6/7/1999 4:55:00 PM Design Guidelines for Medical Ultrasonic Arrays ( SPIE International Symposium on Medical Imaging, Feb. 25, 1998, San Diego, CA ) (Invited Paper / Keynote Presentation) Ronald E McKeighen, Ph.D. Acoustic Imaging Transducers Inc. 10027 South 51 St. Street, Phoenix, AZ 85044 Abstract The basics of ultrasonic transducer array design in the frequency range useful for medical imaging ( 1- 10 MHz) are discussed. Performance parameters of importance in transducer design are considered, including sensitivity, coupling constant, band width, frequency downshift, pulse duration, beam focusing properties, and electrical matching. Two and three dimensional effects must also be taken into account. The advantages of computer modeling in three dimensions with finite element analysis code are highlighted. The principles of multi-element array transducers useful for two dimensional real-time scanning are reviewed. Two dimensional arrays provide the opportunity of focusing the elevation beam dimension in the short axis, or make possible full three dimensional scanning volumes. Keywords: ultrasonic arrays, transducer, broadband, beamforming, system, two dimensional arrays 1. Introduction / Basic Principles A one dimensional array, when coupled with a beamformer, allows broad band ultrasonic pulses to be generated and swept through a body for medical diagnostic imaging. The design of the array, including number of elements, pitch, and frequency, are arrived at with the particular clinical application or target organ in mind. Image resolution increases with frequency, however attenuation of energy by body tissue at the rate of 0.5 dB to 0.7 dB / cm / MHz dictates a compromise in final choice of nominal transducer frequency. The basic construction of a one dimensional array is illustrated in figure 1. Illustrated is the crossection of three major array elements, with each major element made up of two sub-diced elements, to maintain desirable aspect ratios. The basic principles(1-3) involved in constructing a transducer to generate a short pressure pulse can be understood by considering a simple diagram illustrating the four basic pressure pulses (figure 2) generated when a piezoelectric ceramic is shock excited by an impulse voltage of approximately 100 V. At each of the major surfaces of the ceramic, two pressure waves are generated traveling in opposite directions, one a positive pressure wave, and one a negative (rarefaction) pressure wave . If the ceramic is surrounded by air or water, most of the wave amplitude will be reflected at the boundaries of the ceramic, causing the pulse to "ring" for a very long time. The fraction of the pulse amplitude reflected at each interface is proportional to the difference in acoustic impedance between the two materials on either side of the boundary. One way to make a broad band ( short pulse) transducer is to attach a backing material to the back side of the ceramic with an acoustic impedance close to that of the ceramic. Then the internal reverberations of the ceramic are eliminated , allowing a very short pulse to be generated. Second Matching Layer F+ First Matching Layer Front Matching Layer o Ceramic Volt B- F- Ceramic o Pulser B+ Backing Backing Figure 1. Sub-diced Linear Array Figure 2. Four basic pressure waves generated by voltage impulse. (+) Compression wave; (-) Rarefaction wave The acoustic impedance is defined as the product of the density of the material and the longitudinal sound velocity. Non destructive testing transducers (NDT) generate very short pulses by this principle of attaching a very dense backing material(4,14) . The backing material chosen should ideally not only closely match the ceramic acoustic impedance but also be very absorptive or "lossy" so that no energy is left to be reflected from the non ceramic end of the backing and create an artifact echo. Although adequate for NDT, this ceramic construction does not have enough sensitivity for medical applications, since so much of the energy applied in creating the pulse is lost into the backing. The acoustic impedance of PZT ceramic is about 33.0 ( units of 106 Kg/M2/sec or mega Rayl). The acoustic intensity in a material may be expressed by equation 1, where I is the intensity in watts/cm2, Z is the acoustic impedance, u is the particle velocity, and P is the pressure: I = 1/2 (Zu2) = 1/2 (P2/Z) (1) As an example, assume that the backing impedance is 15.0 , and that the front side of the ceramic is coupled into water with an impedance of 1.5 . It can be seen that 10 times more power goes into the backing than goes forward into the water. To increase the relative amount of energy transmitted into the front medium, one need a means to make the water or tissue in front of the ceramic (the effective load) appear to have a high impedance which more closely matches that of the crystal. The situation is analogous to an electrical circuit, where maximum power is transferred from a generator to a load when the electrical impedance of the load equals that of the generator. The mechanism for doing this is the impedance transformer. The basic physics is borrowed from transmission line theory, where n stages of transmission line sections of varying impedances, each a quarter wavelength long, are used to match the generator to a load. In the acoustic analog, the concept of acoustic impedance is used, and the transmission line sections become matching layers of intermediate impedance values between those of ceramic and water or body tissue, and of thickness dimensions nominally one-quarter of a wavelength at the center frequency. Thus, for medical ultrasonic applications, an optimum combination of broad band width, short pulse, and good sensitivity can be achieved by using a light impedance backing ( usually 3 to 5 mega Rayls), with two quarterwave matching layers on the front of the piezoelectric element. One measure of the sensitivity of a transducer is insertion loss. Insertion loss is defined as the ratio of the power delivered to that part of the system following the transducer, before insertion of the transducer, to the power delivered to that same part of the system after insertion of the transducer. The term transducer loss is defined as the ratio of the available power of the specified source to the power that the transducer delivers to the specified load under specified operation conditions. The available power is the quotient of the mean square of the open-circuit terminal voltage of the source divided by four times the resistive component of the impedance of the source. To measure insertion loss one must account for beam spreading diffraction losses, electrical impedance mismatches, etc. This usually requires having the target reflector very close to the face, or a special shaped target to return all the pulse echo energy to the transducer. For medical ultrasonic transducers, insertion