Microsyst Technol (2015) 21:401–414
DOI 10.1007/s00542-013-2030-6
TECHNICAL PAPER
Modeling and analysis of hybrid piezoelectric and electromagnetic
energy harvesting from random vibrations
Ping Li • Shiqiao Gao • Huatong Cai
Received: 5 November 2013 / Accepted: 3 December 2013 / Published online: 15 December 2013
Ó Springer-Verlag Berlin Heidelberg 2013
Abstract We illustrate electroelastic modeling, analysis and
simulation solutions, and experimental validation of hybrid
piezoelectric (PE) and electromagnetic (EM) energy harvesting from broadband random vibration. For a more practically
available ambient source, the more compact expressions of
mean power and spectral density (SD) involving dimensionless parameters are derived when the harvester is subjected to
random excitation. In the study, it is assumed that the base
excitation is white noise. Then, the effect of acceleration SD,
load resistance, coupling strength on harvester performances
are analyzed by numerical calculation and simulation, and the
results are validated by the experimental measurements. It is
founded that, only if the load resistance of PE and EM element
meet the impedance matching can the hybrid energy harvester
output the maximal mean power and spectral density at the
resonant frequency, which increases with PE load resistance
increasing, but hardly affected by load resistance of EM element; the variation extent of mean power with SD of acceleration increasing varies with the load resistance, and it is up to
the maximum under the condition of optimal load; moreover,
the stronger the coupling strength is, the wider the frequency
band becomes, and the greater the mean power and power
spectral density are, while the increasing extent decreases with
the coupling strength increasing. Besides, the coupling
strength can affect the internal resistance of harvester. Furthermore, with coupling strength increasing, the decreasing
degree of mean power falls when the load resistance is greater
than the optimal load.
P. Li (&) S. Gao H. Cai
State Key Laboratory of Explosion Science and Technology,
School of Mechatronical Engineering, Beijing Institute of
Technology, Room.109, Teaching Building 3#,
Haidian District, Beijing 100081, China
e-mail: gstwliping@126.com
1 Introduction
As wireless sensor network and low power devices have
been growing remarkably in recent years, it makes possibility to harvest energy of environment to replace chemical
batteries that raise maintenance, environment issues and
size (Ling et al. 2013; Karami and Inman 2012; Nicholas
and Natarajan 2013). Therefore, it is a research focus on
converting the vibration energy into electrical energy, and
three mechanisms have been proposed: piezoelectric,
electromagnetic and electrostatic (Harne and Wang 2013;
Mitcheson et al. 2007). Electrostatic energy harvesters
need the initial voltage to start harvesting normally, so it
can not work independently. Then, piezoelectric energy
harvester and electromagnetic energy harvester are got
much more attention as they have high electromechanical
coupling effect and no external voltage source requirement;
furthermore, they can be fabricated by MEMS technology.
Shu and Lien (2006) Much of work has been done for
piezoelectric and electromagnetic energy harvesters, for
example, the structure design (Liua et al. 2011; Liu et al.
2012), fabrication process by MEMS or traditional technique (Hatipoglu and Urey 2010; Wang et al. 2012),
mathematics modeling (Williams and Yates 1996; Erturk
and Inman 2008; Karami and Inman 2011; Cheng et al.
2007), power optimization methods (Cho et al. 2005; Wu
et al. 2009; Guyomar and Lallart 2011), and energy storage
and manage circuit (James and Paul 2012; Yogesh and
Anantha 2010).
However, from the results in the former study, although
the voltage of piezoelectric energy harvester can up to
several or tens of volts, the output current is only few
microamperes. On the contrary, the electromagnetic energy
harvest can output hundreds of microamperes, but its
voltage is too small to meet the needs of most devices. The
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402
disadvantages of the two kinds of energy harvesters limit
the application ranges seriously. Therefore, researchers
proposed the vibration energy harvester that combined
piezoelectric and electromagnetic harvesting mechanism,
which reaps the benefits of advantages of single harvesting
mechanism simultaneously (Torsten and Armaghan 2010;
Wu et al. 2008). For hybrid PE and EM energy harvester,
the research works mainly focused on the structure design
and performance validation at present. Wacharasindhu and
Challa tested their designed hybrid piezoelectric-electromagnetic energy harvester respectively and concluded that
output power of hybrid energy harvester is more than that
of single piezoelectric or electromagnetic energy harvester
(Wacharasindhu and Wkwon 2008; Challa et al. 2009).
Yang et al. assumed that power of hybrid energy harvester
was equivalent to the power consumed by electrical
damping in their researches, and they also studied the
output characteristics of hybrid energy harvester and proposed several optimization design methods, but they did
not consider the electromechanical coupling effect in the
analysis (Yang et al. 2010; Tanesse et al. 2009; Robert
et al. 2008).
The above researches are mostly based on the harmonic
excitation, which is a simple and rather idealized representation of real-world ambient vibrations, but the practical
application environments of vibration energy harvester
mostly are broadband random excitation (Cottone et al.
