2 Gyroscope
2.3 Mathematic Model of Gyroscope
Coriolis acceleration could be counted out from relative motion equation as below.
Total force on mass (m) is below (relative to a rotating frame):
f⃑ = m ∙ a�⃑ = m �𝑑𝑑𝑡2𝑅�⃑2 +𝑑𝑑𝑡2𝑟⃑2+𝑑𝛺𝑑𝑡��⃑× 𝑟⃑ + 𝛺�⃑ × �𝛺�⃑ × 𝑟⃑� + 2𝛺�⃑ ×𝑑𝑟⃑𝑑𝑡� (2.1)
Figure 2.8: Chart of mass in rotating frame and fixed frame
Figure 2.8 shows a chart of a mass in rotating frame and fixed frame at the same time. The f �⃑ is total force on mass (m); a�⃑ is defined as the absolute acceleration of mass; the position vector 𝑅�⃑ is relative to rotating frame from fixed frame. Denoting the rotating frame by
F
for short; 𝑟⃑ is the position vector of mass relative toF
denoting by 𝛺�⃑ the angular velocity of the
F
The first term in (2.1) is the linear acceleration ofF
relative to fixed frame, the second term is the linear acceleration of mass relative toF
, the third term is the linear acceleration of mass relative toF
, thefourth term is the centripetal acceleration of mass relative to
F
, and the last term is14
called the complementary acceleration, or Coriolis acceleration, after the French mathematician de Coriolis (1792-1843), it’s also the value that gyroscope wants to measure. Notice that, on purpose of getting Coriolis acceleration, the last term in (2.1) must be existed, and the vector is cross product of angular velocity (𝛺�⃑) and the linear velocity (𝑑𝑟⃑
𝑑𝑡), of mass relative to
F
, hence Coiolis acceleration is vertical to both terms.If the terms of location vectors and angular velocity in (2.1) are replaced by Cartesian coordinates, it can be rewritten as:
{
F
} = {𝑒̂𝑥 𝑒̂𝑦 𝑒̂𝑧} (2.2)(2.3) (2.4) (2.5) also can be replaced by Cartesian coordinates as below:
𝑓𝑆
The d and k represents the coefficient of dampness and elasticity separately; u is the
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input.
Generally, mass of single-axis vibratory gyroscope is fixed on a plane. In this study, it assumes motion of the mass is restricted within x-y plane, but only detects the angular velocity on z-axis. In this assuming (2.1) can be rewritten into 2 axes dynamic equations by (2.2) (2.3) (2.4) (2.5) and (2.6) (2.7) (2.8):
m𝑎x+ mẍ + 𝑑1ẋ + �k1− m�Ωy2+ Ωz2�� x + m�ΩxΩy− Ω̇z�y = 𝑢𝑥+ 2𝑚Ω𝑧𝑦̇ (2.9) m𝑎y+ mÿ + 𝑑2ẏ + �k2− m(Ωx2+ Ωz2)�y + m�ΩxΩy− Ω̇z�x = 𝑢𝑦− 2𝑚Ω𝑧𝑥̇ (2.10)
The two forces, 2𝑚Ω𝑧𝑥̇ and 2𝑚Ω𝑧𝑦̇ are resulted from Coriolis effect, from here, if wants the accuracy of angular velocity measuring more accurate the two Coriolis forces need to as greater as they can be. That depends on the mass and linear velocity.
However the most characteristic of MEMS device is the micro-size, small size comes along with slight mass, about 10−6 ~ 10−8 kg. That also results in the difficulty of high accuracy of measuring.
