Estimating ultrasound transducer parameters using KLM equivalent circuit model

Transcription

1 Estimating ultrasound transducer parameters using KLM equivalent circuit model Kristian Jambrosic, Bojan Ivancevic, Antonio Petosic Faculty of Electrical Engineering and Computing, Zagreb, Croatia, Calculation scheme for estimating ultrasound piezoelectric transducer parameters in form of KLM equivalent circuit has been implemented in MATLAB and used to characterize transducer before its physical construction. Input parameters for equivalent circuits have been found using impedance data measurements of piezoceramic samples as free resonators. Equivalent circuits enable flexible estimation of input impedance of transducer system and gives approximately transfer function of the system in frequency domain near thickness extensional (TE) mode. Transfer function can be given as a ratio of radiated pressure, surface velocity or force (on active surface) and applied voltage in frequency domain. Several configurations of transducers have been made from PZT-A ceramic of different thickness and diameter for studying influence of ultrasound field on human tissue. Transducers were constructed as systems with active piezoelectric element, n λ/4 matching element section in front of active element face (made from perspex), air-backing on back face, and cable. Estimated input impedance of ultrasound system radiating in water depending on frequency using KLM equivalent circuit model is compared with measurement results in frequency range near TE mode of constructed ultrasound transducers configuration. Measured and theoretical conductance is used for calculating available bandwidth of constructed transducers. Transfer function as ratio of force (or pressure) on surface and input voltage is estimated with theoretical KLM model. 1 Introduction Analytical solutions to the wave equations in piezoelectric materials are quite difficult to derive from first equations because it is complicated to write multiphysics equations and set up appropriate boundary conditions. Mason (ref. [4]) had shown that for 1-D analysis most of the problems can be overcome using electrical network theory. He made an equivalent circuit (fig. 1) that separates piezoelectric material in an electrical port and two acoustical ports. Acoustical and electrical side of transducer are connected through electromechanical transformer. One of the problems was that it perceived negative capacitance at the electrical side of circuit. Krimholtz, Leedom and Matthae made new alternative circuit which removes circuit elements between top of the transducer and the node of the acoustic transmission line. This model is called KLM model and it is often used in medicine for construction of high frequency transducers, arrays or transducers in NDT. The models will be used here to estimate input impedance and transfer functions of few ultrasound transducer configurations made from PZT- A as active element, perspex front layer and air backing, and compare them with experimental results obtained with measurements. Loading medium is water because transducers are made in purpose to investigate influence of high power ultrasound energy in mostly continous mode of working (pulse mode is also possible) on different kinds of tumours and healthy tissue. Material properties needed as input parameters for equivalent circuits are calculated from impedance data measurements using IEEE Standard 176 (lossless assumption) and Sheritt method for each individual sample (moderate losses assumption). If the acoustic ports are shorted these models reduce to the free resonator equation derived from the linear piezoelectric equations and the wave equation which has been adopted by the IEEE Standard on Piezoelectricity for determination of the thickness material constants. Equivalent circuit parameters The piezoelectric circular plate sample with diameter D and thickness t elements in equivalent circuits is defined with mathematical relations in table 1. To account for the losses material parameters c 33 D,ε 33 D and k t should be treated as complex quantities by adding an asterisk as the superscript: c 33 D,ε 33 D, k t. Complex material parameters are assumed not to be dependent on frequency and can be calculated using the Sheritt method. It has been shown that using Z(f) at three different frequencies, complex material parameters can be calculated. Attention should be taken in consideration of the sign of imaginary parts of complex parameters for accounting losses in material sample. 773

2 Forum Acusticum Budapest Figure 1: KLM equivalent circuit model of ultrasound transducer Impedance of free piezoceramic sample vibrating in air can be described with eq. 1: ρ m tan( πf ) D L c33 Z( f) = 1 ( kt ) S i πfε33 A ρ m πf D c33 ε 33 S clamped dielectric permittivity k t electromechanical coupling constant for TE mode c 33 D - elastic stiffness constant at constant electric field v P longitudinal velocity in PZT-A e 33 piezoelectric constant δ m density of PZT-A (76kg/m 3 ) t thickness of active element A- surface of active element S ( e33) ( h33) ε ( k ) 33 t = = D S D c33 ε33 c33 k t -electromechanical coupling factor for TE mode D c h kt D ε33 (1) () = (3) Input parameters for KLM equivalent circuit are given with equations 4-14 connected with figure 1: S A C = ε33 (4) t -capacitance of piezoelectric plate v D c33 P δm = () -longitudinal wave velocity in piezoelectric material ω Γ = (6) v P -transmission coefficient in acoustic transmission line Z = δ A v P (7) -wave impedance in piezoceramic sample Γ ZT = iz tan( ) (8) ZS = iz csc( Γ t ) (9) -impedances as in fig. 1 Z M h33 = ω Z 1 (1) X = iz ( M ) sin( Γ t ) (11) 1 Γ φ = csc( ) M Γ Γ ZL + iz TL = Z Γ Γ Z + izl Z Γ Γ ZR + iz TR = Z Γ Γ Z + izr (1) (13) (14) Z L, Z R -loading impedance on the left and right side of active element.1 Calculating PZT-A parameters from free resonator impedance measurement data Impedance data were first imported in MATLAB script for finding Sheritt series and parallel resonance frequencies near TE mode. Desired material parameters have been calculated using modified equations for lossless calculation as prescribed in [1]. D 33 = 4 δm p c t f (1) c D 33 - elastic stiffness coefficient f p - complex parallel frequency proposed with Sheritt method (described bellow) π fs π fp fs t fp fp k = tan( ) (16) 774

