A broadband matching design method for spherical transducers based on finite element modeling

Abstract

The present invention relates to a broadband matching design method for a spherical transducer based on finite element modeling. By using a finite element model in combination with the full electrical equivalent circuit of the transducer, the full electrical equivalent circuit parameters of the spherical transducer are calculated. Then, according to the broadband matching theory, the broadband matching design of the transducer is completed, effectively improving the broadband emission performance of the transducer. Finally, a spherical transducer and its broadband matching network are designed and fabricated. Experiments verify that the broadband matching network designed by this method can effectively increase the emission bandwidth of the spherical transducer and has a high emission voltage response within the working frequency band. The present invention can, through only modeling and simulation calculations, use the finite element method to obtain the full electrical equivalent circuit parameters of the spherical transducer and complete the broadband matching design of the transducer according to the full electrical equivalent circuit model. It can effectively broaden the emission bandwidth of the spherical transducer and improve the broadband emission performance of the spherical transducer.

Description

Technical Field

[0001] The present invention belongs to the field of underwater acoustic transducers, and relates to a broadband matching design method for spherical transducers based on finite element modeling. Background Art

[0002] In the field of underwater acoustic communication, sonars are required to have working characteristics such as high efficiency, broadband, and omnidirectional sound radiation. As the front-end part of a sonar, an underwater acoustic transducer is a key device determining the system performance. This requires that while obtaining a relatively large power, the transmitting voltage response within the band of the underwater acoustic transducer is as flat as possible.

[0003] Generally, techniques such as longitudinal-bending coupling vibration, adding matching layer materials, dual-excitation mode coupling, and impedance matching are used to expand the working bandwidth of the transducer. The longitudinal-bending coupling technique utilizes the multi-mode coupling of the longitudinal vibration and bending vibration of the transducer to expand the bandwidth. The matching layer technique increases the working bandwidth of the transducer by adding one or more matching layers on the front cover plate of the transducer. The impedance matching method adjusts the input impedance of the transducer by using reactive elements, so that the transducer is close to a pure resistance state within a relatively wide frequency range near the resonant frequency, achieving the purpose of expanding the bandwidth of the transducer.

[0004] The longitudinal-bending coupling technique, the matching layer technique, the dual-excitation mode coupling technique, etc. have high requirements for the manufacturing process of the transducer, the transducer structure is complex, and the expanded bandwidth is relatively limited. However, with the development of underwater communication technology, high-resolution imaging technology, and wideband interference technology, the requirements for the bandwidth of underwater acoustic transducers in engineering are getting higher and higher. And the narrowband matching network designed based on the single-frequency point impedance of the transducer also has certain limitations. Therefore, the present invention takes the spherical transducer as the research object, uses the finite element method, combines the full electrical equivalent circuit model of the spherical transducer and the transducer matching theory, and proposes a broadband matching design method for spherical transducers based on finite element modeling, which can effectively broaden the transducer bandwidth and improve the broadband transmitting performance of the transducer. Summary of the Invention

[0005] Technical Problems to be Solved

[0006] In order to avoid the deficiencies of the prior art, the present invention proposes a broadband matching design method for spherical transducers based on finite element modeling.

[0007] Technical Solution

[0008] A broadband matching design method for spherical transducers based on finite element modeling, characterized by the following steps:

[0009] Step 1: Establish a two-dimensional axisymmetric finite element model of half of the spherical transducer in water by the finite element method; apply a load, i.e., an excitation voltage, to the piezoelectric ceramic spherical shell of the spherical transducer model, extract the charge on the electrode nodes of the transducer, and numerically calculate the admittance characteristics of the transducer:

[0010]

[0011] G = Re[Y]

[0012] B = Im[Y]

[0013] where Y is the admittance, ω is the angular frequency, Q is the charge at the electrode, V is the load applied to the transducer electrode, G is the conductance, and B is the susceptance;

[0014] Extract the sound pressure radiated by the transducer in water and numerically calculate the transmit voltage response:

[0015]

[0016] where p a is the sound pressure at a distance r from the equivalent acoustic center of the transducer, r a is the distance from the equivalent acoustic center, and V is the load applied to the transducer electrode;

