Literature Survey

Chapter 1 Introduction

1.4 Literature Survey

It was not a new concept of using surface acoustic wave propagating on medium to detect angular rate of an object in motion. In the early 1974, the research of surface acoustic gyro was started. In 1977, it was reported several types of simulation structures of surface acoustic wave gyro. They used aluminum alloy cylinder or fuse quartz cylinder as propagating medium for surface acoustic wave, which was excited and detected by outboard electromagnetic transducer. Besides, Forst et al. had measured real signal of SAW gyroscopes based on the design above. The operation principle of SAW gyroscopes was similar to optical gyroscopes, i.e. two surface wave propagated in opposite direction along surface of cylinder is subjected to rotation. It would produce phase difference between two surface waves, and the angular velocity is found by measured this phase difference. Due to the velocity of SAW five order smaller than light, it was predicted that the sizes of SAW gyroscopes would better than optical gyro. So far, there was no follow-up paper about this research.

In 1980, B.Y. Lao [5, 7] derived theoretically the dependence of SAW velocity on the rotation rate of wave-propagating medium and it was established that, for an isotropic medium, the rotation rate was a function of Poisson’s ratio. From the relation, he proposed a new design of SAW gyroscope. It was using fuse quartz cylinder which deposited a layer of thin piezoelectric film as propagating medium. The acoustoelectric transformation for excited and detected surface acoustic wave was directly using the piezoelectric effect of material, not outboard electromagnetic transducer. According to the report, with the use of piezoelectric materials, it could increase efficiency of the transformation and greatly reduce the size of SAW gyroscope.

In 1985, a new concept of SAW gyroscope was proposed [5]. It was a planar structure.

The surface acoustic wave was excited by interdigital transducer on a piezoelectric material substrate. In 1998, M. Kurosawa [8] proposed a new type of SAW gyro sensor. He did the design simulations based on the equivalent circuit of the piezoelectric material. This gyro consists of generator interdigital transducer (GIDT), sensor interdigital transducer (SIDT), reflectors and perturbation mass, as shown in Fig. 3. The angular velocity of a body in motion was detected by the standing wave produced by these components. According to the report, the angular velocity could not be experimentally measured yet because the resonant frequency of RIDT and anti-resonant frequency of SIDT were not matched in experiment. In 1999, Varadan and coworkers [9, 10, 11] presented another SAW gyroscope design with a two-port resonator and sensor. The sensing resolution was 1 sec obtained in experiment. In 2002, R. ^o C. Woods et al. [12] reported experimental trials of several SAW devices to evaluate the performance of SAW gyroscope and also made an order-of-magnitude estimate of the sensitivity. His conclusion for this device was that this device was extremely insensitive to measure the angular velocity.

Chapter 2 Interdigital-Electrode Transducers for Surface Waves

Interdigital-electrode transducers were used to excite and detect the waves on the piezoelectric substrate, such that Rayleigh wave, Love wave, and etc. They play key roles in most of the SAW devices. This chapter introduces two methods which are suitable for modeling the behavior of interdigital-electrode transducers for surface wave. In the last of this chapter, we will present a study on the electromechanical interaction between interdigital-electrode transducers and a semi-infinite piezoelectric material. All of these three methods would be utilized in the design process of our SAW gyroscope.

2.1 Coupling-of-Modes Theory

The COM theory [13, 14, 15, 16, 17, 18, 19] has been extensively used since 1950s in various problems related to optical and electromagnetism for the description of wave propagation in periodically perturbed media. It provides an efficient and highly flexible approach for modeling various kinds of SAW devices by a set of transfer matrices. In general, there are three types of the representative elements for a SAW device: IDT, spacing, and reflector, which can be described as transmission matrices [T], [D] and [G]. Depending on the configuration of a SAW devices, any number of [T], [D] and [G] matrices can be used, but their basic form remain the same. For example, the basic elements of SAW devices consist of an IDT and reflector can be modeled as shown in Fig. 4.

Fig. 4(a) shows the basic elements, which has IDT, acoustic spacing and reflector. It can be represented with [T ], [₁ D ], and [₂ G ], as shown as Fig. 4(b). The indexes 1－3 are just for ₃ book-keeping purpose. a and b denote complex electrical input and output signals respectively.

The SAW coming in and out of each representative section is described by the complex amplitudes of forward, W⁺, and backward, W⁻. The amplitudes have dimensions of

Power . In matrix form, amplitudes at an ith reference axis are

[ ]

i ⁱ ^.

