Study on double-sided impedance matching of spherical focusing transducer in ultrasonic abrasive flow polishing system

The impedance matching of spherical focused ultrasonic transducer is one of the primary factors affecting the energy utilization efficiency in focused ultrasonic flow polishing system. To realize the efficient utilization of the vibration energy of the transducer, the front matching layer structure is design to achieve the impedance matching of the transducer and the abrasive flow liquid, thereby improving the transmittance of the transducer vibration energy in the liquid, and the back-matching layer structure is used to reflect the radiated vibration energy on the back of the transducer to reduce dissipation of radiated vibration energy on the back of the transducer. Based on the acoustic impedance matching theory and Kirchhoff’s theory, the influence of the front and back impedance matching structure of spherical focused ultrasonic transducer on the focal point sound pressure is studied theoretically, and experimentally, a novel impedance matching structure for spherical focused ultrasonic transducer is proposed. The experimental results showed that the focal point sound pressure was increased by 72.03% compared with that of the traditional structure.


Introduction
Focused ultrasonic technology has been widely used in the fields of biomedicine, non-destructive testing, and ultrasonic imaging because of its outstanding directivity and high work efficiency [1][2][3]. In recent years, this technology has also been applied in the precision machining of difficult-to-machine materials [4]. With focused ultrasonic technology, the ultrasonic vibration energy can be effectively used to remove material on the specified spot of workpiece surface. Therefore, it has a broad application prospects in industrial fields such as surface processing and polishing. There are three types of focused ultrasonic transducers, including concave spherical focused ultrasonic transducer (SUT), acoustic lens focused transducer, and array focused ultrasonic transducer [5]. The SUT is a commonly used transducer, which focuses the ultrasonic vibration energy of the transducer in its focal point through self-focused of the spherical structure. The focusing efficiency of the transducer is one primary factor affecting its working performance.
In the field of mechanical processing, the high pressure generated by the ultrasonic vibration and ultrasonic cavitation pushes the abrasive flow to remove material on the workpiece surface. The principle of focused ultrasonic processing is shown in Fig. 1. Under the excitation of ultrasonic power supply, the SUT vibrates along its thickness direction and radiates sound waves to the concave (front) and convex (back) surfaces of the spherical shell simultaneously. The vibration energy transmits and focuses at focal point through the medium (liquid) in front direction and dissipates in the medium in back direction. The acoustic impedance matching characteristics of the transducer and the medium directly affect the transmission and reflection of vibration energy. Since the acoustic impedance of the piezoelectric ceramic material and the liquid medium is very different, it is necessary to attach an impedance matching layer between the front of the transducer and the liquid to achieve impedance matching to improve the transmission rate of vibration energy. On the contrary, on the back of the transducer, an appropriate matching layer is needed to increase the difference of acoustic impedance between the transducer and the back medium, so as to achieve high reflectivity of the transducer's back vibration energy. Many researchers have begun to study the structure of ultrasonic transducers in different applications. In the field of ultrasonic testing, Toda and Thompson found that the front matching layer and backmatching layer of the transducer have an impact on the radiated sound wave and successfully improved the energy of the radiated sound wave through multi-layer polymer front matching layer and metal back-matching layer [6]. Fu et al. studied the effect of backing layer thickness on the acoustic impedance and mechanical loss factor, which provided a theoretical basis for the design of ultrasonic transducers [7]. This back-matching structure is used to absorb aftershock signals to improve the ultrasonic detection accuracy. In the ultrasonic radiation application field, Zhang et al. simulated the radiation efficiency of planar piezoelectric transducers under different back-matching materials, finding that the air backing layer had a better radiation effect [8].
It is difficult to make a matching layer on the concave surface of the SUT. Therefore, Liu et al. tried to form a flat matching layer by filling a matching material on the concave side. Through sound field simulation research, they believe that the use of this flat front matching layer can increase the sound pressure and sound intensity at the focal point of the SUT [9]. However, there is no relevant theoretical analysis and experimental research report yet.
To optimize the matching structure of the SUT and achieve high-efficiency energy output, in this paper, the principle of front and back-matching structure of the SUT is systematically studied based on theoretical analysis, numerical calculation and sound field simulation. Finally, the results were verified by experiments.

Calculation of sound pressure
The sound pressure at point Q in the radiation sound field of the SUT can be calculated based on the continuous wave theory [10], as shown in Fig. 2:where the dome center O of the spherical shell is the coordinate origin. R 0 is the curvature radius of the SUT, a is the transducer's caliber, b is the distance from the origin to the sphere edge, point F is the geometric center of spherical transducer, r 0 is the distance from the origin to a point Q, θ is the intersection angle between r 0 and Z axis, ds is the unit area at point Q 1 on the spherical surface, R 1 is the distance from the origin to the point Q 1 , and r is the distance from point Q 1 to Q. Then, the sound pressure at point Q is According to Huygens' theory, when θ = 0°, the point Q is located on the axis Z, and then the sound pressure p along the Z direction can be obtained [11]: where f is the ultrasonic vibration frequency, ρ is the medium density, z is the distance from O to the point Q on the axis Z, u 0 =A/ , A is the ultrasonic vibration amplitude, = 2 f , =c/f , and c is the sound speed in the medium.