[0017] Change the transducer size, simulate the admittance characteristics and transmit voltage response of the spherical transducer under different radii and spherical shell thicknesses, repeat Step 1, numerically calculate the admittance characteristics and transmit voltage response of the transducer, and find the transducer size parameters with the best broadband emission performance;

[0018] Step 2: According to the admittance characteristics of the transducer obtained in Step 1 and the full electrical equivalent circuit model of the spherical transducer, solve for the static capacitance C 0 , dynamic resistance R d , dynamic inductance L d and dynamic capacitance C d :

[0019] The full electrical equivalent circuit model of the spherical transducer is: the dynamic resistance R d , dynamic capacitance C d and dynamic inductance L d are connected in series in sequence and then connected in parallel with the static capacitance C 0 , then the admittance of the dynamic branch is:

[0020]

[0021] On the basis of the series inductance L s matching network of the resonance point matching method, parallel capacitor C P and series capacitor C s, the spherical transducer full - electrical equivalent circuit model after matching is obtained, then:

[0022] The total reactance X of the system e is:

[0023]

[0024] When the system is in resonance, the impedance of the system is:

[0025]

[0026] In the formula, C′ = C 0 + C p , ω s is the resonance angular frequency of the transducer. When the series inductance in the loop

[0027] Step 3: Establish a spherical transducer full - electrical equivalent circuit model based on a broadband matching network in the electrical simulation software OrCAD, optimize the values of the series inductance L s , the series capacitance C s and the parallel capacitance C p to construct two resonance peaks with similar impedance modulus values, and obtain the optimized transducer size.

[0028] A manufacturing method of the spherical transducer based on finite - element modeling, characterized in that: electrodes are coated on the inner and outer surfaces of two hemispherical shells, wires for leading out the internal electrodes are led out, a small groove is arranged along the diameter direction of the spherical shell, after the wires are led out, high - strength epoxy resin is used for sealing at the wire outlet, the spherical transducer is placed in a potting mold for potting treatment to complete the production of the spherical transducer.

[0029] Beneficial effects

[0030] A broadband matching design method of a spherical transducer based on finite - element modeling proposed by the present invention uses a finite - element model combined with the full - electrical equivalent circuit of the transducer to calculate the full - electrical equivalent circuit parameters of the spherical transducer, and then completes the broadband matching design of the transducer according to the broadband matching theory, effectively improving the broadband emission performance of the transducer. Finally, a spherical transducer and its broadband matching network are designed and manufactured, and experiments verify that the broadband matching network designed by this method can effectively increase the emission bandwidth of the spherical transducer and has a high emission voltage response within the working frequency band range.

[0031] The basic principle and design method of the present invention are verified by computer simulation and anechoic tank experiments. The results show that:

[0032] 1. The broadband matching design method for spherical transducers proposed by the present invention can obtain the full electrical equivalent circuit parameters of spherical transducers by using the finite element method through only modeling and simulation calculations, and complete the broadband matching design of the transducers according to the full electrical equivalent circuit model.

[0033] 2. The broadband matching design method for spherical transducers proposed by the present invention can effectively broaden the emission bandwidth of spherical transducers and improve the broadband emission performance of spherical transducers. Description of the Drawings

[0034] Figure 1 : Schematic diagram of the broadband matching design idea of the transducer

[0035] Figure 2 : Full electrical equivalent circuit of the spherical transducer

[0036] Figure 3 : Full electrical equivalent circuit of the spherical transducer based on the broadband matching network

[0037] Figure 4 : Simulation results of the spherical transducer based on the broadband matching network

[0038] (a) Simulation results of adjusting the series inductance (b) Simulation results of adjusting the series and parallel capacitances

[0039] Figure 5 : Pool experiment measurement system

[0040] Figure 6 : Spherical transducer and water area mesh division diagram

[0041] (a) Spherical transducer mesh division diagram (b) Partial view of the water area model mesh division

[0042] Figure 7 : Simulation results of the admittance and transmitting voltage response of the spherical transducer in water

[0043] Figure 8 : Comparison diagram of the calculation results between the finite element method and the full electrical equivalent circuit model

[0044] (a) Admittance curve (b) Impedance curve

[0045] Figure 9 : Electro-acoustic performance of the spherical transducer based on the broadband matching network