loss numbers fall in the range of -6 dB to 10 dB typically. More practically, for comparison purposes in production, when testing the individual elements of a multi-element array, one sets the target a fixed distance close to the elevation axis focal point, and defines a more empirical term called "loop sensitivity". This is defined as the ratio of the peak to peak pulser voltage, to the received (two way) pulse echo voltage peak to peak amplitude. Values for loop sensitivity are typically -25 to -45 dB. 2. Design of Acoustic Stack A variety of references discuss the design and construction of ultrasonic transducers(1-11) . What determines the frequency output of a medical ultrasonic transducer? At what frequency does the transducer resonate ? The frequency output is related to the thickness of the ceramic material from which the transducer is constructed. However, if one attaches leads to the electrodes on a piece of ceramic in air, and measures the electrical impedance as a function of frequency, one observes not one resonance, but two (figure 3). The resonance at which the impedance reaches a minimum is called fr or the resonance frequency, and the higher frequency resonance where the impedance reaches a peak is called fa, the anti resonance. In the literature, these two resonances are also variously referred to as the following: Frequency Definitions Fr=resonant frequency=frequency of zero Fa =Anti-resonant frequency susceptance =frequency of zero reactance 2) Fs=Series resonant frequency=frequency Fp=Parallel resonant frequency of maximum conductance = frequency of maximum resistance 3) Fm =Frequency of minimum impedance Fn=Frequency of maximum impedance The following relationships apply: 1) for the lossless case: Fm=Fs=Fr and Fa=Fp=Fn for the lossy case: Fm<Fs<Fr and Fa<Fp<Fn So, which resonance should one build a transducer around ? A little known phenomena is that the position and shape of the fr resonance depends dramatically on the length and width dimensions, L & W of the ceramic slab, as well as the thickness Tk . There is a strong influence due to modal coupling, of the other dimensions on fr. For example if a piece of ceramic used to dice a one dimensional array has dimensions of approximately 15 mm x 60 mm, the resonance valley near fr will appear washed out, and broad. However, if a smaller piece of dimensions about 15mm x 6 mm is broken off of that same slab, the resonance will be more pronounced and closer to the true resonance behavior because of much lower modal coupling to the later dimensions. If one had observed the anti resonant peak, fa in both of these measurements, one would note that it is very well defined, and repeatable ! This is because the anti-resonance (or parallel resonance) frequency is fundamentally related only to the ceramic thickness, and not influenced by modal coupling. Also, as the amount of poling is increased in a given material, fa will remain nearly constant, while fr will steadily decrease until the material is completely poled. Thus, fa depends on the velocity and thickness, while fr depends on the motional reactance, which varies with the poling state of the material, and cross coupling to other modes. It for these reasons that a more predictable way to specify and construct ultrasonic transducers is to specify ceramic dimensions, and matching layer thickness, related to fa, rather than fr . Figure 3. Fundamental resonance and antiresonance of slab of PZT ceramic Starting with a ceramic that resonates at fa, a broadband transducer design usually has a nominal center frequency much lower than fa. This is because of several downshifting factors, as will be explained below. The final center frequency of an ultrasonic array transducer may be expressed as: fc = Ds * fa = ds1 * ds2 * ds3 * ds4 * fa where, fc = transducer design center frequency, and fa = bulk ceramic antiresonant frequency. Ds = total downshift factor ds1 = downshift factor #1 = lower frequency resonance caused by dicing ceramic into sub-elements, i.e. the acoustic velocity is lower because of side wall boundary conditions (wave guide). = 0.82 ( when kerfs are later filled with polymer resin, ala construction for linear or convex arrays) ds2 = downshift factor #2 = mass loading effect of acoustic stack (matching layers and backing) on ceramic resonance ≅ 0.95 ds3 = downshift factor #3 = due to frequency dependent attenuation of RTV lens or face material ≅ 0.93 ds4 = downshift factor #4 = this phenomenon is due to the fact that pulser circuits usually have very low impedance (5 to 10 ohms). Thus, energy is more easily coupled to the low impedance resonance peak fr, rather than the higher impedance resonance fa. ≅ 0.78 Therefore, the total transducer frequency downshift factor can be DS = ds1*ds2*ds3*ds4 = 0.56 (3) Thus, fc = 0.56 * fa . Therefore, to start a transducer design (linear or convex array), one wants to order ceramic with bulk resonance fa (anti-resonance) in the slab, of frequency approximately 1.77 times the nominal transducer design center frequency. For example, for a 3.5 MHz linear array, one would order ceramic with fa = 6.10 MHz. The final transducer center frequency and bandwidth, can also be heavily influenced by matching layer thickness, and the method and values of electrical tuning chosen. Design of Acoustic Matching to Load; Quarter Wave Matching Layers for Broadband Response: For short impulse response and broad bandwidth, ultrasonic transducers are designed with one or two matching layers on the face. The front matching layers are analogous to anti-reflection coatings on optical lenses. The acoustic impedances of these layers are usually chosen based on transmission line theory. A new way to calculate the optimum quarter wave matching layer impedance values will be shown. The challenge in transducer design is to match the high acoustic impedance of about 33 mega-Rayls typical of PZT ceramics to the relative low impedance 1.5 mega-Rayls characteristic of a water or tissue load on the face of the transducer. Classical transmission line theory may be used to design a multi-stage quarter wave section impedance transformer(12,13). Using the classical approach of Collins 12 for a Butterworth type of filter response, one would calculate the requisite impedances of double quarter wave layers to be applied to the front acoustic port of the ceramic as: (where we assume the ceramic Zc = 33.2, and the front load impedance is water with ZL = 1.50 ) Impedance One section Two section Table I. Quarter Wave Matching Sections predicted by transmission line theory. Formulas Example values to match PZT ceramic Z1 (first layer) Z2 (second layer) Z1 Z2 7.0 Zc • ZL Zc3/4 ZL1/4 Zc1/4ZL3/4 14.8 3.25 Matching layers are typically fabricated using epoxy resins loaded with dense powders such as tungsten or alumina14. However, it is very difficult to achieve dense enough loading