2012; Blystad et al. 2010). Thus, research on performances
of energy harvester under random vibration excitation has
much more practical significance. Zhao established the
model of piezoelectric energy harvest excited by the random vibration for the cantilever beam structure, and tested
the output power under white noise excitation (Zhao and
Erturk 2013). Adhikari et al. (2009) analyzed mean power
of piezoelectric energy harvester in Gaussian white noise
excitation through solving two-dimensional stochastic difference equation, and the results showed that in order to get
the maximum power, mechanical damping should be
reduced and SD of acceleration should be increased. Tang
comparatively studied performances of SDOF (singledegree-of-freedom) piezoelectric energy harvester for the
double mass and single mass structure. It’s illustrated that
their mean power are the same under random excitation
when the mass of block is equal to each other for the two
harvesters, but their power is different under the sinusoidal
excitation (Tang and Zuo 2012). Halvorsen studied
amplitude, power and optimal load of the linear piezoelectric energy harvester under random acceleration excitation, and obtained that the optimal load is not equal under
the conditions of random and sinusoidal excitation. Compared with sinusoidal excitation, optimal load is unrelated
to mechanical damping under the random excitation
(Halvorsen 2008). Moreover, Halvorsen simulated output
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Microsyst Technol (2015) 21:401–414
characteristics of electrostatic energy harvester under the
random excitation by Spice software. Jackson studied
output performances of piezoelectric energy harvester
under random vibration by experimental testing, and
illustrated that mean power is linearly proportional to
acceleration spectral density, and the smaller resonant
frequency, the more output power (Jackson et al. 2013). In
addition, Lefeuvre concluded that output power of the
piezoelectric energy harvester with the standard rectifier
circuit and synchronous charge extraction technology in
random acceleration excitation (Lefeuvre et al. 2007).
Therefore, it can be seen that piezoelectric energy harvester
has been got many researches when it is excited under
random vibration. However, for author’s knowledge, there
are not any related studies for hybrid piezoelectric-electromagnetic energy harvester under the random vibration
excitation.
Thus, in this paper, based on the governing electromechanical equations of hybrid piezoelectric-electromagnetic
energy harvester, the more compact expressions of mean
power and power spectral density involving non-dimension
parameters are derived in the random excitation. Moreover,
the relationship between mean power, spectral density of
power and acceleration spectral density and load resistance
are researched by numerical calculation, simulation and
experimental validation. Besides, the effect of coupling
strength on energy harvesting characteristics is also be
analyzed.
2 Theoretical analysis
2.1 Electromechanical coupling model
Considering a hybrid energy harvester integrated with piezoelectric and electromagnetic conversion mechanisms, as
shown in Fig. 1, magnet is supported by a double-end fixed
beam, piezoelectric layers polarized in the beam thickness
direction are died on the top surface of beams, and coils are
placed under the magnetic. In references (Wang et al. 2012;
Tang and Yang 2011),when the harvester structure vibrates
around at its resonant frequency, the model of PE or EM
energy harvester all can be simplified as a SDOF system.
Similarly, designed hybrid energy harvester in Fig. 1 that
can be fabricated by MEMS process is modeled as a massspring-damper-PE element-EM element system, as shown
in Fig. 2. It consists of the PE harvesting element, EM
harvesting element and the mechanical structure, and PE
and EM element connect with load resistors respectively.
Therefore, when the acceleration is applied to the harvesting system, an effective mass me is bounded on a spring of
effective stiffness K, a damper of coefficient cm, a PE element and an E element.
Microsyst Technol (2015) 21:401–414
403
coil. These parameters are depended on the material
properties and structural parameters of energy harvester,
which can be derived by standard model analysis (Spreemann and Manoli 2012; Erturk and Inman 2011).
According to literature (Serre et al. 2007), for the low
frequency vibration, coil impedance is mainly manifested
as the coil resistance, so the effect of coil inductance can be
negligible. By means of Fourier transform, load current of
EM element and load voltage of PE element in frequency
domain are derived from the Eqs. (2) and (3) respectively,
and the results are shown in Eqs. (4) and (5).
Fig. 1 Designed hybrid PE and EM energy harvester
Iem ðxÞ ¼
1
ge zðxÞix
Rc þ Rm
ð4Þ
Vp ðxÞ ¼
hRp
zðxÞix
1 þ ixCp Rp
ð5Þ
where x
response
excitation
expressed
Fig. 2 SDOF model of hybrid PE and EM energy harvester
In this case, the coupled PE and EM element alter the
vibration response of harvesting system, which affects the
power obtained from energy harvesting system in turn.
