FarnellGw."Ultrasound The Electrical Engineering Handbook Ed. Richard C. Dorf Boca raton crc Press llc. 2000
Farnell, G.W. “Ultrasound” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000
48 trason 18.1 Introduction 48.2 Propagation in Solids 48.3 Piezoelectric Excitation Gerald W. Farnell 48.4 One-Dimensional Propagation McGill University 48.5 Transducers 48.1 Introduction In electrical engineering, the term ultrasonics usually refers to the study and use of waves of mechanical vibrations propagating in solids or liquids with frequencies in the megahertz or low gigahertz ranges. Such waves in these frequency ranges have wavelengths on the order of micrometers and thus can be electrically generated, directed, and detected with transducers of reasonable size. These ultrasonic devices are used for signal processing directly in such applications as filtering and pulse compression and indirectly in acousto- optic processors; for flaw detection in optically opaque materials; for resonant circuits in frequency control applications; and for medical imaging of human organs, tissue, and blood flow. 48.2 Propagation in Solids If the solid under consideration is elastic (linear), homogeneous, and nonpiezoelectric, the components, u, of the displacement of an infinitesimal region of the material measured along a set of Cartesian axes, x,are interrelated by an equation of motion G=∑∑a0x l. (48.1) where p is the mass density of the material and c (i,, k,I= 1, 2, 3)is called the stiffness tensor. It is the set of proportionality constants between the components of the stress tensor T and the strain tensor S in a three dimensional Hooke's law(form: T= cS with S=dul ax). In Eq (48. 1)and in the subse seful for discussion ent equations the form of the equation is shown without the clutter of the many subscripts. The form purposes; moreover, it gives the complete equation for cases in which the propagation can be treated as or dimensional, ie, with variations in only one direction, one component of displacement, and one relevant c In an infinite medium, the simplest solutions of Eq (48. 1)are plane waves given by the real part of UeA(∑-) Form: u= Uej( t-kx) (48.2) where the polarization vector has components U, along the axes. The phase velocity of the wave Vis measured along the propagation vector k whose direction cosines with respect to these axes are given by Lr. Substituting c 2000 by CRC Press LLC
© 2000 by CRC Press LLC 48 Ultrasound 48.1 Introduction 48.2 Propagation in Solids 48.3 Piezoelectric Excitation 48.4 One-Dimensional Propagation 48.5 Transducers 48.1 Introduction In electrical engineering, the term ultrasonics usually refers to the study and use of waves of mechanical vibrations propagating in solids or liquids with frequencies in the megahertz or low gigahertz ranges. Such waves in these frequency ranges have wavelengths on the order of micrometers and thus can be electrically generated, directed, and detected with transducers of reasonable size. These ultrasonic devices are used for signal processing directly in such applications as filtering and pulse compression and indirectly in acoustooptic processors; for flaw detection in optically opaque materials; for resonant circuits in frequency control applications; and for medical imaging of human organs, tissue, and blood flow. 48.2 Propagation in Solids If the solid under consideration is elastic (linear), homogeneous, and nonpiezoelectric, the components, ui , of the displacement of an infinitesimal region of the material measured along a set of Cartesian axes, xi , are interrelated by an equation of motion: (48.1) where r is the mass density of the material and cijkl (i, j, k, l = 1, 2, 3) is called the stiffness tensor. It is the set of proportionality constants between the components of the stress tensor T and the strain tensor S in a threedimensional Hooke’s law (form: T = cS with S = ]u/]x). In Eq. (48.1) and in the subsequent equations the form of the equation is shown without the clutter of the many subscripts. The form is useful for discussion purposes; moreover, it gives the complete equation for cases in which the propagation can be treated as one dimensional, i.e., with variations in only one direction, one component of displacement, and one relevant c . In an infinite medium, the simplest solutions of Eq. (48.1) are plane waves given by the real part of (48.2) where the polarization vector has components Ui along the axes. The phase velocity of the wave V is measured along the propagation vector k whose direction cosines with respect to these axes are given by Li . Substituting r ¶ ¶ ¶ ¶ ¶ r ¶ ¶ ¶ ¶ 2 2 2 2 2 2 2 u t c u x x u t c u x i ijk l j k l j k l = = Â Â Â , Form: u U e u Ue i i jk L x Vt j t kx j j = = Âj Ê Ë ˆ ¯ – – ( – ) Form: w Gerald W. Farnell McGill University
the assumed solutions of Eq (48. 2)into Eq (48.1)gives the third-order eigenvalue equations, usually known ∑∑∑LLU=pVU,Fom:(-pV2=0(483 The three eigenvalues in Eq.(48.3)give three values of pV and hence the phase velocities of three waves propagating in the direction of positive k and three propagating in the negative k direction. The eigenvectors of the three forward solutions give the polarization vector for each, and they form a mutually perpendicular triad. The polarization vector of one of the plane waves will be parallel, or almost parallel, to the k vector, and it is called the longitudinal wave, or quasi-longitudinal if the displacement is not exactly parallel to k. The other two waves will have mutually perpendicular polarization vectors, which will each be perpendicular, or almost perpendicular, to the k vector. If the polarization is perpendicular, the wave is called a transverse or shear wave, if almost perpendicular, it is called quasi-shear. The three waves propagate independently through the solid, and their respective amplitudes depend on the exciting source. In an isotropic medium where there are only two independent values of iu in Eq. (48.1), there are one longitudinal wave and two degenerate shear waves. The phase velocities of these waves are independent of the direction of propagation and are given by (48.4) p The phase velocities in isotropic solids are often expressed in terms of the so-called Lame constants defined by u=G212 and x=Gill -2Gi212. The longitudinal velocity is larger than the shear velocity. Exact velocity values depend on fabrication procedures and purity, but Table 48. 