Fast, Accurate and Full Extraction of Coupling-of-Modes Parameters by Finite Element Method

Abstract

This paper presents a new numerical approach for the full extraction of the coupling-of-modes (COM) parameters by stationary and eigenfrequency analyses in the finite element method (FEM). This is a fast method extracting from the results of static analysis and eigenfrequency analysis. It avoids the long calculation time of admittance frequency response analysis, which is commonly used in extracting COM parameters. In addition to the usual COM parameters (velocity, reflection coefficient, transduction coefficient and capacitance), the phases of reflection and transduction coefficient can be also extracted with this method. The proposed method was applied to different cutting types LiNbO₃ with different types of thicknesses in a varying interdigital transducer (IDT). These examples show that our approach has great potential in extracting all the COM parameters of the Rayleigh SAW for all kinds of IDT structures. Therefore, it is a fast, accurate, general and full extraction approach of COM parameters.

1. Introduction

The design of high-performance SAW devices requires accurate and efficient simulation techniques. The complexity of modeling IDT or functional gradient materials [1,2] make it difficult to obtain a rigorous analytical solution. FEM is an effective method in designing SAW devices because it can directly describe the coupled partial differential equations involving the electric and mechanical fields in a grating consisting of a piezoelectric substrate with electrodes [3]. It can handle arbitrary materials and crystal cuts, different electrode shapes and structures including multiple metal and dielectric layers [4,5]. Compared with phenomenological models, superior performances of FEM on accuracy have been validated when considering a necessary discretization of at least 10 elements per wavelength. That is to say, the accuracy depends on the number of grids. Consequently, it is time-consuming to simulate the whole device, since a huge number of grids are necessary to improve the precision, but which require intensive computation [6,7]. Therefore, it cannot be directly used to optimize the device performance.

Compared with other phenomenological models, such as the equivalent circuit model [8], the COM model is a more efficient model in simulating and designing SAW devices. In addition to the acoustoelectric interactions of the electric current with the surface acoustic waves, the COM model considers the excitation, propagation and mutual reflection of counter-propagating waves in a continuous medium [9]. Since it is a phenomenological model, the heavy dependence of the COM model on the accurate COM parameters is evident. Many researchers have made much effort to acquire these parameters experimentally and theoretically. Hartmann et al. extracted the needed COM parameters from measurements [10,11]. However, this process is both time-consuming and expensive because of the many degrees of SAW structures, such as the relative electrode thickness, the metallization ratio and the electrode profile. Some numerical methods, such as a combination of FEM and boundary element methods [12,13], a full FEM [14] and finite element method–spectral domain analysis [15,16], have also been employed for COM parameters extraction.

Regarding to the reflection coefficient and transduction coefficient, most of researchers are got only in their absolute value. To obtain the best unidirectivity for SPUDT, one must know the accurate COM parameters, including the phases of reflection coefficient and transduction coefficient, to find out if the length between the transduction and reflection centers is 1/8 wavelength or not [17]. Hashimoto et al. used the input admittance to find the two resonances and two antiresonances to extract all the absolute values of the COM parameters except capacitance C and the phase difference between the phase of reflection coefficient and the double phase of the transduction coefficient [15,18,19]. They obtained capacitance C by the fitting methods, with the results of the input admittance calculated using the MSYNC software. However, the sign of the phase difference could not be determined. Yong’an Shui et al. used the periodic Green’s function method to extract all the COM parameters, including the phase of reflection coefficient, but not the phase of transduction coefficient [20].

In this paper, we will demonstrate an approach to extract all the COM parameters. This approach makes full use of the results of static analysis and eigenfrequency analysis obtained by the FEM. It avoids the long calculation time in the frequency response analysis. Subsequently, we will demonstrate the results of several examples of its application. These results show that this approach is powerful to fully extract the COM parameters of the Rayleigh wave for all kinds of IDT structures on arbitrary materials.

2. Theory to Extract COM Parameters

2.1. COM Model

It is assumed that there are two counter-propagating waves which exist in a grating, where one of them propagates toward the +x direction, whereas the other propagates toward the −x direction. The two waves are represented as R(x) and S(x), respectively. There are three common types of COM differential equations. These differential equations vary depending on whether the direction of the current flow is the same or opposite to the coordinate system and the sign of reflection coefficient κ. As shown in Figure 1, R(x) and S(x) represent the two counter-propagating waves, and I(x) is the current caused by the induced charges in the electrodes when the transducer is driven by a voltage V, whose direction is opposite to x. The drive voltage V and current I are defined by their half-power amplitude values.