However, the governing equation for the hybrid energy
harvester in literatures (Wacharasindhu and Wkwon 2008;
Challa et al. 2009) do not consider this electromechanical
coupling effect. For the designed hybrid PE and EM energy
harvester, let z(t) be the amplitude of the magnet, Vp as
output voltage of piezoelectric element, Iem as output
current of EM element, and the governing electromechanical equations can be established as follows (Lallart
and Inman 2010):
me zðtÞ þ cm zðtÞ þ KzðtÞ þ ge Iem ðtÞ þ hVp ðtÞ ¼ l1 me aðtÞ
ð1Þ
Lc
dIem ðtÞ
dzðtÞ
þ ðRc þ Rm ÞIem ðtÞ þ ge
¼0
dt
dt
Vp ðtÞ
dzðtÞ
¼0
þ Cp Vp ðtÞ h
Rp
dt
ð2Þ
ð3Þ
where aðtÞ is the excitation acceleration, and l1 is the
correction factor of single degree of freedom system
compared with the distributed-parameter model (Erturk
and Inman 2011). Moreover, h, ge are PE and EM transfer
factor respectively (Erturk and Inman 2011; Spreemann
and Manoli 2012) (In the reference Spreemann and Manoli
2012, gem is represented by kt), Rp and Rm are load resistor
of PE and EM element, Cp is the capacitance of piezoelectric layer, and Rc, Lc are resistance and inductance of
is excitation frequency. Similarly, frequency
function of amplitude versus acceleration
can be obtained by Fourier transform, which is
as
HðxÞ ¼ ðl1 me ixCp Rp l1 me Þ
8
3
>
>
C
R
ðixÞ
þ
m
þ
cm þ
m
>
e
p
p
e
<
=
>
>
>
:
þ cm þ
g2e
Rc þ Rm
9
g2e
>
Cp Rp ðixÞ2 >
>
=
Rc þ Rm
>
>
;
þ KCp Rp þ h2 Rp ix þ K >
ð6Þ
For a damped linear system of the form
zðxÞ ¼ HðxÞaðxÞ, transfer function of output current of
EM element and output voltage of PE element are
obtained, as illustrated in Eqs. (7) and (8) respectively
HIem ðxÞ ¼
Iem ðxÞ
ixge
¼
HðxÞ
aðxÞ
Rc þ Rm
ð7Þ
HVp ðxÞ ¼
Vp ðxÞ
ixhRp
¼
HðxÞ
aðxÞ
1 þ ixCp Rp
ð8Þ
In order to get more compact expressions of transfer
function, the dimensionless parameters are assumed that
2
h
gp ¼ KC
as piezoelectric coupling coefficient;
p
x g2
n e
as electromagnetic coupling coefficient;
ge ¼ KR
c
x
k ¼ xn as dimensionless vibration frequency;
fm ¼ 2mcemxn as mechanical damping ratio;
rp ¼ Rp Cp xn as dimensionless load resistance of PE
element;
rm ¼ RRmc as dimensionless load resistance of EM element;
where xn is the natural frequency of hybrid energy
harvesting system. Then, the more compact form of Eqs.
(6)–(8) involving the dimensionless parameters can be
derived, such that respectively
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404
Microsyst Technol (2015) 21:401–414
HðkÞ ¼
l1
1 þ rp ðikÞ
h
i
2
3
g
xn rp ðikÞ þ 1 þ 2f þ e rp ðikÞ2 þ 2f þ ge þ rp þ g rp ik þ 1
m
m
p
1þrm
1þrm
HIem ðkÞ ¼ l1
me ge
ik þ rp ðikÞ2
h
i
gem 1 þ rm rp ðikÞ3 þ 1 þ 2f þ ge rp ðikÞ2 þ 2f þ ge þ rp þ g rp ik þ 1
m
m
p
1þrm
1þrm
me
HVP ðkÞ ¼ l1 gp ðik þ rp ðikÞ2 Þ=
h 9
8
ge
1
2ge
4
3
2 >
>
>
>
r
ðikÞ
þ
2
þ
2f
þ
þ
þ
4f
þ
þ
r
þ
g
r
þ
r
ðikÞ
ðikÞ
>
> p
p
p
m
m
p p
=
<
rp
1 þ rm
1 þ rm
>
f
ge
1>
>
>
>
;
:
2 mþ
þ 2 þ gp ik þ >
rp
rp ð1 þ rm Þrp
2.2 Basic equation of broadband random excitation
The more practically available ambient source of energy
harvester is random vibration, where is distributed with in a
certain range of frequency band and acceleration.
According to stochastic theory (Liu 2008), when energy
harvesting system is excited by random acceleration with
spectral density SA(w), the SD of harvester amplitude, EM
output current and PE output voltage are illustrated in Eqs.
(12)–(14) respectively
SZ ðxÞ ¼ jHðxÞj2 SA ðxÞ
ð12Þ
SIem ðxÞ ¼ jHIem ðxÞj2 SA ðxÞ
2
SV ðxÞ ¼ HV ðxÞ SA ðxÞ
ð13Þ
p
Based on Wiener–Khinchin theorem (Liu 2008), for
stationary random process, its spectral density SðxÞ and
autocorrelation function RðsÞ is a Fourier transform pair, as
shown in Eq. (15). Moreover, when s ¼ 0, Rð0Þ is mean
square value of the random signal.
Zþ1
SðxÞeixs dx
ð15Þ
1
Thus, substituting Eq. (12) into Eq. (15), mean square
value of amplitude is
2
ZM
1
¼
2p
Zþ1
1
1
SZ ðxÞdx ¼
2p
Zþ1
jHðxÞj2 SA ðxÞdx
ð16Þ
1
Similarly, we can also get the mean square value of EM
output current and PE output voltage, as shown in Eqs. (17)
and (18) respectively.