3 Forum Acusticum Budapest Sheritt and authors have defined f p and f s as frequencies which correspond the maximum of real parts of f Z(f) and Y(f)/f instead of real parts of Z(f) and Y(f) as in IEEE Standard 176. The complex parallel and series resonance frequencies are using Sheritt method: f f i p p f 1/ f+ 1/ p = p 1 f p (17) f f i s s f 1/ f+ 1/ s = s 1 fs (18) After complex electromechanical coupling factor and stiffness constants had been found complex permittivity coefficient has been calculated substituting c 33 D and k t and the observed value Z (f) at frequency f p in equation which gives the impedance of piezoelectric sample (e.1). Table 1: Evaluated ceramic parameters Ceramic ID k t i i i i i i i c 33 D (Pa) ε 33 S /ε 1.398e e+8i 1.371e e+8i 1.e e+9i 1.63e e+9i e e+9i 1.416e e+8i e e+9i i i i i i i i Figure : Block-scheme of designed transducer Table : Transducers design parameters Ceramic D t (mm) D/t d (mm) ID (mm) KLM equivalent circuit is a three-port transducer model with one electrical and two acoustical ports. The two ports model is useful for considering additional acoustic layers in the transducer. 3 KLM equivalent circuit implementation Except the active element, constructed transducers have housing (backing element is air), cable and front layer with thickness approximately λ/4, as is shown in fig. Equivalent KLM circuit is upgraded with these additional elements which are very important parts of the transducer. Figure 3: Transducer realization (ceramic 363) Figure 4: Two port model of additional layer 77

4 Forum Acusticum Budapest Transfer matrix of individual layer is calculated with sign convention as in fig. 4 (similar as for electrical cabling matrix) and it is frequency dependent: f F1 kf d) iz sin( kfd) F = f (19) v 1 i Z sin( kfd) kf d) v Z f - plane wave acoustic impedance of front layer given: Z f =δ F v F S F () S F - front layer surface (active element surface) δ p - density of front layer material (1kg/m 3 ) v f - lon. wave velocity in front layer (7 kg/m 3 ) k f wave number in front layer ( π/λ F ) d - front layer thickness (table ) 3.1 Radiation impedance When an ultrasonic transducer is used in an ultrasonic measurement system, its acoustic port is always terminated, the output force and velocity are related to one another with radiation impedance. Pressure field on the face of the acoustic port of the transducer assuming uniform velocity distribution can be given using Rayleigh-Sommerfield relation and radiation impedance isn t easily given explicitly. Compressional force on transducer surface is integral of pressure field then the radiation mechanical impedance is given in complicated form: i ωδ exp( i k r) r r Ft( ω) = ( ds( y)) ds( x) v t ( ω) π r S S (1) F( ω) = Z v( ω) () t m t Z m - mechanical impedance which has explicit solution mathematically describing radiation in medium from flanged end (assuming rigid infinite baffle): m [ ( ) ( )] Z = δcπa R k a + ix k a (3) 1 1 J1( ka) H1(ka) R1( k a) = 1, X1( k a) = (4) ka ka a - radius of active element ρ - density of medium where ultrasound is transmitted (1kg/m 3 ) c - longitudinal velocity in medium where ultrasound is transmitted (1m/s) k - wave number in radiation medium J 1 (z) - Bessel function of the first kind H 1 (z) - is the Struve function of the first kind Approximation for Struve function is taken from reference [6] where Struve function is given with: 16 sin( z) 36 1 z) H1( z) J( z) + ( ) + (1 ) π π z π z () The obtained approximation is in agreement with the small as well as large k a approximations known from the literature, but does not require patchwork formulas, since it is accurate for the whole k a range. 4 Comparison theory and measurement results Input parameters for active elements, front layer, backing radiation medium, and cabling have been imported in MATLAB scripts. The output parameters which are estimated are input impedance and transfer function. Theoretical input impedance with input parameters from table 1 for two transmitters (363 and 19) has been compared with experimental impedance results near thickness extensional resonance frequency. Obtained theoretical results are in very good agreement with experimental results. Radiated power depends on input conductance of transmitter system so relative bandwidth is calculated using 3 db points from frequency dependent curve of input conductance. Subscripts in description below figures have meanings: f nt estimated theoretical frequency where input impedance has maximum f mt estimated theoretical frequency where input impedance has minimum f ne measured frequency where input impedance has maximum f mt measured frequency where input impedance has minimum 776

