Modeling, Simulation and Design of a Circular Diaphragm Pressure Sensor
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This document presents the design and analysis of a MEMS pressure sensor utilizing a circular diaphragm, analyzing its performance using ANSYS software. The sensor operates effectively in a pressure range of up to 25 MPa, demonstrating high sensitivity and linearity, and addresses the advantages of circular diaphragms over square ones for high frequency and range applications. The findings underscore the cost-effectiveness and precision of piezoresistive sensors in microelectromechanical systems.
![ﻣﻴﻜﺮوﻣﺎﺷﻨﻴﻜﺎري و رﻳﺰﻓﻨﺎوري اﻟﻤﻠﻠﻲ ﺑﻴﻦ ﻛﻨﻔﺮاﻧﺲ اوﻟﻴﻦICMEMS2014
ﻓﻨﺎوري ﭘﮋوﻫﺸﻜﺪه،اﻣﻴﺮﻛﺒﻴﺮ ﺻﻨﻌﺘﻲ داﻧﺸﮕﺎه ﻧﻮ ﻫﺎي29و30ﺑﻬﻤﻦ1392
Key words Pressure sensor, Circular diaphragm,
MEMS, Terms pressure, Small deflection, Finite
element, ANSYS,
Abstract: This paper aim in design and analysis
of MEMS Pressure Sensor by using ANSYS
software. A diaphragm based MEMS sensor in
the range of 25MPa by measured center
deflection of the circular pressure-sensitive and
using the strain gauge for measurement. First
this paper created the mathematical calculation
and then models the finite element analysis of
the diaphragm using ANSYS and determined the
pressure sensor in the range of 25MPa with
circular diaphragm is designed, which also
reflects the choice of correct design after
analyzing the effect of diaphragm thickness,
location and shapes of piezoresistor. Circular
diaphragm of different thickness and diameters
are simulated to meet the requirement of 25MPa
pressure. The whole fabrication process is
inexpensive and compatible with standard
MEMS process. Measurements show good
results such as high sensitivity 0~25MPa,
linearity 0~11.209MPa and upper pressure we
have 1.25% standard division. The work
indicates pressure sensor with circular
diaphragm could be fabricated for applications
of high range pressure measurement with high
precision.
1. Introduction
Pressure sensors have been developing rapidly
in the last decades with the development of
Micro Electro Mechanical System (MEMS)
technique and high demand of market. Recently,
with the rapid expansion of consumption of
pressure sensors of high range in petroleum
industry and daily applications, great efforts
have been directed towards high pressure
sensors in high temperature environment [1].
These devices can replace bulky actuators and
sensors with micro scale devices that can be
produced in integrated circuit photolithography
[2]. Capacitive sensor and resonator sensor have
been widely used. However, some defects of
them are difficult to overcome. The linear
outputs of capacitive sensor depend on complex
signal processing integrated circuit. The
resonator sensor has limitation to materials of
high quality and frequency. Besides, the cost is
too high. To overcome such disadvantages,
piezoresistive sensor has simple process circuit
Wheatstone bridge and cheap material silicon to
achieve high precision linear output. Many
piezoresistive sensors are fabricated with square
diaphragm of different thickness [3]. Compared
to the square diaphragm, circular diaphragm has
advantage in high pressure range. It has better
high frequency response in high range, although
the sensitivity is lower. The square diaphragm’s
nonlinearity in high range is harmful for the
sensor’s performance. Besides, its frequency
response is much lower than circular diaphragm
of same size. Thus, circular diaphragm with
meander-shaped piezoresistors is utilized for
sensors in high pressure measurement. Circular
diaphragm packaged with glass ring is simulated
by ANSYS software to determine optimal place
for piezoresistors distributed on the surface [4].