123
Zþ1
ð10Þ
ð11Þ
jHIem ðxÞj2 SA ðxÞdx
ð17Þ
jHVP ðxÞj2 SA ðxÞdx
ð18Þ
1
VP2
1
¼
2p
Zþ1
1
The two expressions in Eqs. (17) and (18) will be used
to obtain the average power for the hybrid piezoelectric and
electromagnetic energy harvester, and more details are
shown in Sect. 2.3.
2.3 White noise excitation
ð14Þ
p
1
R½ðsÞ ¼
2p
2
Iem
1
¼
2p
ð9Þ
According to reference (Halvorsen 2007), for linear energy
harvesting system, random vibration can be assumed as
white noise signal when the excitation frequency bandwidth is much bigger than 3 dB bandwidth of energy
harvesting system and the excitation has a flat power
spectral density in frequency domain, which indicates that
SA (w) is a constant S0. Under white noise excitation with
zero mean value, system response will also be stationary
random signal with zero mean value.
2.3.1 PE element performances
For hybrid PE–EM energy harvester excited by white
noise, by substituting SA ðxÞ ¼ S0 into Eq. (18), mean
square value of output voltage of PE element is
VP2
1
S0
¼
2p
Zþ1
1
jHVP ðxÞj2 dx
ð19Þ
Microsyst Technol (2015) 21:401–414
405
In this paper we are interested in the average harvested
power given by (Adhikari et al. 2009)
2 V ðtÞ
E½V 2 ðtÞ
E½PðtÞ ¼ E
ð20Þ
¼
Rload
Rload
so mean power apply on PE load is
Vp2
1 S0
Pp ¼
¼
Rp 2p Rp
Zþ1
2
jHVP ðxÞj dx
ð21Þ
Using the similar analytical method, mean square value of
EM element output current under white noise excitation is
illustrated in Eq. (27) by substituting SA ðxÞ ¼ S0 into Eq.
(17).
2
Iem
Zþ1
xn
S0
¼
2p
jHIem ðkÞj2 dk
ð27Þ
1
1
Using the non-dimension parameters, it can be derived
that
Zþ1
2.3.2 EM element performances
2
jHVP ðxÞj dx ¼ xn
1
Zþ1
And mean power of EM load Rm is
Pem ¼
2
jHVP ðkÞj dk
2
Rm Iem
ð22Þ
1
The integral in Eq. (22) can be obtained using Eq. (39) in
the Appendix. Let
ge
B2 ¼ rp ; B1 ¼ 1; A4 ¼ rp ; A3 ¼ 2 þ 2fm þ
rp ; A 2
1 þ rm
1
2ge
¼ þ 4fm þ
þ r p þ g p r p ; A1
rp
1 þ rm
f
ge
1
¼2 mþ
þ 2 þ g p ; A0 ¼ :
rp
rp ð1 þ rm Þrp
xn
Rm S0
¼
2p
Zþ1
jHIem ðkÞj2 dk
The integral of Eqs. (27) and (28) can be calculated by
Eq. (40) in the Appendix, then mean square value of EM
element output current and mean power can be expressed
in Eqs. (29) and (30) respectively involving the
dimensionless parameters.
me S0 l21 ge
2Rc ð1 þ rm Þ2
ge
rp 2fm þ 1þr
þ rp þ gp rp þ 1
m
h
i
gem
ge
þ rp þ gp rp 1 þ 2fm þ 1þr
rp rp
2fm þ 1þr
m
m
2 ¼
Iem
The solutions of Eqs. (19) and (21) can be simply
expressed as
VP2
S0 me
A1 B22 A3 B21
¼
gp l21
2Cp xn
A0 A23 þ A21 A4 A1 A2 A3
ð23Þ
S0 me gp l21
A1 B22 A3 B21
2 rp A0 A23 þ A21 A4 A1 A2 A3
ð24Þ
Pp ¼
On the other hand, by the physical meaning of power
spectral density (Liu 2008) the SD of output power of PE
element is shown in Eq. (25) when its load resistance is Rp.
Spp ðxÞ ¼
1
S0 jHIem ðxÞj2
SVp ðxÞ ¼
Rp
Rp
ð25Þ
Equation (25) can be expressed in the parameters of the
system as
SPP ðkÞ ¼
me S0 2 gp ðkrp Þ
l
xn
rp 1 þ ðkrp Þ2
1
2 2
2 2
gp k rp
kgp rp
kge
2
1 þ 1þk2 r2 k
þ 2kfm þ 1þrm þ 1þk2 r2
p
ð29Þ
l21 ge rm
me S 0
2 ð1 þ rm Þ2
ge
rp 2fm þ 1þr
þ
r
þ
g
r
þ1
p
p
p
m
h
i
ge
ge
þ
r
þ
g
r
þ
1
þ
2f
2fm þ 1þr
p
p
m
p
1þrm rp rp
m
Pem ¼
ð30Þ
Moreover, when the load resistance is Rm, the SD of EM
element output power is
Spem ðxÞ ¼ Rm SIem ðxÞ ¼ Rm S0 jHIem ðxÞj2
m e S0 r m g e
l2 k2
xn ð1 þ rm Þ2
1þ
gp k2 rp2
1þk2 rp2
1
2 2
kgp rp
kge
k2 þ 2kfm þ 1þr
þ
2 2
m
1þk r
Besides, by Eqs. (24) and (26), we can also obtain the
normalized power and spectral density by further process,
but in this paper we will no longer focus on it.
p
ð32Þ
p
ð26Þ
ð31Þ
Then, Eq. (31) can be expressed as Eq. (32) by using the
non-dimension parameters.
