Chapter 2 Modeling and Simulation of SAW Devices Considering shear wave case, u0 is perpendicular to k gives shear waves and wave vector denoted by kt given by |kt|2 = ω2ρ/µ and the phase velocity of shear waves is denoted by Vt, equal to ω/|kt|, and then transvers wave
𝜕𝜕𝑡𝑡=�𝜇𝜇 𝜌𝜌� (2.27)
For longitudinal wave, u0 is parallel to k then k = ±u0 (|k|/|u0|), using this relation we find (k.u0)k = u0|k|2, and substituting in eq (2.26) gives |kl|2 = ω2ρ/(λ+2µ), and then longitudinal phase velocity is given by
𝜕𝜕𝑖𝑖 =�(𝜆𝜆+2𝜇𝜇)
�𝜌𝜌 (2.28)
Chapter 2 Modeling and Simulation of SAW Devices where, V is the voltage across the IDT electrodes, k0=2π/λ is transducer synchronous wave number, and
2 2 2 2
2 2
2 2
2 2
2 6
9 9
3 3
3 3
2 3 2
3
( ) ( )
T
T
T B
T B
R f f f
E
f f f f
j
R f f
j S
f f f j
R i j
f f f j
S i j
f f
R C j R
v R C R C
e j C R
e j C R
j R e
e j C R
j R e
e j C R
φ
φ
φ φ
φ φ
α ω λ
ω α λ
κ γ
α α
ω α α
ω
α λ
κ κ
ω α λ
κ κ
ω
ω ω
+
−
− +
+
−
= + − + − + +
= +
= +
= + +
+
= + +
+
(2.30)
In the above equations φT = π and φB = π/2 are phase offsets of the grating and the potential respectively, and ω =2πf is the angular frequency. The COM parameters are as follows, v phase velocity of SAW, α transduction coefficient, Rf thin film resistance in one transduction unit, Cf interdigital capacitance in one transduction period, κ the reflection
parameter, γ propagation loss per unit length. The suffix R and S represent the values for the corresponding direction of wave propagation shown in Fig. 2.2. These parameters can be computed through computer simulation such as FEM or using experimental analysis of a test structure. The COM equations can be represented in P matrix form. The acoustic ports are treated as scattering port and electric port as admittance port and
11 12 13
21 22 23
31 32 33
(0) (0)
( ) ( )
S P P P R
R L P P P S L
I P P P V
=
(2.31)
where,
Fig. 2.2. Coordinates of an IDT [3].
λ
~
R(L,ω)
S(L,ω) S(0,ω)
R(0,ω)
L V
I(x, ω)
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Chapter 2 Modeling and Simulation of SAW Devices
11
0 12
13 31
0 22
2
sin( ) cos( ) sin( ) cos( ) sin( )
cos( /2) ( )sin( /2)
sin( /2)
/2 cos( ) sin( )
sin( ) cos( ) sin( )
S
S
jK L
s S R
j K L R
jK DL
P D DL j DL
P D e
D DL j DL
D DL j K DL
P P jL DL
DL D DL j DL
jK DL
P e
D DL j DL
α α α
−
−
= +
+ ∆
= + ∆
+ + ∆
= + ×
+ ∆
= +
+ ∆
α α α
α α α α
α α α α
+ + ∆
= + ×
+ ∆
+ + ∆ + ∆ −
= + × − + ∆
+ + ∆ −
− × +
23 32
33
2 2
3
2 2
3
cos( /2) ( )sin( /2)
sin( /2)
/2 cos( ) sin( )
2 sin( ) (1 cos( ))
2 cos( ) sin( )
2 1 cos( )
2 cos( )
R R S R
S R R S R S
S R R S R S
D DL j K DL
P P jL DL
DL D DL j DL
K K D DL j DL
P j DL
D DL j DL
D
K K DL
D DL j
D
ω λ
ω
×
∆ +
∆ = −
= ∆ −
0 2
3 /
sin( ) 3
f f f
E
R S
j C L
DL j R C
K K
D K K
P11 and P22 are the reflection coefficients, and P12 and P21 are the transmission coefficients.
The remaining terms P13 and P23 correspond to the excitation coefficients of the IDT and the term P33 clearly represents the admittance of the structure relating the current flowing in the electrode (i) and the drive voltage (V). P31 and P32 terms represent the current generated by the waves arriving at the acoustic ports. Admittance or P33 can be calculated from simulations and it is the most useful parameter in the device design.
2.2.2 Discrete source or delta function method
The discrete source method or delta function method is one of the earliest and simplest methods which explain the shape of the IDT frequency response [39]. It does not consider energy, capacitance, and electromechanical coupling coefficient of the material used in the device. The discrete source model is derived from the examination of spatial distribution of charge density or electric field at the surface of piezoelectric substrate. This method associates either two charge spikes of the same sign to each IDT finger or two electric field spikes of the same sign to each interval between adjacent IDT fingers [39]. The amplitude of each delta function is proportional to the corresponding aperture W of the IDT and sign depends on the polarity of the charge density or electric field.
