# Circuit Design of Surface Acoustic Wave Based Micro Force Sensor

Authors: Yuanyuan Li; Wenke Lu; Changchun Zhu; Qinghong Liu; Haoxin Zhang; Chenchao Tang
Journal: Mathematical Problems in Engineering
 (2014)

---

## Abstract

Pressure sensors are commonly used in industrial production and mechanical system. However, resistance strain, piezoresistive sensor, and ceramic capacitive pressure sensors possess limitations, especially in micro force measurement. A surface acoustic wave (SAW) based micro force sensor is designed in this paper, which is based on the theories of wavelet transform, SAW detection, and pierce oscillator circuits. Using lithium niobate as the basal material, a mathematical model is established to analyze the frequency, and a peripheral circuit is designed to measure the micro force. The SAW based micro force sensor is tested to show the reasonable design of detection circuit and the stability of frequency and amplitude.


---

## Body

## 1.Introduction

Wavelet transform finds its application in many disciplines and fields such as in image processing, water-sound, earthquake detection, biomedicine, mechanical vibration, pronunciation recognition, communication, chemical industry, and torrent analysis [1]. SAW devices are early examples of microelectromechanical systems (MEMS) because of the coupling needed between the electrical and mechanical properties as discussed by Ballantine et al. [2]. The method of implementing wavelet transform with SAW devices has been first proposed by Peng et al. [3–5]. The wavelet transform device of SAW can benefit from the excellent properties of the SAW devices, namely, passive, small size, low cost, excellent temperature stability, high reliability, and high reproducibility, which overcomes the complicated algorithms and high power for VLSI [6, 7], and big size and low reproducibility for optical devices [8]. Wave propagation along the surface allows the sensitivity of the wavelet transform device of SAW to change in the external environment and the development of these sensors for applications such as gas detection, changes in fluid viscosity, determination of stiffness constants of mechanical vibration, and detection of the onset of ice formation on aerospace structures [9–11].Nowadays, micromanipulation has performed to design either mobile micro robots or a precise positioning device under the control of mechanical systems. Various tools for manipulating micro parts and assembling micro systems have been developed and integrated. Semiconductor strain gauges are preferred when small forces have been measured. M. Jungwirth has described the micromechanical precision pressure sensor but in delay lines. A silicon based micro force sensor has been developed with larger electromechanical coupling coefficientk
2. The three-axis micro force sensor has been designed by Jungwirth et al. with the problem of measurement uncertainty. The dual axis micro force sensor for robotic manipulations needs to use strain gauges [12–15].This paper proposes to use substrate materials of small electromechanical coupling coefficientk
2 (12
8
°
Y shear LiNb
O
3) in the manufacture to design the wavelet transform device of SAW based micro force sensor, which has high accuracy and sensitivity testing precision. In addition, a mathematical model is established to analyze the frequency of our sensor and a peripheral detection circuit is designed. Within the scope of effective measurement, the SAW based micro force sensor possesses good linearity, consistency, and repeatability in performance. Besides, it uses the piezoelectric properties and the temperature stability of the crystal, as well as the frequency signal instead of the conventional pressure sensor with voltage signal, which makes the signal processing of this device more digital and possesses more stable performances.The succeeding sections are organized as follows. Section2 introduces the fundamental principles of designing interdigital transducers (IDT) for wavelet transform device of SAW. In Section 3, the peripheral detection circuit is explained. In addition, the linear regression model is provided in Section 4. Finally Section 5 delivers the testing and analysis of SAW based micro force sensor which is used for measuring various micro force.


## 2. Design IDT for Wavelet Transform Device of SAW

The wavelet function is(1)
ψ
s
(
t
)
=
1
s
ψ
(
t
s
)
,
where s denotes the scale of wavelet function.The wavelet transform of signalf
(
t
) is
(2)
W
T
s
(
τ
)
=
f
(
t
)
*
ψ
s
(
t
)
=
∫
R
f
(
t
)
1
s
ψ
(
τ
-
t
s
)
d
t
=
1
s
∫
R
f
(
t
)
ψ
(
τ
-
t
s
)
d
t
.Whenψ
s
(
t
) is a Morlet wavelet function, formula (1) is converted into [3–5]
(3)
ψ
s
(
t
)
=
1
s
e
-
(
1
/
2
)
(
t
/
s
)
2
e
j
2
π
(
f
0
/
s
)
t
=
P
s
(
t
)
e
j
2
π
(
f
0
/
s
)
t
,
where P
s
(
t
) is the wavelet-envelope function, P
s
(
t
)
=
(
1
/
s
)
e
-
1
/
2
(
t
/
s
)
2, and f
0
/
s is the center frequency. When s
=
2
k, k is a random number from -
∞ to +
∞. The wavelet function shown in formula (3) is converted into the Morlet dyadic wavelet function [4, 5]. The microwave communication equipment of this paper design needs a single-scale wavelet transform processor of the center frequency f
0, which can be rewritten as
(4)
f
0
=
v
s
λ
=
v
s
2
(
a
+
b
)
,
where λ
=
2
(
a
+
b
) is the wavelength, v
s is the speed of the SAW, and a and b are the width and the interval of IDT.The delay line in wavelet transform device of SAW designed in this paper exhibits the basic structure in Figure1 and two acoustic electric transducers on the piezoelectric characteristic substrate material polishing surface, named input IDT and output IDT.Figure 1
The transducer substrate of wavelet transform device of SAW.As the Morlet wavelet transform of SAW devices is based on formula (3), once all the parameters in (4) are relatively fixed, its center frequency of each scale should be corresponding with the only device. The device has been designed and produced in this paper that is shown in Figure 2.Figure 2
The fabricated single-scale wavelet transform device of SAW.


