Numerical Study of Acoustic-Electric Signal Conversion for GIS Discharge Detection and Structure Optimization for pMUT

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Introduction
Electric insulation is a key problem in the power system.Te accumulation of long-term partial discharge (PD) of insulation media will eventually lead to the electrical breakdown of the entire insulation system, which will cause serious failures.Finding equipment discharge problems as soon as possible through the breakdown testing or realtime monitoring can efectively improve the system reliability and reduce the operation cost [1][2][3].Gas-insulated switchgear (GIS) is a widely used high-voltage sealed switchgear in the current power system.Because GIS is a large fully enclosed device and its internal structure is complicated, the problem of insulation failure and discharge detection of GIS has been receiving great attentions [4,5].In recent years, the ultrasonic detecting method has been gradually mentioned for the discharge detection of GIS, and the common ultrasonic sensors used in engineering are wired piezoelectric sensors.Te basic detection mechanism is to use a piezoelectric ceramic or crystal with the piezoelectric efect to convert the high-frequency acoustic signal generated by the breakdown discharge in the GIS into an electrical signal, which is then used to locate the discharge point by comparing and processing the electrical signal output from multiple sensor terminals [6,7].However, this method requires a certain number of ultrasonic sensors to be arranged regularly on the surface of GIS enclosure.Since the GIS is large and complex, when multiple sensors are arranged, it is difcult to arrange wires or even the wires cannot be arranged.When arranging few sensors, it may not guarantee the efective detection of weak signals or distortion-free detection of high-intensity signals [8].
With the continuous improvement of electromechanical system (MEMS) technology and the continuous progress of wafer level material growth technology, it is possible to use the piezoelectric micromechanical ultrasonic transducer (pMUT) for nondestructive discharge detection of GIS.Te advantages of pMUT, such as miniaturization, wireless transmission, and high efciency of acoustic-electrical conversion, are expected to solve the problems of low sensitivity and wiring difculties of traditional ultrasonic sensors in detecting GIS breakdown discharge [9,10].Compared with traditional piezoelectric sensors, the basic acoustic-electrical conversion mechanism of pMUT has not changed, but the vibration mode of the piezoelectric flm has changed, and the coupling of voltage, deformation, and sound waves is still realized through the piezoelectric efect of the piezoelectric material.As a result, the material of the piezoelectric layer is an important factor afecting pMUT performance.In addition, the element size, materials, and integration process of the piezoelectric layer, elastic support layer, and electrode layer will further afect the mechanical properties of pMUT, which will infuence its multiple response performances [11].Terefore, it is important to design sensors with special structures to meet specifc performance requirements for pMUT applications.Although experiments can efectively verify the sensor performance, while at the design stage, the empirical formula and the equivalent circuit model are often used for the calculations, which are complex and not so accurate, and the infuence of multiphysical felds cannot be well solved.
In recent years, many researchers have designed pMUT and analyzed its performance based on the numerical simulation method.Liu et al. [12] established a fnite element model of the pMUT array based on the AlN flm and simulated the resonant frequency, modal shape, electric impedance, and sound feld under water and air loading, respectively.Te deviation between the simulation results of the resonant frequency and actual measurement was only 1/ 3 of the deviation between the results with the analysis method and actual measurement, and the simulation results of electric impedance were consistent with the measurement values.Tis numerical model is reliable to predict electric and acoustic performances, which is useful to optimize pMUT structures.Wang et al. [13] proposed a piston diaphragm (PD) pMUT with the improved structure.Te geometrical parameters of two diferent piston diaphragms of the PD-pMUT were numerically studied and optimized by using COMSOL software.Teir results showed that the pMUT after structure improvement had better sound transmission performance and high-frequency bandwidth, which would provide a reference for structure improvement of pMUT.An air-pMUT coupling numerical model was established by Massimino et al. [14].In this coupling model, the electromechanical coupling, the thermal-acousticstructure interaction of pMUT, and the thermal-acousticpressure interaction during acoustic waves propagation in the air were considered.Te two-dimensional and threedimensional computational results showed that the acoustic and mechanical responses simulated by the fnite element method were in good agreement with the experimental results.Tis research is meaningful to guide designers to simplify the computational model because, at present, the computational cost for three-dimensional multiphysical feld coupling simulations is still high.Liu et al. [15] designed a square pMUT with an outer frame electrode based on numerical simulations.Tey found that the square pMUT had better performance in receiving ultrasonic signals than the circle pMUT with top electrode arranged in the center, which was also confrmed by the experiments.Terefore, the fnite element simulation should be an efective and convenient way for the structural design and performance analysis of pMUT.However, based on previous application felds of pMUT, the current research is still mainly focused on the electromechanical performance for a single sensor or array or their sound transmission performance in the fuid.Almost no studies were reported on ultrasonic-solid-pMUT coupling for ultrasonic signal detection of GIS breakdown discharge, and the fnite element simulation model for ultrasonic-solid-pMUT coupling has not been established yet.In fact, the solid thickness measurement experiments by Xing et al. [16] showed that pMUT using piezoelectric ceramics (PZT) membranes was a promising ultrasonic-solidstate sensing and detection equipment.In addition, for the numerical simulation of pMUT, the errors caused by the model simplifcation and boundary settings of electrode have also not been systematically studied in the existing literatures.For example, Sammoura et al. [17] established a dual piezoelectric layer pMUT fnite element simulation model, where the actual thickness of the electrode was ignored, and the piezoelectric layer boundary was used for the simulations.Te errors caused by such simplifcations were not discussed, which was directly ignored in the literature.
Terefore, in view of the current lack of simulation research of GIS breakdown discharge detection with pMUT and the low accuracy of pMUT design methods with the empirical formula and equivalent circuit model, in this article, a fnite element simulation model of ultrasonicsolid-pMUT coupling that takes into account the acoustic transmission and the acoustic-electrical conversion process at the same time was established.Ten, the conversion process between GIS breakdown discharge ultrasonic signals and pMUT electric signals was numerically studied, and the acoustic electromechanical performance of pMUT was also analyzed.In addition, the simulation errors caused by the electrode simplifcation were discussed by comparing the simulation results obtained from the unsimplifed top electrode and simplifed top electrode model.Finally, the pMUT structure was optimized by numerical simulations according to the main ultrasonic frequency range of GIS breakdown discharge and key performance parameters of pMUT, and a suitable pMUT structure was found for GIS discharge detection.Tis study would be meaningful for understanding the ultrasonic-solid-pMUT multiphysical feld coupling transfer process and pMUT structure design for GIS breakdown discharge detection.

