IMPEDANCES 



11 



where z ( „ k, A', and Z b are independent of E, I, F, and 

 v n but are in general functions of frequency^ 1 Equa- 

 tions of the type of (1) and (2) have been shown to 

 apply to the majority of electroacoustic transducers 

 now in use, at least for limited ranges of the variables. 



Considering the physical significance of the terms 

 in equations (1) and (2), it is seen that z is the force 

 on the diaphragm divided by the velocity when the 

 current is zero. Let z be the open-circuit mechanical 

 impedance of the transducer. The constant k gives 

 the force developed per unit current into the trans- 

 ducer when the diaphragm is completely constrained 

 (v n = 0). Therefore k is called the electroacoustjc 

 transfer constant. Similarly, />•', the open-circuit volt- 

 age per unit diaphragm velocity, is called the acous- 

 toelectric transfer constant. The term Z h gives the 

 voltage developed for unit current when the dia- 

 phragm is constrained from moving and so is called 

 the blocked electric impedance. 



The equations for a simple electromechanical sys- 

 tem are of precisely the same form as equations (1) 

 and (2). This is to be expected, since the assumption 

 that t'„ is constant reduces the problem to one me- 

 chanical degree of freedom. The acoustic case, how- 

 ever, includes the properties of the sound field, as 

 discussed below. 



3.2 



COUPLING CONDITIONS 



Consider now the relationships which obtain when 

 an electroacoustic transducer is coupled to electric 

 elements or to a medium capable of propagating 

 sound. When the transducer is used to convert acous- 

 tic energy into electric energy, it is terminated in an 

 electric impedance Z L , that is, there is a load imped- 

 ance Z L across its electric terminals. In that case at all 

 times 



E= -Z L I. (3) 



When the transducer is used to convert electric 

 energy into acoustic energy, a source of voltage £ 

 of internal impedance Z int is connected to the electric 

 terminals. Then 



E — E t) — Z int I . 



(4) 



The problem of coupling on the acoustic side may 

 be treated formally in a similar manner. When the 



b The linearity of the equations insures the possibility of 

 treating functions with any time dependence by superposition 

 using Fourier analysis. 



transducer converts acoustic energy into electric en- 

 ergy, we may write 



F = /•„ 



(5) 



The interpretation is somewhat more complicated 

 in this case: The term F really consists of two parts, 

 E« = F inc + F rigid ,, irfr . The first, F inc , is the force on 

 diaphragm that would be present if the transducer 

 had no effect on the incident sound field. The second 

 part, F TigiA diffr , may be considered as representing the 

 force on the diaphragm due to the sound that would 

 be diffracted by the transducer if the latter were per- 

 fectly rigid. The symbol z,. represents the radiation 

 impedance, and the term (— z,. i/„) is the force on the 

 diaphragm due to the additional sound pressure 

 created by the latter's motion in the sound field. 



Finally, when the transducer is used to convert 

 electric energy into acoustic energy, there is no ex- 

 ternal sound field and the equation becomes 



F = - z,.v„. 



(6) 



It should be noted that equation (5) corresponds 

 to the equation for an electromechanical transducer 

 coupled to a source of generated force F and of in- 

 ternal mechanical impedance z,.. 



33 IMPEDANCES 



The next problem is the determination of the effec- 

 tive impedances, electric and acoustic, of the trans- 

 ducer under various conditions of coupling. Begin 

 with the effective electric impedance, defined as the 

 ratio of voltage to current £//. The value of this im- 

 pedance depends on the nature of the acoustic cou- 

 pling. Consider the case of the transducer in an 

 infinite source-free medium. Then the force on the 

 diaphragm is given in terms of the normal velocity 

 t'„ on the surface by equation (6) as F = — z r t'„. 



Substituting (6) in equations (1) and (2), we find 



and 



- Zr v „ = z o V,, + kl 



E = k' v n + Z h I. 



(V) 

 (8) 



Solution of these equations shows that the effective 

 electric impedance Z el is given by 



7 F. _ 7 kk' 



(9) 



