MECHANICAL IMPEDANCE OF TRANSDUCERS 



capacitance^'*. The detailed properties of such a system are outside the 

 scope of the present analysis, but various salient features should be noted : 



(1) mechanical signals propagate along the spring with a finite velocity; 



(2) a phase shift proportional to frequency is thereby introduced; (3) at 

 certain frequencies the signals reflected at the transducer and transmitted 

 back to the input end can cause very severe loading of the preparation. As 

 a result the input impedance of the spring may become very high. 



Figure 33.7 



Equivalent circuit of force transducer 

 fed through a weak spring 



To avoid all these effects the time of transmission of a mechanical signal 

 along the spring must be short compared with the periodic time of the signal. 

 Since the velocity of propagation v along a spring is given by 



1 



where tn^ and c^ are the mass and compliance per unit length, it follows 

 readily that the time of transmission along the spring is 



/ = (MC)i/2 



where M and C are the mass and compliance of the whole spring. It also 

 follows that the frequency at which the input impedance of the spring 

 becomes high is given by 



J' l{MCfl^ 



For example, a spring designed to give a force of 10* dynes (10 g wt) per cm 

 of movement might weigh 50 mg, and be 2 cm long. The propagation time 

 along it would then be about 2 ms. Thus sinusoidal mechanical inputs with 

 a frequency of 120 c/s would suffer a phase shift of about 90 degrees; at a 

 frequency of 250 c/s the input end of the spring would appear almost com- 

 pletely rigid. 



— nnnp — Vv 1| 



© 



1 



Figure 33.8 Equivalent circuit for displacement transducer 

 with added stiff spring 



As mentioned above, a low-impedance transducer can be used to measure 

 force by putting a stiff spring across it : the input member of the transducer 

 is then rigidly attached to the preparation. The equivalent circuit is shown 

 in Figure 33.8. Here m, R, Q are the mass, resistance and compliance 



479 