1 gives typical values for some materials important in ultrasonics In signal processing applications of ultrasonics, the propagating medium is often a single crystal, and thus a larger number of independent stiffness constants is required to describe the mechanical properties of the medium, e.g., three in a cubic crystal, five in a hexagonal, and six in a trigonal. note that while the number of independent constants is relatively small, a large number of the c are nonzero but are related to each other by the symmetry characteristics of the crystal. The phase velocities of each of the three independent plane waves in an anisotropic medium depend on the direction of propagation. Rather than plotting V as a function of angle of propagation, it is more common to use a slowness surface giving the reciprocal of v(or k=o/Vfor a given w)as a function of the direction of k Usually planar cuts of such slowness surfaces are plotted as shown in Figs. 48.1(a) and(b) In anisotropic materials the direction of energy flow (the ultrasonic equivalent of the electroma ynting vector) in a plane wave is not parallel to k. Thus the direction of k is set by the transducer but the energy flow or beam direction is normal to the tangent to the slowness surface at the point corresponding to k. The direction of propagation(of k) in Fig. 48.1 lies in the basal plane of a cubic crystal, here silicon. At each angle there are three waves one is pure shear polarized perpendicular to this plane, one is quasilongitudinal for most angles, while the third is quasi-shear. For the latter two, the tangent to the slowness curves at an arbitrary angle is not normal to the radius vector, and thus there is an appreciable angle between the direction of energy flow and the direction of k. This angle is shown on the diagram by the typical k and P vectors, the latter being the direction of energy flow in an acoustic beam with this k along the cubic axes in a cubic crystal, the two shear waves are degenerate, and for all three waves the energy flow is parallel to k. When the particle displacement of a mode is either parallel to the propagation vector or perpendicular to it and the energy flow is parallel to k, the mode is called a pure mode. The propagation vector in Fig. 48.1(b)lies in the basal plane of a trigonal crystal, quartz. When ultrasonic waves propagate in a solid, there are various losses that attenuate the wave. Usually the attenuation per wavelength is small enough that one can neglect the losses in the initial calculation of the c 2000 by CRC Press LLC
© 2000 by CRC Press LLC the assumed solutions of Eq. (48.2) into Eq. (48.1) gives the third-order eigenvalue equations, usually known as the Christoffel equations: (48.3) The three eigenvalues in Eq. (48.3) give three values of rV2 and hence the phase velocities of three waves propagating in the direction of positive k and three propagating in the negative k direction. The eigenvectors of the three forward solutions give the polarization vector for each, and they form a mutually perpendicular triad. The polarization vector of one of the plane waves will be parallel, or almost parallel, to the k vector, and it is called the longitudinal wave, or quasi-longitudinal if the displacement is not exactly parallel to k. The other two waves will have mutually perpendicular polarization vectors, which will each be perpendicular, or almost perpendicular, to the k vector. If the polarization is perpendicular, the wave is called a transverse or shear wave; if almost perpendicular, it is called quasi-shear. The three waves propagate independently through the solid, and their respective amplitudes depend on the exciting source. In an isotropic medium where there are only two independent values of cijkl in Eq. (48.1), there are one longitudinal wave and two degenerate shear waves. The phase velocities of these waves are independent of the direction of propagation and are given by (48.4) The phase velocities in isotropic solids are often expressed in terms of the so-called Lame constants defined by m = c1212 and l = c1111 2 2c1212. The longitudinal velocity is larger than the shear velocity. Exact velocity values depend on fabrication procedures and purity, but Table 48.1 gives typical values for some materials important in ultrasonics. In signal processing applications of ultrasonics, the propagating medium is often a single crystal, and thus a larger number of independent stiffness constants is required to describe the mechanical properties of the medium, e.g., three in a cubic crystal, five in a hexagonal, and six in a trigonal. Note that while the number of independent constants is relatively small, a large number of the cijkl are nonzero but are related to each other by the symmetry characteristics of the crystal. The phase velocities of each of the three independent plane waves in an anisotropic medium depend on the direction of propagation. Rather than plotting V as a function of angle of propagation, it is more common to use a slowness surface giving the reciprocal of V (or k = v/V for a given v) as a function of the direction of k. Usually planar cuts of such slowness surfaces are plotted as shown in Figs. 48.1(a) and (b). In anisotropic materials the direction of energy flow (the ultrasonic equivalent of the electromagnetic Poynting vector) in a plane wave is not parallel to k. Thus the direction of k is set by the transducer but the energy flow or beam direction is normal to the tangent to the slowness surface at the point corresponding to k. The direction of propagation (of k) in Fig. 48.1 lies in the basal plane of a cubic crystal, here silicon. At each angle there are three waves—one is pure shear polarized perpendicular to this plane, one is quasilongitudinal for most angles, while the third is quasi-shear. For the latter two, the tangent to the slowness curves at an arbitrary angle is not normal to the radius vector, and thus there is an appreciable angle between the direction of energy flow and the direction of k. This angle is shown on the diagram by the typical k and P vectors, the latter being the direction of energy flow in an acoustic beam with this k.Along the cubic axes in a cubic crystal, the two shear waves are degenerate, and for all three waves the energy flow is parallel to k. When the particle displacement of a mode is either parallel to the propagation vector or perpendicular to it and the energy flow is parallel to k, the mode is called a pure mode. The propagation vector in Fig. 48.1(b) lies in the basal plane of a trigonal crystal, quartz. When ultrasonic waves propagate in a solid, there are various losses that attenuate the wave. Usually the attenuation per wavelength is small enough that one can neglect the losses in the initial calculation of the Lk l L c U V U c V U j k l    ij kl j = r r i = 2 2 , Form: ( – ) 0 V c V c 1 s 1111 1212 = = r r and
TABLE 48.1 Typical Acoustic Properties velocity Impedance km/s) kg/m2sx10°) D Material Longitudinal SI ongitudinal Shear (kg/mx10) Alcohol, methanol 1.103 0.872 Lig-25℃C rolled 3.04 8.21 0%Cu,30%Zn 40.6 18.l4 Cadmium sulphide 21.5 82 Piez crys Z-dir or oil 1.507 Liq 20C 4.03 46.6 28.21 5.01 44.6 8.93 1.658 1.845 1.113 Lig-25℃C 13.1 l1.4 Gold, hard drawn 3.24 1.20 63.8 626 2.24 n, cas 5.9 46.4 24.6 Lead 24.6 83 11.2 Lithium niobate, LiNbO 19.17 Piez crys X-dir Nickel 5.6 26.5 8.84 Polystyrene, styron 2.52 121 105 3.1 5.74 Piez crys X-dir 3.5 11.1 44.3 3.99 10.6 teel, mild 46.0 .3 24.2 12.5 Titanium 6.1 Water Liq-20℃C YAG Y,AlsO, 390 4.55 Cryst. Z-axis Zinc Zinc oxide 6.37 2.73 15.47 5.67 Piez crys Z-dir x104 §as5 FIGURE 48.1 (a)Slowness curves, basal plane, cubic crystal, silicon.(b) Slowness curves, basal plane, trigonal cryst quartz