to produce acoustic impedances in the range of 8 to 15 mega-Rayls. Solid materials such as glass and graphite have been considered as matching layers, but there is the classical problem of bonding on pre-fabricated matching layers without the bond line destroying the effectiveness of the matching layer15 . DeSilets and Fraser 16 of Stanford University found that assuming the transducer was air backed, effectively lower target values for the matching layer impedances could be derived, based on the KLM 17 equivalent circuit model, which places a virtual node in the middle of the ceramic. Their analysis predicts : Impedance One section Two section Table II. Quarter Wave Matching Sections predicted by KLM theory. Formulas Example values to match PZT ceramic Z1 (first layer) Z2 (second layer) Z1 Z2 Zc1/3ZL2/3 4.2 Zc4/7 ZL3/7 Zc1/7ZL6/7 8.81 2.33 The optimum impedances for the layers (see Table III.) can also be derived with minimal understanding of the underlying physics, by the means of statistical design of experiment methodology (DOE) and computer simulation models18 . The exact layer thicknesses can also be tailored as another design optimization parameter. An excellent starting point for design optimization is to use the design rule that the layer thickness is 0.350 lambda, at the anti-resonance frequency of the diced ceramic resonance. Table III. Quarter Wave Matching Sections predicted by DOE computer optimization . Formulas Example values to match PZT ceramic Impedance Z1 (first layer) Z2 (second layer) Z1 Z2 One section Two section computer optim.. computer optim.. 8.0 2.55 Influence of back side on front side matching: The formulas for matching layers derived by DeSilets16 et. al. assumed an air load on the back side of the transducer ceramic. The transducer designer has more flexibility in optimizing the design trade off if the impact of the back side loading and impedance on the ceramic are included in choosing the matching layer impedance values applied to the front side of the transducer. One usually assumes that the acoustic impedance of a PZT ceramic is about 33.2 mega-Rayls, and one calculates the requisite matching layer impedances as discussed above. In a new approach 19 , one uses the KLM model or other transmission line model, to first calculate the average acoustic impedance as a function of frequency, looking into the face of the ceramic. This value will be impacted by the load on the back side of the ceramic and by the electrical tuning, because of the electricalacoustic coupling. An example of the acoustic impedance at the face of the ceramic as a function of frequency is shown in figure 4, using a backing impedance of 3.0. Figure 4. Effective acoustic impedance looking into ceramic, as influenced by presence of backing. One may calculate the average effective impedance of the face of the ceramic over the bandwidth of interest, say 80 %, and use this value to plug into the transmission line equations to calculate the desired quarter wave matching layer values. In this example, over a 70 % bandwidth, using an effective average acoustic impedance of 18.54, the resultant matching layer values would be: First matching layer: Z1 = Zc4/7 ZL3/7 = 6.31 Second matching layer: Z2 = Zc1/7ZL6/7 = 2.15 Table IV. Quarter Wave Matching Sections predicted by Modified theory . Formulas Example values to match PZT ceramic Impedance Z1 (first layer) Z2 (second layer) Z1 Z2 One section Zc1/3ZL2/3 3.47 Two section Zc4/7 ZL3/7 Zc1/7ZL6/7 6.31 2.15 This approach has the added advantage that the matching layer impedances predicted are much lower than in the classical approach, and thus much easier to fabricate. Figure 5 shows the pulse echo response (Piezocad) of a transducer with these values for the two front matching layers, and a backing impedance of 3.0 . The design yields a respectable bandwidth of 72 %, a -20 dB pulse length of 0.591 micro-seconds, with a center frequency of 3.73 MHz. Figure 5. Pulse Echo response with double matching layers of Table IV. For broad band width and short pulse ringdown , the thickness of the matching layers should be ground to a tightly controlled dimension. Since repeatable transducer designs are based on the ceramic anti-resonant frequency fa, the matching layers should be ground to a thickness of 0.350 lambda, where lambda is the wavelength of sound in the layer at a frequency given by fa. Optimal performance can usually be achieved with layer thicknesses tailored to within +/- 15% of this design rule starting point. This design rule also coincidentally translates to approximately 0.25 lambda thickness at the net operating center frequency of the total acoustic stack, fc. Hence the layers are effectively "quarter wave layers", and in effect behave as anti-reflection coatings on the face of the transducer to eliminate reverberation of energy in the form of internal reflections that would make the pulse ringdown lengthy. Ted Rhyne20 has analyzed broadband transducer design from the perspective of a filter design problem , and has calculated that with a three matching section design, bandwidths up to 90 % are achievable. 3. Materials Modern ultrasonic arrays for medical imaging are typically constructed from piezoelectric ceramic of the PZT-5H type. PZT ceramic is available in various choices of dielectric constants. Because of the small array element sizes, and subsequent low capacitance and high impedances, one typically chooses a ceramic with a very high relative dielectric constant and high coupling factor. Various additives, such as chromium, iron, nickel, and lanthanum are used to modify the dielectric constant and the piezoelectric properties of the material. Table V lists some typical piezoelectric ceramics and their properties. The constant d33 is the strain developed per applied field and is a measure of the effectiveness of the material as a driver or transmitter. The constant g33 is the electric field generated per applied stress and is a measure of the effectiveness when used as a receiver. A useful figure of merit when choosing materials is the product of d times g. The electromechanical coupling factor (k33) is not actually a measure of intrinsic efficiency of transducer material as its name implies; but the square of k33 does give the relative portion of stored input electrical energy converted to stored mechanical energy, and vice versa. Theory indicates that materials with higher coupling coefficients k33 can give better band widths and pulse response in medical applications. Note that the high dielectric loss of PVDF plastic film limits its signal to noise ratio, achievable in a transducer. Table V: Piezoelectric Material Properties PZT-5A PZT-5H Motorola PZT-7A Lead Quartz PVDF 3203 HD Metaniobate Free Dielectric Constant 1700.00 3400.00 3800.00 425.00 175.00 4.50 12.00 Dielectric Loss Tangent 0.020 0.020 0.020 0.200 d33 (M/V x 10-12) 374.00 