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Mathematical Problems in Engineering

Physical Model and Geometric Parameters.
In order to detect the ultrasonic signal of breakdown discharge in the GIS, the piezoelectric sensors are required to be arranged on the outer surface of the GIS enclosure.In the present article, a coupling physical model of ultrasonic-solid-pMUT is established to analyze the output electric signals generated by the deformation of receiving pMUT, which is caused by the ultrasonic wave, as shown in Figure 1.In the simulation, the ultrasonic signal is generated through voltage excitation of emitting pMUT, which is placed on the inner surface of the GIS enclosure.Te piezoelectric layer of emitting pMUT expands and shrinks due to the inverse piezoelectric efect, then bends and deforms under the binding of the elastic layer, and vibrates to emit acoustic waves.When ultrasonic waves transmit through GIS enclosure and reach the receiving pMUT arranged on the outer wall of the enclosure, the piezoelectric layer of receiving pMUT vibrates and generates induced charges due to the positive piezoelectric efect, and the electrodes output electric signals at the same time.In the present work, a two-dimensional axisymmetric model is established to study the acoustic-electric signal conversion process and a three-dimensional model is established to study the pMUT response characteristics.Te circular pMUT thick flm (lead zirconate titanate) is selected as the acoustic emitter and receiver.Its physical model and typical geometric parameters are shown in Figure 1 and Table 1, respectively.Since the bonding layer (epoxy adhesive) between the electrode and substrate is thin (less than 2 μm) and its Young's modulus is much smaller than that of substrate (Si), the bonding layer is neglected in the numerical model.Te materials and typical physical parameters for each component of pMUT are presented in Table 2 [18,19].Among them, the relative permittivity of the piezoelectric layers is anisotropic, which is formulated as in the following equation [18]: (1)

