Medical Imaging - Introduction to Biomedical Engineering - page 1052

V

− +

d

C 0 = ε A/d f 0 = c/2d

T T

A

poling

(a)

|T(f)|

T(t)

t f 0 3f 0 f

(c)

(b) 0 d/c

FIGURE 16.8 (a) Diagram for a piezoelectric crystal radiating into a medium matched to its impedance, (b)
stress time response, and (c) stress frequency response.

C D , is obtained under a constant dielec- applied electric field. The elastic stiffness constant,

tric displacement, D , and if E is the electric field,

e S AV
dA

e S E ¼ D ¼ ¼ C V = A ð 16 : 22 Þ

0

When a voltage impulse is applied across the electrodes, the piezoelectric effect creates
impulsive forces at the sides of the transducer, given by

F ð t Þ¼ TA ¼ð hC 0 V = 2 Þ½dð t Þþdð t d = c Þ ð 16 : 23 Þ

where the media above and below have the same acoustic impedance, Z c , as the transducer;

p

C D = the speed of sound between the electrodes is given by c ¼ r , and d represents an

impulse (see Figure 16.8b).
Since it can create acoustic waves, the crystal could be regarded as a singing capacitor
with its own unique voice or spectral characteristics and resonant frequency. To obtain
the spectrum of this response, take the Fourier transform of Eq. (16.23),

i p fd
c

F ð f Þ¼ i ð hC V Þ e sin ½pð 2 n þ 1 Þ f = 2 f ð 16 : 24 Þ

0 0

which has maxima at odd harmonics (note n ¼ 0, 1, 2, 3, ... ) of the fundamental resonance

f 0 ¼ c /2 d , as shown in Figure 16.8c.

Transducer Electrical Impedance
Because of the forces generated at the sides of the transducer, the electrical impedance
looking through the voltage terminals is affected. Across the wires connected to the trans-
ducer (Figure 16.8a), a radiation impedance, Z A , is seen in addition to the capacitive reac-

tance, so the overall electrical impedance is
Z T ¼ Z A i ð

1 =o C Þ¼ R A ð f Þþ i ½ X A ð f Þ 1 =o C ð 16 : 25 Þ

0 0