Spem ðkÞ ¼
2
ð28Þ
1
2.3.3 Hybrid energy harvesting system
According to the results in 2.3.1 and 2.3.2, total output
power of hybrid piezoelectric-electromagnetic energy
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406
Microsyst Technol (2015) 21:401–414
harvester is the sum of power of PE energy harvesting
element and EM energy harvesting element, which is
shown as
density of hybrid piezoelectric-electromagnetic energy
harvester are mainly determined by spectral density S0 of
excitation acceleration, dimensionless load rp and rm,
P ¼ Pem þ Pp
3
2
ge
2
r
2f
þ
þ
r
þ
g
r
þ
1
p
p
p
m
p
1þr
l
g
r
m
7
6 1 e m h
i
7
6
ð1 þ rm Þ2 2fm þ ge þ rp þ gp rp 1 þ 2fm þ ge rp rp 7
S0 m e 6
1þrm
1þrm
7
6
¼
7
2 6
7
6
2
2
2
5
4
gp l1
A1 B2 A3 B1
þ
2
2
r p A0 A3 þ A1 A4 A1 A2 A3
So, the optimal load resistance of PE and EM element
can be derived by Eqs. (34) and (35) respectively.
oP
¼ 0 ! rp ¼ rpoptimal
orp
ð34Þ
oP
¼ 0 ! rm ¼ rmoptimal
orm
ð35Þ
Besides, according to the theory of stochastic process
(Liu 2008), the SD of hybrid energy harvesting system is
SðxÞ ¼ Sem ðxÞ þ Sp ðxÞ þ Sp em ðxÞ þ Sem p ðxÞ
ð36Þ
where Sem ðxÞ and Sp ðxÞ are auto-SD of EM and PE element
respectively, and Sp em ðxÞ, Sem p ðxÞ are the cross SD between
EM and PE energy harvesting element. Based on output power
of energy harvesting system shown in Eq. (33), we neglect
cross power spectral density between EM and PE element to
simplify analysis in this paper, so output power spectral density
of hybrid piezoelectric-electromagnetic energy harvester is
"
#
g ðkrp Þ2
me S0
rm ge
2 p
2 2
S¼
l
þ
l k =
xn
rp 1 þ ðkrp Þ2 ð1 þ rm Þ2
2
!2
!2 3
gp k2 rp2
kg
r
kg
p
p
e
4 1þ
5
k2 þ 2kfm þ
þ
1 þ rm 1 þ k2 rp2
1 þ k2 rp2
ð37Þ
Similarly, we can also derive the normalized power and
spectral
density
of
hybrid
piezoelectric
and
electromagnetic energy harvester from Eqs. (33) and (37)
respectively. Moreover, given specific values of the
parameters, the mean power and spectral density of the
harvesters can be predicted.
ð33Þ
dimensionless frequency k, coupling coefficients ge and
gp. Thus, in this section, the effects of above factors on
output performances of hybrid energy harvester are analyzed by numerical calculation and simulation. Table 1
shows model parameters of designed hybrid PE and EM
energy harvester.
In the simulation, use matlab software to simulate output
characteristics of energy harvester under the random
vibration excitation, and compare with the theory results.
Let initial value of energy harvesting system be z(0) = 0,
Iem(0) = 0, Vp(0) = 0, and t [ 0. A white noise signal that
inputs to energy harvesting system is generated by WGN
function in matlab software, and the electric output signal
of harvester under random excitation can be obtained from
the simulation. Then adopting matlab digital signal processing technology to process the signals obtained in the
simulation, and mean square value and SD of output signals can be got. Finally, we can have the RMS (Root Mean
Square) value of output voltage, current, and power,
spectral density of harvester.
3.1 Effect of load and acceleration spectral density
on output characteristics
Based on analysis results shown in Eqs. (33) and (37),
when the excitation frequency is equal to the resonant
frequency, performances of harvester are related with load
resistance and strength of random excitation when the
system parameters of energy harvesting are fixed. Thus,
output characteristics of harvester are analyzed at the different SD of acceleration and load resistance in this part.