Governing Equations and Computational Methods.
In the present study, the piezoelectric constitutive equations in the form of stress-charge are used to describe the relationship between stress, strain, electric feld, and electric displacement of piezoelectric materials, as shown in the following equation [20]: where S is the stress, Pa; c E is the Voigt elastic matrix for anisotropic materials; ε is the strain; e T is the piezoelectric stress constant; D ⇀ is the electric displacement, C/m 2 ; ε 0 is the initial prestrain; ε rS is the relative permittivity; and E ⇀ is the electric feld strength, N/C.
Te piezoelectric equations consist of momentum equation and electrostatic charge conservation equation.Te momentum equation is based on Newton's second law equilibrium equation of solid mechanics, and the stress-charge correlation of the piezoelectric material is formulated with elastic and coupling matrices.Te equations are as follows: where ρ is the density, kg/m 3 ; u ⇀ is the structural displacement vector, m; t is the time, s; ρ v is the volume electric charge density, C/m 3 ; and [e ES ] is the Voigt coupling matrix   for anisotropic materials.In this study, the 6 × 6 elastic matrix [c E ] and 3 × 6 coupling matrix [e ES ] for anisotropic piezoelectric materials of PZT-4 and PZT-5 are presented in Table 3, respectively [18].
In the actual pMUT system, there exists the damping efect.Rayleigh damping is usually used to describe the mechanical loss caused by the system damping in the transient dynamic analysis.Since the damping ratio of pMUT structure is unknown at the design stage, a structured loss factor (η s ) is introduced to describe the mechanical loss, and the original elastic matrix is changed as follows: For the top and bottom electrodes of pMUT, the charge conservation equation is formulated in equation ( 5), and the D-E constitutive relationship based on the dielectric model of relative permittivity is formulated in equation ( 6).
D-E constitutive relationship: where V is the potential, v; c 0 is the vacuum dielectric constant, 10 −9 /36π F/m; and c r is the relative permittivity.Te propagation of ultrasonic waves in GIS enclosure is solved by using the transient pressure acoustic model, as formulated in equation (7), and the ultrasonic waves are coupled to the pMUT by using acoustic-structural boundary conditions, as formulated in equation ( 8) [21].
where ρ c is the structure density, kg/m 3 ; p t is the total sound pressure, Pa; c is the sound velocity, 6300 m/s; q ⇀ d is the dipole domain source; n ⇀ is the surface normal; u ⇀ tt is the structural acceleration, m/s 2 ; and F ⇀ A is the load per unit area of the structure, N/m 2 .
Te boundary conditions are set as follows.For the emitting pMUT, voltage excitation is input on the top electrode surface and the bottom electrode is grounded, which are formulated in the following equation: where V 0 is the input voltage excitation, V.In the present study, one cycle of signal excitation is considered, and the input voltage signal waveform is formed by combining the sine wave and rectangular wave, as shown in Figure 2 [18,22].Te voltage signal amplitude is set to 50 V.In order to observe the transmission process of acoustic waves in the GIS enclosure and reduce the calculation amount of simulation work, the voltage excitation frequency is set to 20 MHz according to the initial pMUT structure size.
For the receiving pMUT, foating potential is set on the top electrode surface and the bottom electrode is grounded, which are formulated in the following equation: For structural mechanics boundary settings, the side boundaries of the pMUT substrate are set as fxed constraint and all other boundaries are free.Te fxed constraint boundary is set as follows: In the present work, since only the process of receiving direct ultrasonic signals by receiving pMUT is considered, the hard sound feld wall boundary setting is adopted on the surfaces, where the GIS enclosure and the pMUT are not directly contacted, which means the acoustic waves are completely refected on these walls.Te governing equation is as follows: All the above equations are coupling solved based on the fnite element software COMSOL Multiphysics.Te MUMPS direct solver is used to solve the equations of pressure acoustics and solid mechanics, and the fuxes at the interface between diferent regions are kept equal.When solving the transient propagation process of ultrasonic, the time step is set as 1/5 of the vibration period, and the computation is considered to be convergent when the iterative calculation residual is less than 10 −4 .For the characteristic frequency and frequency domain analysis of the pMUT, when the iterative calculation residual is less than 10 −6 , the computation is considered to be convergent.