## 3. Design of the Peripheral Detection Circuit

### 3.1. The Principle of Peripheral Detection Circuit

After making the fabricated device, the peripheral detection circuit needs to be designed to getF
m and f which are the pressure and the output frequency of the SAW based micro force sensor. Figure 3 shows the general structure of the SAW micro force sensor with IDTs.Figure 3
General structure of the SAW micro force sensor with two IDTs.If the SAW resonator density of the piezoelectric substrate ism, the unit is m/g2, the dielectric constant is ε, the electrical conductivity is σ, the elastic parameter is c, the unit is n/m2, the environment temperature is t, and the pressure is p [16], v
s based on the theory of surface disturbance is given as
(5)
v
s
=
v
(
m
,
ε
,
σ
,
c
,
t
,
p
)
.After derivation, (5) can be changed to
(6)
Δ
v
s
v
s
=
1
v
s
(
∂
v
∂
m
Δ
m
+
∂
v
∂
ε
Δ
ε
+
∂
v
∂
σ
Δ
σ
+
∂
v
∂
c
Δ
c
+
∂
v
∂
t
Δ
t
+
∂
v
∂
p
Δ
p
)
.Equation (6) has been used to calculate the relationship between the propagation speed of available and the SAW oscillator frequency which is equal to the following equation:
(7)
Δ
f
f
0
=
Δ
v
s
v
s
=
-
1
v
SAW
(
∂
v
∂
m
Δ
m
+
∂
v
∂
ε
Δ
ε
+
∂
v
∂
σ
Δ
σ
aaaaaaa
+
∂
v
∂
c
Δ
c
-
∂
v
∂
t
Δ
t
+
∂
v
∂
p
Δ
p
)
.The change in the oscillator frequency and the measurement of the parameters can be obtained by using (7), which is the working principle of the SAW micro force sensor circuit.


### 3.2. The Actual Oscillator Circuit

The mixing frequency circuit is divided into two oscillator circuits, one for the reference oscillator circuit with a fixed frequency, and the other for the detector circuit. The frequency difference can be received from the mixing circuit. Figure4 shows the circuit framework.Figure 4
The circuit framework of fixed frequency circuit.The oscillator circuit with fixed frequency in the dual-channel structure has an output signal, which is used as the stable signal source during measurement. The digital signal processing circuit is composed of a filter, a mixer, an amplifier, and a shaping circuit.For a surface acoustic generator as the sensitive element of the SAW micro force sensor, the frequency stability directly affects the resolution of the testing precision. The oscillator frequency has depended on the conditions of the feedback loop phase. Therefore, the improvement of the SAW oscillator frequency stability also improves the performance of the micro force sensor.The Pierce oscillator circuit has a better stability compared with Colpitts and Clapp circuits [17]. Figure 5 shows the oscillator circuit of the SAW based micro force sensor based on the principle of Pierce circuit and the combination with the design requirements.Figure 5
The Pierce circuit of the SAW based micro force sensor.The high-frequency triode2
S
C
3357 affects the resonance amplifier in the circuit by ensuring the normal startup of the oscillator circuit. R
1 is the DC bias resistance of this triode, and adjusting the value changes with DC working points. R
2 is the DC bias resistance of the triode s
8050 that changes the DC working points and transforms the bias current of the circuit. This triode functions as the current source and stabilizes the circuit working state. The SAW resonator affects the frequency selection in the feedback loop by inducing the SAW oscillator frequency stability and improving the antijamming ability.To ensure that the feedback loop circuit phase is in balance, the inductanceL has been added to the Pierce circuit to eliminate the effect of the stability of the DC voltage source from the oscillator circuit. The LC parallel resonant circuit also affects the DC bypass circuit. The feedback coefficient F is related to C
1 and C
2 and is given as
(8)
F
=
C
1
C
2
,
where C
3, C
4, and L
2 are the decoupling devices of the DC source that can eliminate the effect of the oscillator circuit from the LC filter function.


### 3.3. Determine the Parameters

The circuit layout, selection of components, and calculated parameters are important. The work frequency of the micro force sensor is 50 MHz; the center frequency of the LC parallel resonant circuit should be(9)
f
0
=
1
2
π
LC
.To ensure that the amplitude of oscillator circuit has better characteristics than the initial conditions,C
2 should be smaller. In addition, the designed oscillator frequency is given; the capacitance is selected as 5.1 pF. From (10), L can be obtained as
(10)
L
=
1
(
2
π
f
0
)
2
C
.Because parasitic parameters in practical circuits affect their performances,L must be adjustable. By changing the number of turns of the air-core coil, the coil inductance can be fine-tuned, and the frequency of the L
C parallel resonant circuit will be in accordance with the working frequency of the SAW based micro force sensor. The air-core coil inductance has been calculated by the following equation:
(11)
L
=
0.01
×
D
×
N
2
(
L
N
/
D
)
+
0.44
,
where L is the coil inductance (in μ
H), D is the diameter of the coil (in cm), N is the number of coil turns, andLN is the coil length (in cm).An oscillator frequency source should have good stability, phase noise, and highQ value [18]. Figure 6 shows the equivalent circuit, with C
2 expressed in picofarads.Figure 6
The equivalent circuit of the SAW device.The SAW resonator can work in series or parallel resonant frequencies upon operating in the feedback loop. The transistor2
S
C
3357 and s
8050 provide the DC bias current in the oscillator circuit design based on the requirements.


## 3.1. The Principle of Peripheral Detection Circuit

After making the fabricated device, the peripheral detection circuit needs to be designed to getF
m and f which are the pressure and the output frequency of the SAW based micro force sensor. Figure 3 shows the general structure of the SAW micro force sensor with IDTs.Figure 3
General structure of the SAW micro force sensor with two IDTs.If the SAW resonator density of the piezoelectric substrate ism, the unit is m/g2, the dielectric constant is ε, the electrical conductivity is σ, the elastic parameter is c, the unit is n/m2, the environment temperature is t, and the pressure is p [16], v
s based on the theory of surface disturbance is given as
(5)
v
s
=
v
(
m
,
ε
,
σ
,
c
,
t
,
p
)
.After derivation, (5) can be changed to
(6)
Δ
v
s
v
s
=
1
v
s
(
∂
v
∂
m
Δ
m
+
∂
v
∂
ε
Δ
ε
+
∂
v
∂
σ
Δ
σ
+
∂
v
∂
c
Δ
c
+
∂
v
∂
t
Δ
t
+
∂
v
∂
p
Δ
p
)
.Equation (6) has been used to calculate the relationship between the propagation speed of available and the SAW oscillator frequency which is equal to the following equation:
(7)
Δ
f
f
0
=
Δ
v
s
v
s
=
-
1
v
SAW
(
∂
v
∂
m
Δ
m
+
∂
v
∂
ε
Δ
ε
+
∂
v
∂
σ
Δ
σ
aaaaaaa
+
∂
v
∂
c
Δ
c
-
∂
v
∂
t
Δ
t
+
∂
v
∂
p
Δ
p
)
.The change in the oscillator frequency and the measurement of the parameters can be obtained by using (7), which is the working principle of the SAW micro force sensor circuit.