[0046] (a) Admittance curve (b) Impedance curve

[0047] Figure 10 : Spherical transducer assembly

[0048] Figure 11 : Bonding model diagram of the spherical transducer

[0049] (a) Mold model diagram (b) Assembly model diagram

[0050] Figure 12 : Sphere transducer potting mold model diagram

[0051] Figure 13 : Sphere transducer prototype

[0052] Figure 14 : Sphere transducer admittance curve

[0053] (a) Admittance curve in air (b) Admittance curve in water

[0054] Figure 15 : Sphere transducer transmit voltage response comparison diagram Specific implementation manner

[0055] Now, in combination with the embodiments and the attached drawings, the present invention will be further described:

[0056] The design idea of broadband matching of the sphere transducer proposed by the present invention is as Figure 1 shown. Through the finite element method, a simulation calculation is carried out on the sphere transducer to obtain the admittance characteristics of the sphere transducer in water, solve the full electrical equivalent circuit parameters of the transducer, construct the full electrical equivalent circuit model of the transducer, design and calculate the broadband matching network parameters of the transducer. Finally, the broadband performance of the sphere transducer based on the matching network is experimentally measured.

[0057] 1. Finite element modeling and simulation of the sphere transducer and size optimization

[0058] Since the sphere transducer is a three-dimensional rotationally symmetric structure, ignoring the influence of the bonding layer with less influence on the transducer performance, a two-dimensional axisymmetric finite element model of half of the sphere transducer in water is established by the finite element method, and a simulation calculation is carried out on the sphere transducer model to study the electro-acoustic performance of the broadband sphere transducer. Modal analysis and harmonic response analysis are carried out on the sphere transducer model to obtain the admittance characteristics and transmit voltage response of the transducer, and the size parameters such as the radius and wall thickness of the sphere transducer are optimized according to the simulation results to expand the working bandwidth of the sphere transducer and improve the transmit voltage response of the sphere transducer.

[0059] A simulation calculation is carried out on the sphere transducer model to study the electro-acoustic performance of the broadband sphere transducer. A load is applied on the piezoelectric ceramic spherical shell, that is, a certain excitation voltage is applied, and modal analysis and harmonic response analysis are carried out on the sphere transducer model to extract the charge on the electrode nodes of the transducer and the sound pressure radiated by the transducer in water. The admittance characteristics and transmit voltage response of the transducer can be obtained through numerical calculation.

[0060]

[0061] G = Re[Y]

[0062] B = Im[Y]

[0063] Where Y is the admittance, ω is the angular frequency, Q is the charge at the electrode, V is the load applied to the transducer electrode, G is the conductance, and B is the susceptance.

[0064] Extract the sound pressure radiated by the transducer in water, and the transmitting voltage response can be calculated by Equation (2):

[0065]

[0066] Where p a is the sound pressure at a certain distance from the equivalent acoustic center of the transducer, r a is the distance from the equivalent acoustic center, and V is the load applied to the transducer electrode.

[0067] Change the transducer size, simulate the admittance characteristics and transmitting voltage response of the spherical transducer under different radii and spherical shell thicknesses, observe the influence of different sizes on the resonant frequency and transmitting bandwidth of the spherical transducer, and find the transducer size parameters with the best broadband transmitting performance.

[0068] 2. Calculation of broadband matching circuit parameters of spherical transducers based on the finite element method

[0069] The full electrical equivalent circuit model of the spherical transducer is as Figure 2 shown. Combining the admittance characteristics of the spherical transducer in water obtained by the finite element method, solve the static capacitance C 0 , dynamic resistance R d , dynamic inductance L d and dynamic capacitance C d .

[0070] The dynamic branch is composed of the dynamic inductance L d , dynamic capacitance C d and dynamic resistance R d connected in series, and its dynamic branch admittance is:[[]]

[0071]

[0072] By adding a parallel capacitance C s and a series capacitance C P on the basis of the series inductance L s matching network of the resonance point matching method, construct two resonant peaks with similar impedance modulus values, reduce the fluctuations between the peaks and valleys and the impedance modulus within the working frequency band, and improve the broadband high-power transmitting performance of the spherical transducer in the frequency range near the resonance point. The full electrical equivalent circuit model of the matched spherical transducer is as Figure 3 shown.