Grid Independence Test and Model Validations
3.1.Grid Independence Test.First, the grid independence test was performed.In this article, the computational grids for the two-dimensional acoustic-solid-pMUT coupling model and the three-dimensional pMUT model are presented in Figure 3.For the two-dimensional coupling model, an unstructured grid is constructed for the computation, the maximum grid size of the GIS enclosure domain (zone 2) is limited according to the ultrasonic wavelength, and the grid of the pMUT piezoelectric layer is locally refned.Te input voltage excitation is shown in Figure 2. Five sets of grids are adopted for the simulation tests, and the specifc grid settings and computational results are presented in Table 4. Te results show that when the total grid number is increased from 26804 (grid 4) to 38426 (grid 5), the central displacement amplitude (u max ) and output voltage amplitude (V max ) of receiving pMUT only change by 1.09% and 0.53%, respectively.Terefore, grid 4 with total element number of 26804 should be good enough for the test, and the similar mesh settings to grid 4 is fnally adopted for the subsequent acoustic-solid-pMUT coupling simulations.For the threedimensional pMUT model, a structured grid is constructed for the substrate and bottom electrode (zone 4), while zone 5 and zone 6 are meshed with free tetrahedral grids.Five sets of grids are adopted for the simulation tests, and the specifc grid settings and computational results are presented in Table 5.It shows that when the total grid number changes from 36669 (grid 3) to 55010 (grid 4), the inherent frequency (f 0 ) and efective electromechanical coupling factor (K 2 eff ) of the receiving pMUT are almost unchanged.Terefore, grid 3 with total element number of 36669 should be good enough for the simulations, and similar mesh settings to grid 3 is adopted for the subsequent frequency analysis of pMUT.

Model Validations.
In order to validate the computational model of pMUT, the performance of circular pMUT based on AlN flm in [15] was restudied in this section.Te physical model and geometric dimensions for the pMUT validation model are presented in Figure 4.It shows that the aluminum electrode is selected in the validation model, the piezoelectric layer is the AlN flm, and the special substrate structure is adopted to improve the pMUT stability and its matching with air.Typical physical parameters of each component for the validation model are presented in Table 6.First, the inherent frequency of the pMUT is simulated and analyzed.Te comparison between the simulation results in this article and the experimental results in [15] is presented in Table 7. Te results show that the current simulation results are in good agreement with the experimental results [15], and the relative deviation of the inherent frequency is   only 0.5%.When 10 V alternating current is applied to the electrode, the piezoelectric layer (AlN) vibrates due to the inverse piezoelectric efect and generates acoustic waves, and then, the acoustic waves propagate in the air domain (when T air � 293.15 K, the acoustic velocity is 343.2 m/s).Liu et al. [15] measured the sound pressure level distribution at a height of 20 mm from the vibration plane.Te comparison between the measurement results in [15] and the simulation results in this article is presented in Figure 5.It shows that the present computational results can agree well with the experimental results as reported in [15].Te maximum deviation of the sound pressure level at the same height is 4.31 dB and its relative deviation is 2.82%.Terefore, it is considered that the numerical models and computational methods adopted in this article are reasonable and reliable.