2. Structure and working principle
The conventional structure of piezoresistive
pressure sensor includes a square diaphragm as
shown in Fig 1. Piezoresistors are located at the
edge of diaphragm to transform pressure to
electrical signal. Stress distribution of one type
Modeling, Simulation and Design of a Circular
Diaphragm Pressure Sensor
ICMEMS2014-2026](https://image.slidesharecdn.com/keywordspressuresensor-151122212334-lva1-app6892/85/Modeling-Simulation-and-Design-of-a-Circular-Diaphragm-Pressure-Sensor-1-320.jpg)
![ﻣﻴﻜﺮوﻣﺎﺷﻨﻴﻜﺎري و رﻳﺰﻓﻨﺎوري اﻟﻤﻠﻠﻲ ﺑﻴﻦ ﻛﻨﻔﺮاﻧﺲ اوﻟﻴﻦICMEMS2014
ﻓﻨﺎوري ﭘﮋوﻫﺸﻜﺪه،اﻣﻴﺮﻛﺒﻴﺮ ﺻﻨﻌﺘﻲ داﻧﺸﮕﺎه ﻧﻮ ﻫﺎي29و30ﺑﻬﻤﻦ1392
of square diaphragm C type is shown in
Fig 2. It can be observed that the stress
concentration region on the edge is long and
narrow. Thus, it’s possible that the piezoresistors
distributed on the edge may not cause changes
equally under corresponding pressure which is
harmful to the sensor’s linearity and life time of
sensor. Meanwhile, Comparing to square
diaphragm, the circular diaphragm has higher
natural frequency which is of great benefit to
dynamic applications. For example, the natural
frequency of circular diaphragm demonstrated in
this paper is 1.4MHZ which is much higher than
that of square diaphragm of similar size. So if
the sensors are used in dynamic measurement,
especially for high frequency applications, the
circular diaphragm should be unutilized.
The design model of circular diaphragm is
shown in Fig 3 and it’s bonded with borosilicate
glass ring to form inverted cup structure as
sensing unit of the sensor. The key parameters
are the diameter of effective area d=2a and the
thickness of piezoresistive element ph . The
theoretical model of effective area is shown in
Fig 4 from which displacement can be observed
and the following theoretical calculation and
analysis are based on the model. The pressure P
is equally distributed and it should satisfy the
following equations with the circular diaphragm
[4].
The behavior of a diaphragm will depend upon
many factors, such as the edge conditions and
the deflection range compared to diaphragm
bornosilicatepiezoresistive
r
Z0
hp
a
z
p
neutral axis
Fig 1: conventional structure of piezoresistive pressure
Fig 2: square diaphragm C type
Fig 3: effective area for sensing pressure
d
Effective area
Fig 4-1: rigidly clamped diaphragm and Fig 4-2: its
associated displacement under uniform pressure.
(1)
(2)
1
2
((1 )(3 ))a ν ν+ +](https://image.slidesharecdn.com/keywordspressuresensor-151122212334-lva1-app6892/85/Modeling-Simulation-and-Design-of-a-Circular-Diaphragm-Pressure-Sensor-2-320.jpg)
![ﻣﻴﻜﺮوﻣﺎﺷﻨﻴﻜﺎري و رﻳﺰﻓﻨﺎوري اﻟﻤﻠﻠﻲ ﺑﻴﻦ ﻛﻨﻔﺮاﻧﺲ اوﻟﻴﻦICMEMS2014
ﻓﻨﺎوري ﭘﮋوﻫﺸﻜﺪه،اﻣﻴﺮﻛﺒﻴﺮ ﺻﻨﻌﺘﻲ داﻧﺸﮕﺎه ﻧﻮ ﻫﺎي29و30ﺑﻬﻤﻦ1392
thickness. The edge conditions of a diaphragm
will depend upon the method of manufacture
and the geometry of the surrounding structure. It
will vary between a simply supported or rigidly
clamped structure, as shown in Fig 4-1 and
Fig 4-2. Simply supported diaphragms will not
occur in practice, but the analytical results for
such a structure may more accurately reflect the
behavior of a poorly clamped diaphragm than
the rigidly clamped analysis. At small
deflections (≤10% diaphragm thickness) the
pressure-deflection relationship will be linear.
As the pressure increases, the rate of deflection
decreases and the pressure-deflection
relationship will become nonlinear. The
deflection Z at radial distance r of a round
diaphragm under a uniform pressure P, rigidly
clamped as shown in Fig 4-2, is given by:
2
2 2 23(1 )
( )
16 p
P
z a r
h
ν−
= −
Ε
(1)
Where ph is the diaphragm thickness, Ε and ν
are the Young’s modulus and Poisson’s ratio of
the diaphragm material, respectively, and a is
the radius of the diaphragm. The maximum
deflection 0z will occur at the diaphragm center
where 0r = . Assuming a common value for
metals of 0.22ν = , the maximum deflection is
given by:
4
0 3
0.1784
p
Pa
z
h
=
Ε
(2)
The stress distribution will vary both across the
radius and through the thickness of the
diaphragm. For example, the neutral axis [shown
in Fig 4-1] experiences zero stress while the
maximum stress occurs at the outer faces. At any
given distance r from the center of the
diaphragm, one face will experience tensile
stress while the other experiences compressive
stress. There are two stress components
associated with a circular diaphragm: radial and
tangential. The radial stress, σr at distance r from
the center of the diaphragm is given by (3). The
maximum radial stress that occurs at the
diaphragm edge (r = a) is given by (4).