Introducing the slow varying fields R₀(x), S₀(x) of R(x) and S(x):

$$\{\begin{array}{c}R\left(x\right)=R_0\left(x\right)\exp \left(-i{k}_{0}x\right)\\ S\left(x\right)=S_0\left(x\right)\exp \left(i{k}_{0}x\right)\end{array},$$

(1)

where k₀ = 2π/λ₀. Moreover, defining a detuning parameter Δ as:

$$\Delta =k-k_0=\frac{\omega }{v}-\frac{2\pi }{{\lambda }_0},$$

(2)

where ω is the angular frequency, ν is the velocity and λ₀ is the electric period of the gratings. Then, the final COM equations can be written as [9,21]:

$$\{\begin{array}{c}\frac{\partial R_0}{\partial x}=-i\Delta R_0\left(x\right)+i\kappa S_0\left(x\right)+i\alpha V\\ \frac{\partial S_0}{\partial x}=-i{\kappa }^{\ast }{R}_0\left(x\right)+i\Delta S_0\left(x\right)-i{\alpha }^{\ast }V\\ \frac{\partial I}{\partial x}=-2i{\alpha }^{\ast }{R}_0\left(x\right)-2i\alpha S_0\left(x\right)+i\omega CV\end{array},$$

(3)

where velocity ν, the reflection coefficient κ, the transduction coefficient α and the static capacitance per unit length C are the four independent COM parameters to be determined. Generally, κ and α are complex numbers.

If we define the reflection coefficient κ = −κ, the resulting COM equations are the second form, which has been widely used in Japan [18,22,23]. If we define I(x) = −I(x) and introduce the thin film resistance of an interdigital transducer, the third form COM equations would be derived. This form was also used in many papers [24,25,26].

No matter which kinds of COM model is used, all the four characteristic parameters need to be extracted. In the next section, we developed a parameters extraction technique for all four COM parameters mentioned above based on the FEM.

2.2. Extraction Technique of COM Parameters

The static capacitance is a measure of the electrostatic energy stored in an IDT and it can be calculated from the charge distribution and electric field distribution excited by the transducer subject to a constant voltage. When the applied voltage is 1V, the static capacitance of one period C_p is the integral of charge distribution on the electrode.

The other parameters can be extracted from the stopband characteristics of the short-circuited and open-circuited grating. For a short-circuited grating, the voltage will be set at 0V, and the COM equations will be reduced to the form [27] given in Equation (4).

$$\{\begin{array}{c}\frac{\partial R_0}{\partial x}=-i\Delta R_0\left(x\right)+i\kappa S_0\left(x\right)\\ \frac{\partial S_0}{\partial x}=-i{\kappa }^{\ast }{R}_0\left(x\right)+i\Delta S_0\left(x\right)\end{array},$$

(4)

The solution is of the form in Equation (5)

$$\{\begin{array}{l}R\left(x\right)=c_1\exp \left[-i\left(k_0-q_{sc}\right)x\right]+c_2\exp \left[-i\left(k_0+q_{sc}\right)x\right]\hfill \\ S\left(x\right)=\frac{1}{\kappa }\left\{c_1\left(\Delta +q_{sc}\right)\exp \left[i\left(k_0+q_{sc}\right)x\right]+c_2\left(\Delta -q_{sc}\right)\exp \left[i\left(k_0-q_{sc}\right)x\right]\right\}\hfill \end{array},$$

(5)

and satisfies the dispersion relation [18] in Equation (6):

$$q_{sc}=\{\begin{array}{c}-\sqrt{{\Delta }^2-{\left|\kappa \right|}^2}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left(\Delta <-\left|\kappa \right|\right)\\ -i\sqrt{{\left|\kappa \right|}^2-{\Delta }^2}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left(\Delta <\left|\kappa \right|\right)\\ -\sqrt{{\Delta }^2-{\left|\kappa \right|}^2}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left(\Delta >\left|\kappa \right|\right)\end{array},$$