[0073] At this time, the total reactance X of the system e is:

[0074]

[0075] When the system is in resonance, the impedance of the system is:

[0076]

[0077] In the formula, C′ = C 0 + C p , ω s is the resonant angular frequency of the transducer. When the series inductance in the loop , the active resistance

[0078] Then, a full electrical equivalent circuit model of the spherical transducer based on the broadband matching network is established in the electrical simulation software OrCAD. The values of the series inductance L s , the series capacitance C s and the parallel capacitance C p are optimized to construct two resonant peaks with similar impedance modulus values. By analyzing the influence of the broadband matching network on the normalized response curve of the system, the normalized response fluctuation and tuning bandwidth between the two resonant peaks are comprehensively adjusted to maximize the broadband emission performance of the spherical transducer. The results of the optimized simulation are as Figure 4 shown.

[0079] 3. Fabrication of the broadband spherical transducer prototype

[0080] According to the optimized transducer size design, the piezoelectric ceramic element of the spherical transducer, as well as the molds for bonding and potting, are designed to fabricate the spherical transducer prototype. The spherical transducer is mainly composed of two hemispherical shells, and electrodes are coated on both the inner and outer surfaces of the spherical shells. At the same time, to lead out the wires of the internal electrodes, a small groove is set along the diameter direction of the spherical shell. After the wires are led out, high-strength epoxy resin is used to seal the wire outlet to prevent gaps from affecting the vibration mode of the spherical transducer. Finally, the spherical transducer is placed in the potting mold, and after the position is determined, potting treatment is carried out to complete the fabrication of the spherical transducer prototype. Finally, based on the broadband matching network designed above, the appropriate matching capacitance and matching inductance are calculated respectively based on the finite element model and the measured transducer admittance to complete the design of the transducer broadband matching network.

[0081] 4. Performance measurement of the spherical transducer in an anechoic tank based on the broadband matching network

[0082] The fabricated spherical transducer and the designed broadband matching network are experimentally measured in an anechoic tank. An anechoic tank experimental measurement system as shown in the figure is built to measure the admittance characteristics and the transmitting voltage response of the spherical transducer before and after matching, and to analyze the resonant frequency and the transmitting bandwidth characteristics of the transducer. Specific embodiments:

[0084] (1) Finite element modeling and simulation of the spherical transducer and size optimization

[0085] Since the spherical transducer is a symmetric structure with three-dimensional rotational symmetry, the influence of the bonding layer with relatively small influence on the transducer performance is ignored, and a two-dimensional axisymmetric geometric model of half of the spherical transducer in water is established. Water areas are added around the transducer model to analyze the vibration characteristics of the spherical transducer in water. Fluid-structure interaction boundary conditions are added to the transducer structure and the surrounding fluid domain to simulate the interaction between the fluid and the solid, and an acoustic absorption boundary is added to the outer periphery of the water area. Figure 6 It is a two-dimensional axisymmetric finite element model after mesh generation.

[0086] The harmonic response analysis is carried out on the finite element model of the spherical transducer. Through numerical calculation, the performance such as the admittance in water and the transmitting voltage response of the spherical transducer can be obtained, so as to optimize the design of the size parameters of the spherical transducer. The thickness of the optimized spherical transducer housing is 3 mm, and the average radius is 41.5 mm. The admittance and transmitting voltage response results of the transducer are as Figure 7 shown. It can be seen from the simulation results that the resonant frequency of the spherical transducer in water is 20.6 kHz, and the corresponding conductance peak value is 8.07 mS. The transmitting voltage response of the spherical transducer at the resonant frequency is 149.81 dB, and the -3 dB bandwidth is 9.84 kHz.

[0087] (2) Full electrical equivalent circuit model of the spherical transducer and broadband matching circuit design

[0088] According to the admittance curve of the spherical transducer in water, the static capacitance C 0 = 59.78 nF, the dynamic resistance R d = 123.87 Ω, the dynamic inductance L d = 2 mH, and the dynamic capacitance C d = 29.91 nF of the spherical transducer in water are calculated. The admittance curve and impedance curve of the full electrical equivalent circuit model of the spherical transducer can be obtained and compared with the finite element calculation results of the spherical transducer, as Figure 8 shown.

