Instrumentation & Measurement Magazine 26-3 - 8

Fig. 4. Theoretical voltage waveform of the piezoelectric material.
Theoretical Analysis of the Proposed SP-OSSHI
To analytically calculate the output power of the proposed circuit,
the theoretical waveform of piezoelectric material voltage
is shown in Fig. 4. The dynamic characteristics of piezoelectric
material can be simplified to the equivalent electrical model of
" equivalent current source + equivalent internal capacitance +
resistance. " Therefore, the current output of piezoelectric material
can be obtained by:
I u CV

 ‐
P
(4)
where I is the current output of the piezoelectric material, α is
the force factor, u and V represent the vibration displacement
and the voltage of piezoelectric material, respectively [44].
During the natural charging stage, i.e., the period t0
-t1
shown in Fig. 4, the piezoelectric material only charges C1
through the D2
tionship can be expressed, as follows:
11
 u C V C V C VC M
t Idt u C Vdt C V V )2 (
t
t00
t
 u C V C V C V V 1,
 
t Idt u C Vdt C V V )2 (
t
t
11
00
t
 


P
1(
Fig. 3. Working process: (a) Natural charging; (b) Extra open circuit; (c) Energy
transfer and voltage inversion.
remains unchanged due to the cut-off effect of the diode D2
the transistor Q2
.
Energy Transfer and Voltage Inversion Stage: When the voltage
value of capacitor CP
drops to the difference of Vbe
with the
voltage VC1, capacitor C1 starts to discharge at this time, so that
transistors Q2 and Q4 are in the forward-biased status, and inductor
L1, capacitors CP‖C1 and Crect form an LC circuit. After
half an oscillation period, the piezoelectric material voltage
VP
flips from VM
to Vm, and capacitor Crect
and Q4
also completes the
charging process. When the voltage inversion process ends,
transistors Q2
rent source iP continues to charge CP
are disconnected and the equivalent curin
the opposite direction.
Then the negative half cycle begins, which is similar to the positive
cycle.
8
and
C M Cm m p peak
1,

1,
Vpeak and Vm

p m) 
1(
C M Cm
1,

are shown in Fig. 4, and VC1,M
and VM
can be expressed as:
cos
VV 

M eak
p
)
and VC1,m
(5)
represent
the initial voltage and end voltage of the capacitor C1
during the voltage inversion stage, respectively. The relation
between Vpeak
(6)
where θ is the phase difference between the displacement extreme
value moment and the moment of voltage inversion.
Because it is relatively small compared to other factors, the influence
of θ is ignored.
In the voltage inversion stage, after 1/2 LC oscillation period,
it can be expressed as:
 
VV
VV
‐
m
M B
B ‐
γ is the voltage inversion factor, which can be obtained by
(8), and VB
is the voltage of the equivalent capacitor in the LC
IEEE Instrumentation & Measurement Magazine
May 2023
(7)
 


P
1(
C M Cm m p peak
1,

1,
 

p m) 
1(
1,

branch, as shown in Fig. 3a. Therefore, the rela
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