## 3.2. The Actual Oscillator Circuit

The mixing frequency circuit is divided into two oscillator circuits, one for the reference oscillator circuit with a fixed frequency, and the other for the detector circuit. The frequency difference can be received from the mixing circuit. Figure4 shows the circuit framework.Figure 4
The circuit framework of fixed frequency circuit.The oscillator circuit with fixed frequency in the dual-channel structure has an output signal, which is used as the stable signal source during measurement. The digital signal processing circuit is composed of a filter, a mixer, an amplifier, and a shaping circuit.For a surface acoustic generator as the sensitive element of the SAW micro force sensor, the frequency stability directly affects the resolution of the testing precision. The oscillator frequency has depended on the conditions of the feedback loop phase. Therefore, the improvement of the SAW oscillator frequency stability also improves the performance of the micro force sensor.The Pierce oscillator circuit has a better stability compared with Colpitts and Clapp circuits [17]. Figure 5 shows the oscillator circuit of the SAW based micro force sensor based on the principle of Pierce circuit and the combination with the design requirements.Figure 5
The Pierce circuit of the SAW based micro force sensor.The high-frequency triode2
S
C
3357 affects the resonance amplifier in the circuit by ensuring the normal startup of the oscillator circuit. R
1 is the DC bias resistance of this triode, and adjusting the value changes with DC working points. R
2 is the DC bias resistance of the triode s
8050 that changes the DC working points and transforms the bias current of the circuit. This triode functions as the current source and stabilizes the circuit working state. The SAW resonator affects the frequency selection in the feedback loop by inducing the SAW oscillator frequency stability and improving the antijamming ability.To ensure that the feedback loop circuit phase is in balance, the inductanceL has been added to the Pierce circuit to eliminate the effect of the stability of the DC voltage source from the oscillator circuit. The LC parallel resonant circuit also affects the DC bypass circuit. The feedback coefficient F is related to C
1 and C
2 and is given as
(8)
F
=
C
1
C
2
,
where C
3, C
4, and L
2 are the decoupling devices of the DC source that can eliminate the effect of the oscillator circuit from the LC filter function.


## 3.3. Determine the Parameters

The circuit layout, selection of components, and calculated parameters are important. The work frequency of the micro force sensor is 50 MHz; the center frequency of the LC parallel resonant circuit should be(9)
f
0
=
1
2
π
LC
.To ensure that the amplitude of oscillator circuit has better characteristics than the initial conditions,C
2 should be smaller. In addition, the designed oscillator frequency is given; the capacitance is selected as 5.1 pF. From (10), L can be obtained as
(10)
L
=
1
(
2
π
f
0
)
2
C
.Because parasitic parameters in practical circuits affect their performances,L must be adjustable. By changing the number of turns of the air-core coil, the coil inductance can be fine-tuned, and the frequency of the L
C parallel resonant circuit will be in accordance with the working frequency of the SAW based micro force sensor. The air-core coil inductance has been calculated by the following equation:
(11)
L
=
0.01
×
D
×
N
2
(
L
N
/
D
)
+
0.44
,
where L is the coil inductance (in μ
H), D is the diameter of the coil (in cm), N is the number of coil turns, andLN is the coil length (in cm).An oscillator frequency source should have good stability, phase noise, and highQ value [18]. Figure 6 shows the equivalent circuit, with C
2 expressed in picofarads.Figure 6
The equivalent circuit of the SAW device.The SAW resonator can work in series or parallel resonant frequencies upon operating in the feedback loop. The transistor2
S
C
3357 and s
8050 provide the DC bias current in the oscillator circuit design based on the requirements.


## 4. The Linear Regression Model of the SAW Based Micro Force Sensor

### 4.1. Establishment of the Linear Regression Model

After getting the pressure of SAW sensor (F
m) and the output frequency (f) from the actual oscillator circuit, the fitting function should be established. The relationship between the pressure of this sensor (F
m) and the output frequency (f) is given as follows:
(12)
F
m
=
k
0
+
k
1
Δ
f
+
k
2
Δ
f
2
+
k
3
Δ
f
3
+
⋯
+
k
n
Δ
f
n
⋯
.IfF
i
,
f
i,  i
=
1,2
,
…
,
n, and n
=
5, the method of least squares can be used to solve this function is given as
(13)
F
m
=
k
0
+
k
1
Δ
f
+
k
2
Δ
f
2
+
k
3
Δ
f
3
+
k
4
Δ
f
4
+
k
5
Δ
f
5
+
k
6
Δ
f
6
.The regression coefficientsk
0,  k
1,  k
2,  k
3,  k
4,  k
5, and k
6 can be calculated through formula (13).


