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: