The Technologies of PEAT Synthesis

Abstract

New methods electrophysical and circuit engineering designing piezoelectric electro-acoustic transducers are described in this chapter. It allows to create tens variants of transducers of the given type from one piezoelement.

Piezoelement of the certain form and sizes usually use at designing of piezoceramic transducers, from defined of piezoceramic material with certain electrophysical properties (characteristics). Thus traditionally vector of force F operating on a piezoelement parallel to a vector of polarisation P.

Simultaneously the vector of force F is parallel to a vector of electric field E of output signal of the sensor, i.e. is perpendicular to electrodes which are put on a piezoelement surface (Fig. 6.1) [1–14].

It is connected, obviously, that these electrodes use for piezoelement polarisation at manufacturing. Simultaneously they are used also for removal of a useful signal at measurement of physical sizes (force, pressure, acceleration, etc.), and also for introduction in a piezoelement of electric voltage at use of a piezoelement as a radiator (projector).

Such type of the sensor is known and named by traditional [6]. Only one transducer with certain characteristics (resonant frequency, sensitivity, a range of working frequencies, etc.) it is possible to receive for this case for a certain piezoelement. Earlier it was necessary to use other piezoelement, other size, other form, from other piezomaterial for production of the sensor with other characteristics.

6.1 Spatial Energy: Force Structure of Piezoceramic Element

Electric field vector E of the sensor output signal (voltage), supplied to the projector, should be considered at piezoceramic transducers designing, as it is suggested in [6, 9, 15].

The location of F, P and E vectors in space characterizes the piezoelement spatial energy power structure.

A rectangular parallelepiped-shaped piezoelement is considered as an example (Fig. 6.1). Its electrodes are attached to the wide faces.

Fig. 6.1

Piezoelement with traditional arrangement of vectors F, P and E

Polarization P, force F and electric field vectors of the output signal E are shown in this Figure. This arrangement of vectors is the most widespread and traditional [4, 16–20].

A piezoelement of other dimensions, shape and material was used in the past to change the characteristics of traditionally designed piezoceramic sensors.

Meanwhile, the sensor characteristics can be changed if polarization P, applied force F and electric field intensity vectors of output signal E are reciprocally rearranged.

This rectangular parallelepiped-shaped piezoelement is considered below (Fig. 6.2).

Let non-interconnected electrodes are attached to all parallelepiped faces and the piezoelement is polarized between 1–1${^{\prime }}$ faces. Let force F measured is applied parallel to polarization vector P, perpendicularly to face 1, while the output voltage is read from faces 1–1$^{\prime }$. Thus, all three vectors are parallel to Z-axis in the case with the given transducer (F $\downarrow$ P $\downarrow$ E $\downarrow )$.

Fig. 6.2

Parallelepiped-shaped piezoelement

It is necessary to notice, that change of a direction of one of vectors on 180${^\circ }$ leads only to change of a phase of a signal.

The above considered transducer with parallel arrangement of three vectors is the most widespread.

This expression is true for it:

$$\begin{aligned} U_{{ OUT}} =\frac{Q}{C_{1\text {--}1^{\prime }} }=\frac{d_{31} \cdot F}{C_{1\text {--}1^{\prime }} }, \end{aligned}$$

(6.1)

where Q—the charge, generated by the piezoelement on faces $1\text {--}1^{\prime }$; $C_{1\text {--}1}^{\prime }$- capacitance between faces 1–1$^{\prime }$; $d_{31}$—piezomodule.

Let polarization vector P do not change its direction, and force F can be applied to both face 1, and faces 2 and 3. Voltage can be read from faces $1\text {--}1^{\prime }, 2\text {--}2^{\prime }$ or $3\text {--}3^{\prime }$. Thus, vectors F and E can be either parallel, or perpendicular to vector P (Fig. 6.3).

The transducer is called transverse [4] in case the force measured is applied to the piezoelement for the angle between the direction of force F and polarization vector P to be 90${^\circ }$ (transducers b and e, in Fig. 6.3).

