I. Introduction

One of the most important requirements in ultrasonic (UT) systems for non-destructive testing and medical diagnostics is a high axial resolution (resolution in depth). This requirement can be met if piezoelectric transducer (transmitter-receiver) used in the system is a wide-band probe or, in other words, can transmit and receive short UT pulses. Many papers, books and patents have addressed this problem due to significant practical benefits of the wide-band UT systems providing short pulses. Moreover, the wide-band transducer can be used in one case as a low-frequency probe and in other case as a high-frequency probe, depending on the requirements.

Various types of wide-band UT transducers have been designed to address the need for high axial resolution. The most commonly used approach in wide-band probe design is to employ a backing (mechanical damper) with the same acoustic impedance that piezoelement has [1-5]. It considerably damps the oscillations of the half-wave resonant piezoelement, allows transmitting and receiving the short UT pulses, and widens the transducer frequency band. However, there is always some mismatch between acoustic impedances of the piezoelement and even the best damper. Moreover, since there are no ideal connections, there is always an interface damper/piezoelement. Subsequently, any mechanically damped probe always has the internally reflected acoustic waves, which increase UT pulse duration and narrow transducer frequency band. At present, due to these reasons, the best wide-band probes excited by short half-cycle electric pulse (shock-wave or spike or d -function excitation) give in pulse-echo (PE) mode the signal consisting of approximately three cycles at –20dB amplitude level [1-4]. It will be further shown that due to physical reasons, even in the "ideal" case, when acoustic impedances of the damper and piezoelement are equal and there is absolutely no interface between them, the shortest signal in PE mode consists of three half-cycles [4].

Alternatively, the wide-band transducers can be based on the electrical damping. Connecting R-, L- and C-circuits to piezoelement one can widen frequency band of the probe.

Acoustically matching layers between piezoelement and working media decrease acoustic wave reflection inside the transducer and increase the bandwidth [5-6].

Method of electrical compensation, based on the transducer excitation by electrical signal of complex shape, allows transmitting and receiving rather short UT pulses, but requires special electrical generator. In general, methods of electrical damping, acoustic matching and electrical compensation are less efficient than mechanical damping [1-4].

Resonant piezoelements with a variable thickness (wedge-shaped, spherically concave, etc) provide rather wide frequency band, but have low efficiency, sensitivity and resolution [4].

The so-called "thick" piezoelements (cylinders, where propagation time of the UT pulse along piezoelement thickness is much greater than pulse duration) are able to transmit and receive short UT pulses [7-11]. However, thick piezoelements cannot generate a single pulse, they always create a sequence of pulses as a response to the single pulse input. It will be explained below.

The non-uniformly polarized piezoelements [4, 11] and different types of piezoelements with an inhomogeneous electric field, including surface-excited transducers, were investigated and described in [4, 12-23]. Such probes provide the widest frequency band and the best ability to transmit and receive short UT pulses.

Each type of wide-band transducers has advantages and disadvantages. The review of different types of probes shows that piezotransducers with non-uniform electric fields are probably the most promising in providing broad frequency band and ability to generate short pulses. In addition, these UT probes are relatively simple, inexpensive and can be manufactured in a variety of sizes to match the application. This paper presents an analysis of wide-band piezoelements with inhomogeneous electric field.

II. General Theory

The general description of piezoelement can be obtained by using system of equations [24] consisting of the Maxwell equations for dielectric (without free charges and magnetic fields), the equation of motion, and the equations of piezo-effect:

(1)

where is the vector of electric displacement; is the vector of electric field, j is the electrostatic potential; T_ik is the tensor of mechanical stress; u_i is the mechanical displacement; r is the density of piezoelectric material, is the tensor of mechanical deformation (strain); l_iklm is the tensor of elastic stiffness constants of the material; S_ijkl is the tensor of elastic compliance constants of the material and and d_ikl are the tensors of piezoelectric constants of the material and is the free space (vacuum) dielectric constant; e_ik is the permittivity tensor of the piezoelectric material; x_i are the coordinates; t is the time.

Solving system (1) with corresponding boundary conditions (mechanical and electric), one can determine time and spatial distributions of the electric potential and field and the mechanical displacement, strain and stress within piezoelement. As a result, it will be possible to find the correlations between piezotransducer input and output signals. Note that the equations in system (1) do not describe nonlinear effects in piezo-materials, electromagnetic forces, dispersion, thermal and magnetic effects, etc. However, even with these simplifications, the solution of the system in general case is a very complex mathematical problem, and it can be solved only for some particular cases. To further simplify the system of equations, we apply a compressed matrix notation for piezoelectric constants and a scalar notation for dielectric and elastic constants. For the most commonly used PZT piezo-ceramic, only the following piezoelectric moduli are not equal to zero in the Cartesian coordinates:

(2)

Substituting these constants into third and forth equations in (1), we obtain the following equations for mechanical displacement u_i in the Cartesian coordinates (x,y,z) for 3D-piezoelement working in the transmission mode:

(3)

where r = 7.6g/cm³ is the density of PZT-5H piezo-ceramic, is the Young modulus, c₁₁ and c₃₃ are the components of compressed matrix of the elastic constants, e₃₃ = 23.3 C/m², e₁₃ = -6.5 C/m²,
e₁₅ = e₂₄= 17 C/m² are the piezoelectric constants of PZT-5H ceramic.

Equations (3) determine time and spatial distributions of three components of the mechanical displacement u_i within the piezoelement in the transmission mode depending on time and spatial distributions of the exciting electric field E_i.