### 4.2. Calculate the Input and Output Variable Regression

The sum of the variance (Q) between the dependent and independent variables is given as [19]
(14)
Q
(
k
0
-
k
6
)
=
∑
i
=
1
N
Δ
i
2
=
∑
i
=
1
N
[
Δ
f
i
-
F
i
]
2
,
where N is the sampling point of the independent variables. Therefore, the least squares estimation calculates the minimum value of Q, which implies that k
∧
0,  k
∧
1,  k
∧
2,  k
∧
3,  k
∧
4,  k
∧
5, and k
6 must suit the following equation:
(15)
Q
(
k
∧
0
,
k
∧
1
,
k
∧
2
,
k
∧
3
,
k
∧
4
,
k
∧
5
,
k
∧
6
)
=
∑
i
=
1
N
[
Δ
f
i
-
F
i
]
2
=
min
⁡
k
∧
0
-
k
∧
6
∑
i
=
1
N
[
Δ
f
i
-
F
i
]
2
,
where k
∧
0,  k
∧
1,  k
∧
2,  k
∧
3,  k
∧
4,  k
∧
5, and k
6 are the respective least squares estimations of k
0,  k
1,  k
2,  k
3,  k
4,  k
5, and k
6.When(16)
∂
Q
(
k
0
,
k
1
,
k
2
,
k
3
,
k
4
,
k
5
)
∂
k
0
|
k
0
=
k
0
∧
=
0
,
∂
Q
(
k
0
,
k
1
,
k
2
,
k
3
,
k
4
,
k
5
)
∂
k
1
|
k
0
=
k
1
∧
=
0
,
∂
Q
(
k
0
,
k
1
,
k
2
,
k
3
,
k
4
,
k
5
)
∂
k
2
|
k
0
=
k
2
∧
=
0
,
∂
Q
(
k
0
,
k
1
,
k
2
,
k
3
,
k
4
,
k
5
)
∂
k
3
|
k
0
=
k
3
∧
=
0
,
∂
Q
(
k
0
,
k
1
,
k
2
,
k
3
,
k
4
,
k
5
)
∂
k
4
|
k
0
=
k
4
∧
=
0
,
∂
Q
(
k
0
,
k
1
,
k
2
,
k
3
,
k
4
,
k
5
)
∂
k
5
|
k
0
=
k
5
∧
=
0
,
∂
Q
(
k
0
,
k
1
,
k
2
,
k
3
,
k
4
,
k
5
)
∂
k
6
|
k
0
=
k
6
∧
=
0
.The equation (17) can be derived by formula (15) and (16)
(17)
∑
i
=
1
N
[
(
k
0
∧
+
k
1
∧
Δ
f
i
+
k
2
∧
Δ
f
i
2
+
k
3
∧
Δ
f
i
3
+
k
4
∧
Δ
f
i
4
+
k
5
∧
Δ
f
i
5
+
k
6
∧
Δ
f
i
6
)
-
F
i
]
=
0
,
∑
i
=
1
N
[
(
k
0
∧
+
k
1
∧
Δ
f
i
+
k
2
∧
Δ
f
i
2
+
k
3
∧
Δ
f
i
3
+
k
4
∧
Δ
f
i
4
+
k
5
∧
Δ
f
i
5
+
k
6
∧
Δ
f
i
6
)
-
F
i
]
Δ
f
i
=
0
,
∑
i
=
1
N
[
(
k
0
∧
+
k
1
∧
Δ
f
i
+
k
2
∧
Δ
f
i
2
+
k
3
∧
Δ
f
i
3
+
k
4
∧
Δ
f
i
4
+
k
5
∧
Δ
f
i
5
+
k
6
∧
Δ
f
i
6
)
-
F
i
]
Δ
f
i
2
=
0
,
∑
i
=
1
N
[
(
k
0
∧
+
k
1
∧
Δ
f
i
+
k
2
∧
Δ
f
i
2
+
k
3
∧
Δ
f
i
3
+
k
4
∧
Δ
f
i
4
+
k
5
∧
Δ
f
i
5
+
k
6
∧
Δ
f
i
6
)
-
F
i
]
Δ
f
i
3
=
0
,
∑
i
=
1
N
[
(
k
0
∧
+
k
1
∧
Δ
f
i
+
k
2
∧
Δ
f
i
2
+
k
3
∧
Δ
f
i
3
+
k
4
∧
Δ
f
i
4
+
k
5
∧
Δ
f
i
5
+
k
6
∧
Δ
f
i
6
)
-
F
i
]
Δ
f
i
4
=
0
,
∑
i
=
1
N
[
(
k
0
∧
+
k
1
∧
Δ
f
i
+
k
2
∧
Δ
f
i
2
+
k
3
∧
Δ
f
i
3
+
k
4
∧
Δ
f
i
4
+
k
5
∧
Δ
f
i
5
+
k
6
∧
Δ
f
i
6
)
-
F
i
]
Δ
f
i
5
=
0
,
∑
i
=
1
N
[
(
k
0
∧
+
k
1
∧
Δ
f
i
+
k
2
∧
Δ
f
i
2
+
k
3
∧
Δ
f
i
3
+
k
4
∧
Δ
f
i
4
+
k
5
∧
Δ
f
i
5
+
k
6
∧
Δ
f
i
6
)
-
F
i
]
Δ
f
i
6
=
0
.Equation (18) can be derived from formulas (14) and (17). Consider
(18)
N
k
∧
0
+
k
∧
1
∑
i
=
1
N
Δ
f
+
k
∧
2
∑
i
=
1
N
Δ
f
2
+
k
∧
3
∑
i
=
1
N
Δ
f
3
+
k
∧
4
∑
i
=
1