It appeared that sensitivity S for this transducer is written like this [6]

$$\begin{aligned} S=\frac{Q}{F}=d_{ij} \frac{h}{a}, \end{aligned}$$

(6.2)

where Q—charge, generated on the corresponding face; h—piezoelement height; a—thickness.

Transverse piezoelements are used in sensors, made by $\ll$Brüel and Kjer$\gg$ (Denmark), and $\ll$Kistler Instrumente AG$\gg$, for example [21, 22].

Transducers with the angle between the electric field vector of output signal E and 90${^\circ }$ polarization vector P (transducers c and d in Fig. 6.3) are called domain-dissipative [4].

The physics of the processes, occurring in these transducers, is insufficiently studied. It is assumed that the following factors influence the transducers characteristics:

• energy dissipation on domains [6, 9];

• change of electric capacitance between electrodes;

• occurrence of other type oscillations in the piezoelement.

The sensor with vectors F and E perpendicular to polarization vector P (transducers d and g in Fig. 6.3) is of a scientific interest. These transducers are called domain-dissipative.

Fig. 6.3

Classification of piezoceramic transducers, depending on directions of vectors F, P, E

When all vectors are perpendicular to each other, transducers are called volume (transducers h and i in Fig. 6.3).

The constructive circuits of the transducers, shown in Fig. 6.3, are only several examples of their designs [6].

More transducer design variants are received if the polarization vector direction becomes perpendicular to parallelepiped faces $2\text {--}2^{\prime }$, 9. Turning polarization vector for it to be perpendicular to piezoelement faces $3\text {--}3^{\prime }$, 9 more variants of transducer designs can be received. 27 transducers variants with various characteristics can be totally received for a rectangular parallelepiped-shaped piezoelement [6, 9].

Experimental dynamic characteristics of the transducers are shown in Fig. 6.3. Characteristics of the transducers in a dynamic mode are called dynamic, i.e. when the value reduced is the time function (process). These parametres characterise transformers internal (own) properties.

All real dynamic systems are theoretically nonlinear and nonstationary to some extent, and their parametres are distributed.

Practically the majority of them can be nominally considered linear stationary dynamic systems with concentrated parametres, those based on nonlinearity are not inclusive.

It is known that linear stationary dynamic system with concentrated parametres is described by the differential equation with constant coefficients [6]:

$$\begin{aligned} a_n \frac{d^{n}y}{dt^{n}}+\cdots +a_1 \frac{dy}{dt}+a_0 y=bm\frac{d^{m}x}{dt^{m}}+\cdots +b_1 \frac{dx}{dt}+b_0 x, \end{aligned}$$

(6.3)

which in the operational form looks like

$$\begin{aligned} \left( {a_n p^{n}+\cdots +a_1 p+a_0 } \right) y(t)=\left( {b_m p^{m}+\cdots +b_1 p+b_0 } \right) x(t), \end{aligned}$$

(6.4)

or shorter

$$\begin{aligned} A_n \left( p \right) \cdot y\left( t \right) =B_m \left( p \right) \cdot x\left( t \right) , \quad m\le n, \end{aligned}$$

(6.5)

whence

$$\begin{aligned} y\left( t \right) =\frac{B_m \left( p \right) }{A_n \left( p \right) }x\left( t \right) =Lx\left( t \right) , \end{aligned}$$

(6.6)

where $p= d / dt$—differentiation operator; L—linear operator of stationary dynamic system.

The differential equation is the exhaustive characteristic of the dynamic system. However, it is hard to calculate its coefficients experimentally.

Using Laplas transformation to the differential equation under initial zero conditions, the following transfer function is received:

$$\begin{aligned} W\left( S \right) =\frac{Y\left( s \right) }{X\left( s \right) }=\frac{b_m s^{m}+b_{m-1} s^{m-1}+\cdots +b_1 s+b_0 }{a_n s^{n}+a_{n-1} s^{n-1}+\cdots +a_1 s+a_0 }, \end{aligned}$$

(6.7)

where s—Laplas operator; Y (s)—Laplas image of output and input values accordingly.