Sound transmission and reflection
When ultrasonic waves are transmitted between different materials, transmission and reflection occur at the interface of the two materials, as shown in Fig. 3.  The transmittance T 12 and T 23 and the reflectivity R 32 and R 21 can be calculated as [12] where Z i (i = 1,2,3) are the acoustic impedances of medium material I, II, and III respectively, Z i = i c i . Obviously, the transmittance and reflectivity depend on the acoustic impedance of the two materials.

Front matching
In focused ultrasound abrasive flow processing, water-based abrasive flow is usually used as the working medium. Due to the large acoustic impedance difference between the SUT and the abrasive flow [13], the sound wave will be reflected at the interface between the abrasive flow and the SUT, which will reduce the energy transmission efficiency of the SUT and even causing the SUT to be damaged. Attaching a matching layer with appropriate acoustic impedance to the front of the SUT (concave surface) to form a front matching layer can effectively reduce the reflection of ultrasonic waves. Two kinds of front matching layer structures will be studied in this paper as shown in Fig. 4: One is flat matching layer, and the other is concave matching layer. Different from the flat matching layer, the concave matching layer is a layer of matching material with equal radial thickness on the SUT's concave side, so it is of the same shape with the SUT.
With these two matching structures, the sound pressure on the axis Z radiated by the SUT in the liquid medium can be numerically calculated without taking the sound attenuation into account. The material parameters used in the calculation are shown in Table 1.

Concave matching layer
The transmittance T of sound wave at interface of transducer and matching layer can be deduced [14]: where 2 =c 2 /f is the wavelength in matching layer and c 2 is the sound velocity of matching layer. As the concave matching layer has equal thickness on the concave surface of the transducer, the sound pressure P can be directly calculated by Formulas (2) and (5) as follows: As shown in Fig. 5, for the material given in Table 1, the sound pressure at point F on the axis Z changes with the thickness of the front matching layer. The maximum sound pressure at point F is obtained, while the thickness of the concave matching layer is equal to an odd multiple of a quarter wavelength.

Flat matching layer
For the flat matching layer, since the radial thickness of the matching layer at different radii of the SUT is not equal, the ultrasonic waves radiated at different radii on the transducer's surface reach the focal point with different phase angles. The sound pressure at point Q on the axis Z can be calculated based on the principle of sound field superposition, as shown in Fig. 6, where H is the axial maximum thickness of the flat matching layer.  The transmittance T ′ changes with the change of r 0 as follows: where, Assume c 2 and c 3 are the sound speed in silica gel and water respectively. θ i is the sound wave incidence angle at interface between silica gel and water. For θ = 0°, point Q is located on axial Z, z = r 0 , and r = [ When silica gel is used as front matching material, c 3 c 2 sin i < 1 , the contribution of the part of the transducer coved by matching layer to the sound pressure at point Q is The radiated sound pressure of other part of the SUT (without matching layer) at point Q is For the SUT without matching layer, the focal point is at about point F. While the flat matching layer is used, the position of focal point F' is offset away from point F and close to the dome center O as shown in Fig. 17a, which is not convenient for practical applications. The sound pressure at point F increases with the thickness H as shown in Fig. 7b. Compared with the result in Fig. 5, although the sound pressure at point F can be increased by using a flat matching layer, the sound pressure at point F is lower than that with concave matching layer.

Traditional back-matching structure
Different from the front-matching, the back matching is used to reflect the vibration radiated from the back of the transducer back to the transducer to enhance the vibration. The  Table 2) with concave spherical end surface which is glued with the back of the SUT, as shown in Fig. 8. Usually, in the traditional backing structure with different material, there always is some part of the sound wave which is still transmitted and dissipated in the backing cylinder, which leads to the loss of the ultrasonic energy. Therefore, further optimization of the structure is needed to increase the acoustic reflectivity.

New double-layer matching structure
To get higher reflectivity, a new back matching with doublelayer matching structure is proposed as Fig. 9. A layer of matching material (layer II) which is of equal radial thickness H 0 and lower acoustic impedance is inserted between the SUT and the steel (or aluminum) backing layer.
The reflectivity R 2 can be deduced from Formula (4) as follows [15]: where Z 4 , Z 5 , and Z 6 are the acoustic impedance of the SUT, layer II, and layer I, respectively; 5 =c 5 /f is the sound wavelength in layer II.
The numerical calculation result of the relationship between H 0 and R 2 is shown in Fig. 10. Obviously, compared with the traditional back-matching structure, the double layer matching structure can significantly change the sound (12) reflectivity. With the suitable material, the maximum reflectivity can be gotten when the thickness H 0 is equal to an odd multiple of a quarter wavelength. Compared with water and silica gel, choosing air as layer II material can obtain more stable and higher reflectivity in a larger thickness range. So, air is the second layer of material in this paper. In summary, the use of an appropriate front and backmatching structure can significantly enhance the ultrasonic vibration of the SUT and increase sound pressure at point F.

