4.3. Frequency Domain Analysis
According to the eigenfrequency analysis of the model shown in
Figure 2a, the approximate resonant frequency
fr of RSAW is 195.1 MHz under the initial structural parameters. Based on this, the frequency scanning of the model shown in
Figure 2b under different structural parameters is carried out around the above frequency, and the scanning step is 0.1 MHz. As shown in
Figure 7a, with the initial structural parameters (i.e.,
Nt is 50), the corresponding frequency of the resonant peak in Rayleigh mode is 200.8 MHz,
IL is 8.37 dB (i.e.,
Vout/
Vin is 38.2%) and
Q is 468.31. For a dual-port SAW resonator with the same material and structural parameters, the experimental results reported by Hongsheng Xu et al. [
9] show that the frequency and
IL are 195 MHz and 10 dB (i.e.,
Vout/
Vin is 31.6%), respectively. Among them, the relative errors of frequency and
IL are about 3% and 16.3%, respectively, and the absolute errors are 5.8 MHz and 1.63 dB (the corresponding
Vout/
Vin is only 6.6%), indicating that our simulation method is reasonable. The small difference may mainly originate from the following: first, the material parameters used by Hongsheng Xu et al. [
9] are not exactly the same as those used in the simulation model; second, the coupling loss and electrode resistance loss are not considered in the simulation; third, the structural parameters of the actually manufactured dual-port SAW resonator are not exactly the same as those of the model used in the simulation due to the process errors.
The variation curves of
IL and
Q of RSAW with respect to the structural parameters of the resonator are shown in
Figure 7 and
Figure 8, where the insets show the resonant peaks of the Rayleigh mode under different structural parameters. As shown in
Figure 7a, when the number of IDT finger pairs
Nt is lower than 60,
Q of this resonator increases with
Nt and becomes to saturate with
Nt greater than 60. Meanwhile, the
Nt of 60 acts as an inflection point of
IL, which declines at a small value of
Nt and increases thereafter. The loss of electric energy of IDTs due to an outward radiation of the SAW energy is represented by the acoustic radiation conductance
, which is calculated as [
34]:
where
C0 is the unit length capacitance of an IDT finger pair,
K is the electromechanical coupling coefficient and
CT is IDTs’ total electrostatic capacity with
Nt pairs of interfinger electrodes in parallel. According to Formula (12),
Nt2 is proportional to the acoustic radiation conductance; therefore, with the increase in the number of IDT finger pairs, the amount of electrical energy converted into mechanical energy increases, more energy is consequently emitted from the outward radiation and the
IL becomes smaller. However, this analysis conflicts with the simulation results. The main reason may be that Formula (12) is derived based on the equivalent circuit model, which ignores the second-order effects as mentioned in the introduction section. It is because of these second-order effects that the
IL increases rather than decreases when the number of IDT finger pairs exceeds a critical value.
The
Q of SAW resonator can be expressed as:
where
Lef is the effective length of the cavity and
δ is the reflection coefficient. It can be seen from Formula (14) that the higher
δ is, the higher the
Q of the resonator will be.
δ can be expressed as:
where
Z is the acoustic impedance in the free surface area between electrodes and the acoustic impedance in the electrode area is
Zm =
Z + Δ
Z. Therefore, the performance of the resonator can be improved by increasing the number of RG pairs
Nr. However, with the increase in
Nr, the increasing of
Q slows down and finally reaches saturation. The simulation results are shown in
Figure 7b, which are in good agreement with the theoretical expectation. In addition, with the change of
Nr,
IL shows the inverse trend as
Q.
It can be seen from
Figure 7c that with the increase in metallization rate
η,
IL first decreases and then increases, and shows a stable value at about
η = 0.5. The
Q increases first and then becomes stable after
η = 0.6.
As shown in
Figure 7d, with the increase in electrode thickness
thAl,
IL first decreases and then increases, and shows a minimum value at about
thAl = 0.2 μm. In contrast,
Q increases monotonically with
thAl.