Thus, any

( )

i−1th SAW amplitudes coming in and out of ith section has following relation, where components of the transmission matrices are T, D and G:

[ ]

2.1.1 The 2×2 Reflector Matrix [G]

Matrix [G] is a 2×2 transmission matrix applied to the SAW reflection gratings, as shown in Fig. 5. The wave amplitudes at x = －L in terms of the wave amplitudes at x = 0 yields the following transmission relation

( ) ( ) [ ] ^{( )} _{( )}

⁰ ^,

where the transmission matrix [G] is

[ ]

frequency-deviation (detuning) parameter from the Bragg frequency f , where ₀ κ₁₁ is the self-coupling coefficient (m^-1) related to velocity shift

(

^{dv v}

)

2.1.2 The 3×3 IDT Matrix [T]

The transmission matrix T of an IDT can be found by manipulating the admittance

matrix based on a Mason equivalent circuit model which described in detail at section 2.2. It is required to relate electrical and acoustic parameters for each IDT. The transmission matrix [T] can be equated to these by

where a and _i b , respectively, denote complex electrical input and output strengths at the _i ith port. Reference planes for the IDT are as shown in Fig. 6.

The IDT matrix elements in Eq. (2.7) are given as [15]

[ ]

¹¹12 22¹² ¹³23 odd number of electrodes N , and the elements in matrix is _t

( ) ( )

where C is the static capacitance per electrode pair, _s Z is load resistance of source _e resistance, N is number of IDT electrodes (not pairs) ,and _t R is combined IDT metal and _s lead resistance. The radiation conductance G can be represented by the unperturbed sinc _a function expression [13]: and K is the electromechanical coupling constant. The radiation susceptance ² B is _a

( ) ( )

2.1.3 The 2×2 Acoustic Spacing [D]

The remaining matrix for the two-port SAW resonator design is a 2×2 one corresponding to each acoustic transmission line separating IDTs and SAW reflection gratings. This is

( ) ( ) [ ] ^{( )} _{( )}

⁰ ^,

where the elements of complex matrix [D] are

[ ]

⁰

in terms of wave number β₀ and acoustic length L between appropriate reference planes.

This matrix (2.2) representation of a lumped system model of SAW devices can be implemented to other SAW structures also since any SAW devices is combination of IDTs, reflectors and spacing. For example, there is a model of a two-port SAW resonator as shown in Fig. 7. Complex SAW structures can also be modeled by adding more transmission matrices at appropriate locations.

The total acoustic matrix [M] of the constituents of the two-port SAW resonator is now obtained as the product of the composite building blocks. Such that [M] is a 2×2 complex acoustic matrix

[ ] [ ][ ][ ][ ][ ][ ][ ]

M = G D t1 2 3 D t4 5 D G5 7 . (2.24) Here,

[ ]

G1 and

[ ]

G7 are relate to two SAW reflectors at the end.

[ ]

D2 and

[ ]

D6 are the

spacing between the grating and adjacent IDTs.

[ ]

D4 is the separation between IDTs.

[ ]

and

[ ]

t5 are the acoustic sub-matrices as shown in Eq. (2.9). From Eq. (2.20), however, the SAW amplitudes associated with transducer T are given by ₃

[ ] [ ]

W2 =t W3 3 + ⋅a3

[ ]

τ3 , (2.25)

which represents two equations with four unknowns W₂^± and W₃^±. Eq. (2.24) is solved by applying the boundary condition

0 7 0.

W⁺ =W⁻ = (2.26)

Thus, the reference axis of transducer T gives ₃

[ ] [ ][ ][ ]

W0 = G D W1 2 2 , (2.27)

and

[ ] [ ][ ][ ][ ][ ]

W3 = D t4 5 D G W6 7 7 . (2.28) By combining Eq. (2.25) through Eq. (2.28), the outward propagating SAW wave W₇⁺ and

W0⁻ are related to the input a by ₃

In addition, a choice of matched conditions at input and output yields voltage values

3 7 0.

b =a = (2.30)

At the output transducer T , ₅

[ ] [ ][ ][ ]

W5 = D G W6 7 7 . (2.31)

Finally, the electrical output voltage V is derived from the scalar product _out

[ ] [ ]

5 ^T 5 .

Vout =b = τ W (2.32)

2.2 Crossed-Field Model

The crossed-field (Fig. 8) model is derived from Mason equivalent circuit model employed for modeling acoustic bulk wave piezoelectric devices [18, 19, 21]. The equivalent circuits also have been widely used as approximate equivalent circuit for SAW IDTs [18, 22, 23].