( ) ( )
2 2
2 2
3
3 1
8
r
Pa r
h a
σ ν ν
= ± + − −
(3)
( )max
2
2
3
1
4
r
Pa
h
σ ν= ± + (4)
Radial stress is equal to zero at a value of r given
by ( )( )( )
1
2
1 3a ν ν+ + (shown in Fig 4-2). The
tangential stress, σt, at distance r from the
center of the diaphragm is given by (5). The
maximum tangential stress that occurs at the
diaphragm center (r = 0) is given by (6).
( ) ( )
2 2
2 2
3
3 1 1
8
t
Pa r
h a
σ ν ν
= ± + − +
(5)
( )max
2
2
3
1
4
t
Pa
h
σ ν= ± + (6)
The diaphragm with embedded piezoresistors is
made by using silicon bulk micromachining
steps. Piezoresistors are made by selectively
doping the silicon diaphragm. The piezoresistors
are connected in the form of Wheatstone bridge
to output electric signal caused by pressure in
Fig 5. They should be located on the maximum
stress area in order to obtain high sensitivity.
The power supply is constant current.
1R R− ∆
4R R− ∆ 3R R− ∆
2R R− ∆
outV
Constant
Current
Fig 5: Wheatstone bridge](https://image.slidesharecdn.com/keywordspressuresensor-151122212334-lva1-app6892/85/Modeling-Simulation-and-Design-of-a-Circular-Diaphragm-Pressure-Sensor-3-320.jpg)
![ﻣﻴﻜﺮوﻣﺎﺷﻨﻴﻜﺎري و رﻳﺰﻓﻨﺎوري اﻟﻤﻠﻠﻲ ﺑﻴﻦ ﻛﻨﻔﺮاﻧﺲ اوﻟﻴﻦICMEMS2014
ﻓﻨﺎوري ﭘﮋوﻫﺸﻜﺪه،اﻣﻴﺮﻛﺒﻴﺮ ﺻﻨﻌﺘﻲ داﻧﺸﮕﺎه ﻧﻮ ﻫﺎي29و30ﺑﻬﻤﻦ1392
Therefore, the functional relationship between
the fractional change in electrical resistance
( )R
R
∆ of the piezoresistor and the transversal
and longitudinal stress components is given by
for tangential resistors:
t t r r
t
R
R
π σ π σ
∆
= +
(7)
For radial resistors:
t r r t
r
R
R
π σ π σ
∆
= +
(8)
Where tπ and rπ are longitudinal and
transversal direction coefficients, respectively.
The power supply is constant current. Therefore,
the changes of resistance in an unbalanced
bridge caused by piezoresistance effect with the
load of pressure can directly transform into
voltage with equation (9).
1 3 2 4
1 2 3 4
o in
R R R R
V I
R R R R
−
= ⋅
+ + +
(9)
Where inI is bridge-input current, and oV is the
differential output voltage. When the pressure is
loaded, 1R and 3R have positive increment, on
the other side, 2R and 4R have negative
increment [6].
3. Analysis and simulation
The diameter and thickness of the diaphragm
and the distribution of piezoresistors affects the
sensor’s range, frequency response, sensitivity
and linearity. The distribution of piezoresistors
is mainly determined by the stress distribution
on the diaphragm and the glass ring packaged
with. Simulations used ANSYS of diaphragm
packaged with different boron glass ring under
the pressure 25MPa are shown in Fig 6. The
outside of the model is restricted by ring and the
pressure is loaded on the opposite of the model.
The stress along x direction is shown in Fig 4.
The sensitivity is mainly determined by the
difference of x stress and y stress at the point
where the resistor is placed. The curve in
Fig 7 and Fig 8 shows the x-y stress of
2 4a mm= and 2 2a mm= along the path of x
direction with the same thickness 0.4h mm= .