593.00 650.00 150.00 65.00 2.00 20.00 g33 (V*M/Newt x 10-3) 24.80 19.70 19.00 39.90 42.00 50.00 210.00 k33 0.71 0.75 0.75 0.66 0.38 0.11 0.10 Acoustic Impedance Z 30.00 30.00 37.00 32.00 23.00 15.00 3.50 Mechanical Q 75.00 65.00 30.00 600.00 650.00 25,000 13.00 The acoustic velocity in materials, both in the ceramic and the matching layers, depends on the aspect ratio w/h of the element geometry. Typically, to avoid spurious resonances, and to allow flexibility of conforming to a convex geometry, the major pitch of a diced array is diced into sub-elements of width to height ratio w/h of between 0.4 to 0.6 . One advantage of this is that the piezoelectric coupling factor for this geometry is higher than for flat plate ceramic slabs. While kp is only 0.5, k33 can be as high as 0.75 . The “material” coupling factor (squared) is: ( Jaffe and Belincourt 21 ): k t = (π / 2 )( Fs / Fp ) tan{(π / 2 )( Fp − Fs ) / Fp } 2 (3) The “effective” coupling constant may be given as: (it is this coupling factor that determines filter or transducer bandwidths !) (4) keff2 = ( Fn − Fm ) / Fn The square of the coupling factor is proportional to the partitioning of the energy available in the electrical domain over that in the mechanical domain, or vice versa. Also, theoretical analysis has shown that the ultimate bandwidth available from a transducer design is proportional to keff squared 16 . 2 2 2 The acoustic materials used to fabricate a transducers front and back layers are typically made from polymer resins. These are heavily loaded with powders to achieve the desired acoustic properties of acoustic impedance, velocity, density, and attenuation. Desired acoustic impedances for backing the transducer range from 3.0 to 7.0 mega-Rayl. Backing materials are typically epoxy, loaded with powders of aluminum oxide, tungsten and its oxides, and additives to increase the absorption of acoustic energy in the backing. A typical formulation has 6 to 9 components. The material used in the backing must be attenuative enough so that any backing echo is at least 90 to 100 dB down from the main bang pulse amplitude. Broadband transducer designs are usually made with double matching layers on the front. The first layer usually needs an acoustic impedance of about 8 mega-Rayls (14,16) and may be made from epoxy loaded with tungsten or aluminum oxide (Kossoff 14). Acoustic impedance is an especially important quality of transducer material that serves as a measure of the pressure required per unit particle velocity, and thus of the stiffness or resistance to passage of pressure waves. The second matching layer materials are usually unloaded epoxy or urethane, with an impedance between 2.2 and 2.9 mega-Rayl. For fabricating matching layers on a transducer, there are two schools of thought . One proposes to grind the precured layers to precise dimensions first and then apply them to the ceramic with thin bond techniques. The other approach is to cast the materials in place on the ceramic, let the thermoset polymer resins cure in place without an intrinsic bond line, and then machine to final dimensions. Casting and grinding requires formulations that do not have significant settling or gradients formed during curing. The grind and glue approach requires that the thin bond lines be less than 1/200 of an acoustic wavelength, in order to be invisible acoustically 15 . The practical problems of doing repeatable ultra thin bond lines in high volume production can be formidable. 4. Thermal Management Another practical issue in good transducer design is to avoid overheating the face of the probe, causing patient discomfort. The temperature at the face of the probe must not exceed 41 degrees centigrade. Phased array probes used for cardiology scanning tend to overheat more so than linear arrays, because the same elements are used to generate every beam firing. Analysis shows that a lot of the heat generated in the probe is caused by self absorption in the materials of the acoustic waves generated. Good thermal management techniques include inserting low profile conductive heat pipes or fins in the backing, and loading the backing and front layer polymers with powders of excellent thermal conductivity, such as aluminum nitride, or industrial grade diamonds. A metallic RF/EMI shield grid placed over the face of the transducer can also serve to quickly carry away the thermal energy. 5. Elevation Focusing: Focusing of the short axis, elevation dimension determines the effective slice thickness of the image plane. There are two generic approaches to focusing the beam elevation in the short axis, internal focusing (by shape of ceramic) , or external focusing (see figure 6.) with a RTV silicone lens. Backing Figure 6. Two alternate geometries to focus short axis elevation. For use with flat ceramic geometry (in short axis), an external convex lens made with material of a lower velocity than water Ceramic Match Layer are used. These materials are in the RTV silicone family, and are available form GE and Dow Corning. RTV typically has Face Lens/Filler Urethane face RTV Lens and acoustic impedance between 1.3 and 1.5 mega-Rayl and Internal Focus External Focus an acoustic loss of about 7 dB/cm/MHz. A non-linear curve for choosing soft or sharp beam focusing has been given by Kossoff 22. The general purpose curves are normalized by the far field transition distance T. T = D2 / 4*lambda with D = the active aperture diameter A formula for determining the requisite radius of curvature for a RTV lens may be derived from optics: ROC = Reff { (Vmedium - Vlens) / Vlens } ~ 0.525 Reff (5) where, Reff is the effective geometric focal point desired, for tissue, Vmedium = 1.540 Km/sec, and for typical RTV silicones, Vlens = 1.010 Km/sec Alternatively, for internally focused array designs, with cylindrical shaped ceramic, it is desirable to have a neutral density face filler with impedance close to tissue or water. These requirements are met by certain families of castable urethane organic polymers. Some moderate level of acoustic absorption is actually desirable in the face material, because it helps attenuate multi-path reverberation from the tissue, which creates near field clutter in the image. To minimize clutter in the image the dimensions of the beam in the elevation focus dimension (out of the electronic focal plane) should also be carefully specified. Note that diffraction theory shows that a lens cannot focus an aperture beyond its farfield Fraunhofer transition distance T, given above. For typical arrays, the short axis (elevation) dimension width is about 30 lambda, and the short axis lens is usually set with a focal distance of about T, for soft focusing with long depth of field. While beam widths are classically quoted as the - 3dB points in optical applications, for ultrasonic imaging, experience has shown that the effective beam thickness in