In crossed-field model, the acoustic wave is represented by an electrical wave on a transmission line, and the piezoelectric energy conversion by a transformer. Moreover, the mechanical force and particle velocity at the acoustic ports are represented by its equivalent voltage and current, respectively. As a result, the equivalent circuits for SAW IDTs have two symmetric acoustic ports and one electric port, as shown in Fig. 9. In the terminology followed here, Port 1 and 2 represent electrical equivalent of “acoustic” ports, while Port 3 is

a true electrical port.

2.2.1 Electroacoustic Equivalences

All of three ports as shown in Fig. 9 are treated in equivalent electrical terms. At Port 1 and 2, the acoustic force F (in newtons) are transformed to electrical equivalent voltages V, while mechanical SAW velocities v are transformed to equivalent currents I. In terms of a common proportionality constant φ these transforms are

V ⁼ϕ ^(2.33)

I v= ϕ, (2.34)

where parameter φ is interpreted as the turns-ratio of an equivalent acoustic-to-electric transformer. This can be written in terms of electromechanical coupling coefficient K , ² frequency f , and total capacitance of the IDT, ₀ C as _T

2f C K A v0 _T ,

ϕ = ρ (2.35)

where ρ is the density of substrate and A is the effective crossed-sectional area.

2.2.2 Admittance Matrix for IDT

Fig. 10 shows the unit cell of IDT and each model can be cascaded depending on number of pairs for sensor IDTs. Essentially, the IDTs are arranged acoustically in cascade and electrically in parallel.

The equivalent current-voltage relations for a unit cell of IDT are given as

[ ]

where the 3×3 admittance matrix [Y] may be expanded as

[ ]

¹¹21 ¹²22 ¹³23

In order to calculate the overall response of the three acoustic transmission sub-matrices, ABCD matrix manipulations must be employed as shown in Fig. 11 [13]. ABCD matrix representation of a transmission line section of length d with equivalent electric impedance Z on the unmetallized region, which is assumed to be a lossless transmission line segment, 0

The corresponding matrix

[ ]

Rm for a lossless metallized-strip may be represented by

[ ]

^cos¹

^{( )} _{( )}

^sin

_{( )} ^{( )}

where the transit angle θ is obtained for the equivalent electric impedance Z as ₀

From these, the total sub-matrix

[ ]

Rt for three cascaded sections of the transmission line in Fig. 10 is obtained as

[ ] [ ][ ][ ]

Rt = Ru Rm Ru ^. (2.42)

To obtain the total 2×2 ABCD matrix [Q] for the equivalent transmission line of the complete IDT with N fingers, as shown in Fig. 12, the matrix in Eq.( 2.42) is cascaded to _t

Working backwards and employing ABCD-to-Y matrix conversions, the acoustic sub-matrix

⎡ ⎤

⎣ ⎦ for the total IDT is

Acoustic reflections from IDT fingers can lead to internal resonances and associated losses. Dealing with such losses required inclusion of an attenuation coefficient α_f in a general ⎡⎣Y^f⎤⎦ -matrix representation for the IDT with finger reflections. In [13], the admittance matrix is approximated as

1 1 And 2×2 acoustic submatrix terms are given by Eq. (2.43). Solving Eq. (2.44) for the electrical admittance of the transducer, Y fL

( )

, with the application of matched boundary conditions such that I₁= −V Z₁ ₀ = −V G₁ ₀ and I₂ = −V Z₂ ₀ = −V G₂ ₀ = yields I₁

We consider the case shown in Fig. 13 where an acoustic wave is incident at Port 1, Port 2 is acoustically terminated, and Port 3 is electrically terminated in ZL

( )

f =¹Y fL

( )

. The transfer function with input and output voltage at port 3 can be obtained by simple electrical network analysis.

2.3 Electromechanical Interaction

The purpose of this section is to derive the electromechanical interaction due to a surface

consider the two-dimensional electroelastic problem concerned with the excitation of harmonic waves in a piezoelectric half-space, as shown in Fig. 14. The electrodes attached to the surface of the piezoelectric are assumed to be weightless, perfectly flexible and conducting, and have length W, which is assumed to be much larger than width

(

^λ ⁴

)

^with

an important consequence that the electroelastic fields in the material are independent of the x -coordinate.2

The electroelastic fields in a piezoelectric are described by the following equation.

