An improved analysis of the SSHI interface in piezoelectric energy harvesting
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Smart Mater. Struct. 16 (2007) 2253–2264 doi:10.1088/0964-1726/16/6/028
An improved analysis of the SSHI
interface in piezoelectric energy harvesting
Y C Shu
1,3, I C Lien
1and W J Wu
21Institute of Applied Mechanics, National Taiwan University, Taipei 106, Taiwan, Republic of China
2Department of Engineering Science and Ocean Engineering, National Taiwan University, Taipei 106, Taiwan, Republic of China
E-mail:yichung@spring.iam.ntu.edu.tw
Received 29 April 2007, in final form 15 August 2007
Published 9 October 2007
Online at
stacks.iop.org/SMS/16/2253
Abstract
This paper provides an analysis for the performance evaluation of a
piezoelectric energy harvesting system using the synchronized switch
harvesting on inductor (SSHI) electronic interface. In contrast with estimates
based on a variety of approximations in the literature, an analytic expression
of harvested power is derived explicitly and validated numerically for the
SSHI system. It is shown that the electrical response using an ideal SSHI
interface is similar to that using the standard interface in a strongly coupled
electromechanical system operated at short circuit resonance. On the other
hand, if the SSHI circuit is not ideal, the performance degradation is
evaluated and classified according to the relative strength of coupling. It is
found that the best use of the SSHI harvesting circuit is for systems in the
mid-range of electromechanical coupling. The degradation in harvested
power due to the non-perfect voltage inversion is not pronounced in this case,
and a new finding shows that the reduction in power is much less sensitive to
frequency deviations than that using the standard technique.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
The need to power remote systems or embedded devices inde-pendently, coupled with advancements in low-power electron-ics, has motivated many research efforts to focus on producing electrical energy from various ambient energy sources. These include solar power, thermal gradients and vibration. Among these energy scavenging sources, parasitic mechanical vibra-tion is a potential power source that is abundant enough to be of use, is easily accessible through microelectromechanical systems (MEMS) manufacturing processes for conversion to electrical energy, and is ubiquitous in applications from small
household appliances to large infrastructure elements [50,52].
While there are several options for transmitting vibra-tion energy into electrical energy, vibravibra-tion-based piezoelec-tric converters have received much attention as
transduc-ers [18,31], since they have high electromechanical coupling,
3 Author to whom any correspondence should be addressed.
require no external voltage source, and are particularly
at-tractive for use in MEMS [10,17, 22, 25, 38]. As a result,
piezoelectric elements for scavenging energy from ambient vibration sources have been used in various types of struc-tures to serve specific purposes, including the use of reso-nant piezoelectric-based structures of cantilever beam
configu-ration [1,11,14,23,40,51,54] as well as plate (membrane)
configuration [7,8,16,29,30,66]. Other harvesting schemes
include the use of long strips of piezoelectric polymers in ocean
or river-water flows [2,61], the use of piezoelectric ‘cymbal’
transducers [27,28], and the use of piezoelectric windmills for
generating electrical power from wind energy [45,46]. The
application to structural health monitoring with vibration
pow-ered piezoelectric sensors can be found in [15,37,41].
Fundamental research on the study of optimal AC power output has been reported extensively in the recent
literature [12, 13, 26, 44, 60, 65]. In most of the work
reported above, the vibration source is typically represented by a single harmonic signal, and an energy scavenger is required
to resonate mechanically at a frequency tuned to that of the driving source to which it is attached, in order to generate
the maximum electrical energy. As a frequency mismatch
of only a few percent may result in a significant drop in power output, methods to adjust the resonance frequency to match the driving vibrations across a range of frequencies have
been proposed by [6, 34, 53]. Fundamental investigations
concerning the conversion efficiency by piezoelectric power
generators include the early works by [19, 62, 63] and the
recent works by [9, 48,49, 56]. Another fundamental study
relating energy harvesting to electrically induced damping can
be found in [36]. It is shown that the damping added to a
vibrating structure is attributed to the removal of electrical energy from the system, and an explicit expression of this induced damping is provided in the case of a weakly coupled electromechanical system. In addition, the power harvesting system is demonstrated to work similarly to a shunt damping
system [35] except that the energy is stored instead of
dissipated.
The research works cited above focus mainly on developing optimal energy harvesting structures so only AC
power output is considered there. However, the electrical
outputs of these devices in many cases are too small to power electrical devices directly. Thus, the methods of accumulating and storing parasitic energy for intermittent use are also the
key to developing self-powered systems [58,59]. In addition,
a vibrating piezoelectric element generates an AC voltage, while the electronic device in many practical applications requires a stabilized DC voltage. As a result, the electrical interface connecting the piezoelectric element and the terminal electrical load is needed to ensure electrical compatibility. Power optimization schemes therefore depend not only on the mechanical solicitation, but also on the specific type of electronic interface circuit chosen in the energy harvesting
system. However, much work has addressed optimal AC
power flow, while little has considered AC/DC power output
until recently. Ottman et al [42, 43] have used the linear
load impedance adaptation to develop highly efficient electrical
circuits to store the generated charge. They have claimed
that at high levels of excitation the power output can be increased by as much as 400%. But the linear assumption implies that the internal current source is independent of the
load impedance [32]. Shu and Lien [55] have proposed
an improved analysis without this assumption to study the optimal AC/DC power generation for a rectified piezoelectric
device. An accurate estimation of AC/DC power output
is proposed explicitly and is related to electrically induced
damping and energy conversion efficiency [56]. It is shown that
the optimization criteria vary according to the relative strength of the electromechanical coupling. Other related studies can be found in [20,24].
