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Sensors and Materials, Vol. 32, No. 7 (2020) 2475–2492
MYU Tokyo
S & M 2276

Equivalent Circuit Model for MEMS Vibrational Energy Harvester
Compatible with Sinusoidal and Nonsinusoidal Vibrations
Hiroaki Honma,* Yukiya Tohyama, Sho Ikeno, and Hiroshi Toshiyoshi
Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan(Received February 3, 2020; accepted March 2, 2020)

Keywords: vibrational energy harvester, LTspice, electret, sinusoidal vibration, nonsinusoidal vibration

We build an equivalent Simulation Program with Integrated Circuit Emphasis (SPICE) model
for a vibrational energy harvester comprising comb electrodes coated with an electret film thatis used to convert the vibrational kinetic energy into electrical output power by electrostaticinduction. In the assembled module, sinusoidal and nonsinusoidal vibrations are imported into
the nonlinear current sources as an inertial force, and the power-generating performances are
simulated. The nonsinusoidal waves observed in an actual environment (highway duct) areused as an input sample. By quantitatively comparing the simulation and experimental results,
we verify the applicability of the equivalent module for various vibrations. When the device
is excited by sinusoidal vibration, the maximum output power is calculated to be 71 µW at
0.044 G (1 G = 9.8 m/s2), which is close to the experimental result of the actual device, 68 µW at
0.045 G. Furthermore, when importing a nonsinusoidal vibration, the two timings at the highest
generating peak are obtained in accordance with the moment of resonance, and the amplitudes
are experimentally and analytically obtained to be 0.80 and 0.63 V, respectively.

1.

Introduction

Interest in Internet of Things (IoT) has been growing since it is expected to work as amaintenance- and operation-assisting apparatus in social infrastructures. Today’s widelyused sensor nodes commonly have the problems of the need for battery replacement. Hence,
the integration of sensor nodes with the autonomous power sources such as environmental
energy harvesters could be a solution to extending their lifetime.(1–6) Ambient vibrations arehighly promising as power source for wireless sensor nodes. However, harvesting from theenvironmental vibrations is difficult because they are usually distributed in a frequency rangeof 200 Hz or lower and in an acceleration range of 0.1 G (1 G = 9.8 m/s2) or less. Recently,
micro-electromechanical systems (MEMS)-based vibrational energy harvesters have beendrawing intense research interest owing to their resonance function to amplify vibrations in
the range of small accelerations.(7–23) The vibrational energy harvesters are grouped into three
types, namely, piezoelectric, electromagnetic, and electrostatic, depending on the mechanismof inducing electrical charges. In particular, an electret-based energy harvester can effectively*

Corresponding author: e-mail: honma-hh@iis.u-tokyo.ac.jp
https://doi.org/10.18494/SAM.2020.2821
ISSN 0914-4935 © MYU K.K.
https://myukk.org/

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Sensors and Materials, Vol. 32, No. 7 (2020)

harvest ambient vibrational energy owing to the flexibility of the MEMS design that allows it toadapt to the various frequency ranges.
The power-generating efficiency of the energy harvesters based on the MEMS technologies
can increase when the frequency of environmental vibrations is equal to its resonant frequency;
however, when the resonance is off, the performance of power generation is severely degraded.
For this reason, for conventional vibrational energy harvesters, the vibration information
relevant to the placement of the sensor node must be analyzed beforehand and must be taken
into account in the design phase. In addition, we must be aware of the difference between
the sinusoidal and nonsinusoidal vibrations. Regardless of the pattern of the environmental
vibration, a reliable simulation method must be developed to foresee the behaviors of energy
harvesters.
Because a general vibrational energy harvester receives kinetic energy and delivers electricalenergy to the subsequent system, a Simulation Program with Integrated Circuit Emphasis (SPICE)
simulation is a helpful multiphysics analysis for handling the physics in two different domains.
Subcircuit models for an electrostatic microactuator and a piezoelectric microactuator have beenreported and can also be used to simulate the performance of the energy-harvesting systems.(24–26)
SPICE simulation results for the PZT-based energy harvester have been presented in a previous
report.(27) Furthermore, the equivalent circuit model of the electrostatic energy harvester has
been presented, and the simulation results are in good agreement with the measurement results
obtained from the developed energy harvesters.(28,29) Subcircuit models of the electrostatic
energy harvester based on electrets have also been developed, and the speculated output
performances have been reported.(30) In this case, the kinetic power input to a harvesting
system is modelled using a constant-voltage source for a SPICE simulation. However, only
sinusoidal waveforms have been used, and accordingly, the output performances when using
nonsinusoidal vibration still have not been analyzed in subcircuit models. Moreover, therelationship between the output power and input kinetic energy still has not been elucidated andneither has the applicability of subcircuit models under various environments.
In this paper, SPICE models for an energy harvester composed of a comb electrode covered
with an electret film are designed. By using such SPICE models, the harvesting system andactuations in the case of sinusoidal and nonsinusoidal vibrations are determined. Moreover, bycomparing the simulated value and the experimental value, we verify the applicability of the
subcircuit models for various vibrations.