It can be seen that the x-y stress of 2 4a mm=
is much higher than that of 2 2a mm= . Thus,
the piezoresistors in Fig 7 will output higher
voltage and show better sensitivity under the
same load. But the piezoresistors in Fig 9 shows
better linearity. Besides, it is in accord with the
theoretical analysis that the deflection and stress
will increase with the increase of radius a within
the allowable pressure load P. Therefore, the
dimensions are determined to meet the
requirement of 25MPa with the consideration of
glass ring as follows: 2 4a mm= , 0.4h mm= .
The deformation on Z direction just the
deflection when P=25MPa is shown in Fig 6,
and the center deflection of the diaphragm is the
maximum shown in Fig 10: Z0 =9.5 µm. Fig 8
show the radial strain on the diameter path of
diaphragm when P=25MPa. Seen from Fig 7 and
Fig 11, the radial stresses and strains at the
a=0.0012mm and a=0.0048mm are much bigger,
where the piezoresistors must be embedded.](https://image.slidesharecdn.com/keywordspressuresensor-151122212334-lva1-app6892/85/Modeling-Simulation-and-Design-of-a-Circular-Diaphragm-Pressure-Sensor-4-320.jpg)
![ﻣﻴﻜﺮوﻣﺎﺷﻨﻴﻜﺎري و رﻳﺰﻓﻨﺎوري اﻟﻤﻠﻠﻲ ﺑﻴﻦ ﻛﻨﻔﺮاﻧﺲ اوﻟﻴﻦICMEMS2014
ﻓﻨﺎوري ﭘﮋوﻫﺸﻜﺪه،اﻣﻴﺮﻛﺒﻴﺮ ﺻﻨﻌﺘﻲ داﻧﺸﮕﺎه ﻧﻮ ﻫﺎي29و30ﺑﻬﻤﻦ1392
Fig 7: X-Y Stress distribution the path of X direction
2a=4mm , h=0.4mm
Fig 10: Z deflection with 2a=2mm, h=0.4mm
Fig 8: X-Y Stress distribution the path of X direction
2a=2mm , h=0.4mm
Fig 11: X-Y Strain distribution the path of X direction
2a=2mm , h=0.4mm
Structure size
2RP=2mm
hP=0.4mm
2RP=2mm
hP=0.2mm
2RP=4mm
hP=0.5mm
2RP=4mm
hP=0.3mm
Theoretical maximum deflection /µm 0.412 1.132 1.329 2.216
Maximum deflection by ANSYS/µm 0.565 1.130 1.326 2.211
Maximum stress by ANSYS /MPa 113.82 118.56 83.508 83.558
[1] J. von Berg, C. Sonderegger, S. Bollhalder, C. Cavalloni,
Piezoresistive SOI-Pressure Sensor for High Pressure and High
Temperature Applications, Sensor 2005, Volume (I), pp. 33-38.
[2] Zhong Z et al,” Calibration of a piezoresistive stress sensor in
(1 00) silicon test chips “in Proc. Elect. Packag. Tech Conf. 2002
pp 323-326.
[3].Yan-Hong Zhang, Chen Yang, A Novel Pressure Microsensor
With 30-um-Thick Diaphragm and Meander-Shaped Piezoresistors
Partially Distributed on High-Stress Bulk Silicon Region, IEEE
Sensors J.,vol.7,no.12, Dec.2007, pp.1742-1748.