According to Equations (12) and (13), the increase in the acoustic aperture
W will lead to the increase in the total electrostatic capacity of IDTs, thus increasing the acoustic radiation conductance and decreasing
IL. On the other hand, if
W is too large, the size of the resonator will become too big, and the loss will also increase. The variation curves of
Q and
IL with respect to
W are shown in
Figure 8a. It can be seen that with the increase in
W,
Q slowly increases, while
IL first decreases and then remains nearly unchanged after
W = 2.2 mm.
For a dual-port SAW resonator to generate standing waves, its
Ltt and
Lrt should satisfy the following relations:
where
m and
n are positive integers. As shown in
Figure 8b,c,
Q and
IL fluctuate in a small range, where
Q shows an overall downward trend and
IL an upward trend with the increase in
Ltt and
Lrt; that is, in general, a smaller
Ltt and
Lrt are better to reduce the transmission loss and acquire a higher
Q. Moreover,
Figure 8b shows that the
IL is relatively small when the distance between the input and output IDTs satisfies an odd multiple of the half-wavelength.
Since the energy of RSAW is mainly confined to the surface of the piezoelectric substrate, RSAW cannot be excited effectively when the substrate is too thin, and its energy will escape through the piezoelectric substrate, resulting in a large energy loss. As shown in
Figure 8d, when the thickness of LN is less than 20 μm (i.e., one wavelength in this paper), the resonant peak of the Rayleigh mode has become insignificant. With the increase in LN thickness,
Q gradually increases and finally reaches a stable value, while
IL decreases first and then slightly increases.
4.4. The Steady State Analysis
After performing the steady state calculation on the model shown in
Figure 3, the distribution of cloud map of the strain tensor component
S11 is plotted in
Figure 9a, which shows obvious strain gradients at both ends of the steel plate as well as the piezoelectric substrate. Therefore, the steel plate needs to be large enough to ensure that the piezoelectric substrate falls in its uniform strain region. In addition, to ensure the accuracy of the measurement of the strain sensor, the electrodes also need to be distributed in the uniform strain region of the piezoelectric substrate. Taking the strain transfer rate as a reference, the strain transfer characteristics of piezoelectric substrates under different thicknesses and lengths (i.e., Z and X directions) are analyzed, and the results are shown in
Figure 9b,c. The expression of strain transfer rate
is:
where
sst and
sLN are the
S11 component of strain tensors which belong to the upper surfaces of the steel plate and LN substrate, respectively. In addition, regions with default strain gradients ≤1% are defined as uniform strain regions (USRs).
As
Figure 9b illustrates, with the increase in LN thickness, the strain transfer rate and the length of USR almost decrease linearly, indicating that the thicker the piezoelectric substrate is, the less favorable the strain transfer and the smaller the sensitivity of the strain sensor and the area available for electrode placement. Combined with the simulation results in
Figure 8d, the thickness of LN piezoelectric substrate is selected as 40 μm, and then the length of LN is analyzed. As shown in
Figure 9c, the strain transfer rate and the length of the USR increase largely linearly with the increase in the length of LN, which indicates that the larger the length of the piezoelectric substrate, the more conducive it will be to strain transfer, thus increasing the sensitivity of the strain sensor and the region available for placing electrodes. To avoid material waste, the size of the piezoelectric substrate should be specified in combination with the area required by the electrodes.
According to the above simulation results, the optimized structural parameters (see
Table 6) of the dual-port SAW resonator based on LN is determined as follows:
The Q and IL of the dual-port SAW resonator used for strain sensing after structure optimization are 538.49 and 5.47 dB, respectively. Compared with before optimization, Q is increased by 15%, IL is decreased by 34.6% and strain transfer rate is increased by 2.4%, which fully demonstrate the significance of optimizing the structural parameters for SAW devices.