The equations of move [24]:

The equation of electrostatics:

The equations of state:

11 C u11 1,1^E C u13 3,3^E e31 ,3, dielectric displacement, the potential, and the mass density, respectively.

Accordingly, the problem reduces to the following system of (two-dimensional electroelastic) equations for the amplitude coefficients of the potential, φ, and displacement components, x and ₁ x ₃

( ) ⁽ ⁾

11 1,11^E 44 1,33^E 13^E 44^E 3,13 31 15 ,13 1, C u +C u + C +C u − e +e φ = && ρu

(

^C⁴⁴^E ⁺^{C u}¹³^E

)

^1,13⁺^{C u}^{44 3,11}^E ⁺^{C u}^{33 3,33}^E ⁻^e^{15 ,11}^φ ⁻^e^{33 ,33}^φ = && (2.50) ^ρ^u³^,

(

e15+e u31

)

1,13+e u15 3,11+e u33 3,33−ε φ11 ,11^S −ε φ33 ,33^S =0.

Here and afterwards the harmonic time dependence e^{j t}^ω is suppressed.

Assume that the displacement and potential vectors are

( )

If the second derivative of the disturbance ^f^"

( )

ξ exists, then we have an eigenvalue problem as follow:

( ) ⁽ ⁾

There exists a non-trivial solution if the determinant of Eq. (2.54a) vanishes, i.e.,

( )

complex. For definiteness of single-value and bounded condition at infinity we assumed,

( )

Im η_k ≥0. (2.56)

The corresponding four eigenvectors satisfy the proportional relation,

( ) ( ) ( )

where the superscript (k) denotes the k-th eigenvalue or eigenvector. The proportional factors

p are defined as ik

Hence, the displacement field may be assumed to be

( 1 1- )

The unknown coefficients C must be determined by boundary conditions. It is a set of _i mechanical and electrical boundary conditions. The surface traction acting on an arbitrary horizontal plane is given by

( ) ( )

2_k 13^E 1_k 1 33^E 2_{k k} 33 3_{k k}, 1, 2, 3.

q =iC p s +iC p η −ie p η k = (2.64)

The C C C satisfy the proportional relations, ₁, ₂, ₃

1 2

12 23 13 22 13 21 11 23 11 22 12 21

C Q.

C C

q q q q =q q q q = q q q q =

− − − (2.65)

Substitute these relations to Eq. (2.59) and the general form of displacement field is obtained.

Then, u and ₃ φ₀ have a general relation (proportional constant) with the first line of Eq.

(2.59) divide by second.

Chapter 3 Design and Simulation of SAW Gyroscope

In this chapter, we present a preliminary design of SAW gyroscope using the COM theory, crossed-field model, and electromechanical interaction. The simulation work is done with MATLAB software package and the simulation results are listed together with the experimental data reported in the work of V. K. Varadan et al. [9] for comparison. For the design simplicity, we neglect the loss and attenuation of SAW propagation due to piezoelectric materials and metallic dots, bulk-wave conversion loss, and reflection and electrical interaction between metallic dots array, and etc.

3.1 Surface Acoustic Wave Gyroscope

When a standing wave is generated on a piezoelectric material surface the anti-node particles would vibrate in the vertical direction ( vv in ± Z-direction). When substrate is subjected to the angular rotation (Ωv

in X-direction) perpendicular to the reference motion

( )

^vv , the Coriolis force ( Fv =2mΩ×v vv in ± Y-direction) is produced in the direction perpendicular to the both vectors as shown in Fig. 15. Due to the wave motion along X-direction, the Coriolis force acting on the surface particles along Y-direction would be distributed in the way of a wave motion.

Base on the above-mentioned, the concept of utilizing SAW for the detection of rotation is described below and illustrated in Fig. 16. It consists of IDTs, reflectors, and a metallic dot array within the cavity, which are fabricated through micro fabrication techniques on the surface of a piezoelectric substrate. The metallic dots of mass m serve as proof mass for this gyroscope. The resonator IDTs create SAW that propagates back and forth between the reflectors and forms a standing wave patternwithin the cavity due to the collective reflection from reflectors. SAW reflection from individual metal strips adds in phase if the reflector periodicity is equal to half a wavelength. For the established standing wave pattern in the cavity as explained in Fig. 16, a typical substrate particle at the nodes of standing wave has no amplitude of deformation in the Z-direction. However, at or near the anti-nodes of standing wave pattern, such particles experience larger amplitude of vibration in the Z-direction, which

serves as the reference vibrating motion for this gyroscope. Metallic dots of mass (m), which serve as the proof mass, are placed in the resonator region where standing waves are formed.