the elevation dimension that influences image quality, is proportional to the -10 dB beam width (one way ) (or -20 dB send / receive beam). The depth of focus of the elevation beam is determined by the mechanical focusing properties of the acoustic lens . Kino gives from optics23 : DOF = 7.1 (f#)2 * lambda (where f# = f number) (6) This expression is the depth of field for -3 dB rolloff in the axial intensity, from the peak value at the focus. For ultrasonic imaging a more useful value is the depth of field for a broadband pulse, for only a 1.5 dB rolloff in sensitivity. This acoustic depth of field would be given by: DOF = 4.7 (f#)2 * lambda (7) Extended Elevation Focusing: The depth of field of the beam can be improved passively by borrowing from the physics of non-diffracting beams, and building apodization into the transducer structure (24-30) .This may be done by one of three ways, 1) patterning of the electrode on the ceramic, 2) gradient attenuation in the lens material, or 3) varying the effective polarization strength of the piezoelectric ceramic across its elevation dimension. The beam may be made more collimated and uniform with a Hamming apodization function applied across the aperture. Figure 7 is a comparative plot of just the -10 dB isocontour beamwidths for the apodized case versus the classical flat aperture case . Experience has shown that the -10 dB one way beamwidth (-20 dB two way), is more representative of the actual effective slice thickness (elevation dimension) seen in the image, than the -3 dB contour classically quoted in optics. Figure 7. Additional depth of focus achieved by apodizing the short axis aperture. 6. System Considerations There must be tight coupling between system design and transducer design, to achieve the best image quality in clinical applications(31-34) .One of the first questions to ask when designing a new ultrasonic array, is what should the major element pitch be ? The number of sub-elements is determined by keeping the sub-element aspect ratio in the range of 0.4 to 0.6 for W/H. A guideline to make a first estimate of a required array pitch, is to decide how many channels are going to be used in beamforming. The lateral image resolution is given by : resol = 1.22 * f# * lambda (8) where f# = Focus depth / Aperture = "f number", and Aperture = N * Pitch ; where N is the number of channels used in beamforming. and lambda = wavelength of sound at image frequency In order to adequately display this resolution on the system monitor, the resolution cell must be sampled 2 to 4 times. This means the number of acoustic lines fired per expected resolution cell in the lateral dimension must be 2 to 4. The density of acoustic lines formed, can be increased by stepping the active aperture across the array in alternate odd then even element groupings. Let's consider designing a convex geometry array, to scan a fan shaped image sector. In terms of angular resolution of beamforming from the face of the array: Θres = arctan( resol / Z) ≅ (resol / Z ) (for small angles) (9) = 1.22 • f# • lambda / Z whereas, the image sampling rate for display, or sampling of fired acoustic lines, is given by the angle subtended by stepping the aperture on ceramic element boundaries and is: [radians] (10) Θtxd = P / R where P = the array element pitch ; and R = the convex radius of the face of the array. For adequate sampling of the image resolution and to avoid aliasing in the image, we want: Number Samples per resolution = Θres / Θtxd ≥ 2 (11) Θres / Θtxd = (R/P) • (1.22 • f# • lambda/Z) ≥ 2 (12) So, (13) or Pitch=P ≤ =0.6 • (R • f# • lambda/Z) and since f#=Z/(N • P) we may also write: P ≤ = 0.6( R • lambda / N ) (14) Figure 8 plots this curve for the cases of a 64, 96 , and 128 channel beamformer for a 3.5 MHz convex array with a 40 mm radius. This pitch value can be used as a starting value in array design, and is adjusted depending on the clinical application and details of the system beamformer . Figure 8: The first estimate of a desired array element pitch for a 3.5 MHz convex array, as function of the convex ROC and number of beamformer channels. Typically to maintain image resolution with depth, the number of beamformer channels active is expanded with depth, up to the maximum number. If the aperture is expanded too slowly, image resolution is sacrificed ; however, if the aperture is expanded too rapidly, acoustic clutter increases and hurts the systems ability to maintain high dynamic range and contrast resolution. This is because of the finite angular acceptance sensitivity of individual array elements. Antenna Angular Sensitivity: The theoretical angular sensitivity of a array element of ceramic width w ( Pitch minus kerf), is given by : P/Po= sinc (π ⋅ x ) • cos( Θ ) = [(sin π ⋅ x ) / π ⋅ x ] • cos( Θ ) (15) Convex Array Element Pitch as function of Convex radius ROC and Number of Beamformer Channels 0.6 Array major element Pitch (mm) 0.5 0.4 0.3 N=64 N=96 N=128 0.2 0.1 0 0 10 20 30 40 50 60 70 80 90 Convex Radius ROC (mm) where P = pressure at angle Θ , Po = pressure on axis Θ = 0, x = (w/lambda)*sin( Θ ) (this equation was derived assuming non-rigid baffle boundary conditions 35 ). Many times a system engineer or transducer designer needs a good estimate of what the element sensitivity function will be before empirical data is available. A useful formula has been derived to give the rolloff angle at various dB criteria. This is a transcendental, iterative relationship, and will give accurate results after the 3rd or 4th iteration. ( formulas for ith iteration estimate) I ) -3 dB One Way Sensitivity Angle (-6 dB two way): Θi = arcsin(0.442 * lambda/w) ( this is the seed value (first estimate of Element (-3db 1way) Sensitivity Angle ) Θ( i + 1) = arctan((1.414/ π )*(lambda/w)*sin( π *(w/lambda)*sin Θi )) [Radians] (17) ( this will converge in 3 to 4 iterations) II ) -6 dB One Way Sensitivity Angle (-12 dB two way): Θi = arcsin(0.60 * lambda/w) (Seed value (first estimate of Element (-6db 1way) Sensitivity Angle ) Θ( i + 1) = arctan((2/ π )*(lambda/w)*sin( π *(w/lambda)*sin Θi )) [Radians] ( this will converge in 3 to 4 iterations) (16) (18) (19) Figure 9 shows the angular sensitivity pattern (directivity ) as calculated by the Pzflex FEA code for a 3.5 MHz phased array element. Comparison is made between air maintained in the kerfs, versus the response if RTV material is allowed to wick into the kerfs, or alternately epoxy resin as a kerf filler. Angular Sensitivity & Kerf FIller 1.000 Figure 9. Plot of angular sensitivity of individual array element, and influence of boundary conditions and kerf fillers (from FEA simulations) 0.900 0.800 Amplitude (Linear) 0.700 Air Kerf Undiced Layers RTV Kerfs Epoxy Kerf Stiffened Baffle Cos Theory 0.600 Experience has shown that a good balance between resolution requirements and keeping down unwanted clutter acceptance , is given by expanding the beamformer aperture at a rate allowed by the element angle rolloff at the -12 dB ( two way send/receive) point. An example of the suggested expanding aperture rate for a 3.5 MHz, convex array with 40 mm radius of curvature is given in figure 11. One can see that having larger number of beamformer channels is only useful at deeper depths. 