FLUID-BASED ENERGY CONVERSION BY MEANS OF A PIEZOELECTRIC
MEMS GENERATOR
Ingo Kuehne*, Matthias Schreiter, Julian Seidel and Alexander Frey
Siemens AG, Corporate Technology, Corporate Research and Technologies, Munich, Germany
*Presenting Author: kuehne.ingo@siemens.com
Abstract: This paper reports the optimum design, fabrication and characterization of a piezoelectric MEMS
generator for fluid-actuated energy harvesting. Depending on the specific load scenario optimum beam shapes of
piezoelectric cantilevers are investigated by theoretical estimations and related experiments. A point load at the
free-end of a cantilever requires a triangular beam shape. Compared to a classical rectangular shape the electrical
area energy density is three times larger. A maximum area energy density value of 35 nJ/mm2 is measured for the
triangular beam shapes. A uniform load as occurring for fluid-actuated energy harvesting calls for a triangularcurved beam shape and is also superior to classical geometries.
Keywords: MEMS generator, piezoelectric, energy harvesting, conversion, fluid-actuated
INTRODUCTION
CONCEPT
State of the art piezoelectric MEMS generators
typically harvest mechanical energy that is based on
environmental vibrations [1, 2, 3 and 4]. These
generators are based on spring-mass configurations
that show the best efficiency when the resonant
frequency of the harvester and the dominant frequency
of the vibration match. Consequently, each generator
design has to be specific and typically requires a
vibration source with dominant spectral acceleration
peaks that are time-invariant regarding amplitude and
frequency. So, drift phenomena of vibrational
frequency spectra caused by environmental variations
(e.g. temperature, humidity) dramatically reduce the
efficiency of such generators.
Moreover, vibrations that are interfered by high
static accelerating forces (e.g. centrifugal forces) are
not efficiently suited for energy harvesting by springmass based generators.
Vibration-based energy harvesters that operate in a
non-resonant mode are able to overcome these
drawbacks. [5] presents a miniaturized non-resonant
inductive generator. Comparable MEMS-based
generators are still be missing.
Piezoelectric MEMS generators introduced in this
work are designed for alternative excitation scenarios.
The primary energy source is not based on an
accelerating force field that is indirectly coupled into
the generator via an inertial mass. Here, the primary
energy is directly coupled into the piezoelectric
structure of the generator. The primary energy can
either be a fluid-based energy (e.g. fluidic pressure
impulse) or a mechanical force that directly deforms
the piezoelectric structure.
In the following, piezoelectric MEMS generator
geometries are presented that are optimized for these
alternative excitation scenarios. The generation of
adequate fluidic respectively mechanical forces and a
reasonable coupling into the energy harvester is not
discussed in this work.
In general, the considered MEMS generators
consist of piezoelectric cantilevers. The excitation
scenarios make an additional inertial mass redundant.
The mass of the cantilever is in the 10-6 gram range
what makes the structure quite insensitive towards
high static accelerating forces.
The mode of coupling the primary energy into the
piezoelectric cantilever has a large impact on an
optimum beam shape. Figure 1 shows two typical
loading conditions for a constant force F.
Figure 1: Typical loading conditions for cantilevers:
a) point load at free-end of the beam, and b) uniform
load over the whole length of the beam.
The bending moment Mp(x) for a point load at the
free-end of the cantilever at x=L is expressed to:
x⎞
⎛
M p ( x ) = F ⋅ L ⋅ ⎜1 − ⎟
L
⎝
⎠
(1)
and the bending moment Mu(x) for a uniform load
results in:
M u (x ) =
1
x⎞
⎛
⋅ F ⋅ L ⋅ ⎜1 − ⎟
2
⎝ L⎠
2
(2)
A classical rectangular beam shape with length L,
width 2⋅w0, and thickness t is presented in Figure 2.
Figure 2: Classical rectangular beam shape.
Here, the curve shape w(x) is constant over the
beam length and can be expressed as:
w(x ) = w0
(3)
The area moment of inertia I(x) results in:
2 ⋅ w(x ) ⋅ t 3
12 ⋅ L
I (x ) =
(4)
The mechanical stress σ(x) can be calculated based
on the equations (1) respectively (2) and (4):
σ (x ) =
M p / u (x )
I (x )
⋅t
(5)
The transversal piezoelectric effect results in the
electrical voltage V(x):
V (x ) = −
d
ε
⋅ t ⋅ σ (x )
On the one hand the clamped region of the
cantilever shows the maximum mechanical stress and
contributes the most to the electrical energy. The
predominant rest of the volume is quite ineffective
regarding the energy harvesting. On the other hand this
effect results in an electrical voltage distribution. But a
real implementation requires planar electrodes which
forces equipotential surfaces respectively a constant
voltage. This can lead to an unwanted back coupling
from the electrical to the mechanical domain especially
for piezoelectric materials with strong electromechanical coupling. Here, the deformation of the cantilever
is retarted resulting in a decreased efficiency.