Recently, Guyomar et al [21] have developed a new
electronic interface circuit to enhance power extraction. It is called synchronized switch harvesting on inductor (SSHI). This approach was derived from the so-called synchronized switch
damping (SSD), a nonlinear technique developed originally
by Richard et al [47] to investigate the effect of vibration
damping on mechanical structures. It is shown that the
electrical harvested power can be increased by as much
as 400–900% over the standard technique [4, 5, 21, 39].
However, the approach they used to evaluate the performance of the SSHI circuit is based on the assumption that the periodic excitation and the speed of mass are in-phase. As a result, the effect of frequency deviation from resonance on the electrical behavior of an SSHI system is not included
in their analysis. As vibration-based energy scavenging
generators achieve the maximum power by requiring the matching of resonance frequency to applied frequency, power reduction due to frequency deviation cannot be ignored. Therefore, this paper revisits performance evaluation of the
SSHI electronic interface based on the improved analysis [57].
A new estimate of average harvested power for the SSHI system is derived explicitly and validated numerically, and the connection between the standard and SSHI techniques is also established here. The electrical performance of the SSHI system is evaluated according to two key requirements: (1) the degradation in harvested power due to the non-perfect voltage inversion, and (2) the sensitivity in power deduction
due to frequency deviation. The analysis reveals that the
best use of the SSHI circuit is for systems in the mid-range of electromechanical coupling, since the performance degradations are the least in these cases.
2. Harvesting circuit: a standard interface
Consider an energy conversion device including a vibrating piezoelectric structure together with an energy storage
system. Suppose the modal density of such a device is
widely separated and the structure is vibrating at around its resonance frequency; we may model the power generator as a
mass+spring+damper+piezo structure [21, 42,48]. The
governing equations of this electromechanical model system can be described by
M¨u(t) + ηm˙u(t) + K u(t) + Vp(t) = F(t), (1)
− ˙u(t) + Cp˙Vp(t) = −I (t), (2)
where u(t) is the displacement of the mass M, Vp(t) the
voltage across the piezoelectric element, ηm the mechanical
damping coefficient,K the effective stiffness,the effective
piezoelectric coefficient,Cpthe clamped capacitance,I(t)the
current flowing into the specified circuit andF(t)the external
forcing function. These effective constantsM, K, andCp
are dependent on the properties of the chosen materials and the specific types of piezoelectric generators. For example,
we refer to [13] and [64] for the explicit expressions of
these effective constants for the piezoelectric bimorph power
generator. This paper considers a harmonic mechanical
excitation given by
F(t) = F0sinwt, (3)
where F0 is the constant magnitude and w (in radians per
second) is the angular frequency of vibration. This case is particularly chosen since many applications of piezoelectric materials for power generation involve the use of periodic straining of piezoelectric elements.
The unknowns in (1) and (2) are u(t), Vp(t) and I(t),
while there are only two equations. An additional equation is required and can be obtained from the consideration of the
Piezoelectric Vibrator Piezoelectric Vibrator
Ce R V (t)c Ce R V (t)c
I(t) I(t)
L
(a) (b)
Figure 1. (a) A standard energy harvesting circuit. (b) An SSHI energy harvesting circuit.
t t ti tf V (t)p Vc -Vc u(t) u0 -u0 T 2 ti tf t t ti tf V (t)p Vc -Vc u(t) u0 -u0 T 2 ti tf VcqI -VcqI (a) (b)
Figure 2. Typical waveforms of displacement and piezoelectric voltage for (a) the standard and for (b) the SSHI electronic interfaces.
chosen electronic interface between the piezoelectric element and the terminal electric load. Since the electronic load needs a stabilized DC voltage while a vibrating piezoelectric element generates an AC voltage, this requires a suitable AC-to-DC
converter to ensure electrical compatibility. For example,
figure1(a) is a standard interface circuit commonly used for
design analysis [42]. It provides an estimation of an upper
bound of the real power that the piezoelectric power generator is able to deliver at a given excitation. The standard interface includes an AC/DC rectifier followed by a filtering capacitance
Ce added to smooth the DC rectified voltageVc. The terminal
load is modeled using an equivalent resistor R with power
assumption equal to the average input power of the actual
terminal electric load. Typically, the filter capacitor Ce is
chosen to be large enough so that the rectified voltage Vc is
essentially constant to have a stable DC output voltage [42]. As
a result, the rectified voltageVcis independent ofCeprovided
that the time constant RCe is much larger than the oscillating
period of the generator [21].