Acoustic-Solid-Electric Signal Conversion and pMUT
Sensing Characteristics Analysis.First, the conversion process between ultrasonic and output electric signals is simulated and analyzed.From the composite waveform of the electric signal (Figure 2), it should be noticed that the voltage excitation is input between 0 and 2 × 10 −7 s.Ten, the acoustic waves are generated due to the diaphragm deformation and vibration of emitting pMUT, and the acoustic pressure distributions in GIS enclosure at diferent moments are presented in Figure 6.It shows that the propagation speed of acoustic waves inside the GIS enclosure is fast, and the direct waves reach the receiving pMUT within 1 μs.Te acoustic waves are difused and transmitted in the form of spherical wave, and the amplitude of sound pressure at the vertical direction of emitting pMUT is maximum.Since the pressure direction of the direct wave is parallel to the polarization direction of the piezoelectric layer, the induced charges are generated in receiving the pMUT piezoelectric layer and electric signals are output through the electrode.Te variations of the input and output electric signals obtained from acoustic-solid-pMUT coupling simulations are presented in Figure 7. Te results show that the direct acoustic wave arrives at the receiving pMUT at about 8 × 10 −7 s and the frequency of the simulated output electric signal is consistent with the input electric signal, while the amplitude of the output voltage (1.88 V) attenuates greatly as compared with the input voltage (50 V).Tis is mainly caused by the large deviation between the input excitation frequency (Figure 2), pMUT resonant frequency (see Table 8), and the low sensitivity of the sensor.In addition, since there exist mechanical losses when the piezoelectric layer vibrates, the energy would be attenuated during acoustic waves transmission.
In the numerical study of emitting pMUT, most researchers have strictly modeled pMUTs containing electrode structures, while some researchers have simplifed the   Mathematical Problems in Engineering electrode structures (top or bottom electrodes) by setting boundary conditions [17,23].In this article, the efect of the top electrode on the acoustic-solid-electric coupling simulation of pMUT is further discussed.Te simulation models of receiving pMUT with the unsimplifed top electrode and simplifed top electrode (using boundary conditions to replace the top electrode) are established, and the comparison of output electric signals for diferent models is presented in Figure 8. Te results show that after simplifying the top electrode, the output response of the receiving pMUT is obviously weakened, the attenuation speed of the output electric signal increases, and the output voltage amplitude decreases to 1.35 V, which is 28.2% lower than that of the model with the unsimplifed top electrode (1.88 V).Tis is mainly because when the upper surface of the PZT is set as the electrode boundary, it is equivalent to set the top electrode completely covered by the PZT membrane, and the energy loss is larger.In addition, due to top electrode simplifcation, the frst-order inherent frequency of the receiving pMUT obtained from numerical simulation changes to 330 kHz, which is 6.45% higher than the actual value (310 kHz).Terefore, simplifying the top electrode will lead to relatively large errors when simulating the acousticsolid-electric coupling transfer process of pMUT.
Ten, the sensing response characteristics of pMUT are analyzed.Te working frequency of the ultrasonic transducer is an important characteristic parameter.When pMUT acts as an acoustic receiver, the closer the vibration frequency of ultrasonic waves to the resonant frequency of membrane, the better response performance of the pMUT.1.2×10 -6 4.0×10 -7  Time (s) In the current study, the frst-, second-, and third-order modal shapes of receiving the pMUT piezoelectric layer are presented in Table 8, and their corresponding inherent frequencies are 310 kHz, 645 kHz, and 1042 kHz, respectively.Te membrane has the maximum amplitude under the frst mode, and the amplitude decreases with the increase of vibration mode order.In addition, the multipoint phase of the piezoelectric membrane is diferent under higher order modes, which is not conducive to the demodulation of ultrasonic signals.Terefore, controlling the frst-order inherent frequency of the receiving pMUT within the optimal frequency range of the ultrasonic signal from GIS discharge can improve its detection sensitivity.Besides, the efective electromechanical coupling coefcient (K 2 eff ) is also an important performance parameter of pMUT, which refects the coupling strength between mechanical and electrical energy through the piezoelectric efect as piezoelectric membrane vibrates.K 2 eff can be expressed by the resonant frequency (f r ) and antiresonant frequency (f a ), as formulated in equation ( 13) [24].When the pMUT piezoelectric membrane resonates, the impedance is minimum and the susceptance is maximum.When the Mathematical Problems in Engineering excitation frequency reaches the antiresonant frequency of membrane, the impedance is maximum and the susceptance is minimum.In this study, the admittance distributions of receiving pMUT at diferent frequencies are presented in Figure 9. Te results show that the resonant frequency of the pMUT is 302.2 kHz, the antiresonant frequency is 320.2 kHz, and the efective electromechanical coupling coefcient (K 2 eff ) is 10.93%.