Complex frequency characteristic is received if Laplas operator is substituted by $j \omega$ in transfer function

$$\begin{aligned} K\left( {j\omega } \right)&=\frac{b_m (j\omega )^{m}+b_{m-1} (j\omega )^{m-1}+\cdots +b_1 (j\omega )b_0 }{a_n (j\omega )^{n}+a_{n-1} (j\omega )^{n-1}+\cdots +a_1 (j\omega )a_0 } \nonumber \\&=P(\omega )+jQ(\omega ) \end{aligned}$$

(6.8)

where $p (\omega )$ and $jQ (\omega )$—real and imaginary part of the complex frequency characteristic.

Whence the amplitude-frequency characteristic (AFC)

$$\begin{aligned} K\left( \omega \right) =\left| {K\left( {j\omega } \right) } \right| =\sqrt{P^{2}\left( \omega \right) +Q^{2}\left( \omega \right) } \end{aligned}$$

(6.9)

and phase-frequency characteristic

$$\begin{aligned} \varphi \left( \omega \right) =arctg\frac{Q\left( \omega \right) }{P\left( \omega \right) } \end{aligned}$$

(6.10)

The pulse transitive characteristic is the dynamic system response to the so-called $\delta$ impulse

$$\begin{aligned} \delta \left( t \right) = \left\{ \begin{array}{l} t\\ 0, \, \text {with}\;\; t \ne 0\\ \infty , \text {with}\; t = 0 \end{array}\right. \end{aligned}$$

and

$$\begin{aligned} \int \limits _{-\infty }^\infty {\delta \left( t \right) } dt=1 \end{aligned}$$

Transitive function is the response of the dynamic system to the input step action in the form of unit function 1(t), the derivative of which equals $\delta$-impulse.

Fig. 6.4

AFC of Transducers from Fig. 6.3

Fig. 6.5

Transient characteristics of transducers from Fig. 6.3

As it is seen from Fig. 6.4, there are several peaks of amplitude-frequency characteristic for a traditional transducer (Fig. 6.4a). In transverse transducers these resonances are partially suppressed (Fig. 6.4b). AFC of domain-dissipative transducers is practically linear (Fig. 6.4c, f, g, h).

In this case transfer coefficient (sensitivity) is smaller in low-frequency area for all transducer types than for traditional. However, considerable increase of transfer coefficient is possible for domain-dissipative transducers in some cases [6, 9].

Transient characteristics of the transducers, represented in Fig. 6.3, are shown in Fig. 6.5. The measurements were made in a piezotransformer mode under the action of meander-shaped voltage on the transducer ($\mathrm{{f}}=500\,\mathrm{{Hz}}$, $\mathrm{{U}}=3\,\mathrm{{V}}$). Pictures were taken by digital photocamera “Nikon-D90”.

As it follows from Figs. 6.4 and 6.5, the change of vectors F, P, E position in space, i.e. change of the spatial energy power structure (SEPS), leads to essential changes of the transducer dynamic characteristics.

This SEPS change is assured by the corresponding electrodes arrangement on the piezoelement surface and the choice of force application site.

As the experiments showed, the change of transducers characteristics occurs also if the angles between the vectors are less than 90$^{\circ }$.

Many sensor designs can be created on the basis of the method offered [6, 11, 12].

6.2 Spatial Arrangement and Conjunction Piezoelement Electrodes

Change of output signal electric field vector E position can be realize by electrodes piezoelement division into parts and connection of these part so that a corner $\alpha$ between vector E and polarization vector P was $0 < \alpha < 90^{\circ }$ (Fig. 6.6).

Fig. 6.6

Piezoelement with the divided electrodes

If piezoelement electrodes divide on two equal parts sensor sensitivity on a charge will decrease twice the charge is proportional to electrodes area. Voltage sensitivity stays same, as for a piezoelement with undivided electrodes.

It is quite obvious, that the capacity between electrodes 1–${2^\prime }$ (or ${1^\prime }$–2) will be less, than capacity between electrodes 1–${1^\prime }$ (2–${2^\prime }$), therefore it is possible to receive electric voltage ($\mathrm{V}_{2})$ several times more than on electrodes 1–${1^\prime }$ or 2–${2^\prime }$ ($\mathrm{V}_{1})$.

