**1. Introduction**

The air-coupled acoustic transducer uses air as the medium to detect the defect in aerospace composites, foods, drugs, etc. [1,2], in which the coupling agent is prohibited or caution used. An air-coupled acoustic transducer commonly uses the piezoelectric composite as the acoustic wave source. When the acoustic wave propagates from the piezoelectric composite to the air directly, a nearly total reflection occurs on the interface due to the tremendous acoustic impedance discrepancy between the piezoelectric composite and air. The high reflection ratio of the acoustic wave limits the energy into the air and tested material, resulting in a low signal amplitude [3,4]. In order to solve this problem, the transition/matching layers are often attached to the piezoelectric composite for the acoustic transmission [5]. The properties of the matching layer, including the geometric and acoustic parameters, ultimately determine the acoustic transmission process [6].

The study of matching layers for the air-coupled transducer has been reported. Toda et al. [7] conducted impedance matching by adjusting the air space and reflectivity by insertion a reflective layer between the transducer and the propagation medium. Kelly et al. [8] added a porous material with extremely low impedance and the low-density rubber material as the matching layer, which induced the amplitude of the received signal to increase by 30 dB compared with the unmatched one. Tomas et al. [9,10] proposed a better matching configuration by studying the acoustic properties of polyethersulfone and other materials to obtain better sensitivity and bandwidth transducer. Saito et al. [11] optimized the acoustic impedance of the matching layer by using the transmission line model, which was proved by experiments to increase the sensitivity by 20 dB by using silicon rubber and thermoplastic hollow microspheres mixture as a matching layer. Botun et al. [12] developed

**Citation:** Zhou, J.; Bai, J.; Liu, Y. Fabrication and Modeling of Matching System for Air-Coupled Transducer. *Micromachines* **2022**, *13*, 781. https://doi.org/10.3390/ mi13050781

Academic Editors: Xiuqing Hao, Duanzhi Duan and Youqiang Xing

Received: 26 April 2022 Accepted: 14 May 2022 Published: 17 May 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

an air-coupled ultrasonic transducer with a simple structure and high sensitivity by using honeycomb polypropylene iron electret film as a matching layer. Kazys et al. [13,14] used low impedance polystyrene foam to improve the efficiency, bandwidth, and radiation pulse waveform of the PMN-32PT crystal transducer. Guo et al. [15] analyzed the effect of the matching layer material on the vibration mode shape of the transducer and found the resin epoxy can improve the transmission ratio. Wang et al. [16] used the superposition of two low impedance matching layers to improve the sensitivity of the designed transducer, which can detect microcracks. Wu et al. [17] developed a single matched layer air-coupled ultrasonic transducer using a hollow polymer microspheres/epoxy resin system, increasing the transducer sensitivity by 20.9 dB. Song [18] introduced the periodic subwavelength apertures that employ coupled resonances to enhance the efficiency and bandwidth of the non-contact ultrasonic transducers. Based on the literature review above, the matching layer properties, including the material and acoustic properties, have a significant influence on the sensitivity of the air-coupled transducer. However, the modeling of the matching layer properties and the bonding material effects on the matching layers is currently missing.

In this study, a double-layer matching system on the air-coupled transducer was investigated. The matching theory of the acoustic impedance is firstly introduced. Then modeling to predict the density, acoustic transmission speed, and the acoustic impedance of the matching layer is proposed based on the raw material. Experiments to validate the modeling results and reveal bonding material effects on the transducer sensitivity are introduced. Lastly, the optimized transducer was applied to detect the defects in two typical non-metallic materials to prove the feasibility.