Structural Optimization and Design of Receiving pMUT.
Based on the abovementioned analysis, it could be known that reducing the diference between the resonant frequency of receiving pMUT and the ultrasonic excitation frequency and increasing the efective electromechanical coupling coefcient of the sensor can improve detection sensitivity for GIS discharge.Te frequency of the ultrasonic signal generated by GIS breakdown discharge is generally between 20 kHz and 200 kHz, and more signals are concentrated around 80 kHz [25].Terefore, this section aims to design a receiving pMUT with a resonant frequency around 80 kHz and a large efective electromechanical coupling coefcient, which would have good response characteristics when detecting GIS discharge.Since the actual damping of the system is unknown at the design stage and the deviation between the resonant and inherent frequencies is small, the inherent frequency is studied instead of the resonant frequency during the process of structural optimization.Te infuence of piezoelectric layer radius (R), piezoelectric layer thickness (d), top electrode radius (R 0 ), and substrate thickness (d si ) on the inherent frequency (f 0 ) and the efective electromechanical coupling coefcient (K 2 eff ) of receiving pMUT are analyzed through fnite element simulations, and a relatively optimal pMUT structure is proposed for GIS discharge detection.First, the inherent frequencies and efective electromechanical coupling coefcients under diferent top electrode radii and diferent piezoelectric layer radii are calculated (d si � 100 μm and d � 40 μm), as presented in Figures 10(a 2 ) on the inherent frequency (f 0 ) is relatively small, while its impact on the efective electromechanical coupling coefcient (K 2 eff ) is relatively large.When the coverage area ratio of the top electrode is between 30% and 80%, the K 2 eff is high and has a maximum value.When the piezoelectric layer radius (R) is 2000 μm and the top electrode radius (R 0 ) is 1000 μm, the K 2 eff is maximum.Since increasing piezoelectric layer thickness (d) will decrease center static displacement of pMUT, in this study, the relationship between receiving sensitivity and piezoelectric layer thickness (d) of receiving pMUT is analyzed to determine the range of piezoelectric layer thickness (d).Te receiving sensitivity is represented by the ratio of the voltage at the output of pMUT to the input sound pressure at center of transducer receiving surface, and its relationship with piezoelectric layer thickness (d) is presented in Figure 11.It shows that when the piezoelectric layer thickness (d) is larger than 100 μm, the receiving sensitivity almost does not increase with the piezoelectric layer thickness (d).Terefore, in this study, only d ≤ 100 μm is considered.Ten, the substrate thickness (d si ) and piezoelectric layer thickness (d) are determined.When R � 2000 μm and R 0 � 1000 μm, the variations of the inherent frequency (f 0 ) for pMUT under Figure 13(a) shows that the inherent frequency (f 0 ) of pMUT is proportional to the piezoelectric layer thickness (d), and the f 0 is around 80 kHz when the substrate thickness (d si ) is 80 μm, 90 μm, and 100 μm with corresponding piezoelectric layer thickness (d) of 100 μm, 60 μm, and 20 μm, respectively.Figure 13(b) shows that the efective electromechanical coupling coefcient (K 2 eff ) of pMUT increases with piezoelectric layer thickness (d) and its increasing rate decreases gradually.When the substrate thickness (d si ) is 80 μm and piezoelectric layer thickness (d) is 100 μm, the efective electromechanical coupling coefcient (K 2 eff ) is maximum, which is 22.59%.Terefore, when the piezoelectric layer radius (R) is 2000 μm, top electrode radius (R 0 ) is 1000 μm, piezoelectric layer thickness (d) is 100 μm, and substrate thickness (d si ) is 80 μm, and the receiving pMUT structure should be optimum.For this case, the inherent frequency (f 0 ) of receiving pMUT is 80.16 kHz, which is suitable for ultrasonic signal detection of GIS breakdown discharge, and the efective 12 Mathematical Problems in Engineering electromechanical coupling coefcient (K 2 eff ) is 22.59%, which is 106.7% higher than the result before structural optimization (K 2 eff �10.93%).