(6)

where c₁ and c₂ are arbitrary coefficients of the two eigenmodes which are determined by the initial and boundary conditions. Each of them may be interpreted as combinations of an incident wave and a reflected wave. It can be seen from Equations (5) and (6) that the eigenmodes form a stopband in the frequency region, |Δ| < |κ|. Here, we use the phase φ_κ and amplitude |κ| to describe the reflection coefficient as κ = |κ|exp(iφ_κ). The amplitude of κ determines the width of the stopband. The lower and upper frequencies of the stopband f_s+ and f_s− can be obtained by considering the zeros of the dispersion relation [28,29]. Then, we have:

$$\{\begin{array}{l}{f}_{s+}=\left(1+\frac{\left|\kappa \right|{\lambda }_0}{2\pi }\right)f_s\\ f_{s-}=\left(1-\frac{\left|\kappa \right|{\lambda }_0}{2\pi }\right)f_s\end{array},$$

(7)

From Equation (7), the center frequency of the short-circuited grating f_s and |κ| can be represented as Equations (8) and (9).

$$f_s=\frac{f_{s+}+f_{s-}}{2},$$

(8)

$$\left|\kappa \right|=\frac{2\pi }{{\lambda }_0}\frac{f_{s+}-f_{s-}}{f_{s+}+f_{s-}},$$

(9)

The Rayleigh mode effective velocity ν in a short-circuited grating can be calculated using the following equation:

$$v=f_s{\lambda }_0,$$

(10)

The field in the grating is a combination of the two waves R(x) and S(x). At the upper frequency of the stopband [20],

$$R\left(x\right)+S\left(x\right)=2\left(c_1+c_2\right)\cos \left(k_{0}x-\frac{{\phi }_k}{2}\right)\exp \left(-i\frac{{\phi }_k}{2}\right),$$

(11)

at the lower frequency,

$$R\left(x\right)+S\left(x\right)=2\left(c_1+c_2\right)\sin \left(k_{0}x-\frac{{\phi }_k}{2}\right)\exp \left[-i\left(\frac{{\phi }_k}{2}+\frac{\pi }{2}\right)\right],$$

(12)

After finding the positions of the nodes of the standing wave during the frequencies at the stopband edges, one can get the phase of κ. In an open-circuited grating, the current density $\partial I/\partial x$ is equal to zero, this yields the following equation

$$V=\frac{2{\alpha }^{\ast }}{\omega C}R\left(x\right)+\frac{2\alpha }{\omega C}S\left(x\right),$$

(13)

Substituting Equation (13) into the remaining COM equations, we have

$$\{\begin{array}{c}\frac{\partial R_0}{\partial x}=-i\left(\Delta -\frac{2{\left|\alpha \right|}^2}{\omega C}\right)R_0\left(x\right)+i\left(\kappa +\frac{2{\alpha }^2}{\omega C}\right)S_0\left(x\right)\\ \frac{\partial S_0}{\partial x}=-i\left({\kappa }^{\ast }+\frac{2{\left({\alpha }^{*}\right)}^2}{\omega C}\right)R_0\left(x\right)+i\left(\Delta -\frac{2{\left|\alpha \right|}^2}{\omega C}\right)S_0\left(x\right)\end{array},$$

(14)

The general solutions of Equation (14) are the same as Equation (4), but with the corresponding equivalent detuning parameter ${\Delta }_{oc}=\Delta -2{\left|\alpha \right|}^2/\left(\omega C\right)$ and the equivalent reflection coefficient ${\kappa }_{oc}=\kappa +2{\alpha }^2/\left(\omega C\right)$. The dispersion relation should be modified by Δ_oc and κ_oc. We also use the phase φ_α and amplitude |α| to describe the transduction coefficient as α = |α|exp(iφ_α). The upper and lower frequencies of the stopband f_o+ and f_o− in an open-circuited grating can be expressed considering Equation (7) as:

$$\{\begin{array}{l}{f}_{o+}=\left(1+\frac{{\lambda }_0}{2\pi }\frac{2{\left|\alpha \right|}^2}{\omega C}+\frac{{\lambda }_0}{2\pi }\left|\kappa +\frac{2{\alpha }^2}{\omega C}\right|\right)f_s\\ f_{o-}=\left(1+\frac{{\lambda }_0}{2\pi }\frac{2{\left|\alpha \right|}^2}{\omega C}-\frac{{\lambda }_0}{2\pi }\left|\kappa +\frac{2{\alpha }^2}{\omega C}\right|\right)f_s\end{array},$$