5 Forum Acusticum Budapest Theoretical (KLM) Impedance Magnitude[Ohms] Impedance Magnitude[Ohms] x x 1 6 Figure : Comparison of experimental and theoretical input impedance magnitude of transducer (ceramic 363) with cabling (f nt =8.kHz, Z maxt =117.Ω, f mt =14.78kHz, Z mint =117.3Ω, f me =3.kHz, Z maxe =1.71kΩ, f me =4.7kHz, Z mine =81.3Ω) Figure 8: Comparison of experimental and theoretical input impedance magnitude of transducer (ceramic 19) with cabling (f nt =194, Z maxt = Ω, f mt =99767, Z mint =16.386Ω, f ne =1116, Z maxe =67Ω, f me =981, Z mine =4.89Ω) Impedance Phase[degrees] - Impedance Phase[degrees] x x 1 6 Figure 6: Comparison of experimental and theoretical input impedance phase of transducer (ceramic 363) with cabling Figure 9: Comparison of experimental and theoretical input impedance phase of transducer (ceramic 19) with cabling 1 1 Conductance[dB (1mSi)] BWexp BWth Conductance[dB (1mSi)] 1 1 BWexp BWth x x 1 6 Figure 7: Comparison of measured and theoretical input conductance (ceramic 363) and available bandwidth (f st =173Hz, G maxt =.83S, BW t =3.3% ; f se =6Hz, G maxe =.119S, BW t =4.99%) Figure 1: Comparison of measured and theoretical input conductance (ceramic 19) and available bandwidth (f st =99817 Hz, G maxt =.6S, BW t =3.1% ; f se =981 Hz, G maxe =.4S, BW t =1.1%) 777

6 Forum Acusticum Budapest Transfer functions based on the KLM model In this section, the pressure radiated by the transducer is considered when it is excited with voltage in phasor domain as shown in fig 11: j( ω + t ϕ V( ω) = V e (6) Figure 11: Complete ultrasound transmitting system Particle velocities in the acoustical transmission line are analogous to currents in an electrical transmission line and total transfer function is given as ratio of force F(ω) and pressure field P(ω) on transducer surfaces and applied voltage V(ω). Fout(w)/V(w)[N/V] ) Medium Backing 6 Conclusion The 1-D modeling of ultrasound transmitters using KLM equivalent circuit has been described, implemented and compared with experimental results on constructed transducer systems. Obtained bandwidth is low because the main purpose of the transducer is transmitting high power ultrasound energy in tissue. This 1-D method gives very good results in case when the transducer is working in continuous mode (individual frequency). Next step will be comparing obtained theoretical with experimental sensitivity and directivity pattern. A good agreement between theoretical and experimental results is obtained when diameter to thickness ratio (D/t) for active is high enough, and in the case when neighboring modes are clearly separated and influence of 3D effects in transducer isn t so expressed. Acknowledgement This paper has been realized by support of the Ministry of Science Education and Sport, The Republic of Croatia, under the project Influence of high energy ultrasound on tissue. Special thanks to dr. Alan Štimac for helping authors with data acquisition using GPIB communication protocol. References 1 - x 1 Figure 1: Sensitivity as ratio of force and applied generator voltage (ceramic 363) with cable and generator impedance R g =Ω Pout(w)/V(w)[Pa/V] Medium Backing 1-1 x 1 Figure 13: Sensitivity as ratio of surface pressure and applied gen. voltage (ceramic 363) with cable [1] IEEE Standard on Piezoelectricity, IEEE/ANSI Std [] Kin Wing Kwok, Helen Lai Wah Chan, Chung Loong Choy, Evaluation of Piezoelectric Parameters by Various Methods, IEEE Transaction of UFFC, Vol 44, pp , (1997) [3] IRE Standards on piezoelectric crystals, Measurements of piezoelectric crystals, Proc. IRE,vol 49,pp , (1961) [4] W..P. Mason, Electromechanical Transducers and Wave Filters, Princeton, NJ, Van Nostrand, (1948) [] Ronald M. Aartsa and Augustus J. E. M. Janssen, Approximation of the Struve function H 1 occurring in impedance calculations, J. Acoust. Soc. Am., Vol. 113, No., May 3 778