© 2000 by CRC Press LLC TABLE 48.1 Typical Acoustic Properties Velocity Impedance (km/s) (kg/m2 s 3 106 ) Density Material Longitudinal Shear Longitudinal Shear (kg/m3 3103 ) Comments Alcohol, methanol 1.103 0.872 0.791 Liq. 25°C Aluminum, rolled 6.42 3.04 17.33 8.21 02.70 Isot. Brass, 70% Cu, 30% Zn 4.70 2.10 40.6 18.14 8.64 Isot. Cadmium sulphide 4.46 1.76 21.5 8.5 4.82 Piez crys Z-dir Castor oil 1.507 1.42 0.942 Liq. 20oC Chromium 6.65 4.03 46.6 28.21 7.0 Isot. Copper, rolled 5.01 2.27 44.6 20.2 8.93 Isot. Ethylene glycol 1.658 1.845 1.113 Liq. 25°C Fused quartz 5.96 3.76 13.1 8.26 2.20 Isot. Glass, crown 5.1 2.8 11.4 6.26 2.24 Isot. Gold, hard drawn 3.24 1.20 63.8 23.6 19.7 Isot. Iron, cast 5.9 3.2 46.4 24.6 7.69 Isot. Lead 2.2 0.7 24.6 7.83 11.2 Isot. Lithium niobate, LiNbO3 6.57 4.08 30.9 19.17 4.70 Piez crys X-dir 4.79 22.53 Nickel 5.6 3.0 49.5 26.5 8.84 Isot. Polystyrene, styron 2.40 1.15 2.52 1.21 1.05 Isot. PZT-5H 4.60 1.75 34.5 13.1 7.50 Piez ceram Z Quartz 5.74 3.3 15.2 8.7 2.65 Piez crys X-dir 5.1 13.5 Sapphire Al2O3 11.1 6.04 44.3 25.2 3.99 Cryst. Z-axis Silver 3.6 1.6 38.0 16.9 10.6 Isot. Steel, mild 5.9 3.2 46.0 24.9 7.80 Isot. Tin 3.3 1.7 24.2 12.5 7.3 Isot. Titanium 6.1 3.1 27.3 13.9 4.48 Isot. Water 1.48 1.48 1.00 Liq. 20°C YAG Y3Al15O12 8.57 5.03 39.0 22.9 4.55 Cryst. Z-axis Zinc 4.2 2.4 29.6 16.9 7.0 Isot. Zinc oxide 6.37 2.73 36.1 15.47 5.67 Piez crys Z-dir FIGURE 48.1 (a) Slowness curves, basal plane, cubic crystal, silicon. (b) Slowness curves, basal plane, trigonal crystal, quartz
propagation characteristics of the material and the excitation, and then multiply the resulting propagating wave by a factor of the form expl-ax] where x is in the direction of k and a is called the attenuation constant. One loss mechanism is the viscosity of the material and due to it the attenuation constant is (48.5) ° in which m is the coefficient of viscosity. It should be noted that the attenuation constant for viscous loss increases as the square of the frequency. In polycrystalline solids there is also loss due to scattering from dislocation and grain structure; thus, for the same material the loss at high frequencies is much higher in a polycrystalline form than in a crystalline one. As a result, in high-frequency applications of ultrasound, such as for signal processing, the propagation material is usually in single-crystal form 48.3 Piezoelectric excitation When a piezoelectric material is stressed, an electric field is generated in the stressed region; similarly, if an lectric field is applied, there will be an induced stress on the material in the region of the field. Thus, there is a coupling between mechanical motion and time-varying electric fields. Analysis of wave propagation in piezoelectric solids should thus include the coupling of the mechanical equations such as Eq(48.1)with Maxwell,s equations. In most ultrasonic problems, however, the velocity of the mechanical wave solutions is low enough that the electric fields can be described by a scalar potential This is called the quasi-static approximation. Within this approximation, the equations of motion in a piezoelectric solid become 0-∑∑AE 22ta2 2 Form: p c ar2 dx dx2 ∑∑Q∑∑∑08 The piezoelectric coupling constants eik form a third-rank tensor property of the solid and are the propor tionality constants between the components of the electric field and the components of the stress. Similarly ey the second-rank permittivity tensor, giving the proportionality constants between the components of the electric field E and of the electric displacement D If the material is nonpiezoelectric e=0, then the first three equations of Eq (48.6)reduce to the corresponding three of Eq (48.1), whereas the fourth equation becomes the anisotropic Laplace equation In a piezoelectric, these mechanical and electrical components are coupled. The plane wave solution of Eq (48.6)then has the three mechanical components of Eq (48.2)and in addition has a potential given by ∑-m) Form:φ=Φejo-) (48.7) Thus, for the quasi-static approximation there is a wave of potential that propagates with an acoustic phase velocity V in synchronism with the mechanical variations. As will be seen in Section 48.5, it is possible to use the corresponding electric field,-Vo, to couple to electrode configurations and thus excite or detect the asonic wave from external electric circuits Rather than substituting Eq (48.7)and Eq (48. 2)into Eq (48.6)to obtain a set of four equations similar to Eq (483), it is frequently more convenient to substitute Eq (48.7)into the fourth equation in the set of Eq (48.6). Because there are no time derivatives involved, this substitution gives the potential as a linear combi nation of the components of the mechanical displacement: c 2000 by CRC Press LLC
© 2000 by CRC Press LLC propagation characteristics of the material and the excitation, and then multiply the resulting propagating wave by a factor of the form exp[–ax] where x is in the direction of k and a is called the attenuation constant. One loss mechanism is the viscosity of the material and due to it the attenuation constant is (48.5) in which h is the coefficient of viscosity. It should be noted that the attenuation constant for viscous loss increases as the square of the frequency. In polycrystalline solids there is also loss due to scattering from dislocation and grain structure; thus, for the same material the loss at high frequencies is much higher in a polycrystalline form than in a crystalline one. As a result, in high-frequency applications of ultrasound, such as for signal processing, the propagation material is usually in single-crystal form. 48.3 Piezoelectric Excitation When a piezoelectric material is stressed, an electric field is generated in the stressed region; similarly, if an electric field is applied, there will be an induced stress on the material in the region of the field. Thus, there is a coupling between mechanical motion and time-varying electric fields. Analysis of wave propagation in piezoelectric solids should thus include the coupling of the mechanical equations such as Eq. (48.1) with Maxwell’s equations. In most ultrasonic problems, however, the velocity of the mechanical wave solutions is slow enough that the electric fields can be described by a scalar potential f. This is called the quasi-static approximation. Within this approximation, the equations of motion in a piezoelectric solid become (48.6) The piezoelectric coupling constants eijk form a third-rank tensor property of the solid and are the proportionality constants between the components of the electric field and the components of the stress. Similarly eij is the second-rank permittivity tensor, giving the proportionality constants between the components of the electric field E and of the electric displacement D. If the material is nonpiezoelectric eijk = 0, then the first three equations of Eq. (48.6) reduce to the corresponding three of Eq. (48.1), whereas the fourth equation becomes the anisotropic Laplace equation. In a piezoelectric, these mechanical and electrical components are coupled. The plane wave solution of Eq. (48.6) then has the three mechanical components of