[4] Xudong Fang, Libo Zhao, Yulong Zhao, Zhuangde Jiang, “A
high pressure sensor with circular diaphragm based on MEMS
technology” , The 2st International Conference of CSMNT, State
Key Laboratory for Mechanical Manufacturing System
Engineering, Xi’an, China
[5] MEMS Mechanical Sensors, Stephen Beeby, Graham Ensell,
Michael Kraft, Neil White, Chapter 6, Pressure Sensors, 2004
British Library Cataloguing in Publication Data
[6] MEMS, Design and Fabrication, edited by Mohamed Gad-el-
Hak., Chapter 7, Fabrication, Characterization, and Reliability
design and fabrication, © 2006 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group](https://image.slidesharecdn.com/keywordspressuresensor-151122212334-lva1-app6892/85/Modeling-Simulation-and-Design-of-a-Circular-Diaphragm-Pressure-Sensor-6-320.jpg)
Modeling, Simulation and Design of a Circular Diaphragm Pressure Sensor
- 1. ﻣﻴﻜﺮوﻣﺎﺷﻨﻴﻜﺎري و رﻳﺰﻓﻨﺎوري اﻟﻤﻠﻠﻲ ﺑﻴﻦ ﻛﻨﻔﺮاﻧﺲ اوﻟﻴﻦICMEMS2014 ﻓﻨﺎوري ﭘﮋوﻫﺸﻜﺪه،اﻣﻴﺮﻛﺒﻴﺮ ﺻﻨﻌﺘﻲ داﻧﺸﮕﺎه ﻧﻮ ﻫﺎي29و30ﺑﻬﻤﻦ1392 Key words Pressure sensor, Circular diaphragm, MEMS, Terms pressure, Small deflection, Finite element, ANSYS, Abstract: This paper aim in design and analysis of MEMS Pressure Sensor by using ANSYS software. A diaphragm based MEMS sensor in the range of 25MPa by measured center deflection of the circular pressure-sensitive and using the strain gauge for measurement. First this paper created the mathematical calculation and then models the finite element analysis of the diaphragm using ANSYS and determined the pressure sensor in the range of 25MPa with circular diaphragm is designed, which also reflects the choice of correct design after analyzing the effect of diaphragm thickness, location and shapes of piezoresistor. Circular diaphragm of different thickness and diameters are simulated to meet the requirement of 25MPa pressure. The whole fabrication process is inexpensive and compatible with standard MEMS process. Measurements show good results such as high sensitivity 0~25MPa, linearity 0~11.209MPa and upper pressure we have 1.25% standard division. The work indicates pressure sensor with circular diaphragm could be fabricated for applications of high range pressure measurement with high precision. 1. Introduction Pressure sensors have been developing rapidly in the last decades with the development of Micro Electro Mechanical System (MEMS) technique and high demand of market. Recently, with the rapid expansion of consumption of pressure sensors of high range in petroleum industry and daily applications, great efforts have been directed towards high pressure sensors in high temperature environment [1]. These devices can replace bulky actuators and sensors with micro scale devices that can be produced in integrated circuit photolithography [2]. Capacitive sensor and resonator sensor have been widely used. However, some defects of them are difficult to overcome. The linear outputs of capacitive sensor depend on complex signal processing integrated circuit. The resonator sensor has limitation to materials of high quality and frequency. Besides, the cost is too high. To overcome such disadvantages, piezoresistive sensor has simple process circuit Wheatstone bridge and cheap material silicon to achieve high precision linear output. Many piezoresistive sensors are fabricated with square diaphragm of different thickness [3]. Compared to the square diaphragm, circular diaphragm has advantage in high pressure range. It has better high frequency response in high range, although the sensitivity is lower. The square diaphragm’s nonlinearity in high range is harmful for the sensor’s performance. Besides, its frequency response is much lower than circular diaphragm of same size. Thus, circular diaphragm with meander-shaped piezoresistors is utilized for sensors in high pressure measurement. Circular diaphragm packaged with glass ring is simulated by ANSYS software to determine optimal place for piezoresistors distributed on the surface [4]. 2. Structure and working principle The conventional structure of piezoresistive pressure sensor includes a square diaphragm as shown in Fig 1. Piezoresistors are located at the edge of diaphragm to transform pressure to electrical signal. Stress distribution of one type Modeling, Simulation and Design of a Circular Diaphragm Pressure Sensor ICMEMS2014-2026