Substituting (2) into first, second and seventh equations in (1), we obtain the following equations for electrical potential j in the Cartesian coordinates (x,y,z) for 3D-piezoelement working in the reception mode:

(4)

where e = e₃₃ =3200 is the dielectric constant (permittivity) of PZT piezo-ceramics.

Equations (4) determine time and spatial distributions of the electric potential j within piezoelement in the reception mode depending on time and spatial distributions of the exciting strain (deformation) field S_ik.

For 2D-case in the Cartesian coordinates or 3D axi-symmetric case in the cylindrical coordinates the equations (3) and (4) will have the simpler form. For instance, for the most commonly used axi-symmetric cylindrical piezoelements we obtain, instead of (3), the following equations in transmission mode in the cylindrical coordinates (z,r) where z is the axial direction and r is the radial direction

(5)

In the reception mode we obtain for the same axi-symmetric piezoelements in cylindrical coordinates, instead of (4), the following equation

(6)

Now we can analyze equations (3)-(4) in general 3D case in the Cartesian coordinates or equations (5)-(6) in axi-symmetric case in the cylindrical coordinates for transmission and reception modes.

III. Transmission Mode

Equations (3) and (5), describing a piezoelement working in the transmission mode, are the wave equations with respect to mechanical displacement u_i of the output acoustic waves created by the input exciting electric field. The first and second equations in (5) describe time and spatial distributions of u_z and u_r displacements, respectively. The fact, that both equations have terms with different displacements in the left-hand sides, is related to internal continuity of the elastic media (piezoelement body). For example, the longitudinal extension (compression) leads to the transverse compression (extension) and vice versa [25]. Such changes are connected by the Poisson’s ratio and present the so-called "barrel-shaped effect".

Terms in the right-hand sides of equations (5) are the driving forces (from mathematical point of view) or the "electrically created mechanical forces", the sources of the output acoustic waves (from physical point of view). Now let us analyze four sources of the acoustic waves in (5), since this analysis is fully applicable to equations (3). These four sources of acoustic wave (the right-hand sides in equations (5)) are represented by four derivatives:

(7)

Since all sources (7) of the acoustic waves are the partial derivatives with respect to coordinates, they all reach maximum in the areas, where the piezoelectric constants e_ik or the exciting electric field E_i have the discontinuities. Source #1 in (7) creates in the cylindrical piezoelement the axial longitudinal acoustic waves with axial mechanical displacement u_z. In a standard piezoelectric disk with two electrodes on the bases, this source reaches maximum value at two bases of the cylinder, where the derivative with respect to z has discontinuity. Source #2 creates shear radial waves with axial mechanical displacement u_z. It has maximum at the cylinder side surface. Source #3 creates shear axial waves with radial mechanical displacement u_r. It has maximum at two bases of the cylinder. Source #4 creates longitudinal radial waves with radial mechanical displacement u_r. It has maximum at the cylinder side surface.

It is clear, that in a standard piezoelectric cylinder with two electrodes on the bases, only axial component E_z of the electric field exists. It means, that only sources #1 and #4 work in such a piezoelement. However, source #4 creates the longitudinal radial waves, which in a typically thin piezoelectric disk are rather small and practically do not affect the piezoelement performance. As a result, only source #1 creates significant UT signals (longitudinal axial waves) in a standard piezoelement. Since within such a piezoelement with two electrodes on the bases, the electric field E_z is almost uniform, source #1 reduces only to two flat sources of UT waves, located at the cylinder faces.

These two sources are very wide-band, i.e. in response to short input exciting electric pulse they generate similar short acoustic pulse. However, these UT pulses generated on the piezoelement faces, propagate in two directions: inside the piezoelement and into the outside medium. Pulses, propagating inside the piezoelement, undergo the multiple reflections from the faces. It explains why thick piezoelement responds to a single input electric pulse with a whole sequence of output UT pulses.

Let us now determine the correlation between output acoustic parameters (mechanical displacement, particle velocity and pressure) in the UT wave radiated by the probe-transmitter and the electric voltage exciting this probe in the transmission mode (electric input). Such a correlation can be obtained from forth equation in system (1) and boundary condition at the transducer front surface. Usually such a boundary is the transducer/fluid interface (for both immersion and contact techniques). It means that this boundary is free, i.e. it has zero stress. Integrating the forth equation in (1) with respect to coordinate z (transducer axial axis), using this boundary condition, and taking into account the relationships between mechanical displacement u_i and strain S_lm, electric field E_l and voltage U, and so on, we obtain the mechanical displacement u_z at the transducer front surface

(8)

By differentiating (8) with respect to time, the correlation between particle velocity v_z and exciting electric voltage U for running acoustic wave, radiated by probe-transmitter, can be written as . Multiplying this expression by characteristic acoustic impedance Z = rc of the medium, where UT wave propagates, we obtain the correlation between pressure p and the exciting electric voltage U for running acoustic wave radiated by probe-transmitter: .

Thus, the axial mechanical displacement u_z at the transducer front surface in the acoustic wave, created in the fluid medium, is linearly proportional to the exciting electric voltage U. In other words, shape of mechanical displacement in the emitted output UT pulse matches to shape of the electric voltage pulse exciting piezoelectric transmitter. The shape of particle velocity and acoustic pressure in the emitted output UT pulse is proportional to first derivative of the exciting electric voltage with respect to time.