0.500 0.400 0.300 \txd\pzdir-k2.xls\ 0.200 0.100 0.000 -50 -40 -30 -20 -10 0 Angle - degrees 10 20 30 40 50 Figure 10: Geometry for determining desirable expanding aperture rate of number of beamformer channels turned on at a given depth, as dictated by individual array element polar sensitivity angle. The aperture subtended by "ni " active elements across a convex array is given by the chord: A = 2 R sin (θ ) The depth at which an given channel ni may first be turned on is determined by its sensitivity angle α . The relation between a given target depth Z, and the allowed aperture and number of channels activated (related to angle θ ), may be shown to be (figure 10): (21) Z = [R sin θ / tan( α - θ )] - R(1-cos θ ) This may be plotted in graphical form as the expanding aperture rate ( number of channels ) Vs image depth Z (mm) . (see figure 11 ) Allowed Expanding Aperture as function of Acceptance Angle 6.5 MHz Convex Array, ROC=10 mm 120 60% Theoretical Directivity 80% Theoretical Directivity 100 80 60 Depth Z 40 100% Theoretical Directivity 20 0 0 10 20 30 40 50 60 70 80 90 Number of Channels Allowed Figure 11 . Plot of expanding aperture of Figure 12. Desired turn on rate of beamformer channels beamformer channels Vs depth. as function of individual element angular sensitivity Another example for a tight radius (10 mm 6.5 MHz) convex array is shown in figure 12. One can see that there is a significant benefit in utilization of available beamformer channels, as the angular sensitivity of individual array elements is improved. The criteria used here is that of the target falling within the -12 dB two-way acceptance angle of the outer array elements included in the aperture. 7. Tuning and Electrical Impedance Matching Another important aspect of relating an array transducer design to the system requirements is electrical impedance matching. Because a multi-element array may have 100 to 200 elements, the individual elements are very small, and subsequently have high electrical impedances ( 100 to 600 ohms). For efficiency of transferring energy, the array elements should be matched to the pulser during transmit, and the pre-amp impedance on receive. For maximum energy transfer between the transducer and pulser or receiver, the load impedance of the transducer should be the complex conjugate of the electrical circuit coupled to it. This is a compromise, since pulser impedances, being voltage sources, have low output impedance of 5 to 10 ohms, whereas pre-amp impedances are usually 50 to 300 ohms. To match the impedances of array elements to the coax cable and system impedances, "tuning" elements(36-43) are typically placed in the probe connector between the coax cable and the system. The equivalent circuits to model send/receive in the finite element code Pzflex are illustrated in figure 13. To solve a consistent set of equations between switching the send and receive circuits in place, each circuit must have the identical number of components, however, their values can be adjusted to emulate the real system. One broad bandwidth tuning component would be a transformer with the proper turns ratio to step the array element impedance down to the system load, and with an intrinsic inductance wound in, to cancel reactive loads. Canceling reactive effects (capacitive or inductive) makes the effective load of the transducer array and cable look primarily resistive to the system pulser / receiver, for more efficient transfer of energy and broad band performance . Because the array contains hundreds of elements, transformer tuning can occupy a large volume of space, and be expensive. Because of space and cost constraints, a more cost effective way of tuning a probe is to use surface mounted miniature inductors. As a design guideline, series inductors will drop the effective element impedances down to 40 to 60 % of their initial value. On the other hand, parallel mounted inductors, will raise the effective net impedance of the elements a factor of approximately 1.5 to 2.0 . Parallel tuning will also help roll off low frequency content in the pulse ( high pass filter), which can sometimes degrade image quality. The choice of series or parallel tuning depends on the specifics of system requirements and the behavior of the particular pulsing circuit. In general, proper tuning of the array elements yields higher sensitivity , aids in broad band response over a range of frequencies, and influences the net center frequency of the transducer design. If an array element with coax attached has an impedance and phase angle at the nominal center frequency of Z ∠ ( Θ ), then a nominal series tuning component can be chosen as: Ls = |Z| * sin ( Θ ) / 2 π Fc , where Fc is the center frequency. A parallel tuning component may be chosen as: Lp = |Z| / (sin( Θ ) * 2 π Fc) . The broadest frequency response can actually be achieved with a distributed tuning network (figure 13) that includes a series inductor at the transducer end and, and a series/shunt inductor at the connector/system end of the coax. Even more broad band frequency response can usually be achieved by skewing the final values of tuning components away from their nominal value. Tuning should be chosen not only to cancel out coax cable reactance, but also the parasitic capacitances of interconnecting system boards, and the inductances of transmit/receive switching circuits. Circuit Models for Transmit / Receive for finite element modeling code PZFLEX date: 2/25/97 c:\visio\txd\coxtune2.vsd PCB + T/R switch + Transmitter System side (Connector interface ) TRANSMIT: Pulser Drive Ls2= 10.0 uH Ls1= 15.0 uH TXD Rtxm= 4 ohm Lp = 10.0 uH Coax Transmission Line Rp = 47 ohm Figure 13: Equivalent circuit schematics used in finite element code (PZFLEX) to model coax cable, tuning, and effects of system reactance loads. RL=10,000 ohm Cap=120pf (PCB) Lt/r=500uh RECEIVE: Ls2= 10.0 uH Ls1= 15.0 uH (Connector interface ) Pre-Amp TXD Lp = 10.0 uH Coax Transmission Line Rp = 47 ohm RL=640 ohm Cap=140pf Lt/r=100uh RL=640 ohm (PCB+pre-amp) . By optimizing the tuning of a new, innovative phased array architecture, bandwidths of over 90 % have been achieved [43], with a two matching layer design. ( see figure 14 ) PCB + T/R switch + Receiver System side PL 20 = 1.04 usec BW = 95.2 % Figure 14 : Experimental results of new transducer architecture for phased array construction, with series-shunt tuning. Bandwidth BW = 95.2 % ; -20 dB pulse length PL20 = 1.04 microsecond. 