- 2. ﻣﻴﻜﺮوﻣﺎﺷﻨﻴﻜﺎري و رﻳﺰﻓﻨﺎوري اﻟﻤﻠﻠﻲ ﺑﻴﻦ ﻛﻨﻔﺮاﻧﺲ اوﻟﻴﻦICMEMS2014 ﻓﻨﺎوري ﭘﮋوﻫﺸﻜﺪه،اﻣﻴﺮﻛﺒﻴﺮ ﺻﻨﻌﺘﻲ داﻧﺸﮕﺎه ﻧﻮ ﻫﺎي29و30ﺑﻬﻤﻦ1392 of square diaphragm C type is shown in Fig 2. It can be observed that the stress concentration region on the edge is long and narrow. Thus, it’s possible that the piezoresistors distributed on the edge may not cause changes equally under corresponding pressure which is harmful to the sensor’s linearity and life time of sensor. Meanwhile, Comparing to square diaphragm, the circular diaphragm has higher natural frequency which is of great benefit to dynamic applications. For example, the natural frequency of circular diaphragm demonstrated in this paper is 1.4MHZ which is much higher than that of square diaphragm of similar size. So if the sensors are used in dynamic measurement, especially for high frequency applications, the circular diaphragm should be unutilized. The design model of circular diaphragm is shown in Fig 3 and it’s bonded with borosilicate glass ring to form inverted cup structure as sensing unit of the sensor. The key parameters are the diameter of effective area d=2a and the thickness of piezoresistive element ph . The theoretical model of effective area is shown in Fig 4 from which displacement can be observed and the following theoretical calculation and analysis are based on the model. The pressure P is equally distributed and it should satisfy the following equations with the circular diaphragm [4]. The behavior of a diaphragm will depend upon many factors, such as the edge conditions and the deflection range compared to diaphragm bornosilicatepiezoresistive r Z0 hp a z p neutral axis Fig 1: conventional structure of piezoresistive pressure Fig 2: square diaphragm C type Fig 3: effective area for sensing pressure d Effective area Fig 4-1: rigidly clamped diaphragm and Fig 4-2: its associated displacement under uniform pressure. (1) (2) 1 2 ((1 )(3 ))a ν ν+ +
- 3. ﻣﻴﻜﺮوﻣﺎﺷﻨﻴﻜﺎري و رﻳﺰﻓﻨﺎوري اﻟﻤﻠﻠﻲ ﺑﻴﻦ ﻛﻨﻔﺮاﻧﺲ اوﻟﻴﻦICMEMS2014 ﻓﻨﺎوري ﭘﮋوﻫﺸﻜﺪه،اﻣﻴﺮﻛﺒﻴﺮ ﺻﻨﻌﺘﻲ داﻧﺸﮕﺎه ﻧﻮ ﻫﺎي29و30ﺑﻬﻤﻦ1392 thickness. The edge conditions of a diaphragm will depend upon the method of manufacture and the geometry of the surrounding structure. It will vary between a simply supported or rigidly clamped structure, as shown in Fig 4-1 and Fig 4-2. Simply supported diaphragms will not occur in practice, but the analytical results for such a structure may more accurately reflect the behavior of a poorly clamped diaphragm than the rigidly clamped analysis. At small deflections (≤10% diaphragm thickness) the pressure-deflection relationship will be linear. As the pressure increases, the rate of deflection decreases and the pressure-deflection relationship will become nonlinear. The deflection Z at radial distance r of a round diaphragm under a uniform pressure P, rigidly clamped as shown in Fig 4-2, is given by: 2 2 2 23(1 ) ( ) 16 p P z a r h ν− = − Ε (1) Where ph is the diaphragm thickness, Ε and ν are the Young’s modulus and Poisson’s ratio of the diaphragm material, respectively, and a is the radius of the diaphragm. The maximum deflection 0z will occur at the diaphragm center where 0r = . Assuming a common value for metals of 0.22ν = , the maximum deflection is given by: 4 0 3 0.1784 p Pa z h = Ε (2) The stress distribution will vary both across the radius and through the thickness of the diaphragm. For example, the neutral axis [shown in Fig 4-1] experiences zero stress while the maximum stress occurs at the outer faces. At any given distance r from the center of the diaphragm, one face will experience tensile stress while the other experiences compressive stress. There are two stress