Now it is clear why even in a "perfectly damped" transducer the UT output response to a very short half-cycle input electric voltage pulse consists of two half-cycles mechanical displacement (or three half-cycles acoustic pressure) pulse in the transmission mode.

Using the analysis, described above, one can formulate now the following conditions for the wide-band piezoelectric transmitter. First of all, such a piezoelement should have only one area with one source of acoustic waves, and secondly, the shape of piezoelement should exclude any possible reflections of UT pulses. Various types of wide-band transducers with non-uniform exciting electric field [4, 7-23] were developed following, to some extent, these rules. The results obtained confirmed the conclusions of theoretical analysis and led to the development of different types of wide-band probes with non-uniform electric field. However, since it is impossible to get rid of all acoustic wave sources except one, absolutely exclude reflections within piezoelement, and provide efficient excitation (large working area), the developed transducers have significant disadvantages: low signal-to-noise level and low response amplitude.

In order to determine the design of wide-band transmitter, we performed the numerical computations for different types of transducers with non-uniform electric field. The numerical solution of non-homogeneous partial differential equations (3)-(6) with non-homogeneous boundary conditions is a very complex mathematical problem. Because of this, we used an alternative approach: numerical calculations (MATLAB Version 6.5) for various acoustic sources and the method of "propagation of the boundary conditions" [26] to simulate and analyze the different designs of wide-band piezoelements with non-uniform electric field.

As a result, we determined two designs, which provide the best realization of conditions formulated above for wide-band piezoelements with non-uniform electric field in the transmission mode. The first is the conical piezoelement polarized in axial direction with one solid electrode on the cone base and the one band electrode on the cone side surface. The second is the conical piezoelement polarized in axial direction with two concentric axi-symmetric electrodes on the cone base: center circle electrode and outer ring electrode.

Such piezoelements will have very strong source #1 of longitudinal axial waves at cone base, all other sources of acoustic waves will be much less in amplitude and emit waves in the other directions. The mentioned conical piezoelements will work like an acoustic traps for all waves propagating inside the cone, so the amplitude of the waves reflected within the cone and coming back to the face will be small. Subsequently, such transducers should be a low-noise wide-band piezoelements. The experimental results presented below confirm this idea.

IV. Reception Mode

Equations (4) and (6), describing piezoelement working in the reception mode, are the Poisson’s equations regarding potential j of the output electric field created by the input exciting acoustic wave. The terms in equations right-hand sides are the driving forces or the "acoustically created electric charge densities", the sources of the output electric field.

Analyze, for example, five sources of the electric charges in (4). Note, that this analysis is fully applicable to equations (6). These five electric charge sources (right-hand sides in equations (4)) are represented by five derivatives:

(9)

Since all sources (9) of the electric charges are partial derivatives with respect to coordinates, they all reach maximum in areas where the piezoelectric constants e_ik or the exciting mechanical strain S_ik have discontinuities. Source #1 in (9) creates in rectangular 3D piezoelement the electrical charge caused by axial longitudinal strain in the incoming acoustic wave hitting the probe face. In a standard piezoelement this source reaches maximum at two faces of the transducer, where derivative with respect to z has discontinuity. Electrical charge #2 is caused by the radial (in x direction) longitudinal strain in the incoming acoustic wave. It has maximum at two piezoelement faces. Electrical charge #3 is caused by the radial (in y direction) longitudinal strain in the incoming acoustic wave. It has maximum at two piezoelement faces. Electrical charge #4 is caused by the radial (in x direction) shear strain in the incoming acoustic wave. It has maximum at piezoelement side surface perpendicular to x direction. Electrical charge #5 is caused by the radial (in y direction) shear strain in the incoming acoustic wave. It has maximum at piezoelement side surface perpendicular to y direction.

Since UT transducer in the reception mode (e.g. a piezoelectric probe) typically records pressure or mechanical displacement in the incoming acoustic wave, but not a particle velocity [27], one should determine the correlation between these input acoustic parameters and the output electric signal. Such a correlation can be obtained from sixth equation in system (1). Integrate this equation with respect to coordinate z (transducer axial axis), take into account the relationships between electric displacement and charge surface density, electric field and voltage, electric charge and current, and so on, and then differentiate the obtained equation with respect to time t. As a result, the final expression, connecting the output electric current I(T), going through piezoelement-receiver, the output electric voltage U(T) on the transducer electrodes and the input acoustic pressure pulse P(t), running within the piezoelement, will be written as

(10)

where C₀ is the transducer capacitance, g₃₃ is the piezoelectric constant and d₃₃»e₃₃g₃₃ (for PZT-5H ceramic ), c is the speed of longitudinal UT wave in piezoelectric material (for PZT-5H ceramic
c = 4560 m/s).

Equation (10) shows that, if piezoelement in reception mode is electrically shorted, i.e. its output voltage U(t)=0, the output current I(t) is linearly proportional to input pressure P(t). If piezoelement is electrically open, i.e. its output current I(t)=0, the output voltage U(t) is proportional to integral of pressure P(t) with respect to time. It means, that shape of the output electric current of electrically shorted piezoelement-receiver, matches to shape of the acoustic pressure and shape of first derivative of the mechanical displacement with respect to time in the input UT wave. Recall that in a running acoustic wave, the pressure is linearly proportional to the particle velocity. The shape of the output electric voltage of electrically open piezoelement-receiver matches to shape of the mechanical displacement and shape of integral of particle velocity or pressure with respect to time in the input running UT wave, coming to transducer front surface from the outer medium.