3.1.1 Mean power
3 Numerical calculation and simulation
As the analysis results illustrated in Eqs. (33) and (37),
under the white noise excitation, mean power and spectral
123
Supposed spectral density of acceleration S0 = 1 (m/s2)2/
Hz, mean power of hybrid piezoelectric and electromagnetic energy harvester with the different load resistances
Microsyst Technol (2015) 21:401–414
407
are shown in Fig. 3. It can be concluded that, when load
resistance of PE element and EM element are 170 kX and
24 X respectively, which are equal to 1/Cpwn and Rc, the
output power of hybrid energy harvester can be up to the
maximum only. Besides, the effect of EM load on voltage
and power of PE element is shown in Fig. 4, where the
Table 1 Physical characteristics of designed energy harvester
Parameters
Values
PZT layer
Length
10 mm
Thickness
8 mm
Width
2 mm
Piezoelectric constant
-100e-12 C/N
Dielectric constant
3.7899 e-8 F/m
Magnet (NdFeB)
Diameter
15 mm
Thickness
40 mm
Beam of one side (stainless steel)
Length
Width
20 mm
8 mm
Thickness
3.8 mm
Coil (copper)
Number of turns
360
Diameter
15 mm
Damping ratio
0.026
voltage and power increase with EM load resistance
increasing. It can be explained that with EM load resistance
increasing, coupling effect of EM element on amplitude of
harvesting system reduces; therefore, output voltage and
power of piezoelectric element will be raised. Moreover, it
can be predicted that voltage and power of PE element
reaches their maximum when the resistance of EM element
is infinity. At this case, hybrid piezoelectric-electromagnetic energy harvester becomes piezoelectric energy
harvester that based on single energy harvesting mechanism. On the other hand, Fig. 5 displays that PE load
almost does not affect the current and power of EM element, which indicates the effect of piezoelectric load variation on vibration characteristics and coupling coefficient
of hybrid energy harvester can be neglected. Besides, the
PE and EM element reach the maximum power at their
optimal load 170 kX, 24 X respectively.
In addition, we also analyze the variation regularity of
mean power with SD of excitation acceleration at the resonant frequency, and the results are shown in Fig. 6. Mean
power of hybrid PE–EM energy harvester linearly increases with the spectral density of acceleration increasing,
which has the same conclusion with Jackson for the piezoelectric energy harvester (Jackson et al. 2013). Besides,
with the load resistance hiking, the variation of mean
power with SD of acceleration rise firstly and fall later, and
reach the maximum at the optimal load of energy harvesting element. Furthermore, the similar regularity happens in output voltage of PE element and output current of
EM element, as shown in Fig. 7. Therefore, for hybrid
piezoelectric and electromagnetic energy harvester, in
order to get the maximal power, load resistance of energy
harvesting element should be equal to their internal resistance respectively, and acceleration excitation should be as
large as possible under the condition of safety structural
strength.
3.1.2 Power spectral density
Fig. 3 Mean power of hybrid PE and EM energy harvester with the
different load resistances
Based on the results in Sect. 3.1.1, output power of hybrid
PE–EM energy harvester is affected by load resistors of PE
and EM energy harvesting element. Thus, power spectral
density of hybrid energy harvesting system in different
Fig. 4 Effect of EM load on
voltage and power of PE
element: a power, b voltage
123
408
Microsyst Technol (2015) 21:401–414
Fig. 5 Effect of PE load on
current and power of EM
element a power, b current
Fig. 6 Variation regularity of
mean power with SD of
excitation acceleration at the
resonant frequency a with the
different load resistors of EM
element, b with the different
load resistors of PE element
Fig. 7 Output voltage of PE
element and output current of
EM element with SD of
excitation acceleration a output
voltage of PE element, b output
current of EM element
vibration frequencies and load resistors are studied in the
several cases. In the analysis, when PE element is in the
state of optimal load, it is assumed that load resistance of
EM element are 1, 10, 24, 50, 200 X respectively; similarly, when EM element meets impedance matching, it is
proposed that the load of PE element are 1, 50, 170, 300 ,
800 kX respectively. Besides, supposed spectral density of
acceleration S0 = 1 (m/s2)2/Hz, and results are shown in
Fig. 8.
In Fig. 8, the corresponding frequency of maximal
spectral density is the resonant frequency of harvesting
system, and the frequency rises from 76 to 78 Hz as piezoelectric load increasing from 1 to 800 kX. Moreover, the
SD reaches the maximum at 170 kX. Therefore, piezoelectric load resistance not only affects the value of spectral
density of hybrid energy harvester but also the distribution
123
in the frequency domain. On the other hand, when EM load
increases from 1 to 200 X, SD maximum is at the 24 X, and
variation of the spectral density reaches the maximum at the
resonant frequency. However, the resonant frequencies do
not vary with the EM load resistance. Thus, it can be concluded that hybrid energy harvester should work at the
resonant frequency, and the effect of load resistance on
resonant frequency should be taken into consideration when
designing the structure of energy harvester.
Furthermore, at the optimal load, power spectral density
of hybrid energy harvester excited with different acceleration is studied, and the result is illustrated in Fig. 9. It can be
concluded that SD of output power increases with spectral
density of acceleration increasing, and the corresponding
frequencies of spectral density peak are all equal to 77.5 Hz.