To amplify the magnitude of the generated Coriolis force in phase, the metallic dots are positioned strategically at the anti-node locations. The rotation ( Ωv

in X-direction)) perpendicular to the velocity ( vv in ± Z-direction) of the oscillating masses (m) produces Coriolis force (Fv =2mΩ×v vv in ± Y-direction) in the direction perpendicular to the both vectors as shown in Fig.16. Since this Coriolis force is applied on a piezoelectric substrate, it generates a secondary SAW in the Y-direction with same frequency as the reference oscillation. The metallic dot array is placed along the Y-direction such that the SAW due to the Coriolis forces adds up coherently. The generated SAW is received by the sensing IDTs placed in the Y -direction [10].

The design of SAW gyroscope is subdivided to two parts, SAW resonator and sensor, as shown in Fig. 17. The former is to generate reference vibrating motion for proof mass; the latter can detect the Coriolis force with voltage output due to rotation from proof mass. It is important to know the characteristics of impedance, admittance, bandwidth and sensitivity near the operating frequency of the SAW gyroscope, because the sensing IDTs have to be designed such that they efficiently pick the SAW wave generated due to Coriolis force. The numerical simulation of the SAW resonator is done by coupling-of-modes (COM) theory because a SAW devices can be easily represented by several basic elements as discussed in Chap. 2. The SAW sensor is modeled using crossed-field model instead of COM theory due to the input to “SAW sensor” is the Coriolis force, not a voltage signal. Furthermore, to estimate the Coriolis force, we need to know the particle vibrating velocity in advance.

Therefore, the “electromechanical interaction” method discussed in Chap. 2 is utilized to convert the transmitted power into exact particle displacement, so as to obtain the velocity term in the Coriolis force.

3.2 PreliminaryDesign of SAW Gyroscope

3.2.1 Operating Frequency

The periodicity of IDTs and reflectors, and the separation between the reflector gratings determine the operation frequency of the device. This device can be operating at higher

frequency with smaller device size or lower frequency with a larger device size. Generally, the frequency range of SAW filters is about 10M ~ 3G Hz. The SAW gyroscope which is similar to SAW filters might also be working at the range of GHz. However, for the comparison purpose, we design the operation frequency of our SAW gyroscope in accordance with data shown in [9] and [10]. In that regard, the SAW gyroscope has the minimum feature of about 6 µm and 75 MHz for the operation frequency.

3.2.2 Substrate

In view of the working principle discussed above, any piezoelectric material such as lithium niobate, lithium tantalite, or quartz can be used as a substrate. For efficient generation and detection of SAWs through IDTs, 128^o YX LiNbO₃ is chosen as a piezoelectric substrate due to its rather high electromechanical coupling. The material properties of

128 YX LiNbO3 are shown in Appendix. A few things to be noted that: Firstly, this electromechanical coupling coefficient ( K ) in X-direction is different from that in ² Y-direction, which is determined by formula [25]. Secondly, the wave velocity is different in the X- and Y-directions due to the anisotropy of 128^oYX LiNbO₃, which measured as 3961m s and 3656m s, respectively. The errors between published data (3992m s, as shown in Appendix) and the experimental results are mainly due to the effect of metallization and the metallic dot array [9], which will be discussed later.

3.2.3 Interdigital Transducer

The design of Interdigital transducer (IDT) of SAW gyroscope is similar to IDT existed in other SAW devices, and they are in charge of exciting and sensing the propagation wave.

The IDTs are electrodes patterned on the piezoelectric substrate and a periodic strain field is generated in the piezoelectric crystal that produces propagating surface acoustic wave, when an alternating voltage is applied. This propagation wave gives rise to a standing wave when the propagating waves are launched in both directions away from the transducer. The transducer electrodes may be either gold or aluminum.

The design of interdigital transducers of both of resonator and sensor is shown in Fig. 18, which utilized aluminum as the material of electrodes. According to [9], to obtain good

overlap or aperture of IDTs should be minimized, but it needs to be large enough to avoid

在文檔中表面聲波陀螺儀的設計 (頁 12-0)

Chapter 1 Introduction

1.4 Literature Survey

Chapter 2

Interdigital-Electrode Transducers for Surface Waves

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Chapter 3

Design and Simulation of SAW Gyroscope

( )

( ) ( ) [ ] ^{( )} _{( )}

( ) ( ) [ ] ^{( )} _{( )}

^{( )} _{( )}

_{( )} ^{( )}

( ) ⁽ ⁾

( ) ⁽ ⁾