In other words, to provide a broad frequency band in the reception mode, a current amplifier with infinitesimal input electric impedance should be used with transducer. In case of any real current amplifier with a small (typically, a few ohms) but not infinitesimal input electric impedance, or voltage amplifier with a large but not infinite input electric impedance, the transducer output electric current and voltage can be calculated using equation (10) and Kirchhoff equation for electric circuitry [4, 21, 23].

To provide a broad frequency band for transducer working in pulse-echo (PE) mode, the voltage amplifier with large input electric impedance should be used. It is necessary because the output voltage in reception mode is proportional to integral of the input acoustic pressure with respect to time, but this pressure (the output signal in transmission mode) is proportional to first derivative of the input electric voltage exciting the transducer (see section III). Thus, the shape of the output voltage in reception mode will match the shape of the input voltage in transmission mode.

However, the output electric voltage has a high level of "piezo-electric noise". It can be explained as follows. Integrate the seventh equation in system (1) with respect to coordinate z (transducer axial axis) and take into account the relationships between electric displacement and charge surface density, electric field and voltage, strain and mechanical displacement, and so on. The obtained equation shows that the output electric voltage is linearly proportional to the mechanical displacements of transducer surfaces with electrodes. And these displacements exist all the time, till the input acoustic pulse propagates inside the piezoelement. As a result, the output electric voltage will have significant low-frequency signals, following the main pulse. On the other hand, the output electric current does not have this "piezo-electric noise", since current is proportional to pressure on the transducer front surface, and there is no pressure on this surface, when acoustic pulse propagates inside the piezoelement. Of course, such a pulse, as a moving electric charge, will induce some output current between piezoelement electrodes, but it will be insignificant. Subsequently, the output electric current should provide rather "clean" response in the reception mode.

When the incoming acoustic pulse hits the face of standard piezoelement with two electrodes on the bases, it partially reflects and partially enters the transducer and propagates inside it, revealing itself as an electric charge. One can see from (9) that in this case the intensity of electric charge sources ##2-5 is negligible. It means that the source of electric charge #1 is practically the only one that works in such a probe and creates a significant output electric signal when this electric charge, generated by the moving acoustic pulse, passes through two piezoelement faces. These two output electric pulses repeat the shape of the input exciting UT pulse. However, the incoming acoustic pulses propagating inside the piezoelement undergo the multiple reflections from the faces. It explains why thick piezoelement responds to a single input acoustic pulse with a whole sequence of output electric pulses. Moreover, now it is clear why even in a "perfectly damped" transducer the output electric response to a very short half-cycle input mechanical displacement (or two half-cycle acoustic pressure) pulse consists of two half-cycle voltage (or three half-cycle current) pulse in the reception mode.

Using this analysis, one can formulate the following conditions for the wide-band piezoelectric receiver. First of all, such a piezoelement should have only one area, where the incoming UT pulse reveals itself as only one source of electric charges. Secondly, the shape of piezoelement should exclude any possible reflections of the incoming UT pulses. And thirdly, electric current of shorted piezoelement should be used as an output electric signal. Different types of wide-band transducers [4, 7-23] were developed following, more or less, these rules. The results obtained confirmed the theoretical analysis and led to the development of various types of wide-band probes. However, since it is impossible to get rid of all electric charge sources except one, absolutely exclude the reflections within piezoelement, and provide high efficiency (large working area), the developed transducers have significant disadvantages: low signal-to-noise level and low response amplitude.

We used numerical calculations for various electric charge sources generated by the incoming UT pulses and method of "propagation of the boundary conditions" to simulate and analyze the different designs of wide-band piezoelements in the reception mode. The results obtained were similar to the ones calculated for the transmission mode. We determined two designs, which provide the best realization of conditions formulated above for wide-band piezoelements with non-uniform electric field in the reception mode. The first is conical piezoelement polarized in axial direction with one solid electrode on the cone base and one band electrode on the cone side surface. The second is the conical piezoelement polarized in axial direction with two concentric axi-symmetric electrodes on the cone base: center circle electrode and outer ring electrode.

Such piezoelements will have very strong electric charge from source #1 at the cone base; all other sources of electric charges will be much less in amplitude. Due to conical shape, the piezoelement will work like an acoustic trap for all waves propagating inside the cone, so the amplitude of the waves reflected within the cone and coming back to the face will be extremely small. As a result, these transducers should be low-noise wide-band piezoelements. The experimental results presented below confirm this idea.

Analyzing the wide-band transducer in sections 2-4, we assumed that it would be able to transform the short electric pulse into short UT pulse and back without distortion. However, it is true only if the transducer is the so-called "minimum-phase system". Such a system should have the transfer function F =(i,w) with no poles and zeros, that is , in the right half-plane i.e. at . For a minimum-phase system its phase response is uniquely determined by the amplitude response.

The described conical transducers with non-uniform electric field are the minimum-phase systems, as it follows from their transfer function analysis, which can be performed in the way it was done in [4]. One can come qualitatively to the same inference, using the approach developed in sections 2-4 and analyzing the ability of conical probes with non-uniform electric field to transmit and receive short UT pulses and harmonic waves with various frequencies. Note that not all UT probes are the minimum-phase systems. For example, the concave transducer is the wide-band probe (due to concave shape of the piezoelement it resonates at different frequencies), but it cannot transmit and receive short UT pulses, since it is the resonant probe. On the other hand, transducer with an electric compensation can transmit and receive short UT pulse, but it is not a wide-band probe since it cannot work with the same efficiency at different frequencies.