When spectral density of random acceleration is S0 =
Microsyst Technol (2015) 21:401–414
409
Fig. 8 SD of hybrid energy
harvesting system at different
vibration frequencies and load
resistors a with the different
load resistors of PE element,
b with the different load
resistors of EM element
3.2 Effect of coupling strength on performance
Fig. 9 Power spectral density of hybrid energy harvester under the
acceleration excitation with different spectral density
Fig. 10 Spectral density of harvester at the different excitation
frequencies
1 (m/s2)2/Hz, Fig. 10 illustrate the spectral density of energy
harvesting element at the different excitation frequencies. In
addition, simulation results of displacement, PE output
voltage and EM output current are shown in Fig. 11.
In Eqs. (33) and (37), it can be got that at the resonant
frequency, performances of energy harvester is mainly
determined by damping and coupling coefficient when load
resistance meets impedance matching, For piezoelectric
energy harvester, literature (Shu and Lien 2006) utilizes the
ratio of coupling coefficients to damping to characterize the
strength of electromechanical coupling effect. Similarly,
for hybrid piezoelectric-electromagnetic energy harvester,
we also use g/fm (g = gp ? gem) to characterize the value
of coupling strength in this paper. Therefore, in this section, we study the influence of coupling effect on power
and spectral density of hybrid energy harvester when SD of
excitation is fixed. For the designed harvester model in
Table 1, ge = 0.0412, gp = 0.0123, fm = 0.026, so g/fm
= 2. In the research, spectral density of random acceleration signal is set at 0.2 (m/s2)2/Hz.
When the values of g/fm are set at 1, 2, 5 respectively, mean
power of hybrid energy harvesters along with EM and PE load
resistance at resonant frequency are shown in Fig. 12a and b
respectively. It can be concluded that the stronger coupling
strength, the larger output mean power. At the both optimal
load of EM and PE element both, the maximal mean power
rise from 0.85 to 2.6 mW when g/fm increase from 1 to 5.
However, the optimal load resistance of EM element increases
with coupling strength enhancing. When g/fm is 1, 2, 5, its
corresponding optimal load resistance is 20, 24, 35 X
respectively, which indicate that coupling effect change the
internal resistance of harvester. Moreover, when load resistance is bigger than optimal load, decreasing degree of mean
power tends to be more slowly along with coupling strength
increasing. On the other hand, the coupling strength does not
affect the optimal load resistance of PE element. When g/fm is
1, 2, 5 respectively, its optimal load are all 170 kX. Therefore,
for vibration energy harvester, electromechanical coupling
strength which can improve output power in practical environments should be enhanced.
Under different coupling strength, spectral density of
output power in different excitation frequencies are shown
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410
Microsyst Technol (2015) 21:401–414
Fig. 11 Simulation results of hybrid energy harvester when S0 = 1 (m/s2)2/Hz a input acceleration, b displacement of magnet, c output current
of EM element, d output voltage of PE element
Fig. 12 Mean power of hybrid
energy harvester at the different
coupling strength a with the
different EM load resistance,
b with the different PE load
resistance
in Fig. 13, which indicates that the stronger coupling
effect, the greater output spectral density, and it reaches the
maximum at the resonant frequency. When g/fm is 1, 2, 5,
power spectral density at resonant frequency is 3 9 10-5,
6.5 9 10-5, 1.8 9 10-4 W/Hz respectively. Besides, the
stronger coupling strength, the wider frequency band of
energy harvesting, the better energy harvesting efficiency.
When the SD of acceleration is 0.2 (m/s2)2/Hz, output
current of EM element and output voltage of PE element for
the different coupling strength are shown in Figs. 14 and 15
respectively.
On the other hand, at the optimal load and resonant
frequency, variation law of output mean power and
spectral density of hybrid piezoelectric-electromagnetic
energy harvester along with the coupling strength are
123
studied. Firstly, the relationship between optimal load
resistance of EM element and coupling coefficient is
shown in Fig. 16, where the optimal load is proportional
to coupling strength. When g/fm increases from 0.35 to
13.38, the optimal load of EM element rises from 20 to
65 X. On this basis, the effect of coupling strength on
mean power and spectral density of hybrid energy harvester are shown in Fig. 17.
From analysis results in Fig. 17, output mean power and
spectral density increase with coupling strength enhancing.
However, the bigger coupling strength is, the smaller
increasing extent becomes. When g/fm is 0.35, 2.68, 13.38,
mean power is 0.44, 1.93, 4.2 mW respectively, and
accordingly its power spectral density is 5.45 9 10-6,
8.57 9 10-5, 2.61 9 10-4 W/Hz.
Microsyst Technol (2015) 21:401–414
Fig. 13 Spectral density of output power for hybrid energy harvester
at the different coupling strength
411
Fig. 16 The relationship between optimal load resistance of EM
element and coupling coefficient
Then, for hybrid piezoelectric-electromagnetic energy
harvester, improving coupling strength can not only
increase output power, but also widen frequency band of
energy harvesting.