(15)

The amplitude of a regeneration reflection term $2{\left|\alpha \right|}^2/\left(\omega C\right)$ can be obtained by Equation (16)

$$\frac{2{\left|\alpha \right|}^2}{\omega C}=\frac{2\pi }{{\lambda }_0}\left(\frac{f_{o+}+f_{o-}}{f_{s+}+f_{s-}}-1\right),$$

(16)

Moreover, we will get the amplitude of the reflection coefficient in an open-circuited grating:

$$\left|\kappa +\frac{2{\alpha }^2}{\omega C}\right|=\frac{2\pi }{{\lambda }_0}\frac{f_{o+}-f_{o-}}{f_{s+}+f_{s-}},$$

(17)

According to the method mentioned above, we can obtain the phase of the reflection coefficient in an open-circuited grating φ_κo. Making full use of Equations (9), (16) and (17), and the phase of φ_κ and φ_κo, we can get

$$\{\begin{array}{l}\cos \left(2{\phi }_{\alpha }\right)=\frac{\left|\kappa +\frac{2{\alpha }^2}{\omega C}\right|\cos \left({\phi }_{\kappa o}\right)-\left|\kappa \right|\cos \left({\phi }_{\kappa }\right)}{\frac{2{\left|\alpha \right|}^2}{\omega C}}\\ \sin \left(2{\phi }_{\alpha }\right)=\frac{\left|\kappa +\frac{2{\alpha }^2}{\omega C}\right|\sin \left({\phi }_{\kappa o}\right)-\left|\kappa \right|\sin \left({\phi }_{\kappa }\right)}{\frac{2{\left|\alpha \right|}^2}{\omega C}}\end{array},$$

(18)

So far, we have solved the four COM parameters including the phases of κ and α. We can get another relation from the edge frequencies of the stopband in a short-circuited grating and an open-circuited grating

$$\cos \left({\phi }_{\kappa }-2{\phi }_{\alpha }\right)=\frac{{\left(f_{o+}-f_{o-}\right)}^2-{\left(f_{s+}-f_{s-}\right)}^2-{\left(f_{o+}+f_{o-}-f_{s+}-f_{s-}\right)}^2}{2\left(f_{s+}-f_{s-}\right)\left(f_{o+}+f_{o-}-f_{s+}-f_{s-}\right)},$$

(19)

The Equation (19) is similar to (7.154) of reference [18], and it can be used to verify the phase difference between the phase φ_κ and the double phase φ_α.

2.3. FEM Model Description and Simulation

The wave of the Rayleigh-type SAW propagates on the surface of the substrate with its amplitude exponentially decaying. This feature enables us to build a model with only a few wavelength depths of the substrate. The waves, excited by infinite numbers of the IDT fingers with the same grating shape on a piezoelectric substrate, are spatially periodic. Therefore, simulating a periodic structure of the device might suffice. With the use of periodic boundary conditions (PBC) in x- and y-directions, this model is reduced to one period λ₀ in the x-direction and λ₀/10 in the y-direction. Static analysis and eigenfrequency analysis are conducted in the FEM to extract the COM parameters. Here, we build the FEM model of a standard EWC SPUDT cell on a substrate to demonstrate the approach to extract the COM parameters.

As shown in Figure 2, a three-dimensional model with a period of λ₀ = 50 μm, in which the IDT structure is a standard EWC SPUDT, is built. This standard EWC SPUDT has three fingers, with two λ₀/8 narrow fingers and one λ₀/4 wide finger, and the gaps between the fingers are λ₀/8 and 3λ₀/16. The first finger is active. In our model, we set the origin at the center of the period structure. A perfectly matched layer (PML) is applied to suppress the spurious resonances caused by the bottom reflection [30]. The thickness of electrodes is denoted by h and the thickness of the substrate and the PML are set as 5λ₀ and λ₀, respectively. Two electrodes on the right in the model are grounded, while the terminal one is 1V voltage or 0C charge, according to the request of the model analysis.

In Equation (3), the COM parameters can be interpreted as parameters for a unit length. Actually, the parameters for a unit period of length λ₀ are more practical. Instead, it is convenient to define the normalized parameters listed in

Table 1. Normalized COM parameters.