[0089] It can be seen that the admittance curve and impedance curve obtained by using the full electrical equivalent circuit model of the spherical transducer are consistent with the finite element calculation results. Therefore, we combine the full electrical equivalent circuit model of the spherical transducer to obtain the circuit parameters of the broadband matching network of the spherical transducer. When the matching network is designed based on the finite element method, the series inductance L s = 0.7 mH, the series capacitance C s = 702 nF, and the parallel capacitance C p = 6.8 nF. The admittance and impedance curves of the spherical transducer based on the broadband matching network are obtained by simulation calculation using the electrical simulation software OrCAD as shown in Figure 9 .

[0090] At this time, due to the introduction of the matching network of the spherical transducer, two resonance frequencies appear in the admittance curve and impedance curve of the transducer. Among them, when the two resonance frequencies of the admittance curve are 17 kHz and 28.6 kHz respectively, the corresponding conductance peaks are 23.19 mS and 21.7 mS. When the two resonance frequencies in the impedance curve are 16.4 kHz and 29.6 kHz respectively, the corresponding impedance modules are 40.87 Ω and 42.08 Ω.

[0091] (3) Fabrication of the broadband spherical transducer prototype

[0092] According to the optimized design dimensions of the transducer, the piezoelectric ceramic hemispherical shell required for the spherical transducer is designed and processed, as shown in Figure 10 . The outer diameter of the spherical transducer is 86 mm.

[0093] After the transducer components are designed, the spherical transducer is bonded and the wires are welded. First, the inner and outer surfaces of the two hemispherical shells are welded together through silver wires to form a whole, and a wire is led out from the groove of the spherical shell from the inside. Then, the spherical transducer is bonded with high-strength epoxy resin glue. In order to ensure the vibration consistency of the transducer, it is necessary to bond it firmly. The bonding mold is designed using the SolidWorks modeling software as shown in Figure 11 . The spherical groove of the mold has the same outer diameter size as the spherical transducer. The mold is fixed by bolts and nuts to prevent the two spherical half shells from being misaligned during bonding and to squeeze out the excess epoxy resin glue. After the glue reaches the maximum bonding strength, the two outer surfaces of the spherical shell are connected, and another wire is led out from the outer surface of the spherical transducer.

[0094] After the transducer bonding is completed, the spherical transducer is connected to the cable. After measuring the admittance of the transducer in air and confirming that it is correct, the spherical transducer is potted. According to the structural characteristics of the spherical transducer, the one shown in Figure 12The transducer potting mold shown. It adopts a split symmetrical structure and pottings the spherical transducer by the way of secondary potting. In order to reduce the influence of the potting layer on the electroacoustic performance of the spherical transducer, the thickness of the potting layer is determined to be 5 mm. The inner side of the mold is evenly sprayed with a release agent to facilitate demolding. After the mold is fixed with bolts, the gaps at the joints of the mold are sealed with waterproof silicone grease to prevent the potting material from flowing out of the gaps and affecting the potting effect. At the same time, in order to prevent the appearance of air cavities after the potting material is cured, the potting material should be slowly injected along one side of the inner wall of the potting port as much as possible during the potting process.

[0095] The prototype of the spherical transducer after potting is as Figure 13 shown.

[0096] (4) Performance measurement of the broadband spherical transducer in an anechoic tank based on the matching network

[0097] Use an impedance analyzer to measure the admittance of the spherical transducer prototype in air and water. The measurement results are as Figure 14 shown. It can be seen from the measurement results that the resonance frequencies of the transducer in air are 20.85 dB respectively, and the corresponding conductance peaks are 0.981 mS respectively; the resonance frequencies of the transducer in water are 20.2 kHz respectively, and the corresponding peaks are 8.70 mS respectively; the measured results are basically in agreement with the simulation results. Based on the measured admittance of the spherical transducer, a full electrical equivalent circuit model is constructed. Among them, the static capacitance C 0 = 57.89 nF, the dynamic resistance R d = 114.96 Ω, the dynamic inductance L d = 2.1 mH, the dynamic capacitance C d = 29.86 nF. In the broadband matching network of the spherical transducer based on the measured admittance, the series inductance L s = 0.68 mH, the series capacitance C s = 950 nF, and the parallel capacitance C p = 13.6 nF.

[0098] Place the spherical transducer prototype at a depth of 2 m in the anechoic tank, and measure the electroacoustic performance of the spherical transducer prototype based on the broadband matching network in the anechoic tank. The experimental measurement system mainly consists of an experimental computer, a signal transmitting and collecting device, an ATA-L8 power amplifier, a B&K8104 standard hydrophone, and the spherical transducer to be measured, etc. The measurement system is as Figure 4 shown. The comparison diagram of the measured transmitting voltage response of the spherical transducer with the broadband matching network based on two methods and the transmitting voltage response of the spherical transducer before matching is as Figure 15 shown.