2.

Electret-based MEMS Energy Harvester

2.1

Actual device

Figure 1 shows a photograph and a scanning electron microscopy (SEM) image of thedeveloped vibrational energy harvester with a comb electrode coated with an electret film.
Details of the device structures and fabrication techniques have been reported elsewhere.(31)The device has symmetric comb-electrode structures that effectively cancel the electrostatic
constraint force. A total of 900 pairs of comb fingers with a gap length of 14 µm, a tip length of

Sensors and Materials, Vol. 32, No. 7 (2020)

(a)

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(b)

Fig. 1. (Color online) Photograph of developed three-port harvester device and a close-up SEM view of the comb
electrodes.

700 µm, and a height of 100 µm have been made. The comb electrode is designed to handle anelectrode stroke of ±350 µm and is coated with electrets charged at about −250 V with respect tothe moving shuttle. The electret film consists of a thin silicon oxide film including potassiumions. The fabrication process of the electret film has been reported elsewhere.(32) Supporting
hinges are designed to have a spring constant of 925 N/m. Owing to the added mass made oftungsten of 1.5 g on the movable electrode, the resonant frequency is experimentally observed
at 125 Hz. Moreover, mechanical damping of the developed harvester was calculated to be 0.0015
from the experimental result of the quality factor.
2.2

Design of subcircuit model of electret comb structure

A module for a fully developed energy harvester, that is, a subcircuit model of a comb
electrode coated with an electret film, is designed and prepared in this work. This subcircuit
model is used to represent the electrostatic induction current from the electret, as well as theelectrostatic force acting on the comb electrodes. Figure 2 shows the schematic structure of the
comb electrode with an electret that is coated on the side of the comb finger. In this figure, thetop electrode is movable, and the bottom is fixed. Each comb pair is described by a model withlength Tt, height H, and air gap length g. The movable electrode is already overlapped with thefixed electrode by the length L, and therefore, the total overlap length of electrodes becomes
L + x when the movable electrode is further displaced by x. As illustrated in Fig. 2, the air andelectret films of this model have the relative dielectric constants ε1 and ε2, respectively. The
fixed charge is located from the film surface to the depth R in the electret layer. The detailed

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Sensors and Materials, Vol. 32, No. 7 (2020)

Fig. 2.

(Color online) Analytical model of comb electrodes with electret charges.

study of the analytical model has been reported elsewhere,(30) but we discuss the procedure ofmodeling here again since its final implementation as a SPICE module is different.
Applying Gauss’s law to the electrical field E1 in the lower side of the silicon oxide, E2 in theupper side, and E3 in the air gap, we write−ε 0 ⋅ ε1 ⋅ E1 + ε 0 ⋅ ε1 ⋅ E2 =σ,0
−ε 0 ⋅ ε1 ⋅ E2 + ε 0 ⋅ ε 2 ⋅ E3 =(1)

where ε 0 is the dielectric constant of vacuum. These electrical fields build up a voltage
difference equal to the applied DC voltage V, and thus,

E1 ⋅ (T − R ) + E2 ⋅ R + E3 ⋅ g =
−V .

