From Eq.(46.5) the transmitting current response is rOz From these simple considerations a number of principles of practical transducer design can be deduced. The mechanical impedance Z,m is in general given by Z=-+ioM+r (46.13) where Km is an effective spring constant, M the mass, and Rm the mechanical resistance. For a piezoelectric transducer Eq (46.9)] Tom is inversely proportional to frequency; hence from Eqs. (46.10)and(4611)we se that a piezoelectric transducer will have a flat receiving sensitivity below resonance(i.e, where its behavior is controlled by stiffness). On the other hand, a moving coil microphone must have a resistive mechanical pedance to have a flat response. From Eq.(4612)we derive the fundamental tenet of loudspeaker design, that a moving coil loudspeaker will have a flat transmitting current response above resonance (i.e, where it mass controlled). Accordingly, moving coil loudspeakers are designed to have the lowest possible resonant frequency(by means of a high compliance since the output is inversely proportional to the mass) and piezo- electric hydrophones are designed to have the highest possible resonant frequency An interesting and important consequence of electromechanical coupling is the effect of the motion of the transducer on the electrical impedance In the absence of external forces(including radiation reactance) from Eqs.(466)and(46.7) E TT (46.14) That is, the electrical impedance has a"motional"component given by Tem Tme /Zm. The motional component can be quite significant near resonance where Zm is small. This effect is the basis of cr ystal-controlled oscillators. 46.6 Radiation Impedance An oscillating surface produces a reaction force FR on its surface given by FR=-ZRV (46.15) where Zg is the radiation impedance. We can thus rewrite Eq. (46.7)as Fev= tem I +(Zr+ zm)v (46.16) where Fat now includes only external forces. For an acoustically small baffled circular piston of radius a, ZR=πap00)2/2c-i(8/3)op0a3 (46.17) The radiation impedance thus has a mass-like reactance with an equivalent radiation mass"of(8/3)poa and a small resistive component proportional to o responsible for the radiated power. A transducer will thus have a lower resonant frequency when operated underwater than when operated in air or vacuum. The total radiated power of the piston transducer is given by c 2000 by CRC Press LLC© 2000 by CRC Press LLC From Eq. (46.5) the transmitting current response is (46.12) From these simple considerations a number of principles of practical transducer design can be deduced. The mechanical impedance Zm is in general given by (46.13) where Km is an effective spring constant, M the mass, and Rm the mechanical resistance. For a piezoelectric transducer [Eq. (46.9)] Tem is inversely proportional to frequency; hence from Eqs. (46.10) and (46.11) we see that a piezoelectric transducer will have a flat receiving sensitivity below resonance (i.e., where its behavior is controlled by stiffness). On the other hand, a moving coil microphone must have a resistive mechanical impedance to have a flat response. From Eq. (46.12) we derive the fundamental tenet of loudspeaker design, that a moving coil loudspeaker will have a flat transmitting current response above resonance (i.e., where it is mass controlled). Accordingly, moving coil loudspeakers are designed to have the lowest possible resonant frequency (by means of a high compliance since the output is inversely proportional to the mass) and piezo￾electric hydrophones are designed to have the highest possible resonant frequency. An interesting and important consequence of electromechanical coupling is the effect of the motion of the transducer on the electrical impedance. In the absence of external forces (including radiation reactance) from Eqs. (46.6) and (46.7) (46.14) That is, the electrical impedance has a “motional” component given by TemTme /Zm. The motional component can be quite significant near resonance where Zm is small. This effect is the basis of crystal-controlled oscillators. 46.6 Radiation Impedance An oscillating surface produces a reaction force FR on its surface given by FR = –ZRV (46.15) where ZR is the radiation impedance. We can thus rewrite Eq. (46.7) as Fext = Tem I + (ZR + Zm)V (46.16) where Fext now includes only external forces. For an acoustically small baffled circular piston of radius a, ZR = pa4r0w2/2c – i(8/3)wr0a3 (46.17) The radiation impedance thus has a mass-like reactance with an equivalent “radiation mass” of (8/3)r0a3 and a small resistive component proportional to w2 responsible for the radiated power. A transducer will thus have a lower resonant frequency when operated underwater than when operated in air or vacuum. The total radiated power of the piston transducer is given by S T A R Z em m = r w p 0 0 4 Z K i m iM R m =+ + m w w E Z T T Z I e em me m = Ê Ë Á ˆ ¯ ˜ –