N
Δ
f
4
+
k
∧
5
∑
i
=
1
N
Δ
f
5
+
k
∧
6
∑
i
=
1
N
Δ
f
6
=
∑
i
=
1
N
F
i
,
k
∧
0
∑
i
=
1
N
Δ
f
+
k
∧
1
∑
i
=
1
N
Δ
f
2
+
k
∧
2
∑
i
=
1
N
Δ
f
3
+
k
∧
3
∑
i
=
1
N
Δ
f
4
+
k
∧
4
∑
i
=
1
N
Δ
f
5
+
k
∧
5
∑
i
=
1
N
Δ
f
6
+
k
∧
6
∑
i
=
1
N
Δ
f
7
=
∑
i
=
1
N
F
i
Δ
f
i
,
k
∧
0
∑
i
=
1
N
Δ
f
2
+
k
∧
1
∑
i
=
1
N
Δ
f
3
+
k
∧
2
∑
i
=
1
N
Δ
f
4
+
k
∧
3
∑
i
=
1
N
Δ
f
5
+
k
∧
4
∑
i
=
1
N
Δ
f
6
+
k
∧
5
∑
i
=
1
N
Δ
f
7
+
k
∧
6
∑
i
=
1
N
Δ
f
8
=
∑
i
=
1
N
F
i
Δ
f
i
2
,
k
∧
0
∑
i
=
1
N
Δ
f
3
+
k
∧
1
∑
i
=
1
N
Δ
f
4
+
k
∧
2
∑
i
=
1
N
Δ
f
5
+
k
∧
3
∑
i
=
1
N
Δ
f
6
+
k
∧
4
∑
i
=
1
N
Δ
f
7
+
k
∧
5
∑
i
=
1
N
Δ
f
8
+
k
∧
6
∑
i
=
1
N
Δ
f
9
=
∑
i
=
1
N
F
i
Δ
f
i
3
,
k
∧
0
∑
i
=
1
N
Δ
f
4
+
k
∧
1
∑
i
=
1
N
Δ
f
5
+
k
∧
2
∑
i
=
1
N
Δ
f
6
+
k
∧
3
∑
i
=
1
N
Δ
f
7
+
k
∧
4
∑
i
=
1
N
Δ
f
8
+
k
∧
5
∑
i
=
1
N
Δ
f
9
+
k
∧
6
∑
i
=
1
N
Δ
f
10
=
∑
i
=
1
N
F
i
Δ
f
i
4
,
k
∧
0
∑
i
=
1
N
Δ
f
5
+
k
∧
1
∑
i
=
1
N
Δ
f
6
+
k
∧
2
∑
i
=
1
N
Δ
f
7
+
k
∧
3
∑
i
=
1
N
Δ
f
8
+
k
∧
4
∑
i
=
1
N
Δ
f
9
+
k
∧
5
∑
i
=
1
N
Δ
f
10
+
k
∧
6
∑
i
=
1
N
Δ
f
11
=
∑
i
=
1
N
F
i
Δ
f
i
5
,
k
∧
0
∑
i
=
1
N
Δ
f
6
+
k
∧
1
∑
i
=
1
N
Δ
f
7
+
k
∧
2
∑
i
=
1
N
Δ
f
8
+
k
∧
3
∑
i
=
1
N
Δ
f
9
+
k
∧
4
∑
i
=
1
N
Δ
f
10
+
k
∧
5
∑
i
=
1
N
Δ
f
11
+
k
∧
6
∑
i
=
1
N
Δ
f
12
=
∑
i
=
1
N
F
i
Δ
f
i
6
.Equation (18) can be written as
(19)
N
k
∧
0
+
A
k
∧
1
+
B
k
∧
2
+
C
k
∧
3
+
D
k
∧
4
+
E
k
∧
5
+
F
k
∧
6
=
M
,
A
k
∧
0
+
B
k
∧
1
+
C
k
∧
2
+
D
k
∧
3
+
E
k
∧
4
+
F
k
∧
5
+
G
k
∧
6
=
P
,
B
k
∧
0
+
C
k
∧
1
+
D
k
∧
2
+
E
k
∧
3
+
F
k
∧
4
+
G
k
∧
5
+
H
k
∧
6
=
Q
,
C
k
∧
0
+
D
k
∧
1
+
E
k
∧
2
+
F
k
∧
3
+
G
k
∧
4
+
H
k
∧
5
+
I
k
∧
6
=
R
,
D
k
∧
0
+
E
k
∧
1
+
F
k
∧
2
+
G
k
∧
3
+
H
k
∧
4
+
I
k
∧
5
+
J
k
∧
6
=
S
,
E
k
∧
0
+
F
k
∧
1
+
G
k
∧
2
+
H
k
∧
3
+
I
k
∧
4
+
J
k
∧
5
+
K
k
∧
6
=
T
,
F
k
∧
0
+
G
k
∧
1
+
H
k
∧
2
+
I
k
∧
3
+
J
k
∧
4
+
K
k
∧
5
+
L
k
∧
6
=
U
,
where
(20)
A
=
∑
i
=
1
N
Δ
f
i
,
B
=
∑
i
=
1
N
Δ
f
i
2
,
C
=
∑
i
=
1
N
Δ
f
i
3
,
D
=
∑
i
=
1
N
Δ
f
i
4
,
E
=
∑
i
=
1
N
Δ
f
i
5
,
F
=
∑
i
=
1
N
Δ
f
i
6
,
G
=
∑
i
=
1
N
Δ
f
i
7
,
H
=
∑
i
=
1
N
Δ
f
i
8
,
I
=
∑
i
=
1
N
Δ
f
i
9
,
J
=
∑
i
=
1
N
Δ
f
i
10
,
K
=
∑
i
=
1
N
Δ
f
i
11
,
L
=
∑
i
=
1
N
Δ
f
i
12
,
M
=
∑
i
=
1
N
F
i
,
P
=
∑
i
=
1
N
F
i
Δ
f
i
1
,
Q
=
∑
i
=
1
N
F
i
Δ
f
i
2
,
R
=
∑
i
=
1
N
F
i
Δ
f
i
3
,
S
=
∑
i
=
1
N
F
i
Δ
f
i
4
,
T
=
∑
i
=
1
N
F
i
Δ
f
i
5
,
U
=
∑
i
=
1
N
F
i
Δ
f
i
6
.Coefficientsk
0 to k
6 in (13) can be obtained, and the input and output variable regression of the SAW based micro force sensor can be calculated by solving (18).


