The one indispensible part of a diagnostic imaging system is the transducer. transducers come in many shapes, frequency and size. In this post I will present a basic intuitive model of a piezoelectric transducer to describe its essential acoustic and electrical characteristic. The simples transducer is apiece of piezoelectric material with electrodes on the top and bottom. Unlike the drawing at the top of this figure, the top has across-sectional area (A) and sides that are much longer (>10 X ) than the thickness ( d ) . Piezoelectric material is dielectric; therefore ., it has a clamped capacitance.
Because of forces generated by the transduce, the electrical impedance looking through the voltage terminals is affected. A radiation impedance, ( Z A ) is added to the capacitive reactance so that an equivalent circuit for the overall electrical impedance is
Z T =Z A- i(1/ωC0 ) = R A (f) + I [X A (f) – 1/ωC0
Here Z A is radiation impedance, of which R A and X A its real and imaginary parts. What is R A ? to first order, it can be found from the total real electical power flowing into the transducer for an applied voltage (V) and current (I)
W E = II* R A / 2 = │I 2│ R A /2
where current is I = iωQ = iωC 0 V. The total power radiated from both sides of the transducer into surrounding medium of spoesific acoustic impedance , Z C = ρ c A (equal to that of the crystal ) is
W A = ATT’ / (2Z C / A ) = A 2 │F (f)/ A│2 / 2Z C = │hC O V sin ( πf/ f0│2/ 2Z C
Setting the Eqs and equal we can solve R A,
R A (f) = R AC sinc 2 ( f / 2 fo )
Where sin C( x ) = sin ( πx) / (πx) and
RAC = k2 T / 4f 0 C 0 = d 2 k 2 T / 2A Ƹ s
The electroacoustic coupling constant is kr , and kr = h / √ C D / Ƹ S , Interesting properties of RAC include an inverse proportionality to capacitance and area of the transducer and a direct dependence on the square of the thickness ( d ). Note that resonance,
RA ( f0) = k 2 / π 2 f o C o
Networks theory requires that imaginary part of an impedance be related to the real part through a Hilbert transform ( Nalamwar and Epstein, 1972)
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