10 Lecture 1 
The piezoelectric equations of state for the length extensional mode of a bar 
excited by a transverse field are 
Si=sh T, + da E3 (5) 
D3;=d3, 7; + 633E3 
When we compare these equations with Eqs. (2), and use Eq. (3), we find that the 
coupling coefficient is given by the equation 
diy 
See (6) 
E 
Sir €33 
Keo 
This equation is frequently used to define the electromechanical coupling co- 
efficient. 
Starting from Eqs. (2), an expression can be derived for the coupling co- 
efficient in terms of the input open- (x2 = 0) and short-circuit (y2 = 0) impedances: 
Zoc —Zsc 
Zoc 
[os (7) 
where the impedances are either all resistances or all reactances of the same 
sign. 
Again starting from Eqs. (2), it can be shown that 
2 Total energy stored unclamped (x2 = 0) — Total energy stored clamped (y = 0) 
RO = Se ee 
Total energy stored unclamped (x2 = 0) 
In the case of a piezoelectric transducer, this is equivalent to 
x2 — Energy stored in mechanical form (9) 
Total electrical energy input 
Using the simple lumped equivalent circuit, Fig. 1.2, the electromechanical 
coupling coefficient derived from Eq. (7) reduces to 
k2 = K 
~T OAK 
(10) 
However, since the equivalent circuit of Fig. 1.2 is not exact, k, is not the true 
electromechanical coupling factor. 
It can be shown that for a longitudinally resonant piezoelectric bar 
2 
a 11 
7?/8 — k? (n?/8 — 1) OM 
A useful approximation in transducer design is 
reel — )= 0m. (12) 
and hence, it can be seen from the previous section, the tuned bandwidth is de- 
pendent upon the coupling factor. 
1.5. TRANSDUCER MATERIALS 
Although some use has been made of the electrodynamic and Vy effects in 
transducers, most underwater acoustic units have made use of the piezoelectric, 