Vibratory MEMS gyroscope used to be operated under the range of frequency from thousands to tens thousands hertz, hence the terms of angular velocities multiplying are far smaller than resonating frequency can be ignored. Therefore, (2.9) (2.10) can be simplified as:
mẍ + d1ẋ + k1x = ux+ 2mΩzẏ (2.11)
mÿ + d2ẏ + k2y = uy− 2mΩzẋ (2.12)
Equation (2.11) (2.12) describes single-axis gyroscope dynamic motions. From (2.11) (2.12), in ideality, there’s only dynamic coupling, in other words, excepting
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itself rotation (Ωz≠0) gyroscope only affected by its own structure design, characteristics. Basic on this equation, angular velocity could be counted from dynamic coupling status.
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Chapter 3
Theory of Piezoelectric Material
3.1 Conspectus of Piezoelectric Material
In 1824, Brewster found that tourmalines generate electrical charge when heated, or what is called “pyroelectricity”.
In 1880, the brothers Pierre Curie and Jacques Curie discovered piezoelectricity.
They showed that crystals of tourmaline, quartz, and Rochelle salt (sodium potassium tartrate tetrahydrate) generated electrical polarization from mechanical stress.
There are many categories of piezoelectric material, and this study classifies them into 7 crystal systems and 32 point groups. 7 crystal systems include triclinic crystal system, monoclinic crystal system, orthorhombic crystal system, tetragonal crystal system, trigonal crystal system, hexagonal crystal system and cubic crystal system.
Point groups are classified according to the structure and crystal symmetry. Crystal structure is directly related to piezoelectricity. For example, when one material is crystallized around a point of symmetry, the material is definitely not piezoelectric material. However, all non-center of symmetry point group, except for 432 point group, are piezoelectric materials. Now a widely used material, piezoelectric ceramics i.e. PZT, is a hexagonal crystal system and in the 6mm point group.
Piezoelectric ceramics are crystallized under high temperature. After polarization, it is like a single crystal with directionality. Another physical characteristic is the dielectric effect. When piezoelectric material is put in an electric field, electric charges are affected by the electric field and move, becoming electric dipoles, which
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causes polarization - hence the dielectric effect. After that, under electric field or mechanical stress influences, piezoelectric materials generally will have a linear response.
Piezoelectricity is an interaction between electrical and mechanical systems.
Piezoelectric materials have two main effects. One is direct piezoelectric effect, and the other is converse piezoelectric effect. The direct piezoelectric effect occurs when piezoelectric materials produce electric polarization by forcing mechanical stress on the material. On the contrary, the converse piezoelectric effect occurs when piezoelectric materials produce deformation by applying electric field on the material.
The electromechanical coupling effect of piezoelectric materials is used widely in engineering. With the two effects, many kinds of actuators and sensor can be made.
However, there are some disadvantages worthy of concern in the application of piezoelectric materials. For example, it usually produces unnecessary coupling effects in any direction perpendicular to driving axis in the process of applying voltage.
In improving the accuracy and stability of piezoelectric devices, the null signal suppression of leakage couplings is significant. These null couplings have two main sources: mechanical and electromechanical couplings. The former is produced from manufacture defects, and the latter is because of the piezoelectric property between the driving and detecting electrode. These cases of leakage couplings can be improved either by changing the designed structure or dealing with control stratagems, in order to improve the resolution of the gyroscopes.
In general, piezoelectric materials have advantages of high strength, high bandwidth, short response time, and are used widely in vibratory gyroscopes.
3.2 Piezoelectric Constitutive Equations
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Piezoelectric Constitutive Equation is described mathematically as below:
d-form:
�𝑆61(𝑇, 𝐸) = 𝑠66𝐸 𝑇61+ 𝑑63𝐸31
𝐷31(𝑇, 𝐸) = 𝑑36𝑇61+ 𝜀66𝑇 𝐸31 (3.1)
The four state variables (S, T, D, and E) can be rearranged arbitrarily to give an additional 3 forms for a piezoelectric constitutive equation by mathematics operation.