Eq. (48.2) and in addition has a potential given by (48.7) Thus, for the quasi-static approximation there is a wave of potential that propagates with an acoustic phase velocity V in synchronism with the mechanical variations. As will be seen in Section 48.5, it is possible to use the corresponding electric field, –¹f, to couple to electrode configurations and thus excite or detect the ultrasonic wave from external electric circuits. Rather than substituting Eq. (48.7) and Eq. (48.2) into Eq. (48.6) to obtain a set of four equations similar to Eq. (48.3), it is frequently more convenient to substitute Eq. (48.7) into the fourth equation in the set of Eq. (48.6). Because there are no time derivatives involved, this substitution gives the potential as a linear combination of the components of the mechanical displacement: a h w r = 2 3 2V r ¶ ¶ ¶ ¶ ¶ ¶ f ¶ ¶ r ¶ ¶ ¶ ¶ ¶ f ¶ ¶ f ¶ ¶ ¶ ¶ ¶ f ¶ 2 2 2 2 2 2 2 2 2 2 2 2 2 2 u t c u x x e x x u t c u x e x x x e u x x e i ijkl j k l ijk j k kjlkj ij i j ijk j i k kjiji -= = = —= ÂÂÂÂÂ ÂÂÂÂÂ –Form: e e u ¶x 2 f f w = = Â Ê Ë ˆ ¯ - F F e e jk L x Vt j t kx j j – – () j Form:
∑∑∑=L ∑∑s,L1 Fom:Φ=2 (48.8) When this combination is substituted into the first three equations of Eq. (48.6)and terms gathered, they become identical to Eq (48.1)but with each ci replaced by e LL E=c(1+K2) (489) Using these so-called stiffened elastic constants, we obtain the same third-order eigenvalue equation, Eq (48.3),and hence the velocities of each of the three modes and the corresponding mechanical displacement components. The potential is obtained from Eq (48.8). The velocities obtained for the piezoelectric material are usually at most a few percent higher than would be obtained with the piezoelectricity ignored. The parameter Kin Eq. (48.9)is called the electromechanical coupling constant. 48.4 One-Dimensional Propagation If an acoustic plane wave as in Eq(48.2)propagating within one medium strikes an interface with another boundary conditions at the interface, it will be necessary in gencral esponding case in optics. To satisfy the medium there will be reflection and transmission much as in the cor enerate three transmitted modes and three reflected modes. Thus, the concepts of reflection and transmission coefficients for planar interfaces tween anisotropic media are complicated In many propagation and excitation geometries, however, one can consider only one independent pure mode with energy flow parallel to k and particle displacement polarized along k or perpendicular to it. This mode (plane wave) then propagates along the axis or its negative in Eq (48.2). Discussion of the generation, propagation, and reflection of this wave is greatly assisted by consid- ering analogies to the one-dimensional electrical transmission line. With the transmission line model operating in the sinusoidal steady state, the particle displacement u; of Eq (48. 2)is represented by a phasor, u. The time derivative of the particle displacement is the particle velocity and is represented by a phasor, v= jou, which is taken as analogous to the current on the one-dimensional electrical transmission line. The negative of the stress, or the force per unit area, caused by the particle displacement is represented by a phasor, (n)=jkcu, which is taken as analogous to the voltage on the transmission line. Here cis the appropriate stiffened elastic constant for the mode in question in Eq (48.3). with these definitions, the general impedance, the characteristic impedance, the phase velocity, and the wave vector, respectively, of the equivalent line are given by (48.10) Vp Some typical values of the characteristic impedance of acoustic media are given in Table 48. 1. The characteristic impedance cor ing to a mode is given by the product of the density and the phase velocity, pV, even in the anisotropic case where the effective stiffness c in Eq.(48.10)is difficult to determine. As an example of the use of the transmission line model, conside wave propagating in n isotropic solid and incident normally on the interface with a second isotropic solid. There would be one reflected wave and one transmitted wave, both longitudinally polarized. The relative amplitudes of the stresses in these waves would be given, with direct use transmission line concepts, by the voltage reflection and transmission coefficients c 2000 by CRC Press LLC
© 2000 by CRC Press LLC (48.8) When this combination is substituted into the first three equations of Eq. (48.6) and terms gathered, they become identical to Eq. (48.1) but with each cijkl replaced by (48.9) Using these so-called stiffened elastic constants, we obtain the same third-order eigenvalue equation, Eq. (48.3), and hence the velocities of each of the three modes and the corresponding mechanical displacement components. The potential is obtained from Eq. (48.8). The velocities obtained for the piezoelectric material are usually at most a few percent higher than would be obtained with the piezoelectricity ignored. The parameter K in Eq. (48.9) is called the electromechanical coupling constant. 48.4 One-Dimensional Propagation If an acoustic plane wave as in Eq. (48.2) propagating within one medium strikes an interface with another medium, there will be reflection and transmission, much as in the corresponding case in optics. To satisfy the boundary conditions at the interface, it will be necessary in general to generate three transmitted modes and three reflected modes. Thus, the concepts of reflection and transmission coefficients for planar interfaces between anisotropic media are complicated. In many propagation and excitation geometries, however, one can consider only one independent pure mode with energy flow parallel to k and particle displacement polarized along k or perpendicular to it. This mode (plane wave) then propagates along the axis or its negative in Eq. (48.2). Discussion of the generation, propagation, and reflection of this wave is greatly assisted by considering analogies to the one-dimensional electrical transmission line. With the transmission line model operating in the sinusoidal steady state, the particle displacement ui of Eq. (48.2) is represented by a phasor, u. The time derivative of the particle displacement is the particle velocity and is represented by a phasor, v = jvu, which is taken as analogous to the current on the one-dimensional electrical transmission line. The negative of the stress, or the force per unit area, caused by the particle displacement is represented by a phasor, (–T) = jkcu, which is taken as analogous to the voltage on the transmission line. Here c is the appropriate stiffened elastic constant for the mode in question in Eq. (48.3). With these definitions, the general impedance, the characteristic impedance, the phase velocity, and the wave vector, respectively, of the equivalent line are given by (48.10) Some typical values of the characteristic impedance of acoustic media are given in Table 48.1. The characteristic impedance corresponding to a mode is given by the product of the density and the phase velocity, rV, even in the anisotropic case where the effective stiffness c in Eq. (48.10) is difficult to determine. As an example of the use of the transmission line model, consider a pure longitudinal wave propagating in an isotropic solid and incident normally on the interface with a second isotropic solid. There would be one reflected wave and one transmitted wave, both longitudinally polarized. The relative amplitudes of the stresses in these waves would be given, with direct use transmission line concepts, by the voltage reflection and transmission coefficients F F = = ÂÂÂ ÂÂ e LLU L L e U ijk i k j kji ij i j ji e e Form: c c e e LL L L cc K K e c ijkl ijkl mij nkl m n nm mn m n nm =+ = ÂÂ ÂÂ e e Form: = (1 + ) with 2 2 2 Z T v Z cV c k V = = == (– ) 0 r r w