(2)

By simultaneously solving these three equations, one obtains the electric fields

 σ gε1 + σ T ε 2 + σ Rε 2 2 − σ T ε 2 2 + V ε 0ε1ε 2 2 
g ε 0 ε 2 2 + T ε 0 ε 2 ε1σ (R −T ) E1  
V+


ε
ε
 
0 1
−
.
 E2  = g ε1
 E3  

T
+
  
ε2



σ Rε 2 − σ T ε 2 + V ε 0ε1ε 2g ε 0 ε1 + T ε 0 ε 2



(3)

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As a result, the total electrostatic energy stored in one unit of the comb tips is=
U ( x)

11
ε 0ε1E12 (T − R )( L + x ) H + ε 0ε1E2 2 R ( L + x ) H
2
2
2

 σ 
1
1
+ ε 0ε 2 E32 g ( L + x ) H + ε 0ε1  (T − R ) (Tt − L − x ) H .
22
 ε 0ε 1 (4)

For virtual displacement Δx, the total energy will increase by ΔU, as defined by∆U = U ( x + ∆ x) − U ( x) .

(5)

At the same time, the upper electrode will gain additional charges ΔQU as∆QU = −ε 0ε 2U (∆ x) E3.

(6)

Considering the conservation of energy, the sum of the increment of the electrostatic energy
and the mechanical work done by the electrostatic force is equalized by the work done by theexternal voltage V that transferred the charge ΔQ against the potential difference as∆U + F ⋅ ∆ x = V ⋅ ∆QU ,

(7)

where

V ⋅ ∆QU ＝V ⋅H (∆ x)ε 0ε 2 (σ Rε 2 − σ T ε 2 + V ε 0ε1ε 2 )(8)g ε 0ε1 + T ε 0ε 2uses the expressions for the electrical fields and derives the electrostatic force as

F= −H ε 2 2 (σ R − σ T + V ε 0ε 1 )2( R − Rε + T ε − 2 gε − 2T ε + gε ε ) .
2222ε 0ε1 ( gε1 + T ε 2 )

2

1

2

2

1 2

(9)

In this work, we define the electret potential by using the electret density asV ε εσ= − e 0 1.
R −T

(10)

The differential voltage V between the comb electrodes is written as V = V b − Vd, where V b
and Vd are the potentials of the upper and lower electrodes, respectively. By using these newparameters, the electrostatic force F is rewritten as2480

Sensors and Materials, Vol. 32, No. 7 (2020)

F= −H ε 0ε1ε 2 2 (Vb − Vd − Ve )2

( R − Rε + T ε − 2 gε − 2T ε + gε ε ) .
22

2 ( g ε1 + T ε 2 )

2

2

1

2

1 2

2

(11)

Because each comb electrode has two sides to generate the electrostatic force and that the
total number of the comb pair in the device is N, the net electrostatic force is F × 2N. The
physics model expressed in Eq. (11) is interpreted in the SPICE as
FSPICE =−2 * V(N) * ((V(H) * V(ep) * V(ep1) * V(ep2) **2 * (V(Vb) − V(Vd) − V(Ve))**2 * (V(R) − V(R) * V(ep2) **2 + V(T) * V(ep2) **2 − 2 * V(g) * V(ep1)(12)− 2 * (T) * (ep2) + V(g) * (ep1) * (ep2))) / (2 * (V(g) *(ep1) + V(T) * (ep2)) **2)),

where V(x) is a reserved function of SPICE that recognizes the argument x as voltage andextracts the value as a unitless number. The parameters used in the model are listed in Table 1.
At the same time, the induction current is determined as the amount of charge change perunit time:I=

2
dQU d  −2 N ε 0ε1ε 2 H (Vb − Vd − Ve ) ( L + x ) 
=

.
dt
dt 
g ε1 + T ε 2


(13)

In a similar manner to FSPICE, the current is translated into the SPICE expression as

ISPICE =ddt  −2 * V(N) * V(H) * V(ep) * V(ep1) * V(ep2) **2 * ( V(Vb) − V(Vd) − V(Ve) ) * ( V(j) + V(x) ) / ( V(g) * V((H) * V(ep) * V(ep1) * V(ep2) **2 * ( V(Vb) − V(Vd) − V(Ve) ) * ( V(j) + V(x) ) / ( V(g) * V(ep1) + V(T) * V(ep2) )  , (14)where the function ddt(x) is another reserved function of SPICE to perform the temporal
differentiation of x. Refer to Ref. 30 for the definition of other parameters.
Table 1
Design parameters and initial conditions for vibrational energy harvester.
Parameter
SymbolValue
Comb’ initial insertion lengthL
350 µm700 µmLength of comb finger
Tt
Dielectric constant of electret film
ep1
3.5
Dielectric constant of air gap
ep2
1.0
Dielectric constantep
8.854 × 10 −12Massm
0.0015 kg (1.5 g)
Height of comb finger
H
100 µmElectret depth in electret film
R
0.79 µm
Thickness of electret film
T
0.8 µmNumber of comb pairsN
900−250 VElectret potential
Ve
Air gap between comb fingersg
14 µm

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2.3

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Design of SPICE module of electret-based MEMS energy harvester

The subcircuit model for the electrostatic force and electrostatic induction current is
implemented as an equivalent circuit that runs on the LTspice platform. The current source ofLTspice is used as a multiphysics behavior model that can be programmed using the unitlessphysical variables in algebraic equations. The fundamental programming technique has beenreported elsewhere.(33) Figure 3(a) shows the circuit diagram of the electret actuator module
unit based on the comb-electrode structure. When the displacement of a movable electrode

(a)

(b)

(c)
Fig. 3. (Color online) (a) LTspice equivalent circuit module for electret comb electrodes. (b) Analytical model implementation of electrostatic force and electrostatic induction current. (c) Internal parameters defined by constantvoltage sources.

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is imported, the subcircuit model relays electrostatic force back to the model of a spring-masssystem. At the same time, the inductive current obtained by the electron flows to an external
circuit. Here, the equations of the electrostatic force [Eq. (12)] and the output current [Eq. (14)]are directly described for the nonlinear current source in the subcircuit model, as shown in Fig.
3(b). In this SPICE simulator, physical constants are defined by the constant-voltage sources
within the module. The imported displacement parameter is terminated with a 1 Ω resistor andis read out as a voltage for internal use, as shown in Fig. 3(c).
The circuit model of the vibrational energy harvester consists of two comb-electrode subcircuit models (left and right) and a velocity-damped resonant generator (VDRG) model asshown in Fig. 4; the design parameters and the initial conditions used are listed in Table 1. The
two comb electrodes are placed symmetrically to cancel out the internal electrostatic forces.
Figures 5(a) and 5(b) respectively show the circuit description for the suspension and the mass
modules; the former reads in the displacements (x1, x2) and the velocities (v1, v2) measured at thetwo ends of the suspension and returns the restoring forces (Fm1, Fm2) as

Fm1 =− Fm 2 =c ⋅ ( v2 − v1 ) + k ⋅ ( x2 − x1 ),Fig. 4.

(15)(Color online) LTspice equivalent circuit model of electret energy harvester with a pair of comb electrodes.

Sensors and Materials, Vol. 32, No. 7 (2020)(a)Fig. 5.

2483(b)(Color online) VDRG model for (a) the suspension and (b) the mass.

where c and k are the viscous damping coefficient and the spring constant, respectively. The
mass module, on the other hand, reads in the electrostatic forces (Fe1, Fe2) and the mechanicalforces (Fm1, Fm2) from both sides and returns the displacement and the velocity by solving the
Newtonian equation of motion. The function idt in Fig. 5(b) is for temporal integration, suchas idt(V(a), 0) for the integration of the acceleration a and idt(V(v1)) is for the integration of thevelocity v1.
The external vibrations are defined as the inertial force (Fm1) acting on the movable mass m,which is expressed using a nonlinear current source. It is also possible to express the incoming
vibrations using time series data of acceleration measured from an actual vibration source suchas a highway duct. The acceleration data could be read as a csv or a text file, and multipliedwith the mass m to calculate the inertial force. This technique could be used for both sinusoidal
and nonsinusoidal waves. The value for the connected stray capacitance Cs is modeled from theelectrical interconnection line area and buried oxide (BOX) thickness, which is calculated to be
100 pF.

3.