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Chapter 2 Modeling and Simulation of SAW Devices 2.2.3 Equivalent circuit model
The mason equivalent circuit model for surface wave transducers is well explained by smith et al. [38]. An IDT with N number of pairs can be modeled using a periodic section of IDT can be represented by an equivalent electromechanical circuit. The equivalent circuit model of one period of IDT is shown in Fig. 2.3. It consists of three ports, viz. two acoustic ports and one electric port. The electrical equivalent of the two acoustic ports is represented as SAW transmission line and third port is represented as electric port where the electric potential is applied and sensed. In this model, θ is expressed as 2πf/f0, where f is the frequency of the applied input potential and f0 is the resonance frequency and is the periodic section transit angle. R0 is the electrical equivalent to Z0 or mechanical impedance given by
𝑅𝑅0= 𝑍𝑍0
𝜙𝜙2 (2.32)
where, ϕ =ςCs/2 is the turns ratio of an acoustic-to-electric circuit transformer, Cs is the static electrode capacitance of one periodic section, and ς is a piezoelectric constant.
In this model, the acoustic forces are converted into electric potentials as En = Fn/θ and SAW velocities are converted into equivalent electric currents In = vn’ θ. These transformations allow the mechanical characteristic admittance (similar to the transmission line characteristic impedance expressed in ohms) to be expressed as an equivalent transmission line characteristic admittance as,
θ = Periodic section transit angle R0 = Mechanical impedence
Cs = Electrode capacitance per electrode j = Imaginary unit
En = Equivalent electric potential In= Equivalent electric current.
j R0 tan(θ/4)
j R0 csc(θ/2)
–Cs/2
Cs/2
–Cs/2
Cs/2
⋅ ⋅
⋅
⋅
⋅⋅
⋅⋅
⋅
⋅
⋅ ⋅
⋅ ⋅
In-1
En
In
En-1
I3n E3n
°
°
°
° Port 2
(Acoustic)
Port 3 (Electric) Port 1
(Acoustic)
One periodic section
Fig. 2.3. Mason equivalent circuit of one period of IDT analyzed by Smith et al. [6]
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Chapter 2 Modeling and Simulation of SAW Devices
G0 = 1/R0 = (ωCsK2)/2π (2.33) where, K2 represents electromechanical coupling coefficient and its values can be approximated by −2Δv/v, where, Δv is the change in SAW velocity when the piezoelectric surface is electrically shorted by a thin metal film [41]. The entire IDT can be realized by connecting acoustic ports in series and electrical ports in parallel as shown in Fig. 2.4.
If the generator IDT and receiver IDT of the SAW device are completely matched, the Y parameters of the 3-port network using the equivalent circuit of a period of an IDT can be expressed as
( ) ( )
( ) ( ) ( )
θ θ θ
θ θ θ
θ θ ω θ
− −
= −
− +
cot csc tan
csc cot tan
tan tan T tan
I jG N jG N jG E
I jG N jG N jG E
I jG jG j C jNG E
1 0 0 0 1
2 0 0 0 2
3 0 0 0 3
4 4
4 4 4 4
where, CT = NCs (total capacitance of IDT) [24].
The overall equivalent circuit of an IDT is shown in Fig. 2.5 [38], [39]. The input admittance can be expressed as
Y (f) = Ga (f) +Ba(f)+jωCT (2.34)
°
°
°
°
1 2 N-1 N
° °
Port 1
Port 3
Port 2
Fig. 2.4. Equivalent circuit for entire IDT, made up with N pairs of IDT finger with acoustic ports in cascade and electrical ports in parallel.
CT Ba(f) Ga(f)
°
°
Fig. 2.5. Overall equivalent circuit of an IDT.
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TH-1583_09610201
Chapter 2 Modeling and Simulation of SAW Devices where, Ga(f) is the radiation conductance and Ba(f) is the susceptance given as
𝐺𝐺𝑎𝑎(𝑓𝑓)≈8𝑁𝑁2𝐺𝐺0�𝑆𝑆𝑖𝑖𝑆𝑆 𝑋𝑋𝑋𝑋 �2 (2.35)
𝐵𝐵𝑎𝑎(𝑓𝑓)≈8𝑁𝑁2𝐺𝐺0�𝑆𝑆𝑖𝑖𝑆𝑆 (2𝑋𝑋)−2𝑋𝑋
2𝑋𝑋2 �2 (2.36)
where, X = Nπ(f−f0)/f0. The radiation susceptance is a reactive parameter that goes to zero at the resonance frequency. Conductance at resonant frequency f0 is given by
𝐺𝐺𝑎𝑎(𝑓𝑓0)≈8𝑁𝑁𝐾𝐾2𝐶𝐶𝑡𝑡𝑓𝑓0 (2.37)
Where N is number of transducer pairs in the IDT structure and f0 is the operating frequency. In the above described models for designing the IDT of a SAW device, the second order effects such as electrode resistance, electrode discontinuities, and propagation losses are neglected.
2.2.4 Device design parameters
In design of SAW devices, certain parameters are very crucial in the performance of device and also the design IDT to match with source impedance (50 Ω). As defined in the above equivalent circuit model, the electromechanical coupling coefficient (K2) which depends on the material for transduction of SAW, frequency of operation is determined by the wavelength of IDT structure and acoustic velocity of the substrate, using these two parameters the admittance of IDT is designed for 50 Ω at operating frequency.