TR (48.11) z02+Z0 When an acoustic wave meets a discontinuity or a mismatch, part of the wave is reflected. For an incident mode, an interface represents a lumped impedance. If the medium on the other side of the interface is infinitely leep, that lumped impedance is the characteristic impedance of the second medium. However, if the second medium is of finite depth h in the direction of propagation and it in turn is terminated by a lumped impedance Zu the impedance seen by the incident wave at the interface is given, as in transmission line theory, by ZL2 cos k,h+ jZoz sin k,h (48.12) Zoz cos k,h+ jZu sin k,h Thus, as with transmission lines, an intervening layer can be used to match from one transmitting medium to another. For example, if the medium following the layer is infinite and of characteristic impedance Zo3,i.e Zu= Zo, the interface will look like Zo to the incident wave if kh= m/2, quarter-wave thickness, and the layer characteristic impedance is Z02=Zo1Zo3. This matching, which provides complete power transfer from medium I to medium 3, is valid only at the frequency for which kh=T/2. For matching over a band of frequencies, multiple matchin 48.5 Transducers Electrical energy is converted to acoustic waves in ultrasonic applications by means of electro-acoustic trans ducers. Most transducers are reciprocal in that they will also convert the mechanical energy in acoustic waves into electrical energy. The form of the transducer is very application dependent. Categories of applications include imaging, wherein one transducer is used to create an acoustic beam, discontinuities in the propagating medium scatter this beam, and the scattered energy is captured by the same or another transducer [see Fig. 48. 4(b). From the changes of the scattered energy as the beam is moved, characteristics of the scatterer are determined. This is the process in the use of ultrasonics for nondestructive evaluation(NDE), flaw detection, for example, and in ultrasonic images for medical diagnosis. These are radar-like applications and are practical at reasonable frequencies because most solids and liquids support acoustic waves with tolerable losses and the avelength is short enough that the resolution is adequate for practical targets By recording both the amplitude and phase of the scattered signal as the transmitter-receiver combination is rotated about a target, one can generate tomographic-type images of the target A second category of transducer provides large acoustic standing waves at a particular frequency and, as a result, has a resonant electrical input impedance at this frequency and can be used as a narrowband filter in electrical circuits. In a third category of transducer, the object is to provide an acoustic beam that distorts the medium, as it passes through, in a manner periodic in space and time, and thus provides a dynamic diffraction grating that will deflect or modulate an optical beam that is passed through it[see Fig. 48.4(c). Such acousto- optic devices are used in broadband signal processing. Another category of transducer uses variation of the shape of the electrodes and the geometry of the electroacoustic coupling region so that the transfer function between a transmitting and a receiving transducer made to have a prescribed frequency response. Such geometries find wide application in filtering and pulse compression applications in the frequency range up to a few gigahertz. Because of the ease of fabrication of complicated electrode geometries, special forms of the solution of the wave equation, Eq (48.1), called surfac acoustic waves(SAW)are dominant in such applications. Because surface acoustic waves are discussed in another section of this handbook, here we will confine the discussion to transducers that generate or detect acoustic waves that are almost plane and usually single mode, the so-called bulk modes The prototype geometry for a bulk-mode transducer is shown in Fig. 48. 2. The active region is the portion of the piezoelectric slab between the thin metal electrodes, which can be assumed to be circular or rectangular c 2000 by CRC Press LLC
© 2000 by CRC Press LLC (48.11) When an acoustic wave meets a discontinuity or a mismatch, part of the wave is reflected. For an incident mode, an interface represents a lumped impedance. If the medium on the other side of the interface is infinitely deep, that lumped impedance is the characteristic impedance of the second medium. However, if the second medium is of finite depth h in the direction of propagation and it in turn is terminated by a lumped impedance ZL2 the impedance seen by the incident wave at the interface is given, as in transmission line theory, by (48.12) Thus, as with transmission lines, an intervening layer can be used to match from one transmitting medium to another. For example, if the medium following the layer is infinite and of characteristic impedance Z03, i.e., ZL2 = Z03, the interface will look like Z01 to the incident wave if kh = p/2, quarter-wave thickness, and the layer characteristic impedance is Z2 02 = Z01Z03. This matching, which provides complete power transfer from medium 1 to medium 3, is valid only at the frequency for which kh = p/2. For matching over a band of frequencies, multiple matching layers are required. 48.5 Transducers Electrical energy is converted to acoustic waves in ultrasonic applications by means of electro-acoustic transducers. Most transducers are reciprocal in that they will also convert the mechanical energy in acoustic waves into electrical energy. The form of the transducer is very application dependent. Categories of applications include imaging, wherein one transducer is used to create an acoustic beam, discontinuities in the propagating medium scatter this beam, and the scattered energy is captured by the same or another transducer [see Fig. 48.4(b)]. From the changes of the scattered energy as the beam is moved, characteristics of the scatterer are determined. This is the process in the use of ultrasonics for nondestructive evaluation (NDE), flaw detection, for example, and in ultrasonic images for medical diagnosis. These are radar-like applications and are practical at reasonable frequencies because most solids and liquids support acoustic waves with tolerable losses and the wavelength is short enough that the resolution is adequate for practical targets. By recording