To understand how the standard AC/DC electronic
interface shown in figure 1(a) works for energy transfer, a
perfect rectifying bridge is chosen for demonstration. It is
open circuited if the piezoelectric voltage|Vp| < Vc, and the
outgoing piezoelectric element currentI in (2) is null. On the
other hand, when |Vp| reaches Vc, the bridge conducts and
the piezoelectric voltage is blocked at the rectified voltage;
i.e., |Vp| = Vc. The conduction in the rectifier diodes
is blocked again when |Vp| starts decreasing. Hence, the
piezoelectric voltageVp(t)either varies proportionally with the
displacementu(t)when the rectifying bridge is blocking, or
is kept equal to Vc when the bridge conducts. As the model
equations (1)–(3) are developed at the resonance mode of the
device, a single-mode vibration of the structure is expected.
Thus, the displacement at the steady-state operation is assumed to be
u(t) = u0sin(wt − θ), (4)
where u0 is the magnitude and θ is the phase shift. This
assumption of choosing the sinusoidal form for displacement
has been made by Guyomar et al [21] excluding the effect of
the phase shiftθ. Shu and Lien [55] have included this effect
and validated it both numerically and experimentally for the
standard interface. The corresponding waveforms ofu(t)and
Vp(t)are illustrated in figure2(a).
To analyze the steady-state response of (1) and (2)
connected to the standard AC/DC interface under the harmonic
excitation (3), first note that it can be shown that the rectified
voltage Vc is related to the magnitude of displacement u0
by [21,42,55]
Vc= 2Rw
2CpRw + π
u0. (5)
This formulation (5) can be derived by integrating (2) over a
semi-period of vibration, and by using the fact that the average
current flowing through the capacitanceCe is null at
steady-state operation. Thus, from (5), u0 has to be determined to
decideVc. There are three approaches in the recent literature
for estimating it [21, 42, 55]. The first one models the
piezoelectric device as the current source in parallel with its
internal electrode capacitanceCp [25,41,42]. It is based on
the assumption that the internal current source of the generator
is independent of the external load impedance. However,
the amplitude of the current source is closely related to that of displacement, which depends not only on the mechanical damping but also on the electrical damping at the resonant
vibration [36, 56]. This assumption is therefore not suitable
result, Guyomar et al [21] have proposed another estimation accounting for the effect of electromechanical coupling. Their estimation is based on the assumption that the external forcing function and the velocity of the mass are in-phase, or in other
words the phase shift effect is neglected in (4). Instead, Shu
and Lien [55] have included this phase factor in their improved
analysis, and derived the analytic expressions of displacement
magnitudeu0, rectified voltageVcand harvested average power
P. Their results are summarized as follows:
u0= u0 F0 K = 1 2ζm+ 2k2er (r+π 2)2 2 2+1− 2+ ke2r r+π 2 212 , (6) Vc= Vc F0 = r r +π2 × ke2 2ζm+ 2ke2r (r+π 2)2 2 2+ 1− 2+ k2er r+π2 212 , (7) P= P F2 0 wscM = 1 r +π 2 2 × ke22r 2ζm+ 2ke2r (r+π 2)2 2 2+ 1− 2+ k2er r+π 2 2, (8)
where several non-dimensionless variables are introduced by
k2e = 2 K Cp , ζm= ηm 2√K M, wsc= K M, = w wsc, r= CpwscR. (9) Above, k2
e is the alternative electromechanical coupling
coefficient,ζmthe mechanical damping ratio,wscthe natural
oscillation frequency (of the piezoelectric vibrator under the
short circuit condition) andandrthe normalized frequency
and electric resistance. Notice that there are two resonances for the system since the piezoelectric structure exhibits both short circuit and open circuit stiffness. They are defined by
sc=1, oc =
1+ k2
e, (10)
wherescandocare the frequency ratios of the short circuit
and open circuit, respectively. Note that the shift in device
natural frequency is pronounced if the coupling factor k2
e is
large.
The improved estimates (6)–(8) have been found to
agree well with experimental observations and numerical
simulations of (1) and (2) [55]. Therefore, these estimates
are suitable for the electrical performance evaluation of the piezoelectric energy harvesting system embedded with the standard electronic interface. Moreover, if the vibration source
is due to the periodic excitation of some base, this givesF0=
M A, whereAis the magnitude of acceleration of the exciting
base. From (8), the harvested average power per unit mass
becomes P M = A2 wscP(r, , ke2, ζm). 0 0.3 0.6 0.9 1.2 1.5 0.8 1 1.2 1.4 1.6 0 0.4 0.8 1.2 1.6 Electromechanical Coupling Factor Normalized Frequency Normalized Power
Figure 3. The normalized power P against the normalized frequency
and the electromechanical coupling factor k2
eat the optimal conditions in the sense that Popt(, k2
e, ζm) = P(ropt(), , ke2, ζm)
and ropt() is determined by solving ∂
∂rP(r, , ke2, ζm) = 0. We use
ζm= 0.04 for the whole simulation. Notice that for large ke2there are
two identical peaks of power evaluated at the frequency ratio close to
sc= 1 and oc=
1+ k2
e. These peaks are saturated for much higher coupling factor k2
e 1.