If electrodes 1 and 2 and $1{^\prime }$ and $2{^\prime }$ carry out from each other (for what it is possible to divide initial electrodes into three parts (Fig. 6.7), sensitivity of piezosensor on voltage will increase even more.

Fig. 6.7

Piezoelement with three systems of electrodes

It is necessary to notice, that a spatial arrangement of electrodes and their switching from each other results not only in capacity change between electrodes and sensitivity, but also to change of dynamic characteristics (AFC, pulse and transient characteristics).

For such transducers piezoelements can be used also of disk form with electrodes in the form of semidisks, disks and rings, piezoelements in the form of hollow cylinders, etc.

6.3 Spatial Electromechanical Feedback

Parameters of automatic control systems (for example, time constants, input and output resistances, frequency and transitive characteristics, etc.) can be widely changed as a result of feedback (FB) introduction [5, 18, 23].

Application of FB in equipment gives excellent results [5, 6]. FB is also widely used and in measuring devices. For example, resonant vibrations are activated in piezoelectric sensors under the affect of positive feedback. Sensors of various physical values can be built on this basis. Negative feedback in resonant piezoceramic sensors gives a chance to linearize their graduation characteristics [6].

Feedback has unique properties, due to which parameters of measuring devices can be essentially improved.

FB in measuring devices is usually introduced along the input action. The general view of the transducer with FB can be represented by the simplified block diagram (Fig. 6.8), where W (p)—direct transform circuit, $\beta (p)$—FB circuit.

Fig. 6.8

Block diagram of piezotransducer with feedback

Using methods of automatic control theory [5, 18], the operational form of the expression for the transducer with FB sensitivity can be written.

$$\begin{aligned} W_{{ FB}} \left( p \right) =\frac{X_2 }{X_1 }=\frac{W(p)}{1\pm W(p)\beta (p)}, \end{aligned}$$

(6.11)

where $X_{1}$ and $X_{2}$—input and output values.

Complex FB is the most common case of FB. Then operator p can be substituted for $j \upomega$ in the Eq. (6.11)

$$\begin{aligned} W_{{ FB}} \left( {j\omega } \right) =\frac{X_2 }{X_1 }=\frac{W(j\omega )}{1\pm W(j\omega )\beta (j\omega )}. \end{aligned}$$

(6.12)

Expressing sensitivity of direct transform and FB circuits in the algebraic form, the module sensitivity $W_{FB} (\upomega )$ and phase displacement $\varphi _{FB}$ after transformations will look like this

$$\begin{aligned} W\left( \omega \right) =\frac{W}{\sqrt{1\pm 2W\beta \cos (\varphi _K +\varphi _\beta )+W^{2}\beta ^{2}}}, \end{aligned}$$

(6.13)

$$\begin{aligned} \varphi _{\text {FB}} \left( \omega \right) = \text {arctg}\frac{tg\varphi _K +W\beta \frac{\sin \varphi _\beta }{\cos \varphi _K }}{1\pm W\beta \frac{\sin \varphi _\beta }{\cos \varphi _K}}. \end{aligned}$$

(6.14)

The expressions received show that both sensitivity module and phase displacement angle depend not only on modules W and $\beta$, but also on values and signs of phase displacement angles in direct $\varphi _{C}$ and inverse $\varphi _{{\beta }}$ transforms.

Influence of Frequency-Dependent Feedback

Working at a close to resonant frequency area, FB is frequency-dependent. A piezoelectric sensor can be represented as series oscillatory element (Fig. 6.9). As it is seen from Fig. 6.9, the oscillatory contour is connected to the amplifier for the amplifier output to be reconnected with it via a phase-shift device [5, 6].

Fig. 6.9

Equivalent circuit of piezoelectric sensor with feedback

If all voltages, operating in the contour, are divided into the current in the contour the voltage diagram (Fig. 6.10a) will change into the resistance diagram (Fig. 6.10b). Feedback introduction can be formally considered as introduction of certain complex resistance into the contour. This can essentially alter both frequency and transitive contour characteristics or any other system, covered by feedback, as the equivalent parameters of this system are changed.

Fig. 6.10

Voltage and resistance diagrams in piezosensor with feedback

Several possible cases are considered below.