both the amplitude and phase of the scattered signal as the transmitter-receiver combination is rotated about a target, one can generate tomographic-type images of the target. A second category of transducer provides large acoustic standing waves at a particular frequency and, as a result, has a resonant electrical input impedance at this frequency and can be used as a narrowband filter in electrical circuits. In a third category of transducer, the object is to provide an acoustic beam that distorts the medium, as it passes through, in a manner periodic in space and time, and thus provides a dynamic diffraction grating that will deflect or modulate an optical beam that is passed through it [see Fig. 48.4(c)]. Such acoustooptic devices are used in broadband signal processing. Another category of transducer uses variation of the shape of the electrodes and the geometry of the electroacoustic coupling region so that the transfer function between a transmitting and a receiving transducer is made to have a prescribed frequency response. Such geometries find wide application in filtering and pulse compression applications in the frequency range up to a few gigahertz. Because of the ease of fabrication of complicated electrode geometries, special forms of the solution of the wave equation, Eq. (48.1), called surface acoustic waves (SAW) are dominant in such applications. Because surface acoustic waves are discussed in another section of this handbook, here we will confine the discussion to transducers that generate or detect acoustic waves that are almost plane and usually single mode, the so-called bulk modes. The prototype geometry for a bulk-mode transducer is shown in Fig. 48.2. The active region is the portion of the piezoelectric slab between the thin metal electrodes, which can be assumed to be circular or rectangular G G R T Z Z Z Z Z Z Z = + = + 02 01 02 01 02 02 01 – 2 and Z Z Z k h jZ k h Z k h jZ k h in L L = + + 02 2 2 02 2 2 2 2 2 cos sin 0 cos sin
AMATCHING ACOUSTIC PORT PORT 2 ACOUSTIC PORT 1 Z。k PORT 2 ACOUSTIC BEAM ELECTROD ACTIVE REGION FIGURE 48.2 Prototype transducer geometry. FIGURE 48.3 Model of active region. in shape. Connections to these electrodes form the electrical port for the transducer and the voltage between them creates a spatially uniform electric field in the active region, and this time-varying electric field couples to the acoustic waves propagating between the electrodes. If the planar electrodes are many wavelengths in ansverse dimensions and the active region is much thinner, and if the axial direction is a pure mode direction for the piezoelectric, the waves in the active region can be considered as plane waves. We then have the one- dimensional geometry considered earlier. The transducer may be in contact with another elastic medium on either side, as indicated in Fig. 48. 2, so that the plane waves propagate in and out of the active regions in the cross-sectional region shown. Thus, the transducer has in general two acoustic ports for coupling to the outside world as well as the electrical port. In the absence of piezoelectric coupling, the active region could be represented by a one-dimensional transmission line as discussed in the previous section and as indicated by the heavy lines in Fig. 48.3. with zoelectricity there will be the stiffening of the appropriate stiffness constants as discussed in Eq (48.9)with the concomitant perturbation of the characteristic impedance Zo and the phase velocity Vp but more important there will also be coupling to the electrical port. One model including the latter coupling is shown in Fig. 48.3 in which the parameters are defined by sin(to/Oo) 2e/∈sin(o/200) (48.13) OAZO Here Co is the capacity that would be measured between the electrodes if there were no mechanical strain on he piezoelectric, A is the cross-sectional area of the active region, and X is an effective reactance. The quantity r is the transformer ratio(with dimensions)of an ideal transformer coupling the electrical port to the center of the acoustic transmission line. K is the electromechanical coupling constant for the material as defined in Eq(48.9). The so-called resonant frequency oo is that angular frequency at which the length d of the active region is one-half of the stiffened wavelength, wo=VId. In the physical configuration of Fig. 48. 4(a), the transducer has zero stress on the surfaces of the active region and hence both acoustic ports of Fig 48.3 are terminated in short circuits and the line is mechanically resonant at the angular frequency oo At this frequency the secondary of the transformer of Fig 48.3 is open circuited if there are no losses, and thus the electrical input impedance is infinite at this frequency and behaves like a parallel resonant circuit for neighboring frequencies. This configuration can be used as a high-Q resonant circuit if the mechanical losses can be kept low, as they are in single crystals of such piezoelectric materials as quartz. It should be noted, however, that the behavior is not as simple as that of a simple L-C parallel resonant circuit, primarily because of the frequency dependence of the effective reactance X and of the transformer ratio in the equivalent circuit. The electrical input impedance is given by oC /1-k2 tan kd/2 (48.14) kd/2 c 2000 by CRC Press LLC
© 2000 by CRC Press LLC in shape. Connections to these electrodes form the electrical port for the transducer and the voltage between them creates a spatially uniform electric field in the active region, and this time-varying electric field couples to the acoustic waves propagating between the electrodes. If the planar electrodes are many wavelengths in transverse dimensions and the active region is much thinner, and if the axial direction is a pure mode direction for the piezoelectric, the waves in the active region can be considered as plane waves. We then have the onedimensional geometry considered earlier. The transducer may be in contact with another elastic medium on either side, as indicated in Fig. 48.2, so that the plane waves propagate in and out of the active regions in the cross-sectional region shown. Thus, the transducer has in general two acoustic ports for coupling to the outside world as well as the electrical port. In the absence of piezoelectric coupling, the active region could be represented by a one-dimensional transmission line as discussed in the previous section and as indicated by the heavy lines in Fig. 48.3. With piezoelectricity there will be the stiffening of the appropriate stiffness constants as discussed in Eq. (48.9) with the concomitant perturbation of the characteristic impedance Z0p and the phase velocity Vp, but more important there will also be coupling to the electrical port. One model including the latter coupling is shown in Fig. 48.3 in which the parameters are defined by (48.13) Here C0 is the capacity that would be measured between the electrodes if there were no mechanical strain on the piezoelectric, A is the cross-sectional area of the active region, and X is an effective reactance. The quantity r is the transformer ratio (with dimensions) of an ideal transformer coupling the electrical port to the center of the acoustic transmission line. K is the electromechanical coupling constant for the material as defined in Eq. (48.9). The so-called resonant frequency v0 is that angular frequency at which the length d of the active region is one-half of the stiffened wavelength, v0 = pV/d. In the physical configuration of Fig. 48.4(a), the transducer has zero stress on the surfaces of the active region and hence both acoustic ports of Fig. 48.3 are terminated in short circuits and the line is mechanically resonant at the angular frequency v0. At this frequency the secondary of the transformer of Fig. 48.3 is open circuited if there are no losses, and thus the electrical input impedance is infinite at this frequency and behaves like a parallel resonant circuit for neighboring frequencies. This configuration can be used as a high-Q resonant circuit if the mechanical losses can be kept low, as they are in single crystals of such piezoelectric materials as quartz. It should be noted, however, that the behavior is not as simple as that of a simple L-C parallel resonant circuit, primarily because of the frequency dependence of the effective reactance X and of the transformer ratio in the equivalent circuit. The electrical input impedance is given by (48.14) FIGURE 48.2 Prototype transducer geometry. FIGURE 48.3 Model of active region. C A d jX j C K r e AZ 0 0 2 0 0 0 0 2 = = = 2 e e ; sin( / ) / / sin( / ) w pw w pw w w pw w Z j C K kd kd in = Ê Ë Á ˆ ¯ ˜ 1 1 2 2 0 2 w – tan / /
TRANSD 默语 COUSTIC WAVE FIGURE 48.4(a) Resonator structure;(b)acoustic probe;(c)acoustic delay line or optical modulator Thus, while the input impedance is infinite as in a parallel resonant circuit at o it is zero as in a series resonant circuit at a slightly lower frequency where the bracketed term in Eq.(48. 14)is zero. When losses are present or there is radiation out of an acoustic port, a resistive term is included in the reactive expression of Eq (48.14) Behavior analogous to that of coupled tuned electrical circuits for multipole filters can be achieved by subdividing the electrodes of Fig. 48.4(a)into different areas, each of which will act separately as a tuned circuit but if they are close enough together there will be acoustic coupling between the different radiators controlling this coupling, narrowband filters of very high Q and of somewhat tailored frequency response can be built in the megahertz and low gigahertz range. The basic geometry of Fig. 48.4(c) gives an electric-to-electric delay line whose delay is given by the lengt of the medium between the transducers divided by the phase velocity of the acoustic wave and would be on the order of 2 us/cm. Since the solid has little dispersion, the bandwidth of the delay line is determined by that of the transducers. Here it is necessary to choose the characteristic impedances and thicknesses of the backing and matching layers in Fig. 48.2 in such a manner that the conversion of the electrical energy incident on the electrical port to the acoustic energy out of acoustic port 2 of Fig. 48.3 is independent of frequency over a large range about the resonant frequency of the piezoelectric transducer itself. Varying the matching and backing .m ers is equivalent to varying the terminating impedances on the acoustic line of Fig. 48.3. The matching is en assisted by lumped elements in the external electrical circuit a The geometry of Fig. 48.4(c)is also the prototype form for acousto-optic interactions. Here the second ansducer is not relevant and can be replaced by an acoustic absorber so that there is no reflected wave present in the active region. An optical wave coming into the crystal as shown in Fig. 48.4(c)sees a propagating periodic perturbation of the medium, and if the photoelastic coefficients of the solid are large, the wave sees appreciable ariations in the refractive index and hence a moving diffraction grating. The angle of deflection of the output optical beam and its frequency as produced by the grating depend on the amplitude of the various frequency components in the acoustic beam when the optical beam traversed it. Thus, for example, the intensity versus angular position of the emerging optical beam is a measure of the frequency spectrum of any information modulated on the acoustic beam As noted previously, ultrasonic waves are often used as probes when the wavelength and attenuation are appropriate. For these radar-like applications, the acoustic beam is generated by a transducer and propagates in the medium containing the scatterer to be investigated as shown in Fig. 48.4(b). The acoustic wave is scattered by any discontinuity in the medium, and energy is returned to the same or to another transducer. If the outgoing ignal is pulsed, then the delay for the received pulse is a measure of the distance to the scatterer. If the transducer is displaced or rotated, the change in delay of the echo gives a measure of the shape of the scatterer. Any movement of the scatterer, for example, flowing blood in an artery, causes a Doppler shift of the echo, and this ft, along with the known direction of the returned beam, gives a map of the flow pattern. Phasing techniques with multiple transducers or multiple areas of one transducer can be used to produce focused beams or beams electrically swept in space by differential variation of the phases of the excitation of the component areas of the transducer c 2000 by CRC Press LLC
© 2000 by CRC Press LLC Thus, while the input impedance is infinite as in a parallel resonant circuit at vo, it is zero as in a series resonant circuit at a slightly lower frequency where the bracketed term in Eq. (48.14) is zero. When losses are present or there is radiation out of an acoustic port, a resistive term is included in the reactive expression of Eq. (48.14). Behavior analogous to that of coupled tuned electrical circuits for multipole filters can be achieved by subdividing the electrodes of Fig. 48.4(a) into different areas, each of which will act separately as a tuned circuit, but if they are close enough together there will be acoustic coupling between the different radiators. By controlling this coupling, narrowband filters of very high Q and of somewhat tailored frequency response can be built in the megahertz and low gigahertz range. The basic geometry of Fig. 48.4(c) gives an electric-to-electric delay line whose delay is given by the length of the medium between the transducers divided by the phase velocity of the acoustic wave and would be on the order of 2 ms/cm. Since the solid has little dispersion, the bandwidth of the delay line is determined by that of the transducers. Here it is necessary to choose the characteristic impedances and thicknesses of the backing and matching layers in Fig. 48.2 in such a manner that the conversion of the electrical energy incident on the electrical port to the acoustic energy out of acoustic port 2 of Fig. 48.3 is independent of frequency over a large range about the resonant frequency of the piezoelectric transducer itself. Varying