However, the discussed negative effects can be
overcome by a suitable geometry. An optimum design
enables a homogeneous mechanical stress distribution
within the lateral beam direction. This directly results
in a fixed electrical voltage distribution – equipotential
surface – so that the entire piezoelectric volume evenly
contributes to the energy transduction.
Equation (5) shows such a solution. Bending moment and area moment of inertia must follow an
identical dependency in x so that the mechanical stress
distribution becomes a constant value. This can easily
be accomplished by choosing the right curve shape
w(x) because it is directly proportional to the area moment of inertia (see Eq. (4)). Figure 3 presents the
optimum beam shapes for both point and uniform load.
(6)
with the piezoelectric constant d and the permittivity ε.
The electrical area energy density wpiezo can be
expressed as:
x= L
w piezo =
1 d2 t⋅L
⋅
⋅
⋅ w(x ) ⋅ σ 2 (x ) dx (7)
A x =0
2 ε
∫
with the cantilever area A.
The mechanical stress, the electrical voltage and
the electrical area energy density of the rectangular
beam shape for both point load and uniform load are
summarized in Table 1. The equations for the
mechanical stress show a spatial dependence. This
results in two drawbacks.
Table 1: Physical quantities for rectangular beam
shape for point load and uniform load.
Point Load
σ (x )
cσ ⋅ (L − x )
V (x )
cV ⋅ (L − x )
w piezo
wp
Uniform Load
cσ
2
⋅ (L − x )
2⋅L
cV
2
⋅ (L − x )
2⋅L
wu
Figure 3: Optimum beam shapes: a) point load at
free-end of the beam, and b) uniform load over the
whole length of the beam.
The curve shape, the mechanical stress, the electrical
voltage and the electrical area energy density of the
optimized cantilevers for both point load and uniform
load are composed in Table 2.
Table 2: Physical quantities for optimum beam shapes
for point load and uniform load.
w(x )
Point Load
x⎞
⎛
w0 ⋅ ⎜1 − ⎟
⎝ L⎠
σ (x )
cσ ⋅ L
V (x )
cV ⋅ L
w piezo
3⋅ wp
2
with cσ=6·F/(w0·t ), cV=-6·d·F/(w0·t),
wp=3·d2·F2·L3/(ε·w02·t3), wu=0.45·d2·F2·L3/(ε·w02·t3)
Uniform Load
x⎞
⎛
w0 ⋅ ⎜1 − ⎟
⎝ L⎠
cσ
⋅L
2
cV
⋅L
2
5 ⋅ wu
2
The optimized triangular beam shape in case of a
point load shows a constant mechanical stress
respectively electrical voltage. Moreover, the area
energy density is three times larger compared to a
rectangular beam shape.
In the event of a uniform load the optimized
triangular-curved beam shape also presents a constant
characteristic regarding mechanical stress and
electrical voltage. The optimized beam shape provides
an area energy density value that is five times larger in
comparison to the rectangular beam shape.
Even assuming that the rectangular beam shape
utilizes the whole device area best the triangular beam
shape with half the area reaches 150% of electrical
energy and the triangular-curved beam shape with a
third of the area achieves 167% of electrical energy
compared to the classical rectangular beam shape.
DESIGN
Figure 4 presents a fully processed 6” SOI-wafer
containing various piezoelectric cantilever structures
for different load scenarios. A single device that is
optimized for a uniform load can be seen in Figure 5.
piezoelectric layer is made of a self-polarized PZT thin
film with a thickness of 1.4 µm that is processed via
sputtering technology [6]. The bimorphic stack can
cope without a carrier layer. Figure 6 presents the layer
compostion of a bimorphic stack.
Figure 6: Detailed SEM picture of a piezoelectric bimorphic stack (optional Si-based carrier layer is not
shown).