Conclusions
In the present article, the ultrasonic-solid-pMUT coupling simulation model was established to investigate the propagation process of ultrasonic waves in GIS enclosure and the conversion between ultrasonic and electric signals through pMUT.Te output electric signals for the unsimplifed top electrode and simplifed top electrode model were compared, and the structure of receiving pMUT was optimized according to the main ultrasonic frequency range of GIS breakdown discharge and the key performance parameters of pMUT.Te major fndings are as follows: (1) In the present ultrasonic-solid-pMUT coupling simulations, the frequency of the simulated output electric signal is consistent with the input electric signal, while the amplitude of the output voltage (1.88 V) attenuates greatly as compared with the input voltage (50 V).During numerical simulations, simplifying the top electrode will lead to relatively large errors.Te output response of simplifed pMUT is obviously weakened, the attenuation rate of the output electric signal increases, and the output voltage amplitude further decreases.Furthermore, simplifying top electrode will also lead to a higher inherent frequency of pMUT as compared with the actual value.
(2) Both the efects of piezoelectric layer radius (R) and substrate thickness (d si ) on the inherent frequency (f 0 ) of pMUT are remarkable.Te f 0 is inversely proportional to R, while directly proportional to d si .
When the coverage area ratio of the top electrode is between 30% and 80%, the efective electromechanical coupling coefcient (K 2 eff ) is high and has a maximum value.When the piezoelectric layer thickness (d) is larger than 100 μm, the receiving sensitivity nearly does not increase with the piezoelectric layer thickness.For the optimum pMUT structure, the inherent frequency (f 0 ) of receiving pMUT is 80.16 kHz, which is suitable for ultrasonic signal detection of GIS breakdown discharge, and the efective electromechanical coupling coefcient (K 2 eff ) is 22.59%, which is 106.7% higher than the result before structural optimization (K 2 eff � 10.93%).Density (kg/m 3 ).ρ v : Volume electric charge density (C/m 3 ).

Figure 4 :
Figure 4: Physical model and geometric dimensions for the pMUT validation model [15].

Figure 5 :
Figure 5: Comparison of the sound pressure level distribution for present simulation and experiment results [15].

Figure 6 :
Figure 6: Acoustic pressure distributions in GIS enclosure at diferent moments.

Figure 7 :
Figure 7: Variations of the input and output electric signals of pMUT.

Table 8 :
Te frst-, second-, and third-order modal shapes and inherent frequencies of receiving pMUT.
) and 10(b), respectively.Te results show that the efect of the piezoelectric layer radius (R) on the inherent frequency (f 0 ) of pMUT is remarkable.With the increase of the membrane and substrate area, the inherent frequency of pMUT decreases gradually.When the piezoelectric layer radius (R) is 2000 μm, the inherent frequency of pMUT is around 80 kHz.Te impact of the coverage area of the top electrode (πR 0

Figure 8 :
Figure 8: Comparison of output electric signals for unsimplifed and simplifed top electrode models.

Table 4 :
Te central displacement amplitude (u max ) and output voltage amplitude (V max ) in acoustic-solid-pMUT coupling simulation for diferent computational grids.