8. Computer Simulation / Modeling Modal Resonances in Array Structures The achieving of broad band performance can also be limited by modal coupling and spurious resonances in the matching layer materials when the geometry has undesirable aspect ratios. Studies of modal coupling lead to a handy rule in transducer design; one should try to maintain aspect ratios in material crossections of approximately 1:2 whether it be for the ceramic pillars, or the matching layer crossection. Therefore, as a general design rule when designing acoustic transducers, it is usually desirable to keep the width to height aspect ratios approximately 1:2 tall and thin, or greater than 3:1 flat and wide. The dynamic behavior of the elements of an array acoustic stack are illustrated in the following figures. The basic geometry was modeled as a mesh and analyzed with the finite element analysis (FEA) code PZFLEX 54, as illustrated in figure 15. The crossection shows the expansion of the center element (two sub-elements) when activated by a pulse voltage. For a linear array, for wider angular response, often the first matching layer is severed. Even with a wider one lambda pitch, maintaining a flat 2:1 aspect ratio still does not guarantee a crossection free from modal coupling. Modal distortions can be set up as seen in figure 15, depending on the relevant material properties and element pitch. In essence, the thickness dimension is working as a quarter wave resonator, whereas the lateral extent on the diced pitch boundary can work as a half wave resonator, and the lateral dimension can thus couple strongly to the thickness dimension and cause undesirable spurious resonances. So aspect ratios greater than 2:1 are desired in this architecture. For this reason, it is sometimes desirable to avoid notching the matching layers when constructing a convex or linear array. However, the design tradeoff dilemma is that the angular sensitivity is improved if the layers are severed. Figure 15. Crossection of expansion and contraction of center element of three element array, as modeled by finite element code PZFLEX. In a phased array transducer, which is usually diced from the front side (matching layer), sub-dicing the matching layers can keep a desirable tall 1:2 aspect ratio. The net effective acoustic velocity in matching layers with a tall , thin aspect ratio is different from the bulk measured value, and must be taken into account when specifying matching layer dimensions to achieve broad band matching. The velocity downshift is due to wave guide and plate mode effects on the sound velocity. Kino 23 gives a graph of downshifted velocity of a "strip guide mode" mode velocity over the bulk longitudinal sound velocity, Vg/Vl as a function of the material’s Poisson ratio. Dispersion curves that give the matching layer velocity downshift as a function of the height to width aspect ratio H/W are based on the coupling mode theory of Onoe and Tiersten44 . Ayter 45 has expressed this in a useful form requiring only knowledge of a materials shear and bulk longitudinal velocities (value of Poisson ratio not needed). The expression is : (Vg/ Vl )2 = ( ( 2 • f • H / Vl ) = 0.5(1 + x − 2 where x = (H/W) 2 (1 + x ) 2 − 4 • x • k ) (24) (25) and k = 1 - [1-2 •(Vs / Vl ) ]2 2 (26) ; with Vs = shear velocity, Vg=longitudinal velocity in strip guide, and Vl = longitudinal sound velocity (bulk), Recently available computer simulation code PZFLEX 54 based on time domain finite element analysis of two dimensional coupling effects, can also used to study these modal resonances46. For a sector phased array probe, to achieve wide angular directivity, it is beneficial to sever the matching layers and dice into the backing (see figure 16 ). However, at a pitch of 0.5 lambda, this leaves a very undesirable aspect ratio for the front matching layers. One solution is to come back and sub-dice the matching layers to achieve the 1:2 tall aspect ratio guide rule. However, plate modes and Lamb waves can still be set up as illustrated in the PZFLEX simulation plotted in figure 16. Figure 16. Modal behavior of the acoustic stack of the center element of a phased array transducer being excited by pulse voltage. The three dimensional FEA code is also useful for evaluating material effects , such as sound speed in the backing and its influence on crosstalk and acoustic coupling . Crosstalk can degrade the ability of the beamformer to focus with precision on the target objects with out additional degradation effects. Crosstalk can occur acoustically in the transducer acoustic stack, and surrounding structures and housing, or electrically due to capacitances and inductances; or crosstalk in the interconnect paths leading through the system to the input points of the preamps. Another form of cross talk is through the backing. Simulations show that when the velocity of the material in the backing is high ( 2.8 KM/sec), the wave emanated into the backing from two sub-elements, can propagate side ways and excite crosstalk energy on nearest neighbor elements, before the pulse has fully left the front of the array. The use of backing material with slower velocity (1.6 KM/sec) can greatly reduce the progression of the wave into the backing relative to the front going waves. The influence of kerf fillers on crosstalk can also be studied. A propagation delay in the occurrence of the crosstalk components indicates, that in this case it is predominately acoustic, rather than electrical. Typical crosstalk isolation for array elements is -28 to -32 dB. A level of -30 dB is quite adequate for most imaging situations. For Doppler shift detection in continuous wave mode (CW), a higher crosstalk isolation of -40 to -50 dB is desired. This can be achieved by leaving a buffer of a few unused element in the array between the groups of elements transmitting versus receiving. Crosstalk can also seriously degrade the achievable wide angle response (element acceptance angle theta) of array elements needed for wide angle beam steering. This can be very important for phased array type transducers, or for linear arrays that need to be steered for improved Doppler shift detection of blood flow. 9. Two Dimensional Arrays A current very active area is the development of two dimensional array. Adding individually phased elements in the orthogonal dimension (elevation) gives the opportunity for focusing the beam for better image slice thickness, and/or arbitrarily steering the beam for volumetric scanning in three dimensions (47-50) . Furthermore, having addressable elements in the elevation dimension will enhance the possiblity of getting aberration correction schemes to work successfully . The design and fabrication of 2-D arrays brings its own formidable challenges. These include the task of a multitude of miniature interconnects, and building an acoustic stack that is well behaved. For good angular sensitivity, the matching layers are typically