### **2. Matching Theory and Modeling of Acoustic Impedance**

### *2.1. Matching Theory of Acoustic Impedance*

A material acoustic impedance *Z* can be calculated by the density ρ and acoustic propagation speed *c<sup>L</sup>* in the material, as shown in Equation (1) [19].

$$Z = \rho c\_L \tag{1}$$

Assuming the piezoelectric composite, medium, and tested sample are half infinite. The acoustic impedance of piezoelectric composite, the Nth matching layer, and medium are represented by *Z*0, *Zn*, and *ZL*, respectively. The theoretical acoustic impedance for single matching layer Z<sup>1</sup> can be calculated by Equation (2) [19]:

$$Z\_1 = \sqrt{Z\_0 Z\_L} \tag{2}$$

The theoretical acoustic impedances for double-layer matching can be calculated as Equations (3) and (4) [19] for the first and second matching layers.

$$Z\_1 = \sqrt[4]{Z\_0^3 Z\_L} \tag{3}$$

$$Z\_2 = \sqrt[4]{Z\_0 Z\_L^3} \tag{4}$$

where *Z*<sup>1</sup> and *Z*<sup>2</sup> are the acoustic impedances for the first and second matching layers.

The ultrasonic wave generated by the piezoelectric composite propagates as a simple harmonic wave to the matching layer. In order to ensure a continuous ultrasonic oscillation, the matching layer thickness should be a quarter of the wavelength, which can also reduce the ultrasonic attenuation. Based on the theory above, the matching layer thickness should be:

$$b = \frac{c}{4f} \tag{5}$$

where *c* is the acoustic propagation speed, *f* is the frequency of the acoustic, and *b* is the thickness of the matching layer.

### *2.2. Modeling of Matching Layer Acoustic Impedance*

Assuming the matching layer consists of the N components, with the weight of *m*1, *m*2, *m*3, . . . , and *m*N. The corresponding densities for the N components are *ρ*1, *ρ*2, *ρ*3, . . . , and *ρ*N, respectively. Generally, the matching layer uses the curing process to form the solid shape from liquid, which may cause the volume change. The volume variation for the N components, described by the shrinkage or expansion ratio, are *k*1, *k*2, *k*3, . . . , and *k*N, where the negative value means the expansion and the positive value means shrinkage. The volume of the matching layer after solidification is

$$V = \sum\_{i=1}^{N} (1 - k\_i) \frac{m\_i}{\rho\_i} \tag{6}$$

The density of the matching layer is

$$\rho = \frac{\sum\_{i=1}^{N} m\_i}{V} = \frac{\sum\_{i=1}^{N} m\_i}{\sum\_{i=1}^{N} (1 - k\_i) \frac{m\_i}{\rho\_i}} \tag{7}$$

In order to model the acoustic velocity of the matching layer, the acoustic velocity in each pure component in solid mode should be measured in advance. Assuming the acoustic velocity in N components are *v*1, *v*2, *v*3, . . . , and *v*N. Based on the probability statistics theory, when the matching layer thickness is *b*, the average prorogation distance, in statistics, of acoustic in the *i* th component is

$$b\_i = b \frac{(1 - k\_i)m\_i/\rho\_i}{V} \tag{8}$$

The propagation time in the *i* th component is *t*<sup>i</sup> = *b*i/*v*i, and the total time used to cross the matching layer is

$$t = \sum\_{i=1}^{N} \frac{b\_i}{v\_i} \tag{9}$$

So the average acoustic velocity in the matching layer is

$$v = \frac{b}{t} = \frac{\sum\_{i=1}^{N} (1 - k\_i) \frac{m\_i}{\rho\_i}}{\sum\_{i=1}^{N} (1 - k\_i) \frac{m\_i}{\rho\_i v\_i}} \tag{10}$$

Based on the Equation (1), the acoustic impedance of the matching layer is

$$Z = \rho v = \frac{\sum\_{i=1}^{N} m\_i}{\sum\_{i=1}^{N} (1 - k\_i) \frac{m\_i}{\rho\_i}} \frac{\sum\_{i=1}^{N} (1 - k\_i) \frac{m\_i}{\rho\_i}}{\sum\_{i=1}^{N} (1 - k\_i) \frac{m\_i}{\rho\_i v\_i}} = \frac{\sum\_{i=1}^{N} m\_i}{\sum\_{i=1}^{N} (1 - k\_i) \frac{m\_i}{\rho\_i v\_i}} \tag{11}$$

Based on Equation (11), with the mass, density, shrinkage/expansion ratio, and acoustic velocity of each component, the acoustic impedance of the matching layer can be calculated.

