Instrumentation & Measurement Magazine 26-3 - 17

coordinate u(t). As a result, the projection of the equation of
motion onto the first bending mode ɸ(x) reduces it to:
12.7331 12.3624 10.6371 0.0666
11
 

 4.0063 10  
VV t

1
4.0063 10
2Co7.1049 Z Ω cos Ω
2
2
u uu u
11 2
3

 (4)
where ω = 0.977 is the nondimensional fundamental frequency
(which corresponds to 2.76 kHz), and Ω is the nondimensional
base excitation frequency.
Electrical Conditioning Circuit
The proposed electrical conditioning circuit based on Bennet's
machine influence is shown in Fig. 4 [2]. The electrical
model contains two variable capacitors (C1
and C2
, D2 and D3
movable electrode. Consequently, VS is observed with an exponential
increase.
The variable capacitances C1
and C2
can be used to calculate
the governing equation of the conditioning circuit shown
in Fig. 4. The current flowing into the diodes is estimated using
Shockley model that estimates the reverse current losses IS
. The
corresponding equations are given as:
1
i I Exp
12 
  
   1
 VV
SS
th
  
th
i I Exp
) working in
an out-of-phase manner, and representing the one formed by
the beam and the stationary electrodes 1 or 2. The circuit also
includes three diodes (D1
itor CS
) and a large storage capachaving
a fixed value that is used to collect the harvested
electrical energy.
To better understand the working principle of the circuit,
we suppose that at the beginning, the capacitance C1
()Max
is
at its maximum value C and C2
min
1

2
C . To initiate the circuit, C1
and CS
is at its minimum value
have to be initially
charged with an external power source and using a bias voltage
V0
= V1 − V2 should decrease and VS
. When the system is mechanically excited, we assume
that the beam starts moving so that C1
starts decreasing while
C2 starts increasing. Considering the charge conservation law,
V1 should increase, V2c
should
this case in a series configuration and a charge ΔQ flows from
C1
augmented each by ΔQ. When the movable electrode reverses
direction, C1
increases again and C2
crease of V1, an increase of V2C
diodes D1 and D2
, and an unvaried VS
become ON and D3
recovers a charge of 2ΔQ from C2
and CS
decreases, resulting in a de.
Here, the
turns OFF. The system
switches to a parallel configuration of the capacitances. Therefore,
C1
. As a result,
the system increases its total charge by ΔQ at each cycle of the
where i1, i2 and i3 are the current flowing into diodes D1
D3, respectively. V1
C1, D2 and CS
, V2 and VS
, respectively. Also, the currents i4
spond to those passing through C1, C2 and CS
, D2
and
are the electrical potential across
, i5 and i6
. The term Exp
represents the exponential function, ƞ = 1 is the ideality factor
of the diodes, the reverse saturation current is IS
= 0.1 nA,
the storage capacitor sets to CS = 10 pF, and the thermal voltage
sets to Vth = 25 mV.
Applying the Kirchhoff law to the three nodes conditioning
remain constant. At this stage, the diode D3 turns ON, while  
 dt
circuit shown in Fig. 4, we end up with:
 
I Exp1VV d CV dC V V1  2
11
D1 and D2 are OFF, the three capacitances (C1, C2 and CS) are in 
S 
Vth
process, C1 loses a charge ΔQ, while the charge into C2 

IS Exp
 V 

th
IS Exp




Vth
2 
  dt
  
Vth
 0
Vth
Exp
 1
CS

Results and Discussion
We numerically solve the coupled governing equation of the
electrostatic MEMS EH given by (4) and (6) using a RungeKutta
discretization. To better understand the dynamics of the
system, we start by simulating the frequency-response curves
of the device at fixed voltages (V1
= 0 V and V2C
= 0 V). The applied
excitation, in this case, is Zo = 2 μm and Ω = 5.1 kHz. The
result is shown in Fig. 5 for the simulated time response of the
beam tip. The capacitances 1 and 2 are varied with large values
according to Fig. 3c. In the literature, it was demonstrated that
a minimum ratio of 2 should be achieved between the maximum
and minimum values of the capacitance to be sure that
the energy can be transduced between the kinetic and the electrical
form [1].
Fig. 4. Electrical conditioning circuit based on the Bennet doubler.
May 2023
The steady-state behavior of the response, obtained in
Fig. 5, can be further analyzed using different excitation amplitudes
and frequencies. For this, in Fig. 6, we represent
the frequency-response curves of the beam when different
amplitudes Zo
IEEE Instrumentation & Measurement Magazine
17

 
to each of the capacitances C2 and CS. During this part of the   0
and CS is 
 
 V 2   Exp
 dt
  
VV dC V V21 2
2  S  
 

VV VV dV
    SS S
 S 10
2

dt
(6)
correi

2
  

VV 

S 



11
34
Vth
S 

VVS
dC V1 V2


56 ;
dt
 d CV
  1; i
2 
 dt
  i  CS
dVS
dt
  
 1; i I Exp2V
(5)
are used. A global hardening behavior of
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