V. Experimental results

5.1 Transducers and testing equipment
A few different types of wide-band transducers with non-uniform electric field were developed, design and manufactured in UTX Inc. (Holmes, NY). In one design we used a conical piezoelement with one solid electrode on cone face and one band electrode on cone side surface, as shown schematically in Fig. 1.

Fig 1: Schematic design #1 of wide-band conical piezoelement.

The piezoelements were made from different piezoelectric materials (including PZT-5H piezo-ceramic and 1-3 piezo-composite) with various sizes of cone height and diameter and band electrode width and location. Changing these dimensions it is possible to control the transducer bandwidth. The cone piezoelement was put into a standard cylindrical metal case with connector on the top. Thin layer of epoxy coating protected the cone base.

In other design we used a conical piezoelement with one circle electrode in the center of cone face and one ring electrode on the cone face, as shown schematically in Fig. 2.

Fig 2: Schematic design #2 of wide-band conical piezoelement.

The transducer frequency responses were measured in water tank in PE mode in accordance with a standard technique [28]. Pulser-receiver UTEX UT-340, SONIX STR-8100 digitizer card and WINSPECT software were used for experiments.

5.2 Frequency responses
The PE responses obtained on wide-band conical piezoelement (design #2) in water tank with 3.2mm diameter ball-reflector are presented in Fig. 3 for output voltage and in Fig. 4 for output current. In the last case, in order to measure the current, going through piezoelement in reception mode, 3W resistor was connected in parallel to the transducer. Such a connection, as one can see in Figs. 3 and 4, dramatically decreases "piezo-electric noise", but changes the response shape. It happens because the voltage output is proportional to mechanical displacement in the input UT pulse, but current output is proportional to the acoustic pressure. Subsequently, the current output of the shorted transducer is the derivative of the voltage output of electrically open transducer with respect to time.

Fig 3: PE response (output voltage) of wide-band PZT-5H conical transducer (design #2) in water tank. Reflector is a metal ball located at 33mm from probe face (at the middle of focal zone). Cone piezoelement diameter 9.5mm, cone height 21mm.

Fig 4: PE response (output current) of wide-band PZT-5H conical transducer (design #2) in water tank. Reflector is a metal ball located at 33mm from probe face (at the middle of focal zone). Cone piezoelement diameter 9.5mm, cone height 21mm.

Figs. 3 and 4 clearly demonstrate that the output electric voltage has a high level of "piezo-electric noise", while the output electric current provides rather "clean" response. The "noise" pulse located at ~ 12ns in Fig. 4 is probably a response from some radial wave propagating within the piezoelement. The change of transducer shape (electrode position, cone angle, and so on) should eliminate it. Similar responses were obtained for both designs #1 and #2 with various dimensions. These results confirm theoretical analysis and show that current should be used as an output electric signal in the reception mode. Note that tuning the transducer, e.g. by connecting an inductor in parallel to probe, filters out most of the noise (i.e. "cleans" the signal), increases response amplitude due to better matching between electric impedances of the generator and transducer, but increases the pulse duration and decreases transducer bandwidth.

In order to estimate the bandwidth of the developed transducers, the following technique was used. If we had a shock pulse (spike) generator, which could excite transducer with very short d -function pulse with a broad spectrum, than only the response spectrum in PE mode would be sufficient to characterize the transducer bandwidth. Of course, it is true only if the amplifier of pulser-receiver has broad band and the reflection from ball-target does not distort the signal spectrum. In our experimental setup these two conditions were satisfied, therefore we will further neglect the amplifier and ball-reflector contribution in the formation of system spectrum. However, the required shock pulse generator was not available. Three pulser-receivers, UTEX UT-340, MATEC SR-9000, and Panametrics Ultrasonic Analyzer 5052-UA, which were available, provided a short excitation pulse only at 50W load, and rather distorted pulse when loaded by transducer in PE mode.

Because of this, we used the following method, based on the analysis of system transfer function, to determine the transducer bandwidth. At first, using Winspect software, we obtained in time domain the excitation pulse of electric voltage from generator loaded with transducer, and calculated its spectrum A₁(w). The results for transducer designs #1 and #2 are presented in Figs. 5 and 8, respectively. Then, using Winspect software again, we obtained in time domain the transducer PE voltage response reflected from ball-target in water tank, and calculated its spectrum A₂(w). (Recall, that for wide-band transducer in PE mode, shape of the output voltage pulse in time domain in reception mode should match with shape of the input voltage pulse, exciting transducer in transmission mode, see section IV). The results for transducer designs #1 and #2 are presented in Figs. 6 and 9, respectively. And at last, using MATLAB, we determined transducer spectrum A₀(w) using formula , because we assume that system consists only of generator and transducer working in PE mode, with the later participating twice (as a transmitter and as a receiver) in the formation of the spectrum. The results for transducer design #1 and design #2 are given in Figs. 7 and 10, respectively.

Fig 5: Measured voltage of UTEX excitation pulse in time domain (a) and its spectrum (b) when UTEX is loaded with wide-band PZT-5H conical transducer (design #1) in PE mode. Cone piezoelement diameter 9.5mm, cone height 21mm.