severed, but can lead to spurious mode coupling, and difficulty in connecting the signal ground return path. For 2-D arrays, new design rules for element aspect ratios, kerf widths, and accounting for long range coupling/composite effects will have to be developed. The electrical matching problem for 2-D arrays is even more challenging, because of the very large electrical impedance of very small elements. This creates some interesting phenomena when running computer simulations with modeling codes, such as PZFLEX. A curious baseline drift, or sagging of the pulse echo is sometimes observed (figure 17.) For very large receiver impedances, the effect is even more exacerbated (figure 18 ). In effect, the capacitance of the coax cable swamps the capacitance of very small elements, and terminated with a pre-amp of high input impedance, the coax cable acts as a capacitor being charge up by the returning pulse. The effect is particularly noticeable for pulse echo ringdown shapes that are not symmetrical above and below baseline. The RC time constant of the equivalent circuit creates a low frequency response super imposed on the pulse. In the FEA code (PZFLEX) this phenomena can be compensated for by use of a pass band filter. However the software algorithm used by the code, itself imposes a slight DC base line shift to the pulse, and will give erroneous results for the -20 dB and -40 dB pulse ringdown, unless compensated for by an additional correction for DC offset. When processed in this manner, the strange looking pulse of figure 18, can be filtered and adjusted to look like the more normal pulse echo shown in figure 19 ( along with its frequency spectrum ). Figure 17. Sag and baseline drift of returning pulse/echo, R=1000 Figure 18. Sag and base line drift due to Figure 19. Restoration of integrity of pulse charging of coax, with Rcvx=14,000 ohms shape by band pass filtering & DC offset 10. Issues / Challenges A particular challenge is to be able to fabricate arrays that achieve the predicted performance, and the ability to reproduce them in high volumes, with consistent, repeatable performance specifications. One must be able to create in the laboratory, acoustic materials with the requisite properties called for by the optimized design. Conversely, for accurate simulation results, one needs careful characterization of the material properties to be called upon by the computer simulation. Typical material properties that need to be carefully measured are: density, longitudinal velocity, Poisson’s ratio or shear wave velocity, acoustic impedance, and the frequency dependent attenuation of an acoustic wave passing through the sample. There are at least three ways to model the frequency dependent attenuation of materials within the PZFLEX code. The accuracy of the simulations depends on using the best fit to the empirical data. Excellent agreement can be obtained between empirical results and the predictions of FEA computer simulations. The fabrication yield of good arrays with very fine pitch in production, depends on the fundamental strength of materials, particularly in the piezoelectric ceramic. Pinhole defects, porosity, and incipient fault lines or stress fracture can wreck havoc on good yields. The recent development of ceramics with very fine grain structure promises better transducer results and the ability to fabricate transducers of much higher frequencies 51 Another important area of active research, is the growth of single crystal forms of the relaxor-lead titanate (PT) materials, in pieces large enough to fabricate arrays. These new materials have the performance advantage of much larger coupling constants, which should facilitate the fabrication of high sensitivity arrays, having useable bandwidths(52) over 100 % . A very important, practical challenge in transducer development is cost reduction, concurrent with performance improvement. The signal micro-interconnect challenge for arrays with large numbers of elements is significant. The majority of the cost of materials in a probe is in the coax cable and system connectors ! So innovations here will be welcomed. 11. Summary: The principles of designing and fabricating high sensitivity, broadband, short impulse response array transducers for medical ultrasonic imaging have been reviewed. The most commonly used ceramic piezoelectric for multi-element arrays is the ceramic PZT-5H (or equivalent). Design rules for selection and optimization of front matching layer impedance values and backing were reviewed. New design rules taking into account the influence of the backing on the effective front impedance of the ceramic, reveal that impedances for the matching layers much lower than previously estimated can be effective. Design rules for a good place to a start a broadband design with two front matching layers are: First matching layer ( acoustic impedance): Z1 ≅ 7.0 Second matching layer: Z2 ≅ 2.2 Layer thickness: Thk ≅ 0.350 ( Vlayer/ Fa ) where Vlayer is the layer longitudinal acoustic velocity (downshifted) , and Fa is the diced ceramic element anti-resonant frequency. For selecting the pitch of the array elements, typical design rules are: Linear/Convex array: Pitch = P ≅ 1.00 to 1.50 lambda Sector Phased array: Pitch = P ≅ 0.50 to 0.70 lambda Principles for making broad band transducer designs, well matched to the system requirements were discussed. Since a transducer operates in both mechanical and electrical domains, electrical tuning or impedance matching of the probe is as important as the clever design of acoustic matching layers and backing, on achieving short impulse response, over a broad range of frequencies, with good sensitivity. Recently available computer simulation tools such as Pzflex 54 greatly assist in studying and understanding two and three dimensional effects on array performance. This code is based on finite element analysis in the time domain, and thus allows modeling of broadband pulses with computational efficiency 10 to 100 faster than traditional FEA codes that operate in the frequency domain. Subtle transient effects that would be missed by a frequency domain code can be predicted accurately by the time domain FEA code43. The two dimensional and 3-D effects, if not controlled can cause unwanted mode coupling and spurious responses that seriously degrade pulse length and frequency bandshape. These phenomena cannot be modeled directly in simpler computer simulation models such as Piezocad53, based on the KLM model of a transducer 17 . 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Shrout, “Characteristics of relaxor-based piezoelectric single crystals for ultrasonic transducers”, Proc IEEE Ultrasonics Symposium, Vol 2, pp. 935-942, IEEE Cat. # 96CH35993, 1996 53) PIEZOCAD code, from Sonic Concepts, Woodinville, Wa ph (206) 485-2564 54) PZFLEX code, from Weidlinger Associates, Los Altos, Ca, ph (415) 949-3010