## 4.1. Establishment of the Linear Regression Model

After getting the pressure of SAW sensor (F
m) and the output frequency (f) from the actual oscillator circuit, the fitting function should be established. The relationship between the pressure of this sensor (F
m) and the output frequency (f) is given as follows:
(12)
F
m
=
k
0
+
k
1
Δ
f
+
k
2
Δ
f
2
+
k
3
Δ
f
3
+
⋯
+
k
n
Δ
f
n
⋯
.IfF
i
,
f
i,  i
=
1,2
,
…
,
n, and n
=
5, the method of least squares can be used to solve this function is given as
(13)
F
m
=
k
0
+
k
1
Δ
f
+
k
2
Δ
f
2
+
k
3
Δ
f
3
+
k
4
Δ
f
4
+
k
5
Δ
f
5
+
k
6
Δ
f
6
.The regression coefficientsk
0,  k
1,  k
2,  k
3,  k
4,  k
5, and k
6 can be calculated through formula (13).


## 4.2. Calculate the Input and Output Variable Regression

The sum of the variance (Q) between the dependent and independent variables is given as [19]
(14)
Q
(
k
0
-
k
6
)
=
∑
i
=
1
N
Δ
i
2
=
∑
i
=
1
N
[
Δ
f
i
-
F
i
]
2
,
where N is the sampling point of the independent variables. Therefore, the least squares estimation calculates the minimum value of Q, which implies that k
∧
0,  k
∧
1,  k
∧
2,  k
∧
3,  k
∧
4,  k
∧
5, and k
6 must suit the following equation:
(15)
Q
(
k
∧
0
,
k
∧
1
,
k
∧
2
,
k
∧
3
,
k
∧
4
,
k
∧
5
,
k
∧
6
)
=
∑
i
=
1
N
[
Δ
f
i
-
F
i
]
2
=
min
⁡
k
∧
0
-
k
∧
6
∑
i
=
1
N
[
Δ
f
i
-
F
i
]
2
,
where k
∧
0,  k
∧
1,  k
∧
2,  k
∧
3,  k
∧
4,  k
∧
5, and k
6 are the respective least squares estimations of k
0,  k
1,  k
2,  k
3,  k
4,  k
5, and k
6.When(16)
∂
Q
(
k
0
,
k
1
,
k
2
,
k
3
,
k
4
,
k
5
)
∂
k
0
|
k
0
=
k
0
∧
=
0
,
∂
Q
(
k
0
,
k
1
,
k
2
,
k
3
,
k
4
,
k
5
)
∂
k
1
|
k
0
=
k
1
∧
=
0
,
∂
Q
(
k
0
,
k
1
,
k
2
,
k
3
,
k
4
,
k
5
)
∂
k
2
|
k
0
=
k
2
∧
=
0
,
∂
Q
(
k
0
,
k
1
,
k
2
,
k
3
,
k
4
,
k
5
)
∂
k
3
|
k
0
=
k
3
∧
=
0
,
∂
Q
(
k
0
,
k
1
,
k
2
,
k
3
,
k
4
,
k
5
)
∂
k
4
|
k
0
=
k
4
∧
=
0
,
∂
Q
(
k
0
,
k
1
,
k
2
,
k
3
,
k
4
,
k
5
)
∂
k
5
|
k
0
=
k
5
∧
=
0
,
∂
Q
(
k
0
,
k
1
,
k
2
,
k
3
,
k
4
,
k
5
)
∂
k
6
|
k
0
=
k
6
∧
=
0
.The equation (17) can be derived by formula (15) and (16)
(17)
∑
i
=
1
N
[
(
k
0
∧
+
k
1
∧
Δ
f
i
+
k
2
∧
Δ
f
i
2
+
k
3
∧
Δ
f
i
3
+
k
4
∧
Δ
f
i
4
+
k
5
∧
Δ
f
i
5
+
k
6
∧
Δ
f
i
6
)
-
F
i
]
=
0
,
∑
i
=
1
N
[
(
k
0
∧
+
k
1
∧
Δ
f
i
+
k
2
∧
Δ
f
i
2
+
k
3
∧
Δ
f
i
3
+
k
4
∧
Δ
f
i
4
+
k
5
∧
Δ
f
i
5
+
k
6
∧
Δ
f
i
6
)
-
F
i
]
Δ
f
i
=
0
,
∑
i
=
1
N
[
(
k
0
∧
+
k
1
∧
Δ
f
i
+
k
2
∧
Δ
f
i
2
+
k
3
∧
Δ
f
i
3
+
k
4
∧
Δ
f
i
4
+
k
5
∧
Δ
f
i
5
+
k
6
∧
Δ
f
i
6
)
-
F
i
]
Δ
f
i
2
=
0
,
∑
i
=
1
N
[
(
k
0
∧
+
k
1
∧
Δ
f
i
+
k
2
∧
Δ
f
i
2
+
k
3
∧
Δ
f
i
3
+
k
4
∧
Δ
f
i
4
+
k
5
∧
Δ
f
i
5
+
k
6
∧
Δ
f
i
6
)
-
F
i
]
Δ
f
i
3
=
0
,
∑
i
=
1
N
[
(
k
0
∧
+
k
1
∧
Δ
f
i
+
k
2
∧
Δ
f
i
2
+
k
3
∧
Δ
f
i
3
+
k
4
∧
Δ
f
i
4
+
k
5
∧
Δ
f
i
5
+
k
6
∧
Δ
f
i
6
)
-
F
i
]
Δ
f
i
4
=
0
,
∑
i
=
1
N
[
(
k
0
∧
+
k
1
∧
Δ
f
i
+
k
2
∧
Δ
f
i
2
+
k
3
∧
Δ
f
i
3
+
k
4
∧
Δ
f
i
4
+
k
5
∧