1. Angle $\psi$, formed by FB resistance $Z_{FB, }$and active resistance R, is in these $0 <\psi < 90^{\circ }$ limits, as it is shown in Fig. 6.10c.

Resolving $Z_{FB}$ into active and reactive components, one can see that the reactive component of FB resistance directionally coincides with $X_{L}$. As a result, equivalent inductive resistance of contour $X_{LE }$increases. This is equal to equivalent inductance increase. Equivalent active resistance $R_{E}$ also increases in this case.

This implies, FB introduction leads to resultant own frequency reduction of an electric contour or mechanical system. In this case its attenuation increases while Q factor decreases.

2. Angle $\psi$ satisfies inequalities 90$^{\circ } <\psi < 180^{\circ }$. Now, as it is seen from Fig. 6.10d, equivalent inductive resistance $X_{LE}$, and, consequently, equivalent inductance increases. However, active component $R_{FB}$ has a negative value. This leads to decrease of equivalent contour active resistance. This resistance may equal zero if equality $R_{FB} = R$ exists.

Thus, the resultant own frequency of the electric contour or mechanical system decreases in the case considered, while Q factor increases. It can become infinitely big. As a result, continuous oscillations appear in the contour under the action of any charge fluctuation.

3. The following case is possible if angle $\psi$ equals $180^{\circ } <\psi < 270^{\circ }$. As it follows from Fig. 6.10e, this FB increases contour or system equivalent capacitive resistance. It means that equivalent capacitance is reduced. In this case resultant active resistance decreases simultaneously. Resultant own frequency or Q factor increases if contour or mechanical system equivalent parameters decrease.

4. And finally, inequality $270^{\circ } <\psi < 360^{\circ }$ is true when vector $Z_{FB}$ is in the fourth quadrant, as is shown in Fig. 6.10f. Equivalent capacitive resistance is increased by feedback in this case. The contour equivalent capacity is decreased, while active resistance increases under its action.

Thus, contour or corresponding mechanical system resultant own frequency is increased by FB while its Q factor is decreased.

The analysis shows that FB can essentially alter the system properties, its frequency and transitive characteristics. Then own frequencies, attenuation values, etc. can be either increased, or decreased.

No power compensators are used if the method offered is used. As a result, the design of piezoceramic sensors with NFB becomes simpler. In this case the piezoelement is also a power compensator. Besides that, as summation of direct transform and FB signals is made in the piezoelement volume, this type of FB was called spatial electromechanical.

A circuit of a piezoceramic sensor with FB, based on the method discussed, is shown in Fig. 6.11.

The sensor, represented in Fig. 6.11, is a closed static follow-up system [5, 18]. It consists of piezoelement PE and matching voltage preamplifier A. Three electrodes 1, 2 and 3 are attached to the piezoelement. Electrode 1 is connected to the preamplifier input, electrode 2—to the common wire of the circuit, and electrode 3—an additional piezoelement electrode—to the preamplifier output.

Fig. 6.11

Piezoceramic sensor with FB

Because electrodes can settle down on various sides of a piezoelement, the feedback in this case is named by spatial.

Transfer function of such device looks like

$$\begin{aligned} W_{FB} (p)=W_1 (p)\frac{W_{TR} (p)}{1+W_{TR} (p)\beta (p)}, \end{aligned}$$

(6.15)

where $W_{1}(p)$—transfer coefficient of not direct transform circuit with NFB;

$W_{TR}(p)$—transfer coefficient of direct transform circuit with NFB;

$\beta (p)$—transfer coefficient with NFB circuit.

Relative error devices, represented on Fig. 6.11, is possible to define under the formula [6]:

$$\begin{aligned} {\gamma }_{{ FB}} =\gamma _{{ W}} \frac{1}{1+W(p)\beta (p)}-\gamma _{W} (1-\frac{1}{1+W\left( p \right) \beta (p)}), \end{aligned}$$

(6.16)

where $\gamma _{W}$—relative error of direct transfer subcircuit with NFB.

