Chapter 3 



GENERALIZED THEORY OF ELECTROACOUSTIC 



TRANSDUCERS 



By Leslie L. Foldy and Henry Primakoff 



3.1 



INTRODUCTION 



Following the qualitative discussion of various 

 simple idealized transducers given at the begin- 

 ning of the preceding chapter, a generalized theory of 

 linear passive electroacoustic transducers is now de- 

 veloped. Attention is centered on deriving relation- 

 ships true for all linear passive transducers, rather 

 than on a detailed analysis of particular types. There 

 is a rather complete analogy between the theory of 

 electroacoustic transducers, taking into account the 

 properties of the sound field, and that of electro- 

 mechanical transducers, 39 where only one mechani- 

 cal degree of freedom is present. Consequently, this 

 discussion is carried through in fairly abstract terms, 

 insofar as this can be done without undue mathemati- 

 cal complexity. Analogies are pointed out as they 

 occur. 



An electroacoustic transducer is a device for trans- 

 forming electric energy into acoustic energy, or vice 

 versa. If all the energy delivered by the transducer to 

 the electric or acoustic systems to which it is con- 

 ' nected is derived from power absorbed by the trans- 

 ducer from these systems, the transducer is said to be 

 passive. This does not prohibit the presence of active 

 internal sources of power such as are used to provide 

 polarizing voltages and currents in some types of 

 transducers, provided that these internal sources do 

 not supply power to the electric or acoustic systems 

 to which the transducer is connected. 



Schematically, an electroacoustic transducer may 

 be represented as a pair of electric terminals, by 

 means of which connection to electric systems is 

 made, and a closed surface" which is in contact with 

 a medium capable of propagating sound. The sim- 

 pler electromechanical transducer, which may be 

 represented schematically by a box with a pair of 

 electric terminals and a pair of mechanical terminals, 

 is shown in Figure 1. 



Only those transducers are considered in which the 

 acoustically active part of the surface, the diaphragm, 



ELECTRO- 

 MECHANICAL 

 TRANSDUCER 



DIAPHRAGM 



Figure 1. Electroacoustic transducer represented as a 

 four-terminal electromechanical network. 



vibrates in such a manner that its normal velocity is 

 the same at all points (rigid vibration). It is possible 

 to remove this restriction and develop the theory for 

 any type of vibration, as discussed briefly later. The 

 former case, however, is considerably simpler mathe- 

 matically and the treatment of it given here contains 

 the essential physical principles of the problem. 



The quantities of interest, expressed as functions 

 of time t, are the following: the voltage E(t) across the 

 electric terminals of the transducer, the current I(t) 

 into the electric terminals, the normal velocity v n (t) 

 of the diaphragm, and the total force F(t) on the dia- 

 phragm. This total force may be considered as the 

 integral over the diaphragm of the pressure at each 

 point on its surface. The pressure and particle veloc- 

 ity of the sound field at any point in the medium are 

 also of interest. 



Consider only the steady state, where all quantities 

 vary harmonically with the time with the same fre- 

 quency: £(/) = E e' ut , v n (t) = v„ e' at , etc. For electro- 

 acoustic transducers in which the diaphragm vibrates 

 rigidly, it is found that any two of the four quantities 

 E, I, F, and i> n determine the values of the other two. 

 Take / and r'„ as independent variables. The most 

 general equations for a linear transducer of this type 

 are then 



and 



a Only part of this surface need be acoustically active. 



F = z v n + * I 



E = k' v n + Z b I, 



(1) 



(2) 



10 