Thus, the optimization scheme is closely related to the tuning of the electric resistance, the selection of suitable operation points, and the magnitudes of the coupling coefficient
and mechanical damping ratio. Basically, from (8) the
harvested average power increases significantly for smaller
mechanical damping ratio ζm or larger electromechanical
coupling coefficient k2
e. It is consistent with that found
by Badel et al [3], who have performed an interesting
experiment by comparing the performances of vibration-based piezoelectric power generators using a piezoelectric ceramic and a single crystal. Under the same operating conditions, the power generated using the single crystal is much higher than that using the ceramic, since according to their measurements
the coupling factork2
e of the former is 20 times larger than
that of the latter. However, one has to be cautious that the average harvested power approaches its saturation value for
much largerk2
e, as illustrated in figure3.
The improved estimates have also been compared with the un-coupled and in-phase estimates according to the relative magnitudes of electromechanical coupling coefficient and
mechanical damping ratio. The results of [55] show that
the conventional un-coupled solution and in-phase estimate
are suitable, provided that the ratio ke2
ζm 1, while
the discrepancies among these distinct approaches become
significant when k2e
ζm increases. If the shift in device natural
frequency is pronounced and the mechanical damping ratio of the system is small, i.e. k2e
ζm 1, the harvested power is shown
to have two optima evaluated at(r1opt, opt1 ) and(r2opt, opt2 ),
where opt1 is close to sc and the electric loadr1opt is very
small, while opt2 is close to oc and r2opt is large. Indeed,
table1summarizes the relation between the system parameters
k2
e and ζm and the normalized load, displacement, voltage
and power at these two optimal conditions. The first optimal
pair is designed at the short circuit resonance sc with the
optimal loadrscopt ∝ k12e
ζm
10–4 10–2 100 102 104 0.8 1 1.2 1.4 1.6 Normalized Frequency Normalized Resistance Normalized Power 0 0.4 0.8 1.2 1.6 10–4 10–2 100 102 104 0.8 1 1.2 1.4 1.6 Normalized Frequency Normalized Resistance Normalized Power 0 0.4 0.8 1.2 1.6 (a) (b)
Figure 4. The normalized power against the normalized electric resistance and frequency ratio. (a) A strongly coupled electromechanical
system using the standard AC/DC electronic interface (k2
e = 1.0, ζm= 0.04,k
2 e
ζm = 25). (b) A weakly coupled electromechanical system using
the ideal SSHI electronic interface (k2
e= 0.01, ζm= 0.04,k
2 e
ζm = 0.25, QI= ∞). Notice that (a) and (b) provide identical peaks of harvested
power evaluated at different conditions.
Table 1. The relation between the system parameters k2
eandζmand the normalized electric resistance, displacement, voltage and power operated at the short circuit (sc) and open circuit (oc)
resonances [55]. Note that the conditionk2e
ζm 1 is implied in the analysis. Optimal conditions sc oc Resistance ropt sc ∝ 1 ke2 ζm < ropt oc ∝ 1 (1+k2 e) k2 e ζm Displacement uopt0 ∝ 1 ζm > u opt 0 ∝ 1 ζm(√1+k2 e) Voltage Voptc ∝ 1 < V opt c ∝√1 1+k2 e k2e ζm Power Popt∝ 1 ζm = P opt ∝ 1 ζm
the open circuit resonanceoc with the optimal loadrocopt ∝
1 (1+k2
e) k2
e
ζm. They give identical values of maximum harvested
power, which depends only on the mechanical damping ratio
ζm. Unlike the power, the displacement is higher atscthan at
oc, while the voltage operating at the first peak is one order
of magnitude smaller than that operating at the second peak.
Finally, figure4(a) gives the dependence of the normalized
harvested power on the normalized resistance and frequency ratio for the case of strong electromechanical coupling. While such a strong coupling is not commonly observed in the conventional piezoelectric power generators, we particularly emphasize it here since we will show in the next section that the behavior of an ideal SSHI system is similar to that of a strongly coupled electromechanical standard system excited at around the short circuit resonance. This finding is generally valid no matter whether the real electromechanical system is weakly or strongly coupled.
3. Harvesting circuit: an SSHI interface
An SSHI electronic interface consists of adding a switch and
an inductanceLconnected in series and is in parallel with the
piezoelectric element as shown in figure1(b). The electronic
switch is triggered according to the maximum and minimum of the displacement of the mass, causing the processing of piezoelectric voltage to be synchronized with the extreme values of displacement.
To illustrate the electrical behavior of this nonlinear processing circuit, consider the harmonic excitation given
by (3). In view of the single-mode excitation, the mechanical
displacement u(t) is assumed to be sinusoidal as in (4) in
steady-state operation. The validation of this assumption will
be examined in section4.1by considering the output voltage.