It is possible to transform piezoelectric constitutive data from one form to another form. In addition to the coupling matrix d, they contain the other coupling matrices e, g, or h in another 3 forms. What follows another 3 piezoelectric constitutive equations and their mutual transformations：
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T, S, E, D, d, g, h, 𝑘𝑑2 and e represent respectively stress, strain, electric field, electric displacement, piezoelectric strain/coefficient of electric charge, piezoelectric strain/coefficient of voltage, piezoelectric stress/ coefficient of voltage, coefficient of mechanical-electric coupling and piezoelectric stress/ coefficient of electric charge.
Those codes, such as 𝑠𝐸 with superscript means when the coefficient of function of S with constant variable E. 𝑠𝐷, 𝜀𝑇, 𝛽𝑇, 𝛽𝑠, 𝐶𝐸 represent respectively the coefficient of softness with constant electric displacement, coefficient of dielectric with constant stress, coefficient of converse dielectric with constant stress, coefficient of converse dielectric with constant strain and coefficient of stiffness with constant electric field.
In order to determine the piezoelectric material’s status, it needs 45 independent coefficients of material, but most piezoelectric materials have characteristic of symmetry in structure, so there are not as many coefficients as the hypothesis. The following diagram discribes piezoelectric ceramic as example, which is polarized in
x3 direction.
(3.5)
The T, D, S, E, 𝐶𝐸 in (3.5) are stress, electric displacement, strain, electric field and coefficient of stiffness with constant electric field respectively 𝜀𝑇, e, are
21
coefficients of dielectric and piezoelectric with constant strain. Table 3.1 shows parameters of PZT-5, which is also the material used in this study.
Parameters Values
Table 3.1: Parameters of piezoelectric ceramic (PZT-5)
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Chapter 4
Design and Simulation Analysis
4.1 Shape Design and Vibration mode Analysis
Existing piezoelectric gyroscopes have various designs - beam-shaped, fork-shaped etc. This study proposes a design that separates the driving and detecting part at two end of the device (Figure 4.1) to reduce the capacity effect. Based on the fork-shaped gyroscope and H-shaped gyroscope, this one is designed in a flat shape (Figure 4.2) in order to reduce installing-volume in vertical direction. Four hammer-shaped masses at four ends of driving and detecting arms are for coefficient of stiffness and resonating frequencies. At the ends of two stretched out parts from center mass are fixed points.
Figure 4.1: Chart of gyroscope this study proposes
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Figure 4.2: Flat piezoelectric gyroscope
This gyroscope uses piezoelectric effects to drive (Figure 4.3). When there is input of positive volt at the outer electrode (PZT) of the driving arm and negative volt at the inner electrode of the driving arm, the outer one will extend and the inner part will contract, then the driving arm will curve (Figure 4.4).
Figure 4.3: Flat piezoelectric gyroscope working principle
24
Figure 4.4: Input voltages on electrodes (PZT)
Meanwhile, if the gyroscope is driven under resonating frequency then the left half part will resonate in opposite directions (Figure 4.5).
25
Figure 4.5: Detecting part resonates in opposite phase
Furthermore, if the gyroscope is rotating around the Z-axis, then the detecting arm will curve (Figure 4.6) and then the deformation on detecting arm will make PZT electrodes output signals. Depending on the signals, Coriolis force and angular velocity can be measured.
In figure 4.1, the driving arm is for the driving input. The detecting driving arm is used to sense the amplitude of driving arm of being driven under resonating frequency. When the driving arm is driven under resonating frequency, the detecting driving arm will resonate in opposite direction with same amplitude. The detecting arm is for sensing. The feedback driving part will be introduced in section 4.2.
26
Figure 4.6: Flat piezoelectric gyroscope is driven and rotating at angular velocity 20 rad/sec.
After the prototype shape has been decided, vibration modes were close to being determined. Figure 4.7 shows 10 vibration modes. Sizes of every part are still important parameters to decide structure characteristics i.e. coefficient of stiffness, natural resonating frequency and performance etc.