Verification of Applicability of Built Energy Harvester

3.1

Sinusoidal vibration

To demonstrate the applicability of the previously described subcircuit model in the SPICE
simulation, energy equivalence based on the input energy and the output energy is calculated,2484

Sensors and Materials, Vol. 32, No. 7 (2020)as shown in Fig. 6. The imported sinusoidal vibration is set to have an acceleration of 0.1 G at
the resonant frequency of 125 Hz, and is defined using the nonlinear current source. Figure6(a) shows the calculated amplitude, which gradually grows with resonance of the movable
electrode. At this time, kinetic energy and strain energy in the subcircuit model are reciprocallytransferred in accordance with the motion of the movable electrode, as shown in Fig. 6(b). Here,quantities of energy losses due to several factors in the subcircuit model are calculated and are
indicated together in Fig. 6(c). It is commonly known that the factors that cause energy loss areof the following three types: mechanical loss, electrical loss, and energy stored in a spring–masssystem. The total potential indicates the amount of energy stored in the spring–mass system,
which is calculated by summing the kinetic energy and the elastic energy presented by Fig.

(a)

(b)

(c)

(d)

Fig. 6. (Color online) Simulation results of energy balance of MEMS vibrational energy harvester. (a)Displacement of movable electrode when shaken by 0.1 G vibrations at 125 Hz. (b) Potential energy stored in thesuspension and kinetic energy of the mass. (c) Mechanical loss and electrical loss (output). (d) Input energy is equal
to total output energy.

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6(b). Mechanical energy loss VMechanical is calculated as an integrated value in which viscous
resistance is time-integrated as
VMechanical=

2∫ c ⋅ vel dt ,(16)

where vel is the velocity of the movable electrode in Fig. 5(b).
In the same manner, electrical loss VElectrical is also calculated as a time-integrated value ofan electrostatic force as
VElectrical=∫ Fe1 ⋅ vel dt .

(17)

As a result, total output energy can be plotted as the total amount of energy consumed by theabove-described three losses, as shown in Fig. 6(d). Moreover, the time-integrated workload
as the input energy of the spring–mass system is also plotted together and is compared with thetotal output energy in the same figure. As can be seen from Fig. 6(d), the two curves indicating
input and output energies are in complete agreement; therefore, it is evident from the calculated
results that the energy balance of the subcircuit model is completely satisfied. Thus, the validityof the subcircuit models has been quantitatively verified.
To verify the applicability of the subcircuit model operated under the condition of receiving
sinusoidal vibration, calculated values of the power generation performances obtained from the
input sinusoidal vibration are quantitatively compared with the experimental values measured
under the same condition. The electrical circuit used to measure the short-circuit current of
the developed energy harvester(31) is shown in Fig. 7(a). We used a current–voltage converter(Hamamatsu, C7319) with a large input impedance to isolate the measurement from the input
impedance of the oscilloscope. The fixed electrode on the right was directly grounded, while
the one on the left was connected to the virtual ground point of the operational amplifier.
The experimentally observed short-circuit current waveform for the device operated with thesinusoidal vibration of 0.01 G acceleration at 125 Hz is shown by the dots in Fig. 7(b), alongwith the calculated waveform indicated by the solid line. As can be seen in Fig. 7(b), theexperimental current waveform is in good agreement with the calculated one.
The short-circuit current as a function of the input acceleration is calculated and quantitively
compared with the experimental value to verify how well the relationship between output
performance and input vibrational energy can be reproduced. Figure 8 indicates the shortcircuit current obtained when the movable electrodes oscillate within the stroke limit atresonance. The experimental value is represented by the dots and the calculated value is thesolid curve. Here, the input acceleration is set to handle the maximum stroke (±350 µm) of the
movable electrode. It is clear that short-circuit current is proportional to input acceleration sinceboth functions are proportional to velocity.(31–34) As shown in Fig. 8, both plotted currents areproportional to input acceleration. The developed device reached the maximum stroke at 0.020 G,
while the modeled device in SPICE reached it at 0.022 G. Also, the obtained maximum shortcircuit current under the same condition was 5.0 µA in the experiment and 5.2 µA in the2486

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(a)

(b)

Fig. 7. (Color online) (a) Circuit for measuring the short-circuit current. (b) Simulated and experimental shortcircuit current excited by 0.01 G acceleration at 125 Hz.

Fig. 8.

(Color online) Simulated and experimental short-circuit currents as a function of acceleration at resonance.

simulation. Thus, the experimental values are in good agreement with the calculated values
even when oscillated at maximum displacement.
For another verification, a comparison between the input and output energies is performed,
under the condition that generated power is electrically consumed by the external load applied
for the verification. Figure 9(a) shows the electrical circuit used to measure the output power
for the device connected to the optimal load and operated at its resonant frequency (125 Hz).
Here, the electrical power consumed as the external electrical loss is estimated from the output
current flowing through the load resistance of 7 MΩ. Hence, the output power as a functionof the input acceleration is experimentally obtained and shown in Fig. 9(b). The output power

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(a)

(b)

Fig. 9. (Color online) (a) Electrical circuit for measuring the output voltage across the load resistance. (b)Simulated and experimental output voltages at resonance.

calculated using the subcircuit model under the same conditions as the actual device is also
shown by the solid curve. When the external vibration frequency matches the harvester’s
eigenfrequency ω, then the output power is generally described as(35)

=
Pout

2

1
1
 G × 9.8 =mω 3 Q y 2 EHmω 3Q 
 EH ,
22
 ω2 

(18)

where EH indicates the harvester effectiveness. Here, as is clear from this equation, the power
is proportional to the square of the acceleration G. As seen in Fig. 9(b), this relation was clearlyshown by both curves of output power. When the device was excited at its resonance suchthat the movable electrode oscillated at its limit of ±350 µm, the maximum output power was
calculated to be 71 µW at 0.044 G using the simulation module. On the other hand, the output
power of the actual device reached 68 µW at 0.045 G. Thus, the experimental values are ingood agreement with the calculated ones. When we take a close look at the horizontal axis, theacceleration necessary for the maximum stroke increases compared with that under the no-loadcondition (0.020 G). This result can be explained by the kinetic energy stored in the springmass system to move the electrode being electrically consumed by an energy conversion systemwith an electret. Consequently, in this subcircuit model, degradation of the quality factor
in accordance with the electrical loss by the electret was also accurately reproduced in thesimulation results.

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The comparison of the simulation and experimental results revealed that the assembled
SPICE simulator reproduces the relationship between the amplitude of the movable electrodeand the change rate of inductive charges. An excellent agreement was found in all cases evenwhen input acceleration was changed. Thus, the subcircuit model and its applicability in casesof sinusoidal vibration were validated.
3.2

Nonsinusoidal vibration

In this paper, the output-generating performance of a developed device under nonsinusoidal
vibration is also experimentally observed and quantitatively compared with the performance
calculated under the same conditions using SPICE. To verify the output voltage in the caseof nonsinusoidal vibration, the typical environmental vibration was measured using an
accelerometer (Emic, P51CAC57) at a highway duct. At the same time, the output voltage of thedeveloped harvester placed at the same spot was observed.
Figure 10(a) shows the time-series pattern of acceleration on the measured ambientenvironmental vibration. Figure 10(b) shows the results of wavelet analysis of the acceleration.
Here, the operating direction of the energy harvester corresponds to the measured axis of theaccelerometer. The energy harvester is modified with additional mass to be 2.8 g with anelectret potential of −200 V. Thus, the resonant frequency is down-shifted to 91 Hz owing to
the increase in the additional mass, and the estimated mechanical damping is 0.0073. For thismeasurement setup, the fixed electrode on the left is directly connected to the input terminal in
the oscilloscope, and the output voltage is observed across the 1 MΩ resistance.
Figure 10(c) shows the collected pattern of the output voltage that is synchronized with thetime of acceleration in Fig. 10(a). By comparing Figs. 10(a) and 10(c), we see that the graphpattern of the observed output voltage does not resemble the pattern of the time-series data ofacceleration. Since the energy harvester is responsive to the input vibrations corresponding to
its resonance only, the two graph patterns will not necessarily match. The timing of the wavelet
peak in Fig. 10(b) naturally matches the timing of the surge of the acceleration in Fig. 10(a).
On the other hand, the timing of the measured peak voltage in Fig. 10(c) does not necessarily
match those of the acceleration surge in Fig. 10(a) but to the timing of the appearance ofthe frequency component near the mechanical resonance of the harvester (91 Hz). In otherwords, the waveform of the harvester voltage differs from that of the acceleration because
the characteristics of the gained voltage have been filtered and enhanced by the mechanicalresonance of the harvester.
Figure 10(d) shows the output voltage estimated using the subcircuit model input with themeasured time-series data of acceleration. The waveforms of the measurement in Fig. 10(c)and the simulation in Fig. 10(d) do not give a perfect match. In both patterns of the timeseries, however, the timing of the highest generated peak was 5.6 s. At the same time, theexperimental and simulated amplitudes were obtained to be 0.80 and 0.63 V, respectively, in thealmost similar range. Furthermore, other peaks of the collected output voltages were in good
agreement with the calculated one. The results indicate that the developed simulation model
can reproduce the harvester behavior at a frequency near the mechanical resonance.

Sensors and Materials, Vol. 32, No. 7 (2020)

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(a)

(b)

(c)

(d)

Fig. 10. (Color online) (a) Acceleration signal measured on an actual vibration source (highway duct) and (b)wavelet analysis of the acceleration. (c) Output voltage of MEMS energy harvester measured simultaneously on site. (d)Simulation result of the equivalent circuit model excited by the same vibrations.

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However, the calculated voltage pattern is not completely the same as the experimentallyobserved voltage pattern, as shown in Figs. 10(c) and 10(d). Figure 11 shows the Fouriertransformed time-series data of the experimentally measured acceleration. The measured
vibration is mostly distributed in the frequency range below 400 Hz. On the other hand, Fig.
12 shows the experimental frequency response spectrum from the actual device that defines theresonant frequency to be in the range of 90.5–91.5 Hz.
To verify the responding frequency range of the energy harvester, the wavelets of the timeseries data of the observed and calculated output voltages were transformed, as shown in Figs.
13(a) and 13(b), respectively. It was expected that the wavelet of the actual device would act asshown in Fig. 13(b), but only at resonant frequency. However, the collected data was analyzed
and the results are shown in Fig. 13(a). There is a slight response around the resonant frequency
with a tendency of increase toward the high frequency.

Fig. 11. (Color online) Results of FFT analysis of excitation vibrations.

Fig. 12. (Color online) Frequency spectrum of short-circuit current of the energy harvester used for the on-sitemeasurement.

Sensors and Materials, Vol. 32, No. 7 (2020)

(a)

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(b)

Fig. 13. (Color online) Wavelet analysis results for (a) experimentally obtained output voltage and (b) simulated
output voltage.

The possible reason for this discrepancy is that the analytical model only responded to input
vibrations corresponding to its resonance; however, the real device also responded to input
vibrations other than those of its resonance. The actual device responded to the vibrations invarious directions. Moreover, because the assembled model in SPICE only defines the vibrationin the axis corresponding to the operation direction of the vibrational energy harvester, it cannot
respond to other axes. Thus, there is a slight difference between the two patterns of output
voltage. This effect could be included in the simulation by expanding the one-dimensionalmodel into two or three dimensions, which is to be performed in future work.

4.

Conclusions

Analytical models for the vibrational energy harvester based on the comb structures coatedwith the permanent charge or electret were represented using a SPICE simulator. To verify theapplicability of the subcircuit model for various vibrations, several sinusoidal and nonsinusoidal
vibrations were input as nonlinear current sources emulating the inertial force, and each outputgenerating performance was simulated. The simulation results and experimental results
obtained from the actual device under the same conditions were compared. For sinusoidal
vibration, excellent agreement was found in all cases even when the input acceleration waschanged. Furthermore, peaks of the collected output voltages were in good agreement with thecalculated one even when the external vibration was nonsinusoidal. Therefore, the applicabilityof the model to cases of sinusoidal and nonsinusoidal vibrations was confirmed. However,the calculated voltage patterns were not completely the same as the experimentally observed
voltage patterns under the condition of inputting nonsinusoidal vibration. The assembled
model still did not respond well to multiple axes because it only defines vibration in the axis
corresponding to the operation direction of the vibrational energy harvester. In future work,
the assembled subcircuit model corresponding to multiple axes must also be implemented in the
SPICE module.

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Sensors and Materials, Vol. 32, No. 7 (2020)

Acknowledgments
This work was supported by JST CREST Grant Number JPMJCR15Q4. This work was also
supported by JSPS Grant-in-Aid for Scientific Research (B), Grant Numbers 18H01490 and
19K15038. We also thank Prof. Hidehiko Sekiya of Tokyo City University for providing thedata of nonsinusoidal vibrations.

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