Δ
f
i
5
+
k
6
∧
Δ
f
i
6
)
-
F
i
]
Δ
f
i
5
=
0
,
∑
i
=
1
N
[
(
k
0
∧
+
k
1
∧
Δ
f
i
+
k
2
∧
Δ
f
i
2
+
k
3
∧
Δ
f
i
3
+
k
4
∧
Δ
f
i
4
+
k
5
∧
Δ
f
i
5
+
k
6
∧
Δ
f
i
6
)
-
F
i
]
Δ
f
i
6
=
0
.Equation (18) can be derived from formulas (14) and (17). Consider
(18)
N
k
∧
0
+
k
∧
1
∑
i
=
1
N
Δ
f
+
k
∧
2
∑
i
=
1
N
Δ
f
2
+
k
∧
3
∑
i
=
1
N
Δ
f
3
+
k
∧
4
∑
i
=
1
N
Δ
f
4
+
k
∧
5
∑
i
=
1
N
Δ
f
5
+
k
∧
6
∑
i
=
1
N
Δ
f
6
=
∑
i
=
1
N
F
i
,
k
∧
0
∑
i
=
1
N
Δ
f
+
k
∧
1
∑
i
=
1
N
Δ
f
2
+
k
∧
2
∑
i
=
1
N
Δ
f
3
+
k
∧
3
∑
i
=
1
N
Δ
f
4
+
k
∧
4
∑
i
=
1
N
Δ
f
5
+
k
∧
5
∑
i
=
1
N
Δ
f
6
+
k
∧
6
∑
i
=
1
N
Δ
f
7
=
∑
i
=
1
N
F
i
Δ
f
i
,
k
∧
0
∑
i
=
1
N
Δ
f
2
+
k
∧
1
∑
i
=
1
N
Δ
f
3
+
k
∧
2
∑
i
=
1
N
Δ
f
4
+
k
∧
3
∑
i
=
1
N
Δ
f
5
+
k
∧
4
∑
i
=
1
N
Δ
f
6
+
k
∧
5
∑
i
=
1
N
Δ
f
7
+
k
∧
6
∑
i
=
1
N
Δ
f
8
=
∑
i
=
1
N
F
i
Δ
f
i
2
,
k
∧
0
∑
i
=
1
N
Δ
f
3
+
k
∧
1
∑
i
=
1
N
Δ
f
4
+
k
∧
2
∑
i
=
1
N
Δ
f
5
+
k
∧
3
∑
i
=
1
N
Δ
f
6
+
k
∧
4
∑
i
=
1
N
Δ
f
7
+
k
∧
5
∑
i
=
1
N
Δ
f
8
+
k
∧
6
∑
i
=
1
N
Δ
f
9
=
∑
i
=
1
N
F
i
Δ
f
i
3
,
k
∧
0
∑
i
=
1
N
Δ
f
4
+
k
∧
1
∑
i
=
1
N
Δ
f
5
+
k
∧
2
∑
i
=
1
N
Δ
f
6
+
k
∧
3
∑
i
=
1
N
Δ
f
7
+
k
∧
4
∑
i
=
1
N
Δ
f
8
+
k
∧
5
∑
i
=
1
N
Δ
f
9
+
k
∧
6
∑
i
=
1
N
Δ
f
10
=
∑
i
=
1
N
F
i
Δ
f
i
4
,
k
∧
0
∑
i
=
1
N
Δ
f
5
+
k
∧
1
∑
i
=
1
N
Δ
f
6
+
k
∧
2
∑
i
=
1
N
Δ
f
7
+
k
∧
3
∑
i
=
1
N
Δ
f
8
+
k
∧
4
∑
i
=
1
N
Δ
f
9
+
k
∧
5
∑
i
=
1
N
Δ
f
10
+
k
∧
6
∑
i
=
1
N
Δ
f
11
=
∑
i
=
1
N
F
i
Δ
f
i
5
,
k
∧
0
∑
i
=
1
N
Δ
f
6
+
k
∧
1
∑
i
=
1
N
Δ
f
7
+
k
∧
2
∑
i
=
1
N
Δ
f
8
+
k
∧
3
∑
i
=
1
N
Δ
f
9
+
k
∧
4
∑
i
=
1
N
Δ
f
10
+
k
∧
5
∑
i
=
1
N
Δ
f
11
+
k
∧
6
∑
i
=
1
N
Δ
f
12
=
∑
i
=
1
N
F
i
Δ
f
i
6
.Equation (18) can be written as
(19)
N
k
∧
0
+
A
k
∧
1
+
B
k
∧
2
+
C
k
∧
3
+
D
k
∧
4
+
E
k
∧
5
+
F
k
∧
6
=
M
,
A
k
∧
0
+
B
k
∧
1
+
C
k
∧
2
+
D
k
∧
3
+
E
k
∧
4
+
F
k
∧
5
+
G
k
∧
6
=
P
,
B
k
∧
0
+
C
k
∧
1
+
D
k
∧
2
+
E
k
∧
3
+
F
k
∧
4
+
G
k
∧
5
+
H
k
∧
6
=
Q
,
C
k
∧
0
+
D
k
∧
1
+
E
k
∧
2
+
F
k
∧
3
+
G
k
∧
4
+
H
k
∧
5
+
I
k
∧
6
=
R
,
D
k
∧
0
+
E
k
∧
1
+
F
k
∧
2
+
G
k
∧
3
+
H
k
∧
4
+
I
k
∧
5
+
J
k
∧
6
=
S
,
E
k
∧
0
+
F
k
∧
1
+
G
k
∧
2
+
H
k
∧
3
+
I
k
∧
4
+
J
k
∧
5
+
K
k
∧
6
=
T
,
F
k
∧
0
+
G
k
∧
1
+
H
k
∧
2
+
I
k
∧
3
+
J
k
∧
4
+
K
k
∧
5
+
L
k
∧
6
=
U
,
where
(20)
A
=
∑
i
=
1
N
Δ
f
i
,
B
=
∑
i
=
1
N
Δ
f
i
2
,
C
=
∑
i
=
1
N
Δ
f
i
3
,
D
=
∑
i
=
1
N
Δ
f
i
4
,
E
=
∑
i
=
1
N
Δ
f
i
5
,
F
=
∑
i
=
1
N
Δ
f
i
6
,
G
=
∑
i
=
1
N
Δ
f
i
7
,
H
=
∑
i
=
1
N
Δ
f
i
8
,
I
=
∑
i
=
1
N
Δ
f
i
9
,
J
=
∑
i
=
1
N
Δ
f
i
10
,
K
=
∑
i
=
1
N
Δ
f
i
11
,
L
=
∑
i
=
1
N
Δ
f
i
12
,
M
=
∑
i
=
1
N
F
i
,
P
=
∑
i
=
1
N
F
i
Δ
f
i
1
,
Q
=
∑
i
=
1
N
F
i
Δ
f
i
2
,
R
=
∑
i
=
1
N
F
i
Δ
f
i
3
,
S
=
∑
i
=
1
N
F
i
Δ
f
i
4
,
T
=
∑
i
=
1
N
F
i
Δ
f
i
5
,
U
=
∑
i
=
1
N
F
i
Δ
f
i
6
.Coefficientsk
0 to k
6 in (13) can be obtained, and the input and output variable regression of the SAW based micro force sensor can be calculated by solving (18).


