and Imaginary parts of the optimal impedance *

C, eih ZZ of the harvesting

circuits for the (a) the electromagnetic and (b) the piezoelectric seismic harvesters.

Figure 5: Spectra of the power harvested with reference to a 1 g base acceleration for the (a)

electromagnetic and (b) the piezoelectric seismic harvesters with damping ratio � = . 6 (solid black line)

and � = .6 (dashed blue line) and optimal harvesting impedances*

C, eih ZZ .

Often, practical harvesters are equipped with purely resistive electric loads in which case the harvesting

electrical impedance hZ is bound to purely real. In this case, the purely real impedance that maximizes the

harvested power at each frequency is obtained by setting only the condition given in Eq. (26), which

gives:

eih ZZ R, . (32)


Thus, in this case the maximum for the harvested power hP is produced when the electric impedance of

the harvesting circuit RhZ , equals the modulus of the electrical impedance of the freely suspended seismic

transducer eiZ given in Eq. (5). Substituting the optimal condition for

hZ given in Eq. (32) into Eq. (25),

leads to the following expression for the maximum harvested power:

2

2

Re4

1b

eiei

ew

h wZZ

TP

, (33)

Recalling Eqs. (5) and (6), this equation can be specified respectively for the electromagnetic and for the

piezoelectric seismic harvesters as follows:

2

2

2222

2

4

1b

mmfiewaeaefiewmaefiewm

mewh w

kmZLZRcZRc

mP

,

(34)

2

2

22222

2

4

1b

mmapmm

mh w

kmZCcc

mP

, (35)

where aZ in Eqs. (34) and (35) is defined from Eq. (11). The two plots in Figure 6 show the spectra of the

real part of the harvesting impedances for the electromagnetic and piezoelectric seismic harvesters. Plot

(a) shows a peak in correspondence to the fundamental natural frequency of the electromagnetic seismic

harvester at about 20 Hz. Also, above the cut off frequency at about 800 Hz where the inductive effect

dominates the electrical impedance of the transducer, the real part of the harvesting impedance tends to

rise proportionally to 1 . Instead, Plot (b) is characterised by a rather large capacitive effect at low

frequencies that drops proportionally to 1 , apart from a small glimpse in correspondence to the

fundamental natural frequency of the piezoelectric seismic transducer.

Figure 6: Spectra of the optimal real impedance eih ZZ R, of the harvesting circuits for (a) the

electromagnetic and the piezoelectric (b) seismic harvesters.

Figure 7 shows the 1 Hz – 2 kHz spectra of the power harvested by the electromagnetic seismic harvester

(Plot a) and by the piezoelectric seismic harvester (Plot b), with reference to a 1 g base acceleration, when

the harvesting circuits are characterised by the optimal complex impedance eih ZZ R, . The two plots

show that, in this case, both the electromagnetic and the piezoelectric seismic harvesters are particularly

effective when they are operated at frequencies close to the fundamental natural frequency of the seismic

transducers, i.e. 20 Hz. At this frequency, the electromagnetic seismic harvester produces about W04.0 ;

indeed, the same value obtained when the optimal complex harvester impedance is implemented, which,

as discussed above, at 20 Hz is purely real and has the same value as that found in this analysis. The


piezoelectric seismic harvester produces instead a lower amount of power, which is about W02.0 . Thus,

in order to effectively extract power with the piezoelectric seismic transducer it is important to have the

complex impedance load given in Eq. (28). As discussed in Section 2 and shown in Figure 3c, the

piezoelectric harvester is characterised by a strong reactive electrical impedance due to the inherent

capacitive effect of the piezoelectric transducer, which need to be compensated in the harvesting circuit to

maximise the power harvesting.

Figure 7: Spectra of the power harvested with reference to a 1 g base acceleration for (a) the

electromagnetic and (b) the piezoelectric seismic harvesters with damping ratio � = . 6 (solid black line)

and � = .6 (dashed blue line) and optimal harvesting impedances eih ZZ R, .

4 Conclusions

This paper has presented a comparative study on the energy harvested by electromagnetic and

piezoelectric seismic harvesters. The two systems were modelled with consistent electro-mechanical

lumped parameter models to allow the derivation of a unified formulation for the energy harvesting and

thus a direct comparison of the electro-mechanical response and energy harvesting properties of the two

harvesters. The study was limited to energy harvesting from tonal ambient vibrations.

The study on the dynamic response of the two harvesters has shown that they are characterised by typical

mechanical impedance of seismic systems. The electrical impedance of the electromagnetic transducer is

dominated by the resistive-inductive effects of the coil except in the vicinity of the fundamental natural

frequency of the harvester, where the mechanical to electrical impedance effect produced by the

transducer becomes relevant. The electrical impedance of the piezoelectric transducer is instead

characterised by a capacitive effect and the mechanical to electrical impedance effect is negligible apart at

the fundamental natural frequency of the harvester where in any case it produces a very small effect. The

electromechanical transduction of the electromagnetic seismic harvester becomes relevant at the

fundamental natural frequency of the seismic transducer where it reaches a maximum value and, at higher

frequencies, it remains constant. Instead, the electromechanical transduction of the piezoelectric seismic

harvester is relevant only at the fundamental natural frequency of the seismic transducer where it peaks.

The study on the power harvesting has considered two configurations of the harvesting circuit, which are

characterised either by a complex or a purely real impedance set to maximise the harvested power. The

complex impedance necessary to maximise the harvested power was found to be given by the complex

conjugate of the electrical impedance of the seismic transducers. This is line with the maximum power

transfer theorem, which states that a load collects the maximum amount of power from a source when its

resistance is equal to the internal resistance of the source and when the imaginary part of the impedance of

the load is counter phase with the imaginary part of the impedance of the source. Alternatively, the purely

real impedance necessary to maximise the harvested power was found equal to the modulus of the

electrical impedance of the seismic transducers.

The study on the harvested power has considered a base acceleration equal to 1 g. When the harvesters

implement the optimal complex impedance, the electromagnetic seismic harvester generates a peak level


of power of W04.0 at about the fundamental natural frequency of the seismic transducer while the

piezoelectric seismic harvester generates a constant level of power equal to W05.0 . For the

electromagnetic seismic harvester, it was noted that the impedance for the maximum power harvesting at

about the fundamental natural frequency is actually purely real. Alternatively, when the harvesters

implement the optimal real impedance, both the electromagnetic and piezoelectric seismic harvester

generate a peak level of power respectively of W04.0 and W02.0 at about the fundamental natural

frequency of the seismic transducers.

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