Figure 4.7: (a) 1120 Hz Figure 4.7: (b) 1149Hz
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Figure 4.7: (c) 1161 Hz Figure 4.7: (d) 1179Hz
Figure 4.7: (e) 1222 Hz Figure 4.7: (f) 2125Hz
Figure 4.7: (g) 3433 Hz Figure 4.7: (h) 4654 Hz
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Figure 4.7: (i) 4674 Hz Figure 4.7: (j) 5784 Hz Figure 4.7: 10 vibration modes of gyroscope
This study picks figure 4.7(h) and figure 4.7(i) as the driving and detecting mode respectively. As the driving mode and detecting mode resonating frequencies become much closer, the device will produce a greater signal output. These two modes resonating frequencies are 4654 and 4674, so the input frequency in this study used is 4664 Hertz.
Silicon makes up the whole structure, and compared to PZT thickness, the thickness of silicon affects performance much more. Table 4.1 shows the displacement when it is driven under operation frequency and its resonating frequencies. The displacement is an objecting point on driving arm, and this is a harmonic simulation with damping ratio 0.005.
The first row of table was the very first idea of design, and this table shows that the thicknesses of silicon and PZT does not affect resonating frequencies as much, but performance (the displacement of driving) is strongly affected by thickness of the silicion. In addition, it shows linear tendency. The thicker the silicon, the poorer the performance, while the thicker the PZT, the better the performance.
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Table 4.1: Performance and resonating comparisons of different thickness
After a lot of tests and optimization of every part sizes, and taking into
consideration the cost, the final thickness and resonating frequency decided are shown in Table 4.2, the silicon basement is 300μm and PZT is 1.875μm.
Input voltage
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Table 4.2: Performance and operation frequency of flat piezoelectric gyroscope
Detail sizes are showed in figure 4.8:
Figure 4.8: Detailed sizes of flat piezoelectric gyroscope Input voltage
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4.2 Harmonic Simulation Analysis
In this study, harmonic simulation inputs a sinusoidal signal, which is within the resonating frequency to drive gyroscope. This section shows the extent of the deformation it could have. When the gyroscope is put it in a rotating frame, the experiment can also simulate how heavy the Coriolis force is and how much signal it can output.
Figure 4.9 was driven by a sinusoidal signal ranged from 20 ~ -20 volt:
Figure 4.9: Flat gyroscope is driven under operating frequency (4664 Hz)
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The structure of design is on one side of wafer, and hence when it is operated, there may be structure deflection along out-plane axis caused by stress concentration.
Table 4.3 shows 3 axes displacements of 4 nodes in figure 4.9. In consideration of fabrication process, the PZT is only designed on one side, hence the unsymmetrical structure may cause z-axis deformation, so the displacement in z-axis is a key point needed to observe. Table 4.3 shows that the displacement in z-axis is much smaller than two other axes, thus this displacement is insignificant. Damping ratio here is 0.005
Node Ux (m) Uy (m) Uz (m)
29524 5.80963E-6 -3.26782E-7 1.31608E-9
34374 5.73231E-6 5.39366E-7 1.27389E-9
13634 -5.73259E-6 -5.38841E-7 1.13422E-9
17531 -5.80965E-6 3.27264E-7 1.04556E-9
Table 4.3: Displacements of 4 nodes on gyroscope arms
In figure 4.1 there is one part of the gyroscope named ‘feedback driving arm’, which is used for feedback driving. Because of its micro size, if the device undergoes a small velocity condition, the Coriolis force would be very weak, and in this condition, the feedback driving arm can be used. Input signal on the feedback driving arm can help the detecting arm reach the predicted amplitude, then, subtracting the amplitude from feedback input, the amplitude caused from Coriolis force can be derived. Figure 4.10 shows the diagram with 2 nodes that be observed.
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Figure 4.10: Driving on feedback driving arm under operation frequency (4664Hz)
Table 4.4 shows displacement of 2 nodes in figure 4.10. Apparently, the displacements in y-axis of 2 nodes are almost the same, hence once the gyroscope is driven on feedback driving arm, the left part can resonate in opposite phase. In addition, displacements in z-axis are much smaller, so the displacement of z-axis can be ignored. Damping ratio here is 0.005.