The waveform of the piezoelectric voltageVp(t), however, may
not be sinusoidal and is dependent on the specific type of the interface circuit connected to the piezoelectric element. To
see this, letT = 2π
w be the period of mechanical excitation
andti andtf be two time instants such that the displacement
u(t)undergoes from the minimum−u0 to the maximumu0
as illustrated in figure2(b). The switch is turned off most of
the time during this semi-period(ti+, tf). When it is turned on
at the time instantti, |Vp(t)|remains lower than the rectified
voltage Vc. So the rectifying bridge is open circuited, and
an oscillating electrical circuit composed by the inductance
L and the piezoelectric capacitanceCp is established, giving
rise to an inversion process for the piezoelectric voltageVp.
Specifically, let tbe the half electric period of this oscillating
L–Cpcircuit. It is equal to [21]
t = π LCp.
We assume that the inversion process is quasi-instantaneous in the sense that the inversion time is chosen to be much
smaller than the period of the mechanical vibration; i.e., t =
ti+− ti T. The switch is kept closed during this small
time period t, resulting in the reverse of voltage on the
piezoelectric element, i.e.,
Vp(ti+) = −Vp(ti)e
−π
2QI = VcqI, qI=e
−π
as illustrated in figure2(b). Above,QIis the inversion quality factor due to the energy loss mainly from the inductor in series with the switch. As a result, the current outgoing from the piezoelectric element through the rectifier during a half
vibration period can be obtained by integrating (2) from time
ti+totf [3] tf ti+ − ˙u(t) + Cp˙Vp(t) dt= −2u0 + Cp 1−e−2QπI Vc = −T 2 Vc R
since the rectifier bridge is blocking during the inversion
process and the inversion time t T. The relation between
the magnitude of displacementu0and the rectified voltageVc
is therefore obtained by
Vc= 2Rw
(1− qI)CpRw + π
u0. (12)
The rest of the problem is to estimate the magnitude of
displacementu0and the phase shiftθ. To solve them, consider
the balance of energy first. Let (1) be multiplied by˙u(t)and (2)
be multiplied byVp(t). Integration of the addition of these two
equations from timeti+totf gives the equation of the energy
balance tf ti+ F(t) ˙u(t)dt = tf ti+ ηm˙u2(t)dt+ tf ti+ Vp(t)I (t)dt +1 2M˙u 2(t)|tf ti++ 1 2K u 2(t)|tf ti++ 1 2CpV 2 p(t)| tf ti+ = tf ti+ ηm˙u2(t)dt+ tf ti+ Vp(t)I (t)dt+12Cp(1− qI2)V 2 c (13) where tf ti+ F(t) ˙u(t)dt= π 2F0u0sinθ, tf ti+ ηm˙u2(t)dt= π 2ηmwu 2 0, tf ti+ Vp(t)I (t)dt = π w V2 c R.
There are now three unknowns,u0, θ andVc, while only
two equations, (12) and (13), are found. The third equation
connecting these unknowns can be obtained as follows. Notice that from (2)
˙Vp(t) =
Cp[−I (t) + ˙u(t)].
(14)
Differentiating (1) with respect to timetand substituting (14)
into it, we find
M d dt¨u(t) + ηm d dt˙u(t) + K + 2 Cp d dtu(t) − Cp I(t) = d dtF(t). (15)
Integrating (15) with respect to time t from ti+ to tf and
using (4) provides the third equation
K− Mw2+ 2 Cp u0− π 2CpwR Vc= F0cosθ. (16)
Thus, the unknownθ can be eliminated from (13) and (16).
This gives F02= ηmwu0+ 2 wR+ (1− qI2) Cp π V2 c u0 2 + K− Mw2+ 2 Cp u0− π 2CpwR Vc 2 . (17)
The above equation (17) can be further simplified to findu0by
using the relation betweenu0and Vc given by (12), and this
gives u0= F0 ⎧ ⎨ ⎩ ηm+2 1+C p Rw2π (1−qI2) 2R ( 1−qI) 2 CpwR+π2 2 2 w2+ K−w2M+ ( 1−qI) 2 w2R (1−qI) 2 CpwR+π2 2 ⎫ ⎬ ⎭ 1 2 .
The average harvested power is therefore obtained onceu0is
determined since from (12)
P= V 2 c R = 4R2w2 {(1− qI)CpRw + π}2 u20. (18) The following summarizes our main results for the piezoelectric element connected to the SSHI interface circuit.
The normalized displacement magnitude uSSHI
0 , rectified
voltage VSSHIc and average harvested power PSSHI are given
respectively by uSSHI0 =u SSHI 0 F0 K = 1 ⎧ ⎨ ⎩ 2ζm+2[1+ r 2π(1−qI2)]k2er (1−q I) 2 r+ π 2 2 2 2+ 1−2+ ( 1−qI) 2 k2er (1−qI) 2 r+π2 2 ⎫ ⎬ ⎭ 1 2 , (19) VSSHIc = V SSHI c F0 = r (1−qI) 2 r + π2 × ke2 ⎧ ⎨ ⎩ 2ζm+2[1+ r 2π(1−q2I)]k2er (1−q I) 2 r+π2 2 2 2+ 1−2+ ( 1−qI) 2 k2er (1−qI) 2 r+ π 2 2 ⎫ ⎬ ⎭ 1 2 , (20) PSSHI= P SSHI F2 0 wscM = 1 (1−qI) 2 r +π2 2 × ke22r 2ζm+2[1+ r 2π(1−q2I)]ke2r ( 1−qI) 2 r+ π 2 2 2 2+ 1− 2+ ( 1−qI) 2 k 2 er (1−qI) 2 r+π2 2. (21) All above are expressed in terms of non-dimensionless
parameters defined in (9).