## 5. Testing and Analysis of the SAW Based Micro Force Sensor

### 5.1. Actual Circuit Detection Results

The output frequency of the SAW based micro force sensor is conducted by using the network analyzer equipment E5061A (Figure7). Force-measuring elements employ a cantilever beam loaded with 0–20 kPa pressure and add 2 kPa to this beam at each time [20].Figure 7
Schematic of the SAW based micro force sensor.Figure8 shows the test schematic. The SAW based micro force sensor has been designed with a single channel. For the inductive components, the electromagnetic field interference is more sensitive; two inductors have been placed at 90° to reduce interference between these components. Figure 9 shows the actual circuit. In the process of real production, the device location and the connection between our devices are more important.Figure 8
Test schematic diagram.Figure 9
The actual circuit testing diagram.


### 5.2. The Fitting Curve for the Measured Frequency

Figure10 shows the actual test waveform diagram [21].Figure 10
The actual test waveform diagram.To eliminate the system instability, the experimental data has been averaged in Table1.Table 1
Difference in frequency data for various pressures.
(kPa)
F
m
0
2
4
6
8
10
12
14
16
18
20
f (MHz)
49.255389
49.255391
49.255378
49.255393
49.255434
49.255459
49.255474
49.255481
49.255447
49.255486
49.255497
f (MHz)
49.255321
49.255312
49.255298
49.255296
49.255338
49.255343
49.255357
49.255359
49.255319
49.255361
49.255363
Δ
f (Hz)
68
79
80
97
96
116
117
122
128
125
134Frequency experiment data samples have been generated from (18) according to the different micro pressure reading, and the estimated regression coefficients k
∧
0,  k
∧
1,  k
∧
2,  k
∧
3,  k
∧
4,  k
∧
5 and k
∧
6 can be calculated. The method of least squares can reduce the error due to measurement inaccuracies caused by the SAW device based micro force sensor. Through the minimization of squared errors of our actual data, the least squares method has the ability to find the best matching function data. Additionally, the fitting curve of the experimental data shows that the circuit design is reasonable and the device exhibits good linearity (Figure 11). The pressure F
m is proportional to the output frequencyf, which has proved that the SAW detection and pierce oscillator circuit have correct logic function.Figure 11
The experimental data fitting curve.Based on (13), the available in different micro force sensor output difference frequency transfer function is
(21)
F
m
(
Δ
f
)
=
3.1441
×
1
0
-
1
+
5.9057
×
1
0
-
2
Δ
f
i
-
2.3012
×
1
0
-
4
Δ
f
i
2
+
5.4127
×
1
0
-
7
Δ
f
i
3
-
6.0364
×
1
0
-
10
Δ
f
i
4
+
3.0200
×
1
0
-
13
Δ
f
i
5
-
5.0540
×
1
0
-
17
Δ
f
i
6
.Because of the load impedance having affected the amplitude output of the micro force sensor, the impedance of the oscilloscope probe during testing is set at1
m
Ω to stabilize the oscillator amplitude. The working circuit voltage and current are 3
V and 10
mA. Using the difference in frequency data for various pressures in Table 1, the Oscilloscope measurement result has been calculated in Table 2.Table 2
Oscilloscope measurement results.
Measurement parameter
Average value
Minimum value
Maximum value
Voltage output (mV)
320
314
326
Frequency output (MHz)
49.2553965
49.255296
49.255497


## 5.1. Actual Circuit Detection Results

The output frequency of the SAW based micro force sensor is conducted by using the network analyzer equipment E5061A (Figure7). Force-measuring elements employ a cantilever beam loaded with 0–20 kPa pressure and add 2 kPa to this beam at each time [20].Figure 7
Schematic of the SAW based micro force sensor.Figure8 shows the test schematic. The SAW based micro force sensor has been designed with a single channel. For the inductive components, the electromagnetic field interference is more sensitive; two inductors have been placed at 90° to reduce interference between these components. Figure 9 shows the actual circuit. In the process of real production, the device location and the connection between our devices are more important.Figure 8
Test schematic diagram.Figure 9
The actual circuit testing diagram.


## 5.2. The Fitting Curve for the Measured Frequency

Figure10 shows the actual test waveform diagram [21].Figure 10
The actual test waveform diagram.To eliminate the system instability, the experimental data has been averaged in Table1.Table 1
Difference in frequency data for various pressures.
(kPa)
F
m
0
2
4
6
8
10
12
14
16
18
20
f (MHz)
49.255389
49.255391
49.255378
49.255393
49.255434
49.255459
49.255474
49.255481
49.255447
49.255486
49.255497
f (MHz)
49.255321
49.255312
49.255298
49.255296
49.255338
49.255343
49.255357
49.255359
49.255319
49.255361
49.255363
Δ
f (Hz)
68
79
80
97
96
116
117
122
128
125
134Frequency experiment data samples have been generated from (18) according to the different micro pressure reading, and the estimated regression coefficients k
∧
0,  k
∧
1,  k
∧
2,  k
∧
3,  k
∧
4,  k
∧
5 and k
∧
6 can be calculated. The method of least squares can reduce the error due to measurement inaccuracies caused by the SAW device based micro force sensor. Through the minimization of squared errors of our actual data, the least squares method has the ability to find the best matching function data. Additionally, the fitting curve of the experimental data shows that the circuit design is reasonable and the device exhibits good linearity (Figure 11). The pressure F
m is proportional to the output frequencyf, which has proved that the SAW detection and pierce oscillator circuit have correct logic function.Figure 11
The experimental data fitting curve.Based on (13), the available in different micro force sensor output difference frequency transfer function is
(21)
F
m
(
Δ
f
)
=
3.1441
×
1
0
-
1
+
5.9057
×
1
0
-
2
Δ
f
i
-
2.3012
×
1
0
-
4
Δ
f
i
2
+
5.4127
×
1
0
-
7
Δ
f
i
3
-
6.0364
×
1
0
-
10
Δ
f
i
4
+
3.0200
×
1
0
-
13
Δ
f
i
5
-
5.0540
×
1
0
-
17
Δ
f
i
6
.Because of the load impedance having affected the amplitude output of the micro force sensor, the impedance of the oscilloscope probe during testing is set at1
m
Ω to stabilize the oscillator amplitude. The working circuit voltage and current are 3
V and 10
mA. Using the difference in frequency data for various pressures in Table 1, the Oscilloscope measurement result has been calculated in Table 2.Table 2
Oscilloscope measurement results.
Measurement parameter
Average value
Minimum value
Maximum value
Voltage output (mV)
320
314
326
Frequency output (MHz)
49.2553965
49.255296
49.255497


## 6. Conclusion

This paper has addressed three problems in designing the SAW based micro force sensor, namely, the envelope of IDT, the variable regression, and the fitting curve analysis for the measured frequency, which are according to the wavelet transform method, SAW detection, and pierce oscillator circuits.The paper has also proposed the mixing frequency circuit for the reference oscillator and the detector circuit. The changes between the oscillator frequency and the measurement of the parameters can be obtained through the pierce oscillator circuits, and the difference in frequency data for various pressures also can be established by the linear regression model. By affecting the production of amplitude of impedance and parasitic capacitance, the output frequency of 50 MHz still has stabilized features and performance. The parameters of frequency, amplitude, frequency stability, and amplitude stability have been measured during the device response and the sensor simulation.According to the Morlet wavelet transform, SAW detection, and pierce oscillator circuits, the experimental results have confirmed that the SAW based micro force sensor can implement high reproducibility. When the input transducer of our device has been designed with the envelope of the conducting strips, our device performs well linearity. In addition, the device uses the piezoelectric properties and the temperature stability of the crystal, as well as the frequency signal instead of the conventional pressure sensor with voltage signal, which has made the signal processing of this device more digital and possessed more stable performances. The SAW based micro force type can be fabricated and has high performance in either mobile micro robots or a precise positioning device under the control of mechanical system.



---

*Source: 701723-2014-04-29.xml*