Node Ux (m) Uy (m) Uz (m)
32907 2.63623E-9 -8.77733E-7 9.81261E-10
17827 -2.66178E-9 8.77252E-7 9.97343E-10
Table 4.4: Displacement of 2 nodes by driving on feedback driving arm
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4.3 Motion Simulation Analysis
4.3.1 Displacement and Electric Potential Simulation
The harmonic simulation in section 4.2 without considering Coriolis effect, but in this section, Coriolis effect will be considered.
Figure 4.11 shows that, when gyroscope is driven on right arm then left arm will resonate in opposite phase, when it is in rotation frame (undergo angular velocity), Coriolis force will occur then bend detecting arm.
Figure 4.11: Flat piezoelectric gyroscope undergo angular velocity 20 rad/sec and Coriolis force chart
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After Coriolis force bend the detecting arm, because of PZT electrodes deformation then they will output signals. Figure 4.12 shows that, when detecting arm is bended by Coriolis force, the PZT electrodes on detecting arm will deform. The below PZT electrode will extend a little bit and the one above will be contract slightly.
Once deformation occurs, piezoelectric material will output electric signal. Hence according to the signals, Coriolis force and angular velocity can then be measured.
Figure 4.12; Bending arm chart
Because the bending curve quantity is small, it is assumed that the bending curve can be seen as linear displacement. Following Table 4.5 takes this assumption.
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Figure 4.13 and table 4.5 show that, there is linear relation between displacement of the end of detecting arm and electric potential. Damping ratio here is 0.005:
Figure 4.13: Diagram of displacement – electrical potential
Voltage (v) 1.11E-04 2.22E-04 5.56E-04 1.11E-03 2.22E-03 5.56E-03 1.11E-02 Displacement
(m) 6.99E-09 1.40E-08 3.50E-08 6.99E-08 1.40E-07 3.50E-07 6.99E-07 Table 4.5: Data of displacement of the end of detecting arm and electric potential
4.3.2 Relation between Angular Velocity and Electric Potential
After determining the linear relation between deformation and output electric potential, it is important to find out the link between angular velocity and output voltage.
Table 4.6 shows voltage outputs of flat piezoelectric gyroscope undergo different angular velocities:
37
ω (°/sec) 1 2 5 10 20 50
Voltage
(volt) 1.11E-04 2.22E-04 5.56E-04 1.11E-03 2.22E-03 5.56E-03
ω (°/sec) 100 150 200 250 300 350
Voltage
(volt) 1.11E-02 1.67E-02 2.22E-02 2.77E-02 3.32E-02 3.87E-02 Table 4.6: Angular velocity – Output voltage
Figure 4.14 plots the table 4.6 data to a diagram:
Figure 4.14: Diagram of angular velocity – output voltage
Table 4.6 and figure 4.14 show the linear tendency between output voltage and angular velocity.
4.3.3 Coriolis force simulation
Another important purpose for the gyroscope is to determine how great the
38
Coriolis force is. Hence this study tries to address this. In steady state condition, inputting a constant electric field on PZT will cause deformation. Table 4.7 shows the displacements of Node 29524 (Figure 4.15) by input different voltages on driving arm:
Figure 4.15: Nodes for observation
Input voltage (v) Node 29524 x-axis displacement (m)
0 0
1 -1.12E-08
2.5 -2.80E-08
5 -5.60E-08
7.5 -8.40E-08
10 -1.12E-07
15 -1.68E-07
20 -2.24E-07
30 -3.36E-07
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Table 4.7: Table of input voltage – displacement
After that, a force is input on Node 29524 to match the displacement. Table 4.8 shows the conclusion:
Input force (N) Node 29524 x-axis displacement (m)
Table 4.8: Force – displacement chart
Tables 4.7 and 4.8 is combined into table 4.9:
Node 29524 x-axis Input
voltage Displacement (m) Force/voltage N/V
Input
force Displacement (m) K- driving (N/m)
Table 4.9: Relations of voltage, displacement and force on driving arm
40
Because the displacements are small, this study assumes the bending displacement can be seen as linear displacement, and according to basic hook’s law:
𝐹 = 𝑘𝑥, k is coefficient of stiffness, x is displacement, hence the coefficient of stiffness (K-driving, in table 4.9) of driving arm can be derived.