the matching and backing layers is equivalent to varying the terminating impedances on the acoustic line of Fig. 48.3. The matching is often assisted by lumped elements in the external electrical circuit. The geometry of Fig. 48.4(c) is also the prototype form for acousto-optic interactions. Here the second transducer is not relevant and can be replaced by an acoustic absorber so that there is no reflected wave present in the active region. An optical wave coming into the crystal as shown in Fig. 48.4(c) sees a propagating periodic perturbation of the medium, and if the photoelastic coefficients of the solid are large, the wave sees appreciable variations in the refractive index and hence a moving diffraction grating. The angle of deflection of the output optical beam and its frequency as produced by the grating depend on the amplitude of the various frequency components in the acoustic beam when the optical beam traversed it. Thus, for example, the intensity versus angular position of the emerging optical beam is a measure of the frequency spectrum of any information modulated on the acoustic beam. As noted previously, ultrasonic waves are often used as probes when the wavelength and attenuation are appropriate. For these radar-like applications, the acoustic beam is generated by a transducer and propagates in the medium containing the scatterer to be investigated as shown in Fig. 48.4(b). The acoustic wave is scattered by any discontinuity in the medium, and energy is returned to the same or to another transducer. If the outgoing signal is pulsed, then the delay for the received pulse is a measure of the distance to the scatterer. If the transducer is displaced or rotated, the change in delay of the echo gives a measure of the shape of the scatterer. Any movement of the scatterer, for example, flowing blood in an artery, causes a Doppler shift of the echo, and this shift, along with the known direction of the returned beam, gives a map of the flow pattern. Phasing techniques with multiple transducers or multiple areas of one transducer can be used to produce focused beams or beams electrically swept in space by differential variation of the phases of the excitation of the component areas of the transducer. FIGURE 48.4 (a) Resonator structure; (b) acoustic probe; (c) acoustic delay line or optical modulator
Defining Terms Characteristic impedance: Ratio of the negative of the stress to the particle velocity in an ultrasonic plane wave Form: Term used to indicate the structure and dimensions of a multiterm equation without details within component terms Phase velocity: Velocity of propagation of planes of constant phase Piezoelectric transducers: Devices that convert electric signals to ultrasonic waves, and vice versa, by means of the piezoelectric effect in solids. Pure longitudinal and shear waves(modes): Ultrasonic plane waves in which the particle motion is paralle or perpendicular, respectively, to the wave vector and for which energy flow is parallel to the wave vector. Slowness surface: A plot of the reciprocal of the phase velocity as a function of direction in an anisotropic crystal. Related Topics 15.2 Speech Enhancement and Noise Reduction.49.2 Mechanical Characteristics B.A. Auld, Acoustic Fields and Waves in Solids, 2nd ed, Melbourne, Fla. Robert E. Krieger, 1990 E.A. Gerber and A. Ballato, Precision Frequency Control, vol 1, Acoustic Resonators and Filters, Orlando, Fla: cademic Press, 1985 G.S. Kino, Acoustic Waves: Devices Imaging and Analog Signal Processing, Englewood Cliffs, N J Prentice-Hall, Landolt-Bornstein, Numerical Data and Functional Relationships in Science and Technology: Gp III Crystal and Solid State Physics, vol. 11, Elastic, Piezoelectric, Pyroelectric and Piezooptic Constants of Crystals, Berlin Springer-Verlag, 1979. W.P. Mason and R.n. Thurston(Eds ) Physical Acoustics, Principles and Methods, multivolume series, New York: Academic Press H B. Meire, Basic Ultrasound, New York: Wiley, 1995 J.E. Rosenbaum, Bulk Acoustic Wave Theory and Devices, Boston: Artech House, 1988 Further information The main conferences in the ultrasonics area are the annual Ultrasonics Symposium sponsored by the IEeE Ultrasonics, Ferroelectrics and Frequency Control Society and the biannual Ultrasonics International Confer ence organized by the journal Ultrasonics, both of which publish proceedings. The periodicals include the Transactions of the IEEE ULtrasonics, Ferroelectrics and Frequency Control Society, the journal Ultrasonics pub- lished by Butterworth Co, and the Journal of the Acoustical Society of America. The books by Kino and by Rosenbaum in the References provide general overviews of the field c 2000 by CRC Press LLC
© 2000 by CRC Press LLC Defining Terms Characteristic impedance: Ratio of the negative of the stress to the particle velocity in an ultrasonic plane wave. Form: Term used to indicate the structure and dimensions of a multiterm equation without details within component terms. Phase velocity: Velocity of propagation of planes of constant phase. Piezoelectric transducers: Devices that convert electric signals to ultrasonic waves, and vice versa, by means of the piezoelectric effect in solids. Pure longitudinal and shear waves (modes): Ultrasonic plane waves in which the particle motion is parallel or perpendicular, respectively, to the wave vector and for which energy flow is parallel to the wave vector. Slowness surface: A plot of the reciprocal of the phase velocity as a function of direction in an anisotropic crystal. Related Topics 15.2 Speech Enhancement and Noise Reduction • 49.2 Mechanical Characteristics References B.A. Auld, Acoustic Fields and Waves in Solids, 2nd ed., Melbourne, Fla.: Robert E. Krieger, 1990. E.A. Gerber and A. Ballato, Precision Frequency Control, vol.1, Acoustic Resonators and Filters, Orlando, Fla.: Academic Press, 1985. G.S. Kino, Acoustic Waves: Devices Imaging and Analog Signal Processing, Englewood Cliffs, N.J.: Prentice-Hall, 1987. Landolt-Bornstein, Numerical Data and Functional Relationships in Science and Technology: Gp III Crystal and Solid State Physics, vol. 11, Elastic, Piezoelectric, Pyroelectric and Piezooptic Constants of Crystals, Berlin: Springer-Verlag, 1979. W.P. Mason and R.N. Thurston (Eds.), Physical Acoustics, Principles and Methods, multivolume series, New York: Academic Press. H.B. Meire, Basic Ultrasound, New York: Wiley, 1995. J.F. Rosenbaum, Bulk Acoustic Wave Theory and Devices, Boston: Artech House, 1988. Further Information The main conferences in the ultrasonics area are the annual Ultrasonics Symposium sponsored by the IEEE Ultrasonics, Ferroelectrics and Frequency Control Society and the biannual Ultrasonics International Conference organized by the journal Ultrasonics, both of which publish proceedings. The periodicals include the Transactions of the IEEE Ultrasonics, Ferroelectrics and Frequency Control Society, the journal Ultrasonics published by Butterworth & Co., and the Journal of the Acoustical Society of America. The books by Kino and by Rosenbaum in the References provide general overviews of the field