### **3. Fabrication of Transducer**

From the literature [8,11,13,17], the first matching layer is commonly fabricated by the mixture of hollow glass microspheres with epoxy, and the second matching layer is often made of microcellular foam polypropylene. Their acoustic impedances can be adjusted by changing the ratio between hollow glass microspheres and epoxy and the foaming rate, respectively. In this study, the matching layer used the same raw materials.

### *3.1. 1-3 Piezoelectric Composite*

The piezoelectric composite used in this study is the 1-3 type, which consists of one dimension piezoelectric ceramic column and a 3D polymer structure. The embed polymer helps to reduce the density and acoustic impedances of the piezoelectric composite due to the low density and acoustic impedances. The reduced acoustic impedance of the composite makes it an ideal material for an air-coupled piezoelectric transducer [9,20] to ensure more wave energy can propagate into the air. The 1-3 type piezoelectric ceramic/polymer has the following advantages for the air-coupled transducer:

(1) Low acoustic impedance, which is between the pure piezoelectric ceramic and polymer, and easier to achieve the impedance matching.

(2) High electromechanical coupling factor, which is almost the same as the longitudinal electromechanical coupling coefficient of piezoelectric ceramic when the composite is working under the thickness mode.

(3) The mechanical quality factor is low, which is suitable for the broadband transducer.

(4) Weak lateral coupling effects due to the separation of the polymer, which is suitable for the longitudinal transducer.

For the machining of piezoelectric composite, the solid piezoelectric ceramic need to dice or wire saw some kerfs on the surface to separate the ceramic block into the small individual columns. Then the polymer is immersed and cured on the kerf to connect the individual columns to a whole part. After the curing, the top and bottom of the composite surface are ground to remove the uncut ceramic layer and the excess polymer layer. Then continuing grinding is conducted to obtain the specified thickness of the composite.

### *3.2. Fabrication of the First Matching Layer*

The first matching layer consists of hollow glass microspheres, epoxy, curing agent, and diluent. The density of the hollow glass microsphere changes with the diameter; the smaller the diameter of the hollow glass microsphere, the larger density is. The density and acoustic velocity of the first matching layer can be adjusted by changing the hollow glass microsphere's diameters and weight ratios. The fabrication method is mold casting in a vacuum chamber, which is shown in Figure 1:

Step 1. The epoxy and curing agents are weighted in 6:1, suggested by the vendor (Shanghai Aotun Chemical Technology Co., LTD, Shanghai, China), by analytical balance and poured into a glass beaker for a mixture. Then the hollow glass microspheres are weighted based on the designed weight ratio and added to the beaker. If needed, the diluent weighted in the ratio of the epoxy and curing agent is added to assist the air bubble release.

Step 2. After all raw materials are put in the beaker, a stirring of about 5 min is taken for the mixture. The vacuum-pumping process under the pressure of −0.1 Mpa is applied for 5 min to help the evacuation of air bubbles.

Step 3. The mixture is then poured into the rectangular mold, followed by another 5 min vacuum to remove the bubbles and moisture.

Step 4. Curing on a thermostat at 60 °C temperature for 12 h.

Step 5. After curing, the composite is taken from the mold and cut into the same size as the 1-3 piezoelectric composite.

Step 6. The density and acoustic velocity of the matching layer are determined by using drainage and pulse insertion

Step 7. Cut the matching layer thickness to a quarter of the acoustic wavelength from the calculation.

Step 6. The density and acoustic velocity of the matching layer are determined by

Step 7. Cut the matching layer thickness to a quarter of the acoustic wavelength from

**Figure 1.** The steps to fabricate the matching layer. **Figure 1.** The steps to fabricate the matching layer.