In the same way, the same process was done on Node 32907 then arranged data to table 4.10:
Node 32907 y-axis Input
voltage Displacement (m) Force/voltage N/V
Input
force Displacement (m) K- detecting (N/m)
Table 4.10: Relations of voltage, displacement and force on detecting arm
Tables 4.9 and 4.10 show the coefficients of two arms, and the coefficient for the transformation between force and voltage. That means when detecting arm outputs a signal, it can be measured then transformed to an equal force, which is seemed apply on observing node. Therefore, the Coriolis force can then be measured.
41
Chapter 5
Fabrication Process
5.1 Fabrication process
The fabrication process will be discussed in this chapter step by step. During the fabrication process underwent changes twice because of unforeseen problems. The problems will also discussed in this chapter.
In order to simplify the process, PZT is only deposited on one side of silicon wafer, hence a single polished silicon wafer is used in this study. PZT is the actuating and detecting material, it is deposited by spin-coating [16] [17] [18] [19] [20] [21] [22]
[23] [24] [25]. However, PZT is not an electric conductor, so it needs other metal layer as its electrodes. Platinum (Pt) is the most compatible candidate with PZT, and their crystal sizes are similar. As for electrodes, platinum has low electronic resistance, high melting point and high chemical stability. Because of the high chemical stability, that leads to challenges when patterning Pt by wet-etching, therefore, in order to pattern Pt, lift-off is used as the solution to this problem [25] [26]. Lift off process is a method of patterning of a target material on the surface of a substrate (ex. wafer) using a sacrificial material (ex. Photoresist). It is an additive technique as opposed to the more traditional subtracting technique like etching.
5.2 Experiments of Fabrication
5.2.1 First failed fabrication process The first process is shown below (Figure 5.1):
42
43
Figure 5.1: Fabrication process I
At very beginning, a layer of silicon nitrite is deposited by PECVD (plasma-enhanced chemical vapor deposition) as a seed layer, then a layer of photoresist(FH-6400L) is spin coated for lift-off, in order to pattern platinum (bottom electrode). Next 0.1μm thick platinum is evaporating deposited on it. Right after that, the whole wafer is dipped into acetone and an ultra sonic cleaner is used to remove photoresist. After these steps, the Pt is patterned on wafers.
The following chart shows the steps to spin coat PZT. (Figure 5.2):
Figure 5.2: Steps for spin-coating PZT
The purpose of baking at 150℃ is to evaporate water molecules. Baking at 350℃
44
is to evaporate organic compounds. Baking at 700℃ is to crystallize PZT. The was smooth, but PZT layer was unable to function.
In figure 5.3, it is obviously can seen that, where the bottom is platinum (Pt), PZT could crystallize well on it, but apparently not on silicon nitride (Si3N4).
Figure 5.3: PZT sample by microscope
45
During the process, a main issue that surfaced is where PZT cannot be crystallized on a silicon nitride well, and cavities grow between the crystals. The rough surface then gradually grows and extends to a smooth surface (Figure 5.4). In the end, smooth surface will be covered by a rough surface. (Figure 5.5)
During the process, a main issue that surfaced is where PZT cannot be crystallized on a silicon nitride well, and cavities grow between the crystals. The rough surface then gradually grows and extends to a smooth surface (Figure 5.4). In the end, smooth surface will be covered by a rough surface. (Figure 5.5)