Fig 8: Measured voltage of UTEX excitation pulse in time domain (a) and its spectrum (b) when UTEX is loaded with wide-band PZT-5H conical transducer (design #2) in PE mode. Cone piezoelement diameter 9.5mm, cone height 21mm.

As one can see from Figs. 5 and 8, the excitation signal from UTEX pulser, loaded with transducer, really does not look like a spike, and its spectrum is not, of course, the spectrum of d -function. Two other pulsers, MATEC SR-9000, and Panametrics 5052-UA, loaded with transducer, have similar or even worse shape of the excitation signal.

Comparing Fig. 5 with Fig. 6 and Fig. 8 with Fig. 9, it is easy to notice the satisfactory similarity in signal shapes in time domain and in signal spectra between the excitation pulse and the PE response for both transducers (designs #1 and #2). It means, that both piezoelements are really the wide-band transducers. The calculated spectra of both probes, presented in Figs. 7 and 10, confirm this inference. The mismatch between excitation pulses and PE responses and "the non-ideal" shape of spectra in Figs, 7 and 10 can be attributed to non-optimized shape of the piezoelements and measurement errors. However, generically, the obtained results confirm the validity of initial assumptions, the correctness of general approach, and the conclusions of theoretical analysis. Thus, the conical piezoelements with non-uniform electric field are really able to transmit and receive short UT pulse (unipolar pulse ~ 100ns duration) and have a wide working frequency range (from ~ 100kHz to ~30 MHz). Of course, the manufactured experimental prototypes of probes, despite the theoretical predictions, are not the ideal wide-band transducers: they have some radial oscillations, mode converted UT waves, internal reflections, induced electric noise, and so on.

Fig 6: Measured voltage of PE response of wide-band PZT-5H conical transducer (design #1) in water tank in time domain (a) and its spectrum (b). Reflector is a flat metal plate located at 30mm from probe face (at the middle of focal zone). Cone piezoelement diameter 9.5mm, cone height 21mm.

Fig 7: Calculated spectrum of wide-band PZT-5H conical transducer (design #1). Cone piezoelement diameter 9.5mm, cone height 21mm.

Fig 9: Measured voltage of PE response of wide-band PZT-5H conical transducer (design #2) in water tank in time domain (a) and its spectrum (b). Reflector is a flat metal plate located at 30mm from probe face (at the middle of focal zone). Cone piezoelement diameter 9.5mm, cone height 21mm.

Fig 10: Calculated spectrum of wide-band PZT-5H conical transducer (design #2). Cone piezoelement diameter 9.5mm, cone height 21mm.

The manufactured conical probes provide reasonably high signal amplitude and signal-to-noise ratio. By changing the cone dimensions, shape, and electrode locations and sizes, the transducer bandwidth, efficiency and signal-to-noise ratio can be controlled. Note that these transducers can work in both immersion and contact methods of inspection. In both cases they can transmit and receive the short UT pulses, thus providing the high axial resolution of the system. Moreover, such a probe can be used as both low-frequency transducer (with frequency ~ 200kHz) and high-frequency transducer (with frequency ~ 20MHz), depending on the application. Of course, the efficiency (amplitude response) of conical wide-band probes is lower than that of standard resonant transducers, since their working area (cone base area around the shortest distance between electrodes) is much smaller. However recall that, although due to a different reason (lack of resonant standing wave in the piezoelement), the efficiency of any wide-band highly damped probe is always significantly lower than efficiency of a non-damped transducer.

It has already been explained in section IV that shape of the output voltage pulse in reception mode matches to shape of the input voltage pulse in transmission mode. At the same time, shape of the output current response in reception mode matches to shape of first derivative of the input voltage with respect to time in transmission mode. In order to provide high axial resolution of UT system, the short pulse of acoustic pressure should propagate from transmitter to receiver. It means, that input voltage in transmission mode should generate this short pressure pulse, and the output electric response in reception mode should reproduce this pressure pulse. Subsequently, we come to the conclusion, that step-function of voltage should be placed across the electrodes of the transmitter. Such input voltage in transmission mode will create a short pressure pulse propagating in the test object, since acoustic pressure in the emitted output UT pulse is proportional to first derivative of the exciting electric voltage with respect to time (see section III). In the reception mode, the current should be used as an output electric response, since shape of the current pulse replicates shape of the incoming pressure pulse. As a result, in order to transmit and receive a short pressure pulse, i.e. to provide high axial resolution, a step-function generator should be used in transmission mode and a current amplifier should be employed in reception mode.

5.3 Acoustic fields
One more interesting and useful characteristic of these probes is their acoustic field, see Figs.11-13. Transducer acoustic fields were measured in water tank in PE mode accordingly to a standard technique [28] with 1.5mm ball-reflector and UTEX UT-340 pulser-receiver. The water tank had a rig, which allowed motion of ball-reflector in three dimensions with 0.25mm accuracy.

	Fig 11: Axial-radial cross-section of acoustic field of conical transducer with non-uniform electric field, design #1. Measured in water tank in PE mode with 1.5mm ball-reflector. Probe from 1-3 piezo-composite, cone diameter 12.7mm, cone height 12.7mm.
	Fig 12: Axial-radial cross-section of acoustic field of wide-band conical transducer, design #2. Measured in water tank in PE mode with 1.5mm ball-reflector. Probe from PZT-5H, cone diameter 9.5mm, cone height 21mm. Color scale is in Fig. 11.
	Fig 13: Axial-radial cross-section of acoustic field of wide-band conical transducer, design #2. Measured in water tank in PE mode with 1.5mm ball-reflector. Probe from PZT-5H, cone diameter 9.5mm, cone height 21mm. Color scale is in Fig. 11.