[0099] When the spherical transducer prototype is unmatched, the maximum measured transmitting voltage response is 149.2 dB, and the corresponding resonance frequency is 19 kHz. When the broadband matching network is designed based on the finite element method, the maximum transmitting voltage response of the spherical transducer is 152.3 dB, and the corresponding frequency is 21.1 kHz. The fluctuation of the transmitting voltage response is less than 3 dB in the frequency range of 14.2 k - 30.9 kHz, and the transmitting voltage response of the spherical transducer is greater than 150 dB in the frequency range of 14.5 k - 30.4 kHz. When the broadband matching network is designed based on the measured admittance, the maximum transmitting voltage response of the spherical transducer is 152.9 dB, and the corresponding frequency is 21.3 kHz. The fluctuation of the transmitting voltage response is less than 3 dB in the frequency range of 14.1 kHz - 29.7 kHz.

[0100] It can be seen from the experimental results that the matching network designed based on the finite element model proposed in the present invention can expand the transmitting bandwidth of the spherical transducer and enable the spherical transducer to obtain excellent broadband transmitting performance.

Claims

1. A broadband matching design method for a spherical transducer based on finite element modeling, characterized in that the steps are as follows: Step 1: Establish a two-dimensional axisymmetric finite element model of half of the spherical transducer in water through the finite element method; apply a load, i.e., an excitation voltage, to the piezoelectric ceramic spherical shell of the spherical transducer model, extract the charge on the transducer electrode nodes, and numerically calculate the admittance characteristics of the transducer: G = Re[Y] B = Im[Y] where Y is the admittance, ω is the angular frequency, Q is the charge at the electrode, V is the load applied to the transducer electrode, G is the conductance, and B is the susceptance; Extract the sound pressure radiated by the transducer in water and numerically calculate the transmit voltage response: where p a is the sound pressure at a distance from the equivalent acoustic center of the transducer, r a is the distance from the equivalent acoustic center, and V is the load applied to the transducer electrodes; Change the transducer size, simulate the admittance characteristics and transmit voltage response of the spherical transducer under different radii and spherical shell thicknesses, repeat Step 1, numerically calculate the admittance characteristics and transmit voltage response of the transducer, and find the transducer size parameters with the best broadband emission performance; Step 2: Solve for the static capacitance C, dynamic resistance R, dynamic inductance L, and dynamic capacitance C of the spherical transducer according to the admittance characteristics of the transducer obtained in Step 1 and the full electrical equivalent circuit model of the spherical transducer: 0 , dynamic resistance R d , dynamic inductance L d and dynamic capacitance C d : The all-electric equivalent circuit model of the spherical transducer is as follows: dynamic resistance R d , dynamic capacitance C d , and dynamic inductance L d are connected in series in sequence and then connected in parallel with static capacitance C 0 . Then the admittance of the dynamic branch is as follows: Based on the series inductance L of the resonance point matching method s On the basis of the matching network, a parallel capacitor C P and a series capacitor C s are obtained to get the fully electric equivalent circuit model of the matched spherical transducer, then: The total reactance X of the system e is as follows: When the system resonates, the impedance of the system is: where C′ = C 0 + C p , ω s is the resonant angular frequency of the transducer. When the series inductance in the circuit is , the active resistance is Step 3: Establish a full electrical equivalent circuit model of the spherical transducer based on the broadband matching network in the electrical simulation software OrCAD, and optimize the values of the series inductance L s , the series capacitance C s and the parallel capacitance C p to construct two resonant peaks with similar impedance modulus values, and obtain the optimized size of the transducer.

2. A manufacturing method for the spherical transducer based on finite element modeling according to claim 1, characterized in that: Both the inner and outer surfaces of the two hemispherical shells are coated with electrodes, lead wires for the internal electrodes are led out, a small groove is provided along the diameter direction of the spherical shell, after the lead wires are led out, high-strength epoxy resin is used for sealing at the wire outlet, the spherical transducer is placed in a potting mold for potting treatment to complete the manufacture of the spherical transducer.

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