The condition under which the error of a piezosensor with NFB will equal zero can be easily seen from this equation, i.e. $\gamma _{{ FB}}=0$:

$$\begin{aligned} W(p)\beta (p)=1 \end{aligned}$$

(6.17)

It is necessary to notice also, that in this case a various arrangement of vectors F, P, $\mathrm{E}_\mathrm{IN}, \mathrm{E}_\mathrm{FB}$ allows to receive, for example, for a piezoelement in the form of a parallelepiped, tens variants of sensors with various characteristics.

Fig. 6.12

AFC of piezoelectric transducer: a without FB; b with FB

For FB radiators it is possible to enter by means of the additional chain which has been not connected galvanic with a chain of passage of a signal of excitation (Fig. 6.13). This circuit allows to create necessary dynamic characteristics of the transducer.

Fig. 6.13

Projector with FB: G generator; A amplifier of voltage;PE piezoelement

6.4 Inclusion of Piezoelements in Schemes of Electric Filters

The idea is put in a basis of creation of such transducers that if piezoelement (piezoelements) to include in the scheme of the electric filter amplitude-frequency responses (AFC) transducers will correspond AFC of the filter [24].

Electric filters are well enough studied and described in the literature [16, 17, 25]. As the electric filter is called the device serving for allocation (or suppression) electric voltage or currents of the set frequency. Depending on characteristics some types of filters from which the greatest interest for the given case is represented by filters of the bottom and top frequencies are known.

Filters of the bottom frequencies (FBF) pass fluctuations of all frequencies from a direct current and to some top boundary frequency $\upomega _\mathrm{t}$.

Filters of the top frequencies (FTF) pass fluctuations from some bottom boundary $\upomega _\mathrm{b}$ to infinitely high.

Two variants of the transducer with piezoelements in scheme FBF and FTF are shown in Fig. 6.14 [9, 15, 17, 24].

Fig. 6.14

Sensors with piezoelements in schemes of electric filters: a in scheme FBF; b in scheme FTF

Lack of these sensors is necessity of use for some schemes of two piezoelements or a piezoelement and a capasitor. To eliminate this lack, it is offered to use in schemes of sensors piezotransformers, that is piezoelements with two systems of electrodes. Besides, it is offered to have electrodes on a piezoelement so that the electric field vector between these electrodes was under a corner $\alpha$ to a polarisation vector ($0< \alpha \le 90^{\circ }$).

Two schemes of the sensors, realising these ideas, are shown in Fig. 6.15 [11, 24].

Fig. 6.15

Sensors with piezotransformers in schemes with piezotransformers in schemes: a in scheme FBF; b in scheme FTF

The idea of inclusion of piezoelements in schemes of filters can be realised and for radiators.

6.5 Technology of Additional Elements

The essence of this technology consists in addition to a piezoelement of elements which change sensor characteristics. Here two variants are possible, at least. In the first case the second piezoelement, a metal plate or the ultrasonic concentrator mechanically joins with a piezoelement [9, 11, 26].

In the second case the capacity joins a piezoelement electrically inductance, an oscillatory contour, a piezoelement or a piezoelement part.

Bimorph and trimorph elements

Two piezoelements connected among themselves mechanically and electrically (symmetric bimorph piezoelement), allow to increase sensitivity at 10–20 times and in as much time to reduce resonant frequency. Connection of a piezoelement and metal plate (asymmetric bimorph piezoelement) also leads to increase 10 times to sensitivity and reduction of resonant frequency.

In bimorph piezoelements arise curving fluctuations that allows to use them in microelectromechanical systems and devices (MEMS), for example, in scanners of nanomicroscopes [10, 20].

Ultrasonic concentrators

Ultrasonic concentrators are devices for ultrasound (US) intensity increase, i.e. the amplitude of particles vibration displacement [6, 14].

An ultrasonic concentrator is a mechanical transformer of vibrations. It means that displacement amplitude on the output concentrator side is in K times bigger than on the input, where K—concentrator transfer coefficient.

Two types of concentrators, based on different action principle, are distinguished: focusing or high-frequency, and rod, or low-frequency. In the given section the influence of US rod concentrators on the piezoelectric sensors parameters are studied.