As one can see from Figs. 11-13, the conical wide-band probes with non-uniform acoustic field create the narrow weakly diverging acoustic beam. Beam diameter is ~ 1mm at –6dB level within axial ranges 35-80mm (Fig. 11), 12-38mm (Fig. 12), and 28-43mm (Fig. 13) for different transducers.

This feature is a result of non-uniform excitation. The ability of transducers with non-uniform excitation field to form narrow weakly diverging acoustic beams was explained and analyzed in [29-32]. The developed conical transducers with band electrode on side surface (design #1) have large exiting electrical field in the peripheral areas of cone face and small electric field in the central area. It happens due to small distance between band electrode and peripheral area of face electrode and large distance between band electrode and central area of face electrode. The developed conical transducers with two concentric electrodes (circle and ring) on the cone base (design #2) have large exiting electrical field only in the area between electrodes. Because of this non-uniformity of the exciting electric field, all developed conical transducers create narrow collimated acoustic beam, see for details [30-32]. The depth of focal zone and the beam diameter can be controlled by varying cone dimensions and electrode sizes and locations. However, these changes entail the variation of frequency response too. To control independently the transducer acoustic field, one can use e.g. the lens at the cone base made from epoxy coating. The influence of such a lens on the probe acoustic field will significantly exceed the effect of non-uniform electric field. The acoustic fields of wide-band conical transducers (design #1) with acoustic "logarithmic" lenses [32], designed and manufactured in UTX Inc. (Holmes, NY), are shown in Figs. 14 and 15.

	Fig 14: Axial-radial cross-section of acoustic field of wide-band conical transducer (design #1) with "logarithmic" lens. Measured in water in PE mode with 1.5mm ball-reflector. Probe from 1-3 piezo-composite, cone diameter 12.7mm, cone height 12.7mm.
	Fig 15: Axial-radial cross-section of acoustic field of wide-band Fig 15: Axial-radial cross-section of acoustic field of wide-band conical transducer (design #1) with "logarithmic" lens. Measured in water tank in PE mode with 1.5mm ball-reflector. Probe from PZT-5H, cone diameter 9.5mm, cone height 21mm.

Both transducers, whose acoustic fields are presented in Figs. 14 and 15, had logarithmic lenses focusing beam within axial range ~ 20-50mm. As one can see, UT beams of both probes are really concentrated approximately in this range. It means that acoustic field of conical wide-band transducer can be easily controlled by acoustic lens made from epoxy coating at the cone base, which does not affect the transducer frequency response.

VI. Conclusions

1. General equations (3)-(6), describing piezoelectric transducer in the transmission and reception modes of operation, are derived. From these, one can calculate the acoustic (electric) output depending on the electric (acoustic) input for the transmission and reception modes, respectively.
2. The sources of the acoustic wave (7) and the electric charge (9) are analyzed for the transmission and reception modes for various types of transducers. It is shown that low-noise wide-band piezoelements should have conical shape with solid electrode on the face and band electrode on the side surface or two concentric electrodes (circle and ring) on the face.
3. Shape of the mechanical displacement in the emitted output UT pulse in transmission mode matches to shape of the electric voltage pulse exciting the transducer. The acoustic pressure in the emitted pulse is proportional to first derivative of the exciting electric voltage with respect to time.
4. Shape of the output electric current pulse of electrically shorted piezoelement in the reception mode matches to shape of the acoustic pressure pulse and shape of first derivative of the mechanical displacement pulse with respect to time in the incoming UT wave. Shape of the output electric voltage pulse of electrically open piezoelement-receiver matches to shape of the mechanical displacement pulse and shape of the integral of pressure with respect to time in the input UT signal.
5. A numerical computation algorithm, based on simulation of the acoustic and electric sources and "the boundary condition propagation method", was developed for calculating the optimum parameters of conical transducer with non-uniform electric field.
6. The obtained results of theoretical analysis were used to develop, design and manufacture a low-noise wide-band conical piezotransducers with non-uniform electric field. These probes have either solid electrode on the face and band electrode on the side surface or two concentric electrodes (circle and ring) on the face.
7. Experimental testing demonstrated that such probes are really low-noise wide-band piezo-transducers, thus confirming the correctness of the general theoretical approach, the initial assumptions, method and accuracy of calculations. By changing the piezoelement dimensions (cone diameter and height) and electrode sizes and locations, one can control the transducer bandwidth.
8. The developed conical piezotransducers with non-uniform acoustic field are wide-band UT probes. They provide high axial resolution during UT testing. Moreover, depending on the application these probes can be used with approximately equal efficiency as both the low and high frequency transducers.
9. In order to transmit and receive a short pressure pulse, i.e. to provide high axial resolution of the UT system, a step-function voltage generator should be used in the transmission mode and a current amplifier should be employed in the reception mode.
10. Due to a non-uniform electrical excitation (strong electric field at the peripheral area of cone face and weak field in the centre), such transducers form the narrow weakly diverging acoustic beams.
11. Transducer acoustic field can be controlled independently of its frequency response, by the acoustic lens made at the cone base.

