D. Schofield 9 
tively. It is to be noted that in this circuit the motional capacity and inductance 
are lumped constants while in the mechanical system the stiffness and mass are 
distributed. The immediately obvious difference is that, whereas the mechanical 
system can resonate at higher harmonics, the electrical circuit has only one 
resonant frequency. It is, nevertheless, a very useful circuit for any piezoelec- 
tric transducer provided attention is limited to the region about the fundamental 
resonant frequency. Similar equivalent circuits can be developed for transducers 
based on other force effects. 
One of the important characteristics of a transducer is the bandwidth. It has 
become customary to consider two Q factors in transducers: the mechanical or 
motional Q, Q,, which can be calculated from the mechanical characteristics of 
the unit or from the motional arm of the equivalent electrical circuit and the 
electrical Q, Q., which from Fig. 1.2 is w oCS;, where neglecting the dielectric 
loss resistance $;=S,;+Sr. Q,controls the frequency response in both projecting 
and receiving modes of operation if either the driver or the receiver has a high 
impedance. However, in transmission the electrical storage element is usually 
tuned out and the amplifier approximately matched to the motional resistance. 
From Fig. 1.2 it can be shown that, parallel-tuning a piezoelectric transducer 
with an inductance, neglecting the dielectric loss resistance, and matching the 
impedances of the amplifier and transducer, the bandwidthis given by 2/(0,, + 0.). 
This quantity is a maximum when 0,,=9,. It will be shown later that the product 
Qn0- is related to the electromechanical coupling factor, and that for a wide 
bandwidth a high coupling factor is required. 
1.4. ELECTROMECHANICAL COUPLING FACTOR 
The importance of the ratio of the clamped capacity to motional capacity as 
a measure of activity in a piezoelectric transducer was first noted by Dye [3] in 
1926, but it was not until 1935 that Mason [4] introduced the idea of an electro- 
mechanical coupling coefficient closely related to this capacitance ratio. Since 
that time there have been a number of definitions and approaches [1, 2, 3, 6-9] 
almost all of which were both consistent and correct, but there remained a need 
for a general definition from which all such definitions could be derived. Hersh 
[10] presented such a general approach in 1957: if the equations defining a coupled 
reciprocal system can be written as 
¥1= 411 X%1 + 412 X2 (2) 
Y2= 421 X1 + 222 X2 
and are homogeneous in the variables of generalized force and displacement, 
where the coefficients a,,, are coefficients or ratios of coefficients from the 
same energy function, the coupling coefficient k of the coupled system is defined 
by 
K2= 412 Ga (3) 
4i1 422 
If the equations of state are mixed in the variables of generalized force and dis- 
placement, the coupling factor is defined by 
412 42 
i = —— (4) 
441 Ag + 442 Ari ; 