So, rod US concentrator (RUSC) is a device for increase of particles vibration displacement amplitude (or particles oscillatory velocity) in a low-frequency range. RUSC is a hard rod of variable section or variable density, attached to the radiator by its wider end or by the part of greater material density.

RUSC action principle is based on increase of particles vibration displacement amplitude as a result of its cross-section reduction or density, according to the momentum conservation law. Then, displacement amplitude increases with the rod opposite ends diameters or densities difference.

RUSC is widely used in ultrasonic technology. RUSCs are components of ultrasonic vibration systems. RUSC can be considered as an acoustic waveguide in which a zero vibrations mode is propagated. It is characterized by the constant section amplitude. The maximum linear size of the concentrator D wide should be less than $\frac{\lambda }{2}$, where $\lambda$—wave length in the concentrator material.

RUSCs usually work at resonant frequency, therefore length L of the concentrator should be multiple to

half-waves integer: $L = \frac{n\lambda }{2}$, where n $=$ 1,2,3. If the frequency is intended $\lambda$ depends on RUSC form as a result of US propagation velocity in the wave guides with variable cross-section.

RUSC with variable density is usually made of two interconnected rods of various materials $\frac{\lambda }{4}$long with identical variable cross-section.

RUSCs are usually classified according to the following features [53]:

• (a) longitudinal section form (Fig. 6.16);

• (b) cross-section form (round, sphenoid, etc.);

• (c) amount of elements with various longitudinal section profile (simple, compound—Fig. 6.17);

• (d) amount of tandem resonant concentrators of half-wave length (one–two–etc., stage—Figs. 6.18, 6.19);

• (e) mean line form (rectilinear, bent);

• (f) concentrator vibrations type (longitudinal, shear, torsion).

Fig. 6.16

Sections of round simple one-step concentrators of longitudinal vibrations: a stepwise; b conic; c exponential; d catenoidal; e gausses (ampoule); curves show the distribution of amplitude oscillatory velocity $\upsilon$ and deformations U’ along the concentrator length

Fig. 6.17

Compound Concentrator: I—big diameter cylinder; II—conic or exponential-shaped rod part; III—small diameter cylinder

Fig. 6.18

Two-step Concentrator: I—step concentrator; II—ampoule concentrator

RUSC cross-section change can occur as a result of rod external and internal profile change (Figs. 6.16 and 6.9) accordingly.

Fig. 6.19

Concentrators with variable internal profile: a exponential; b and c stepwise

RUSC force coefficient $C = \frac{\xi }{\xi _0 }$, where $\xi$ and $\xi _{0}$—displacement amplitudes on its narrow and wide ends accordingly. Under the influence of harmonious vibration with circular frequency $\omega$ the vibrational speed amplitude $V = \omega \xi$ and, consequently, $C = \frac{V_{\mathrm{{t}}}}{V_0}$. For step RUSC $C = N^{2}$, where $N = \frac{R_0 }{R_e }$, and $R_{e}$ and $R_{0}$—radiuses of narrow (output) and wide (input) ends accordingly. For exponential $C=$ N, catenoid $C = \frac{N}{\cos \frac{2 \pi \cdot l}{\lambda }}$, and for conic $C < N$ and always $C < 4.6$ [14].

The vibrational speed $V_{m}$ maximum amplitude, received on the narrow RUSC end, depends on the concentrator material properties, destroying the fatigue stress F, and wave resistance pc (where p—density, c—US waves propagation velocity), and on dimensionless function T, depending only from the concentrator form:

$$\begin{aligned} V_{m}=\frac{F}{\rho c} T. \end{aligned}$$

(6.18)

RUSC are widely used in US technology as various US tools with US machine working, soldering, crushing, dispersion, clearing, in medicine, etc. [6].

The sensitivity increase effect of mechanical values resonant piezoceramic transducers with RUSC use is discovered and partially studied by I.G. Minaev and V.M. Sharapov in 1976 [56–58]. Some RUSC uses in piezosensors are described in Trofimov’s [50] and Sharapov’s [6] papers.