components associated with a circular diaphragm: radial and tangential. The radial stress, σr at distance r from the center of the diaphragm is given by (3). The maximum radial stress that occurs at the diaphragm edge (r = a) is given by (4). ( ) ( ) 2 2 2 2 3 3 1 8 r Pa r h a σ ν ν = ± + − − (3) ( )max 2 2 3 1 4 r Pa h σ ν= ± + (4) Radial stress is equal to zero at a value of r given by ( )( )( ) 1 2 1 3a ν ν+ + (shown in Fig 4-2). The tangential stress, σt, at distance r from the center of the diaphragm is given by (5). The maximum tangential stress that occurs at the diaphragm center (r = 0) is given by (6). ( ) ( ) 2 2 2 2 3 3 1 1 8 t Pa r h a σ ν ν = ± + − + (5) ( )max 2 2 3 1 4 t Pa h σ ν= ± + (6) The diaphragm with embedded piezoresistors is made by using silicon bulk micromachining steps. Piezoresistors are made by selectively doping the silicon diaphragm. The piezoresistors are connected in the form of Wheatstone bridge to output electric signal caused by pressure in Fig 5. They should be located on the maximum stress area in order to obtain high sensitivity. The power supply is constant current. 1R R− ∆ 4R R− ∆ 3R R− ∆ 2R R− ∆ outV Constant Current Fig 5: Wheatstone bridge
- 4. ﻣﻴﻜﺮوﻣﺎﺷﻨﻴﻜﺎري و رﻳﺰﻓﻨﺎوري اﻟﻤﻠﻠﻲ ﺑﻴﻦ ﻛﻨﻔﺮاﻧﺲ اوﻟﻴﻦICMEMS2014 ﻓﻨﺎوري ﭘﮋوﻫﺸﻜﺪه،اﻣﻴﺮﻛﺒﻴﺮ ﺻﻨﻌﺘﻲ داﻧﺸﮕﺎه ﻧﻮ ﻫﺎي29و30ﺑﻬﻤﻦ1392 Therefore, the functional relationship between the fractional change in electrical resistance ( )R R ∆ of the piezoresistor and the transversal and longitudinal stress components is given by for tangential resistors: t t r r t R R π σ π σ ∆ = + (7) For radial resistors: t r r t r R R π σ π σ ∆ = + (8) Where tπ and rπ are longitudinal and transversal direction coefficients, respectively. The power supply is constant current. Therefore, the changes of resistance in an unbalanced bridge caused by piezoresistance effect with the load of pressure can directly transform into voltage with equation (9). 1 3 2 4 1 2 3 4 o in R R R R V I R R R R − = ⋅ + + + (9) Where inI is bridge-input current, and oV is the differential output voltage. When the pressure is loaded, 1R and 3R have positive increment, on the other side, 2R and 4R have negative increment [6]. 3. Analysis and simulation The diameter and thickness of the diaphragm and the distribution of piezoresistors affects the sensor’s range, frequency response, sensitivity and linearity. The distribution of piezoresistors is mainly determined by the stress distribution on the diaphragm and the glass ring packaged with. Simulations used ANSYS of diaphragm packaged with different boron glass ring under the pressure 25MPa are shown in Fig 6. The outside of the model is restricted by ring and the pressure is loaded on the opposite of the model. The stress along x direction is shown in Fig 4. The sensitivity is mainly determined by the difference of x stress and y stress at the point where the resistor is placed. The curve in Fig 7 and Fig 8 shows the x-y stress of 2 4a mm= and 2 2a mm= along the path of x direction with the same thickness 0.4h mm= . It can be seen that the x-y stress of 2 4a mm= is much higher than that of 2 2a mm= . Thus, the piezoresistors in Fig 7 will output higher voltage and show better sensitivity under the same load. But the piezoresistors in Fig 9 shows better linearity. Besides, it is in accord with the theoretical analysis that the deflection and stress will increase with the increase of radius a within the allowable pressure load P. Therefore, the dimensions are determined to meet the requirement of 25MPa with the consideration of glass ring as follows: 2 4a mm= , 0.4h mm= . The deformation on Z direction just the deflection when P=25MPa is shown in Fig 6, and the center deflection of the diaphragm is the maximum shown in Fig 10: Z0 =9.5 µm. Fig 8 show the radial strain on the diameter path of diaphragm when P=25MPa. Seen from Fig 7 and Fig 11, the radial stresses and strains at the a=0.0012mm and a=0.0048mm are much bigger, where the piezoresistors must be embedded.