Parameters	Symbol	Dimension (SI)
Velocity in short-circuited grating	ν	m/s
Reflection coefficient	κp = κλ0	%
Transduction coefficient	αp = αλ0	${\Omega }^{-\frac{1}{2}}$
Normalized transduction coefficient ${\alpha }_n$	${\alpha }_p/\sqrt{\omega C_p}$	%
Capacitance	Cp = Cλ0	F
Normalized capacitance	Cn= Cp/W	F/m

.

In static analysis, a voltage 1V is applied on the terminal electrode. After the model is solved, we can get the surface charge density σ_s(x₁) and the tangential component of the electric field on the substrate surface E₁(x₁) and the surface potential Φ(x₁), as shown in Figure 3.

The capacitance can be calculated by summing the electrostatic charges on the electrodes connected to the terminal of voltage.

The other COM parameters can be obtained from the results of the eigenfrequency analysis. In this analysis, the edge frequencies of the stopband f_s+, f_s−, f_o+, f_o− can be obtained. The two frequencies (f_s+ and f_s−) correspond to two eigenfrequencies (symmetric frequency and antisymmetric frequency) of the eigenfrequency analysis by setting the terminal voltage to 0V. Here, we also donate the lower of the two eigenfrequencies as f_s− and the higher one as f_s+, and they are 77.646 MHz and 78.079 MHz for 128° YX LiNbO₃, respectively. The displacement distributions at the lower and upper stopband edges also can be obtained from the postprocessing of the eigenfrequency analysis in the FEM, and the normalized displacements are shown in Figure 4.

It can be seen that there is a π/2 phase difference between the two displacements distribution. One can get the phase φ_κ after finding the position of the displacement distribution at the lower edge frequency. Then, the COM parameters of ν, |κ| and φ_κ can be derived using Equations (8)–(12).

Table 2. COM parameters for EWC SPUDT on 128°YX LiNbO₃ for h = 0.

Parameters	Values
Normalized capacitance (pF/m)	546.33
Velocity in short-circuited grating (m/s)	3892.7
Reflection coefficient |κp| (%)	1.73
Phase of reflection coefficient	−157.56°
Normalized transduction coefficient ${\alpha }_n$ (%)	14.68
Phase of transduction coefficient	−37.84°

gives the calculated COM parameters for EWC SPUDT on 128°YX LiNbO₃ with h = 0.

We have extracted the COM parameters [31] for EWC SPUDT on YZ-LiNbO₃, bidirectional IDTs on 128°YX LiNbO₃ and Y-Z LiNbO₃. Moreover, these results are compared with those of well-known researchers in the field of surface acoustic waves (SAW), and that shows this approach is powerful to fully extract the COM parameters of the Rayleigh-type SAW for all kinds of IDT structures on arbitrary materials [20,32,33].

We modify the directivity D given in reference [18], which can be simplified to Equation (20):

$$D=\left|\frac{1+i\exp \left(i\phi \right)\tanh \left(\frac{N\left|{\kappa }_p\right|}{2}\right)}{1+i\exp \left(-i\phi \right)\tanh \left(\frac{N\left|{\kappa }_p\right|}{2}\right)}\right|,$$

(20)

where φ = φ_κ − 2φ_α, we call it unidirectional angle, and N is the number of electrodes.

In Equation (20), the defined directivity is the ratio of the excitation efficiency of the IDT in positive x to the excitation efficiency in negative x. This means that the wave excitation in direction of positive x is favored over the negative x if D > 1, and the wave excitation in positive x is weaker than the negative x if D < 1, or else the period of an IDT cell on a given substrate is bidirectional. When fixed for the value N|κ_p|, one can find that the condition of the best right directivity is φ = −π/2. The sign of φ is important to decide the direction of the wave excited by the drive voltage is applied on IDTs. Definitely, one can get the sign of φ since all the COM parameters, including the phases of φ_κ and φ_α, can be calculated by our method.