A simple sensor with RUSC is shown in Fig. 6.20 [4]. Here stepwise concentrator 2 is attached to piezotransformer 1 surface. The piezotransformer is connected to oscillations generator 3 and measuring device 4. The dependence of piezotransformer output voltage on the force for a sensor without concentrator (curve 1) and for the sensor in Fig. 6.20 (curve 2) is shown in Fig. 6.21. As it is seen from Fig. 6.21, the transducer sensitivity was 10 times increased because of the concentrator use.

Fig. 6.20

Resonant piezosensor with ultrasonic concentrator: 1—piezotransformer; 2—concentrator; 3—generator; 4—measuring device

Fig. 6.21

Piezosensor static characteristics: 1—without concentrator; 2—with concentrator

One more variant of the sensor design is shown in Fig. 6.22 [6]. Here the piezoelement is made as a stepwise concentrator. It can be practically realized if two piezoelements of different diameters and lengths, satisfying the creation condition of resonant vibrations in piezoelements, are interconnected. The connections can be glued with epoxy compound or soldered (Rose’s and Wood’s alloys, etc.). Sensitivity increases for the given design can reach $D^{2}/d^{2}$, where D and d—piezoelements diameters.

Fig. 6.22

Piezosensor: 1—stepwise concentrator-shaped piezoelement; 2—generator; 3—measuring device

Not only considerable sensitivity increase, but also accuracy improvement can be referred to concentrators use advantages. This can be reached if the concentrator is made of material with better than piezoceramics elastic characteristics (steel, quartz and bronze). In addition, the force can be applied to the point, through spherical elements, for example, as it is done in force measuring equipment. Fastening the transducer to the concentrator vibration node the losses in environment can be avoided completely. Finally, the force can be transferred through a precision resilient element, rigidly welded or soldered to the concentrator. This allows contact rigidity influence avoidance and the static characteristic linearization.

Joining of the ultrasonic concentrator to a piezoelement increases amplitude of oscillatory displacement (or speeds), that allows to use such devices for ultrasonic are sharp, sinks, dispersions of a liquid, and also in measuring devices on the basis of resonant piezoelements [6, 14]. The use of concentrators for increase in capacity of low-frequency ultrasonic radiators is not less perspective.

Electric elements and circuits

Because the piezoelement is the electromechanical device to which there corresponds an electric circuit (in that specific case—a consecutive oscillatory contour), connection to it of electric elements (resistors, condensers, inductance) can change characteristics piezosensors (see also Chap. 7–9).

For example, inductance connection between electrodes piezotransformer the sensor allows to increase level of output voltage and acoustic power piezoradiator, to expand a pass-band of the sensor, etc. (see Chap. 7–9).

Connection is consecutive with a resistor piezoelement, reduces good quality of a piezoelement and expands a working strip of frequencies.

Capacity inclusion between input and output systems of electrodes of piezotransformers the sensor also allows to expand a working range of frequencies.

Inductance connection between electrodes piezotransformer the sensor allows to increase level of output voltage and acoustic power piezoradiator, to expand a pass-band of the sensor, etc. Results of researches in this area will be published in separate work [15, 27–40] (see Chap. 7–9).

Acoustic resonators

The use of acoustic resonators for work PEAT in air, for example, of resonator Helmholtz allows to raise level of sound pressure (see Chap. 7–9).

6.6 Technology of Synthesis of the Transducers, Considering Electric signals

The change of the form of the electric signal arriving on the transducer, can lead to change of its technical characteristics. For example, if the tax on the electro-acoustic transducer electric voltage in the form of a meander, AFC such transducer extends towards low frequencies [7, 11].

At giving on a piezoelement simultaneously two signals new properties and transducer functions can be received. For example, if the tax on a piezoelement with two inputs two signals of the sinusoidal form, can be received a low-frequency signal enough high capacity [11, 21, 22, 28, 30, 32, 33, 42, 43].

6.7 The Combined Technology

In this case the technologies described above are used simultaneously or in various combinations. It is easy to see, that in this case from one piezoelement hundreds (!) of variants of sensors with various characteristics among which it is possible to choose a variant with necessary or best characteristics (increase of accuracy, stability, sensitivity, expansion of a working range of frequencies, etc.) can be received. Some devices mentioned in the given chapter, are described in patents [34–39].