Finally, Guyomar et al [21] have used the in-phase
assumption to analyze the electrical performance of the power generator using the SSHI interface. To be precise, they have assumed that the external forcing function and the velocity of
Normalized Frequency Normalized Power r : 0.13 r : 0.28 r : 0.48 r : 5.5 r : 10 r : 20 r : 0.82 r : /2 r : 3 Normalized Frequency r : 0.13 r : 0.28 r : 0.48 r : 0.82 r :π/2 r : 3 r : 5.5 r : 10 r : 20 N ormalize dP o w e r (a) (b) 0 0.3 0.6 0.9 1.2 1.5 0.9 0.95 1 1.05 1.1 1.15 0 0.3 0.6 0.9 1.2 1.5 1.1 1.15 0.9 0.95 1 1.05 π
Figure 5. Normalized harvested power versus frequency ratio for various load resistances based on (a) the in-phase estimate PSSHIin−phaseand
(b) the improved analytic estimate PSSHI. We take k2
e= 0.09, ζm= 0.04 and QI= ∞ for the whole simulation.
the mass are in-phase, giving rise to no phase shift effect in their formulation. The following summarizes their results for future comparisons: uSSHIin−phase=u SSHI in−phase F0 K = 1 2ζm+2[1+ r 2π(1−qI2)]k2er (1−q I) 2 r+π2 2 , (22) VSSHIin−phase= V SSHI in−phase F0 = r (1−qI) 2 r +π2 × ke2 2ζm+2[1+ r 2π(1−q2I)]ke2r (1−q I) 2 r+π2 2 , (23) PSSHIin−phase= P SSHI in−phase F2 0 wscM = 1 (1−qI) 2 r +π2 2 × k2er 2ζm+2[1+ r 2π(1−q2I)]ke2r ( 1−qI) 2 r+ π 2 2 2. (24)
4. Discussion
4.1. In-phase versus improved analysis
The in-phase estimate shows a lack of frequency dependence. To see this, we consider a case where the ratio of the electromechanical coupling factor to the mechanical damping
ratio is in the medium range; i.e., k2e
ζm is of order one. We
take k2
e = 0.09 andζm = 0.04, and therefore
k2 e
ζm = 2.5. It
can be shown that other parameters provide similar contrasting comparisons between the in-phase and improved estimates.
Figure 5(a) describes the average harvested power versus
frequency ratio for various load resistances based on the
in-phase estimate while figure 5(b) is based on our improved
analysis. For fair comparison, the ideal inversion of voltage
is implied in this case; i.e., QI = ∞. We will discuss
this condition in more detail later. Clearly, the two estimates predict identical values of the optimal power. However, the in-phase estimate is unable to predict the system behavior when the applied driving frequency deviates from the system resonance frequency. As the reduction in power is significant due to frequency deviation, such an effect cannot be ignored in practical design.
We next validate our improved estimates numerically by
transforming (1)–(3) to an equivalent circuit with R∗ = ηm
2
as resistance, L∗ = M2 as inductance and C∗ =
2 K as
capacitance as shown in figure 6(a). We use the software
PSpice to simulate this equivalent circuit connected to the
SSHI interface. We takeke2 =0.09, ζm= 0.04 and consider
the non-ideal voltage inversion with quality factor QI =
2.6 [21]. The results are illustrated in figure 6(b), where
the normalized power is plotted against the frequency ratio
evaluated at the optimal electric load,ropt =1.01 in this case.
The predicted results from the in-phase and analytic estimates
are represented by dashed and solid lines in figure6(b), while
the numerical results are marked using open circles there. Apparently, the numerical simulation favors results predicted based on our analytic improved estimate. Therefore, from
now on, we will use (19)–(21) as the fundamental tool for the
performance evaluation of the piezoelectric energy harvesting system embedded with an SSHI interface circuit.