4 Experimental validation
Fig. 14 Output current of EM element when the SD of acceleration is
0.2 (m/s2)2/Hz
Fig. 15 Output voltage of PE element when the SD of acceleration is
0.2 (m/s2)2/Hz
In order to test output characteristics of hybrid piezoelectric-electromagnetic harvester and validate the numerical calculation results, the meso hybrid piezoelectricelectromagnetic harvester is fabricated, and material
properties and structure parameters are illustrated in
Table 1. Experimental setup is shown in Fig. 18. The
whole setup of the device is mounted on the vibrating
shaker which is connected to a signal generator through a
power amplifier. The signal generator is used to provide the
source vibration frequency and amplitude excitation. Lead
wires from the piezoelectric cantilever beam are connected
across a variable resistor to maximize the piezoelectric
power output. Similarly, the current generated in the coil
due to electromagnetic power generation is also captured
across a different variable resistor. In addition, an accelerometer is used to record vibration acceleration, highspeed camera test the displacement of magnet and dynamic
signal analyzer is used to record output voltage of piezoelectric and electromagnetic energy harvesting element.
In the experiment, random excitation, which input into
the energy harvester, and its power spectral density are
shown in Fig. 19a and b respectively. It can be got that
spectral density of acceleration is 1.3 9 10-4 (m/s2)2/Hz.
At this case, output voltage of piezoelectric and electromagnetic energy harvesting element are tested at the resonant frequency when the load resistors change. Using the
matlab algorithm studied in Sect. 3, mean power of energy
harvester can be analyzed.
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412
Microsyst Technol (2015) 21:401–414
Fig. 17 Effect of coupling strength on mean power and spectral density of hybrid energy harvester a power, b spectral density
harvester at the different spectral density of acceleration is
shown in Fig. 21. It illustrates that output power is linearly
proportional to the acceleration spectral density, which is
consistent with theoretical analysis results.
5 Conclusion
Fig. 18 Experimental setup
The experimental results, as shown in Fig. 20, indicate
that with load resistance of PE and EM element increasing,
their mean power all increase first and decrease later, and
the maximum can be output at optimal load respectively,
which has the same conclusion with theoretical analysis
results in Sect. 3. However, because of uncertainty of random acceleration, there are errors between experimental
test results and theoretical analysis results. For random
signal process, the appearances of errors are in the expected
range (Jackson et al. 2013). At the optimal load resistance,
mean power of hybrid piezoelectric-electromagnetic energy
Aiming at designed hybrid piezoelectric-electromagnetic
energy harvester, electromechanical coupling state equations are set up. Using non-dimension parameters, the more
compact expressions of output mean power and spectral
density of energy harvester under random excitation are
derived. By means of numerical calculation, simulation and
experimental test, the effect of acceleration, load and
coupling strength on performances of hybrid energy harvester are studied. It can be obtained as follows.
1.
Mean power and power spectral density of hybrid
piezoelectric-electromagnetic energy harvester are
mainly related to vibration frequency k, damping ratio
fm, coupling coefficients ge, gp and load resistance
rp,rm.
Fig. 19 Random acceleration excitation, which input into the energy harvester a measured random acceleration, b its power spectral density
123
Microsyst Technol (2015) 21:401–414
413
to acceleration spectral density, but increasing degree
is affected by the value of load resistance. When the
load meets the impedance matching, enhancing extent
reaches the maximum.
Only at the optimal load and resonant frequency can
output mean power and spectral density reach the
maximum, and the resonant frequency of energy
harvesting system rises with the load resistance of
PE element increasing, but not influenced by the EM
load resistance.
Enhancing coupling strength can not only improve
output power and power spectral density, but also
widen frequency band of energy harvesting. The
smaller coupling strength is, the greater increase extent
of output power and spectral density will be. In
addition, optimal load of EM element varies increases
with coupling strength enhancing.
3.
4.
Acknowledgments This work is supported by the National High
Technology Research and Development Program of China (Grant No.
SS2013AA041104).
Appendix
By using the table in reference (Gradshtenyn and Ryzhik
1994), for the system if the complex frequency response
function is of the form
Fig. 20 Effect of load resistance on power of harvester a effect of PE
load resistance on power of PE element, b effect of EM load
resistance on power of EM element
HðxÞ ¼
ixB1 þ ðixÞ2 B2
A0 þ ixA1 þ ðixÞ2 A2 þ ðixÞ3 A3 þ ðixÞ4 A4
ð38Þ
Their integral of the square of the absolute value of
complex frequency response can be computed as
respectively
Z1
A1 B22 A3 B21
HðxÞ2 dx ¼ p
ð39Þ
A0 A23 þ A21 A4 A1 A2 A3
1
Also the integral for the transfer function in equation is
2
Zþ1 ixB1 þ ðixÞ2 B2
dx
2
3
A0 þ ixA1 þ ðixÞ A2 þ ðixÞ A3 1
¼p
Fig. 21 Output power of hybrid energy harvester at different spectral
density of acceleration
2.
Output mean power, voltage, and current of energy
harvester increase as SD of acceleration increasing;
moreover, mean output power is linearly proportional
B22 A1 þ B21 A3
A3 ðA1 A2 A0 A3 Þ
ð40Þ
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