In contrast to Figure 5 where the cantilever is
spacious released the cantilever can also be released in
a way that the structure is inherently enclosed within a
proper fluidic channel. Figure 7 presents a
corresponding detail of a bimorphic cantilever with a
narrow fluidic channel.
Figure 4: Fully processed 6” SOI-wafer containing
piezoelectric cantilever structures of various shapes.
Figure 7: Detailed SEM picture of a piezoelectric
bimorphic beam tip enclosed within a narrow (50µm)
fluidic channel.
EXPERIMENTS
Figure 5: SEM picture of a single triangular-curved
piezoelectric cantilever which is an optimum for a
uniform load.
There are two species of piezoelectric beam stacks
– monomorphic and bimorphic. This refers to the
number of piezoelectric layers. The monomorphic
stack has a single piezoelectric layer and requires an
additional carrier layer (e.g. 5 µm Si) in order to fix the
neutral fiber outside the piezoelectric material. The
First measurements of monomorphic rectangular
and triangular beam shapes have been done. The
piezoelectric cantilevers were excited by a mechanical
deflection at the free-end and then abruptly released in
order to see the decay of the natural oscillations.
Based on these measurements the optimum
external resistive load was determined. Figure 8 shows
an optimum impedance matching between the
piezoelectric impedance and an external resistance of
6.67 kΩ.
Furthermore, the dependency of the electrical
quality factor regarding the ambient pressure is
characterized and displayed in Figure 9. For this
particular cantilever structure quality factors of more
than 100 were found in case of atmospheric pressure.
Triangular Beam Shape
Theory
35.0
80
60
40
20
(w0 = 1.5 mm / L = 3.0 mm)
Rectangular Beam Shape (w0 = 1.5 mm / L = 3.0 mm)
P l
30.0
i
h (T i
l
Mechanical
Destruction
Electrical Energy Density [nJ/mm2]
Normalized Electrical Energy [%]
40.0
Measurement
100
25.0
20.0
15.0
10.0
5.0
0
100
1000
Resistive Load [Ω]
10000
Mechanical
Destruction
0.0
100000
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Free-End Deflection [mm]
Figure 8: Normalized electrical energy depending on
an external resistive load (impedance matching).
Figure 11: Electrical energy density depending on
beam deflection of the free-end.
600
CONCLUSION
Electrical Quality Factor
500
400
300
200
100
0
0.01
0.1
1
10
100
1000
Ambient Pressure [mbar]
Figure 9: Electrical quality factor of a piezoelectric
cantilever depending on ambient pressure.
Figure 10 shows the maximum electrical voltage
for both triangular and rectangular beam shapes
depending on the initial beam deflection of the freeend of the cantilever. It can be clearly seen that the
triangular beam shape tolerates a considerably larger
deflection. Furthermore, at a comparable deflection
level the achieved electrical voltage is larger in the
case of the triangular beam shape.
Li
(T i
l
This work is supported by the “Bundesministerium für
Bildung und Forschung”, Germany, (reference
16SV3336) and contributes to the project „ASYMOF Autarke Mikrosysteme mit mechanischen Energiewandlern für mobile Sicherheitsfunktionen“.
[1]
B
Mechanical
Destruction
Maximum Electrical Voltage [V]
(w0 = 1.5 mm / L = 3.0 mm)
Rectangular Beam Shape (w0 = 1.5 mm / L = 3.0 mm)
0.5
ACKNOWLEDGEMENT
REFERENCES
0.6
Triangular Beam Shape
Depending on the load scenario there is an
optimum beam shape where the resulting mechanical
stress respectively the generated electrical voltage is
constant over the whole lateral beam area. A point load
at the free-end of a cantilever requires a triangular
beam shape. Compared to a classical rectangular shape
the electrical area energy density value is three times
larger. This was successfully proven by suitable
experiments on piezoelectric cantilever test structures.
The maximum area energy density value for triangular
beam shapes was measured to 35 nJ/mm2. A uniform
load calls for a triangular-curved beam shape and is
also better performing than rectangular beam shapes.
0.4
0.3
[2]
0.2
0.1
[3]
Mechanical
Destruction
0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Free-End Deflection [mm]
Figure 10: Maximum electrical voltage depending on
beam deflection of the free-end.
The corresponding electrical area energy density
values are presented in Figure 11. The area energy
density values of the triangular beams are between two
and three times larger compared to the rectangular
beam. This is in accord with the theoretical
estimations. The maximum area energy density for
triangular beam shapes was measured to 35 nJ/mm2.
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