4.2. Ideal inversion of piezoelectric voltage
To see how the SSHI electronic interface boosts power extraction, consider an ideal case where the inversion of the
piezoelectric voltageVpis complete; i.e.,QI= ∞. From (11)
Ce Vc I(t) Cp L Piezoelectric Vibrator R R* L* C* (a) 0.9 0.95 1 1.05 1.11 1.15 Normalized Frequency 0 0.4 0.8 1.2 1.6 Normalized P ower Simulation Analytic Estimate In-Phase Estimate (b)
Figure 6. Numerical validation with parameters k2
e= 0.09, ζm= 0.04 and QI= 2.6. (a) The equivalent circuit model for the system of
equations (1)–(3) connected to the SSHI interface. (b) The simulation results compared with those predicted by analytic and in-phase estimates. becomes PSSHI= 4 π2 r k2 e2 4 ζm+4k2 er π2 2 2+ (1− 2)2 . (25)
The optimal electric load resistance and the normalized power
operated atscare therefore
ropt= π 2 4 1 k2 e ζm , PSSHI|r=ropt,=1= 1 16ζm. (26)
From (26), the optimal load resistance is inversely proportional
to the ratio k2e
ζm while the corresponding optimal power depends
only on the mechanical damping ratioζmand is independent of
the electromechanical coupling coefficientk2
e. Comparing all
of these features with table1suggests that the behavior of the
power harvesting system using the SSHI interface is similar to that of a strongly coupled electromechanical system using the standard interface and operated at the short circuit resonance
sc. Indeed, comparing (5) with (12) suggests that the original
capacitanceCpis replaced by the effective capacitance ¯Cp =
(1−qI)
2 Cp. Therefore, the effective electromechanical coupling
coefficient¯ke2can be realized by
¯k2 e = 2 K ¯Cp = 22 K(1− qI)Cp . (27)
For the ideal inversion,qI → 1, and this gives ¯ke2
ζm → ∞no
matter what the original value of the ratio k2e
ζm is. The response
using the SSHI interface is therefore similar to that using the standard interface in a strongly coupled electromechanical
system operated atsc. In addition, according to table1, there
exists another identical peak of power operated at the open circuit resonance. But now this frequency ratio is realized as
¯oc =
1+ ¯k2
e → ∞. Hence, the second peak of power is
moved to the infinite point in the(r, )space, and therefore
there is only one peak of power for the SSHI electronic interface, no matter whether the real electromechanical system is weakly or strongly coupled, as schematically shown in figure4(b).
In addition, we particularly takek2
e =0.01 andζm=0.04
in figure4(b) so that the electromechanical generator itself is
weakly coupled(k2e
ζm = 0.25). The harvested power obtained
using the standard harvesting circuit is pretty small in this case,
since it has been shown that [55]
P ropt= π 2, =1, k 2 e, ζm ≈ 2 π ke2 ζm 1 16ζm 1 16ζm = PSSHI max if k2e
ζm 1. But the inclusion of the SSHI circuit boosts
the average harvested power, whose maximum is the same as that using a strongly coupled electromechanical generator
(k2
e =1.0,ζm=0.04 and
k2 e
ζm =25). Therefore, the harvested
power increases tremendously for any weak coupling SSHI system at the cost of using a much larger optimal electric load,
which is proportional to 1
ke2
ζm
according to (26).
5. Comparisons
We now compare the electrical performance of a vibration-based piezoelectric power generator using the standard and SSHI electronic interfaces according to the different ratios of ke2
ζm. As in many practical situations, the inversion of the
piezoelectric voltage Vp is not perfect (QI = ∞), which
accounts for a certain amount of the performance degradation
using the SSHI electronic interface. We take QI = 2.6
for comparison here [21]. It is possible to have a larger
value of quality factor QI by requiring the use of a low loss
inductor [33].
First, consider a weakly coupled electromechanical
system; i.e., the ratio k2e
ζm 1. We take k
2
e = 0.01
and ζm = 0.04 for demonstration. This gives k2e
ζm =
0.25. The harvested power versus frequency ratio for various
normalized resistances is shown in figure 7(a) based on the
standard interface and in figure 7(d) based on the SSHI
interface. The maximum normalized power generated for the
ideal voltage inversion is around PSSHI|QI=∞ = 1.56, while
PSSHI|QI=2.6 = 0.67 in the non-ideal case. In spite of the significant performance degradation in this case, the achieved optimal power is three times larger than that using the standard
interface (P=0.23) at the cost of using a larger matching load
resistance fromropt = π/2 to 5.2 by comparing figure7(a)
with figure7(d).
Next, suppose the electromechanical coupling is in the
medium range; i.e., the ratio of k2e
ζm is of order one. We take
k2
e = 0.09 and ζm = 0.04. This gives
k2 e
ζm = 2.25. The
harvested power versus frequency ratio for various normalized
resistances is shown in figure 7(b) based on the standard
interface and in figure 7(e) based on the SSHI interface.
The maximum normalized power for the non-ideal voltage
inversion is PSSHI|QI=2.6 = 1.38, which is slightly smaller
than the ideal case (PSSHI|QI=∞ = 1.56), but it is slightly
larger than that using the standard electronic interface (P =
1.20). While there is no significant increase of power output
using the SSHI electronic interface in this case, figure 7(e)
demonstrates that the harvested power evaluated at around the optimal load is less sensitive to frequency deviating from the resonant vibration. For example, the amount of normalized
harvested power P evaluated atr = π2 in the standard case
drops from 1.2 to 0.6 for about 5% frequency deviation, and from 1.2 to 0.2 for about 10% frequency deviation. However, under the same conditions, the normalized harvested power
PSSHI in the SSHI circuit drops from 1.3 to only 1.0 for
about 5% frequency deviation, and from 1.3 to 0.5 for about 10% frequency deviation. This frequency-insensitive feature is much more pronounced in the case of ideal voltage inversion,
as can be seen by comparing figure5(b) with figure7(b).