3. Results and Discussions

In this section, we compute the COM parameters versus the thickness of the electrode for the bidirectional IDT on the 128°YX LiNbO₃, bidirectional IDT on Y-Z LiNbO₃, EWC SPUDT on 128°YX LiNbO₃ and EWC on Y-Z LiNbO₃, respectively. The material constants of LiNbO₃ are obtained from reference [34], and the metal electrodes are assigned with aluminum (Al) material with a Young’s modulus of 70 GPa, a Poisson’s ratio of 0.33 and a density of 2700 kg/m³.

Figure 5a–d illustrates the results of the COM parameters for the bidirectional IDT on the 128°YX LiNbO₃. As shown in Figure 5a, with an increase of h/λ₀, the velocity ν decreases, while the capacitance $C_n$ has almost no change. Figure 5b shows the amplitude and the phase of κ. It is easy to see that |κ| decreases at first and then increases with the increase of h/λ₀, and it becomes zero at h/λ₀ ≈ 2.76%. This is due to the fact that the electrical reflection by the short-circuited grating has the opposite sign to the mechanical reflection [18]. The corresponding experiment results [35] verified its correctness. This phenomenon can be deduced from the phase φ_κ. When h/λ₀ is small, φ_κ = π, namely, κ < 0, and when h/λ₀ > 2.76%, φ_κ = 0, namely, κ > 0. This is to say that κ increases monotonically with an increase in h/λ₀. Figure 5c shows ${\alpha }_n$ and the phase φ_α. We can see that the value of ${\alpha }_n$ has a slowly increasing trend and that the phase remains the same at zero. The directivity and the unidirectional angle versus the thickness of the electrode are displayed in Figure 5d. It can be seen that the directivity has stayed at 1, indicating that the IDT is bidirectional and there is no natural unidirectionality in the material orientation. In Figure 5d, we can see the unidirectional angle is 0 or 180°. Substituting the unidirectional angle into Equation (20), the directivity of 1 also can be deduced.

Figure 6a–d illustrate the results of the COM parameters for the bidirectional IDT on the Y-Z LiNbO₃. The COM parameters of C and ν have the same change trends compared with the bidirectional IDT on 128°YX-LiNbO₃, while the value of ${\alpha }_n$ has an opposite trend. From Figure 6b, the reflection coefficient increases without an inflection point and the phase φ_κ is offset by 0 slightly and increases slowly with h/λ₀. These features demonstrate that directivity is inherent in Y-Z LiNbO₃ despite being very small. Additionally, the directivity can be adjusted slightly by changing the thickness of the electrode. It also can be seen from Figure 6d that the unidirectional angle is offset by 180° when h/λ₀ increases, and the directivity increases as h/λ₀ increases.

Figure 7a–d and Figure 8a–d illustrate the results of the COM parameters for the EWC SPUDT on 128°YX LiNbO3 and Y-Z LiNbO₃, respectively. These COM parameters of C, ν and ${\alpha }_n$ have the same change trends in two cutting types of LiNbO₃. The same is seen in the bidirectional IDT, where the |κ_p| has a zero [36] value in 128°YX-LiNbO₃ when h/λ₀ is approximate to 2.47%, but such a zero value does not exist in YZ-LiNbO₃. In relation to the |κ_p|, it is relatively small in the EWC SPUDT structure compared with the bidirectional IDT structure. From Figure 7d and Figure 8d, we can see that the unidirectional angles are close to −90° or 90°, but not −90° or 90° exactly. This means that the standard EWC SPUDT structure on 128°YX LiNbO₃ and Y-Z LiNbO₃ do not obtain the best directivity. From the directivity curve in Figure 7d, the wave excitation in the right direction is favored over the left wave when h/λ₀ is small, while when h/λ₀ is larger than 2.76%, this is reversed. This phenomenon has not appeared in previous research and is not found in YZ-LiNbO₃.

4. Conclusions

A novel method is proposed to extract all the COM parameters, especially for the phases φ_κ and φ_α. This method fully utilizes the static and eigenfrequency characteristics computed by the FEM. The corresponding relationships between the characteristics obtained by the FEM and the characteristics in the COM theory are established. All the COM parameters for a bidirectional IDT cell on 128°YX LiNbO₃, a bidirectional IDT cell on Y-Z LiNbO₃, an EWC SPUDT cell on 128°YX LiNbO₃ and an EWC SPUDT cell on Y-Z LiNbO₃ are calculated, respectively. These examples show that this is an effective method to extract the COM parameters quickly, accurately, generally and fully.