Sound field simulation
To verify the theoretical analysis, the two-dimensional axisymmetric frequency domain sound field model of the SUT, is established by using simulation software. Specific parameters and model are shown in Table 3 and Fig. 11.

Front-matching simulation
Simulations were performed under different front matching structural, as shown in Fig. 12. The sound pressure at point F with a flat matching layer is significantly lower than that   Fig. 8 Traditional backing matching structure model of concave matching layer under the same double layer backmatching structure. This is because when flat matching layer structure is used, on the one hand, the radial thickness of the matching layer changes with the diameter of the point on the SUT, and the acoustic waves radiated by the transducer have different phases when they reach the interface of the matching layer and the liquid. Therefore, the sound waves have different initial phases when they transmit in the liquid and part of the energy cancels each other out at point F. On the other hand, due to the change of the incident angle, the radial sound wave is refracted at the interface between the matching layer and the liquid, which changes the propagation direction of the sound wave in the liquid and reduces the convergence of the sound wave at the point F. Also, the sound pressure at point F is related to the transmittance of the front matching layer. The relation between the thickness of the matching layer and the sound wavelength in it determines the acoustic transmittance of the material. As shown in Fig. 13, the sound pressure reaches the maximum at point F when the matching layer thickness The tickness of H 0 (mm) layer Ⅱ for air layer Ⅱ for water layer Ⅱ for silica gel 2 3 0.8 The tickness of H 0 (mm) (b) layer Ⅲ for steel Fig. 10 The reflectivity under different thickness of layer II. a Layer III for aluminum. b Layer III for steel is an odd number of times of one quarter of wavelength, which is agreed with those of theoretical calculation, which is agreed with that of the theoretical analysis. Figure 14 shows the sound field simulation results of the SUT under two different back-matching structures. In the traditional back layer reflective structure (Fig. 14a, c), the radiated vibration from the back of the transducer are more easily transmitted into the back layer, so the sound pressure at point F is weak. In contrast, with the double-layer reflective structure (Fig. 14b, d), the radiated sound wave on the back of the SUT is reflected at the interface of the matching material, and only a small amount of sound wave is transmitted into the matching layer. So, most of the vibration energy of the SUT is focused to point F, and the sound pressure is significantly increased.

Back-matching simulation
With the same material, the thickness H 0 is the primary factor affecting the reflectivity of the double-layer backmatching structure. As shown in Fig. 15, when air is used as the material of matching layer II, the sound pressure reaches the maximum at point F in front of the transducer when the thickness H 0 is an odd multiple of a quarter wavelength.

Experimental research
Based on above analysis, a new SUT with front and backmatching layer is fabricated as shown in Fig. 16. A focused ultrasonic vibration experimental system was developed with this new matching structure transducer as shown in Fig. 17.
The signal generator generates a sine wave electric signal with the same resonance frequency as the transducer, which is amplified by the power amplifier and applied to the SUT. The SUT resonates along its thickness direction under the drive of the power electric signal. The SUT is placed in the liquid in the water tank. When the transducer resonates, its front side radiates ultrasonic vibration into the liquid and focuses at point F. A needle-type hydrophone is used to measure the sound pressure  The sound pressure value P on the axis is calculated by the sound pressure conversion formula: where U is the peak to peak voltage recorded by the oscilloscope and M = -228.9 dB the sensitivity of the needle-type hydrophone.
The axial radiated sound pressure of the SUT in the liquid with the traditional back-matching structure and the new back-matching structure (without front matching layer) were measured respectively. The result is shown in Fig. 18. The axially radiated sound pressure of the new matching structure is greater than that of the traditional structure, and the maximum sound pressure at point F is 15.3 MPa, which is 29.7% higher than that of the traditional structure.
By changing the thickness of the air layer, it is found that when the thickness of the air layer is 6 mm (about 71 times a quarter of the wavelength), the sound pressure at point F reaches its maximum value. Based on this back-matching structure, a SUT with a 1.3-mm-thick concave silicone matching layer (about 5 times the quarter wavelength) and another one with a flat silicone matching layer (fulfilling the concave surface) were fabricated and tested experimentally. As shown in Fig. 19, with the concave front-matching layer and the double backing layer, the maximum sound pressure at the point F in the liquid reaches 20.3 MPa, which is 72.03% higher than that of traditional structure. For the flat front matching layer structure, the sound pressure of point F reaches to 17.5 MPa, higher than the traditional structure but lower than that of the concave front matching layer structure.