References

1. Nondestructive Testing Handbook. Ultrasonic testing. Second edition, v. 7. American Society for NDT. 1991.
2. Silk M. G. Ultrasonic Transducers for NDT. Bristol, Adam Hilger Publishing, 1984.
3. Krautkramer J. and Krautkramer H. Ultrasonic Testing of Materials. Third edition. Berlin. Springer-Verlag, 1984.
4. Korolev M. V. and Karpelson A. E. Wide-band Ultrasonic Piezo-transducers. Moscow. Mashinostroenie, 1982 (in Russian).
5. Kossoff G., "The effects of backing and matching on the performance of piezoelectric ceramic transducers", IEEE Transactions on Sonics and Ultrasonics, vol. SU-13, No. 1, p.p. 20-30. 1966.
6. Desilets C. S., Fraser J. D. and Kino C. S., "The design of efficient broad-band piezoelectric transducers", IEEE Transactions on Sonics and Ultrasonics, vol. SU-25, No. 3, p.p. 115-125. 1978.
7. Jacobsen E. H., "Sources of sound in piezoelectric crystals", Journ. Acoust. Soc. Amer., vol. 32, No. 8, p.p. 949-953, 1960.
8. Redwood M., "Piezoelectric generation of an electrical impulse", Journ. Acoust. Soc. Amer., vol. 33, No. 10, p.p. 1386-1390, 1961.
9. Redwood M., "Study of waveforms in the generation and detection of short ultrasonic pulses", Journ. Appl. Mater. Research, vol. 2, p.p. 76-84, 1963.
10. Redwood M., "Experiments with the electrical analog of a piezoelectric transducer", Journ. Acoust. Soc. Amer., vol. 36, No. 10, p.p. 1872-1880, 1964.
11. Korolev M. V., "An aperiodic piezoelectric transducer for ultrasonic testing", Sov. Journ. of NDT, No. 4, p.p. 181-187, 1973.
12. Brown A. F. and Weight I. P., "Generation and reception of wide-band ultrasound", Ultrasonics, vol. 12, No. 4, p.p. 161-167, 1974.
13. Mitchel B. F. and Redwood M., "The generation of sound by non-uniform piezoelectric materials", Ultrasonics, vol. 7, No. 2, p.p. 123-129, 1969.
14. Peterson I. G. and Rosen M., "Use of thick transducers to generate short pulses in specimens", Journ. Acoust. Soc. Amer., vol. 41, No. 2, p.p. 336-345, 1967.
15. Van der Pauw L. J., "The planar transducer – a new type of transducer for exciting longitudinal acoustic waves", Appl. Phys. Lett., vol. 9, No. 3, p.p. 129-131, 1966.
16. Kazhis R. I. and Lukoshevichyus A. I., "Wide-band piezoelectric transducers with an inhomogeneous electric field", Sov. Phys. Acoust., vol. 22, No. 2, p.p. 167-168, 1976.
17. Foster F. S. and Hunt J. W., "The design and characterization of short pulses ultrasonic transducers", Ultrasonics, vol. 16, No. 3, p.p. 116-122, 1978.
18. Hayman A. J. and Weight J. P., "Transmission and reception of short ultrasonic pulses", Journ. Acoust. Soc. Amer., vol. 66, No. 4, p.p. 945-951, 1979.
19. Low G. C. and Jones R. V., "Short pulse ultrasonic probes for NDT", Ultrasonics, vol. 22, No. 22, p.p. 85-95, 1984.
20. Hutchins D. A., Brosse L. F. and Palmer S. B., "Measurements with a thick conical piezoelectric transducer", Journ. Acoust. Soc. Amer., vol. 85, No. 6, p.p. 2417-2422, 1989.
21. Korolev M. V., "Piezoelectric transducers with surface excitation", Sov. Journ. of NDT, No. 9, p.p. 652-658, 1978.
22. Korolev M. V. and Karpelson A. E., "Performance of a surface-excited piezoelectric transducer in the radiation mode of operation", Sov. Journ. of NDT, No. 3, p.p. 145- 154, 1979.
23. Korolev M. V. and Karpelson A. E., "Performance of a surface-excited piezoelectric transducer in the reception mode of operation", Sov. Journ. of NDT, No. 3, p.p. 154-160, 1979.
24. Landau L. D. and Lifshits E. M. Electrodynamics of Continuous Media, 2^nd Edition, Oxford, Pergamon press, 1994.
25. Landau L. D. and Lifshits E. M. Theory of Elasticity, 3^rd Edition, Oxford, Pergamon press, 1986.
26. Tikhonov A. N. and Samarski A. A. Equations of Mathematical Physics, New York, Macmillan, 1963.
27. Shutilov V. A. Fundamental Physics of Ultrasound. New York, Gordon and Breach Science Publishers, 1988.
28. Standard Guide for Evaluating Characteristics of Ultrasonic Search Units. ASTM Standard E-1065, 1999.
29. Kondratiev Y. A. and Karpelson A. E., "Formation of narrow slightly diverging ultrasonic beams", Sov. Journ. of NDT, No. 10, p.p. 938-943, 1978.
30. Kondratiev Y. A., "Investigation of ultrasonic field of flat transducer with non- uniform pressure distribution on its working face", Sov. Journ. of NDT, No. 1, p.p. 824-837, 1981.
31. Korolev M. V., Karpelson A. E., and V. G. Shevaldykin, "Acoustic fields formed by surface-excited piezotransducers", Sov. Journ. of NDT, No. 11, p.p. 250-264, 1979.
32. Karpelson A., "Piezotransducers creating narrow weakly diverging ultrasonic beam", e-journal of NDT, vol. 7, No. 6, June 2002.

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