Finally, we turn to a strongly coupled electromechanical system(ke2
ζm 1). Our numerous numerical simulations show
that the rule of thumb for the appearance of double identical peaks of power in the standard case is when
k2 e ζm 10.
We then take k2
e = 1.0 and ζm = 0.04, and this gives
k2 e
ζm = 25. The harvested power versus frequency ratio for
various normalized resistances is shown in figure7(c) based
on the standard interface and in figure7(f) based on the SSHI
interface. In the standard case, the harvested power has two identical optimal peaks, and the switching between these two
peaks can be achieved by varying the electric loads. The
envelope of these peaks has a local minimum, which is closely related to the minimum proof mass displacement. On the other hand, there is only one peak of power in the SSHI circuit, as explained in the previous section. Unlike the standard case
as illustrated in figure7(c), the peaks of the average harvested
power decrease significantly as the load resistances increase, as
shown in figure7(f). In addition, it can be seen from (26) that
the optimal electric load for the SSHI system is very small, since ke2
ζm 1. Thus, figure7(f) indicates that any deviation
in the load resistance will cause a significant power drop in the SSHI case. Such an effect cannot be ignored in practical design, since there may exist other inherent electrical damping in the whole circuit system; for example, the diode loss is not taken into account in the present analysis. As a result, there seems to be no obvious advantage in using the SSHI electronic
interface from the comparison between figures7(c) and (f).
6. Conclusions
The electrical behavior of the piezoelectric power harvesting system using the SSHI electronic interface is analyzed and
compared to that using the standard interface. Instead of
using the un-coupled or in-phase assumptions to estimate the harvested power, an analytic expression of it based on the improved analysis is proposed explicitly and validated numerically for the SSHI circuit system. It is found that no matter whether the real electromechanical system is weakly or strongly coupled, the electrical response using an ideal SSHI interface is similar to that using the standard interface in a strongly coupled electromechanical system operated at the short circuit resonance. As a result, the harvested power increases tremendously for any weak coupling SSHI system at the cost of using a much larger optimal electric load.
The performance degradation due to the non-perfect voltage inversion is discussed and classified according to the relative strength of the coupling. This effect on power deduction is significant for weakly coupled electromechanical
systems. On the other hand, if the electromechanical
coupling is in the medium range, the degradation in harvested power is not pronounced in this case, and a new finding shows that the reduction in power is much less sensitive to frequency deviations than that using the standard technique. This provides a great advantage in design since the energy scavenger has a wider inherent bandwidth. Moreover, this frequency-insensitive feature is much more conspicuous when the inversion quality factor is improved.
0.9 Normalized Frequency 0.95 1 1.05 1.1 0 0.15 0.3 0.45 0.6 0.75 Normalize d P ower r : 0.4 r : 0.52 r : 0.7 r : 0.1 r : /2 r : 2.8 r : 5.2 r : 8 r : 12 0.9 Normalized Frequency 0.95 1 1.05 1.1 0 0.15 0.3 0.45 0.6 0.75 Normalized P ower r : 0.4 r : 0.52 r : 0.7 r : 0.1 r : /2 r : 2.8 r : 5.2 r : 8 r : 12 (a) (d) 0.9 Normalized Frequency 0.95 1 1.05 1.1 1.15 0 0.3 0.6 0.9 1.2 1.5 Normalize dP ower r : 0.13 r : 0.28 r : 0.45 r : 0.75 r : /2 r : 3 r : 5.5 r : 10 r : 20 r : 0.13 r : 0.28 r : 0.45 r : 0.75 r : /2 r : 3 r : 5.5 r : 10 r : 20 0.9 Normalized Frequency 0.95 1 1.05 1.11 1 .15 0 0.3 0.6 0.9 1.2 1.5 Normaliz ed Power (b) (e) r : 0.1 r : 0.25 r : 0.42 r : 0.7 r : 1.2 r : 2 r : 3.2 r : 5.3 r : 13 0.8 1 1.2 1.4 1.6 Normalized Frequency Normalized Powe r 0 0.25 0.5 0.75 1 1.25 1.5 1.75 r : 0.1 r : 0.25 r : 0.42 r : 0.7 r : 1.2 r : 2 r : 3.2 r : 5.3 r : 13 0.8 1 1.2 1.4 1.6 Normalized Frequency Norma lized Power 0 0.25 0.5 0.75 1 1.25 1.5 1.75 (c) (f ) π π π π
Figure 7. Normalized power versus frequency ratio for different values of normalized resistances. Notice that (a)–(c) are obtained using the
Acknowledgments
We thank Professors K C Wu, C K Lee and U Lei for
their constant encouragement and support. We are glad
to acknowledge partial support from the National Science Council under grant No 96-2628-E-002-119-MY3, and from the Ministry of Economic Affairs under grant No 95-EC-17-A-05-S1-017 (WHAM-BioS).
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