- 5. ﻣﻴﻜﺮوﻣﺎﺷﻨﻴﻜﺎري و رﻳﺰﻓﻨﺎوري اﻟﻤﻠﻠﻲ ﺑﻴﻦ ﻛﻨﻔﺮاﻧﺲ اوﻟﻴﻦICMEMS2014 ﻓﻨﺎوري ﭘﮋوﻫﺸﻜﺪه،اﻣﻴﺮﻛﺒﻴﺮ ﺻﻨﻌﺘﻲ داﻧﺸﮕﺎه ﻧﻮ ﻫﺎي29و30ﺑﻬﻤﻦ1392 Fig 6: Finite element model of diaphragm Fig 7: Deformation on Z direction when P=25MPa Fig 8: Stress on X direction when P=25MPa Fig 9: Von Mises stress when P=25MPa 1 X Y Z FEB 8 2013 09:21:13 ELEMENTS 1 MNMX X Y Z -.134E+09 -.104E+09 -.735E+08 -.434E+08 -.132E+08 .170E+08 .472E+08 .774E+08 .108E+09 .138E+09 FEB 8 2013 09:32:58 NODAL SOLUTION STEP=1 SUB =1 TIME=1 SX (AVG) RSYS=0 DMX =.410E-06 SMN =-.134E+09 SMX =.138E+09 1 MN MX X Y Z 1.163 .173E+08 .345E+08 .518E+08 .691E+08 .863E+08 .104E+09 .121E+09 .138E+09 .155E+09 FEB 8 2013 09:32:17 NODAL SOLUTION STEP=1 SUB =1 TIME=1 SEQV (AVG) DMX =.410E-06 SMN =1.163 SMX =.155E+09 1 MN MX X Y Z 0 .455E-07 .910E-07 .137E-06 .182E-06 .228E-06 .273E-06 .319E-06 .364E-06 .410E-06 FEB 8 2013 09:36:46 NODAL SOLUTION STEP=1 SUB =1 TIME=1 USUM (AVG) RSYS=0 DMX =.410E-06 SMX =.410E-06
- 6. ﻣﻴﻜﺮوﻣﺎﺷﻨﻴﻜﺎري و رﻳﺰﻓﻨﺎوري اﻟﻤﻠﻠﻲ ﺑﻴﻦ ﻛﻨﻔﺮاﻧﺲ اوﻟﻴﻦICMEMS2014 ﻓﻨﺎوري ﭘﮋوﻫﺸﻜﺪه،اﻣﻴﺮﻛﺒﻴﺮ ﺻﻨﻌﺘﻲ داﻧﺸﮕﺎه ﻧﻮ ﻫﺎي29و30ﺑﻬﻤﻦ1392 Fig 7: X-Y Stress distribution the path of X direction 2a=4mm , h=0.4mm Fig 10: Z deflection with 2a=2mm, h=0.4mm Fig 8: X-Y Stress distribution the path of X direction 2a=2mm , h=0.4mm Fig 11: X-Y Strain distribution the path of X direction 2a=2mm , h=0.4mm Structure size 2RP=2mm hP=0.4mm 2RP=2mm hP=0.2mm 2RP=4mm hP=0.5mm 2RP=4mm hP=0.3mm Theoretical maximum deflection /µm 0.412 1.132 1.329 2.216 Maximum deflection by ANSYS/µm 0.565 1.130 1.326 2.211 Maximum stress by ANSYS /MPa 113.82 118.56 83.508 83.558 [1] J. von Berg, C. Sonderegger, S. Bollhalder, C. Cavalloni, Piezoresistive SOI-Pressure Sensor for High Pressure and High Temperature Applications, Sensor 2005, Volume (I), pp. 33-38. [2] Zhong Z et al,” Calibration of a piezoresistive stress sensor in (1 00) silicon test chips “in Proc. Elect. Packag. Tech Conf. 2002 pp 323-326. [3].Yan-Hong Zhang, Chen Yang, A Novel Pressure Microsensor With 30-um-Thick Diaphragm and Meander-Shaped Piezoresistors Partially Distributed on High-Stress Bulk Silicon Region, IEEE Sensors J.,vol.7,no.12, Dec.2007, pp.1742-1748. [4] Xudong Fang, Libo Zhao, Yulong Zhao, Zhuangde Jiang, “A high pressure sensor with circular diaphragm based on MEMS technology” , The 2st International Conference of CSMNT, State Key Laboratory for Mechanical Manufacturing System Engineering, Xi’an, China [5] MEMS Mechanical Sensors, Stephen Beeby, Graham Ensell, Michael Kraft, Neil White, Chapter 6, Pressure Sensors, 2004 British Library Cataloguing in Publication Data [6] MEMS, Design and Fabrication, edited by Mohamed Gad-el- Hak., Chapter 7, Fabrication, Characterization, and Reliability design and fabrication, © 2006 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group