TRANSDUCERS 



Therefore the voltage on the capacitance represents 1/C times the charge; 

 using this result the equivalent circuit may be drawn as in Figure 33.3. 



This circuit has already been analysed in Chapter 5 ; it was shown (Graph 

 23) that for a constant sinusoidal input the output is constant at low fre- 

 quencies, reaches a maximum the height of which depends on the resistance, 

 then falls steeply away at high frequencies. The response of the circuit to a 

 sudden transient input has also been discussed, and is shown in Graph 22. 

 From this analysis it is clear that the apparatus will record faithfully periodic 

 forces only if their frequency is well below the resonant frequency. In 

 addition, sudden changes of force will be recorded at best only after a delay 

 of about half the natural period of the apparatus, or, if the damping is low, 

 with many 'overshoots'. 



Typical values of the mechanical constants are m = 10 g, C = 1 cm per 

 kg wt (i.e. 10~^ cm per dyne). The resonant frequency would then be about 

 50 c/s. If the value of the resistance is appropriate for 'critical damping' 

 (see Chapter 5) the response of the apparatus to periodic forces will be only 

 90 per cent of its expected value at about 20 c/s ; the pen will also take about 

 10 msec to register a sudden change of force. 



In evaluating the response of a transducer it is necessary to take into 

 account not only the mechanical properties of the transducer itself but also 

 those of the devices coupling it to the preparation, and in some cases the 

 properties of the preparation itself. For example, an RCA mechano- 

 electronic transducer (described on page 490) has been used by Dr. G. M. 

 Hughes to measure the propulsive force of the jet efflux from a dragon-fly 

 nymph; the arrangement is shown in Figure 33.4. Here it is necessary to 

 consider not only the inertia and compliance of the transducer but also the 

 compliance of the support arm, the mass of the insect and the drag of the 

 water on it. Figure 33.5 shows the equivalent circuit. Here R is the resistance 

 due to water drag, M^ and M^, the moments of inertia of the insect and trans- 

 ducer plate respectively, and C^ and Cg are the compliances of the support 

 arm and the transducer diaphragm respectively. The input to the circuit 

 is a couple (the product of the required force and the support length), while 

 the output is l/Cg times the angular displacement of the transducer plate; 

 the electrical output of the transducer is proportional to this displacement. 



It is often possible to simplify an equivalent circuit, since the component 

 parts are usually of widely differing magnitudes. In the present case, 71^2 <^ 

 Ml and Q <^ Cg. The circuit reduces to that of Figure 33.6, which is similar 

 to that of Figure 33.3. Now, however, the resonant frequency and damping 

 of the system depend directly on the properties of the biological preparation 

 under investigation. Although the natural frequency of the transducer alone 

 is 12 kc/s, with the insect and support in place it is reduced to 120 c/s, with 

 a damping ratio of about 0-4. In Dr. Hughes' experiments this lowering of 

 the resonant frequency was of no importance, since the jet force varied 

 relatively slowly. However, the analysis of this mechanical system demon- 

 strates the care needed in designing attachments to preparations if the 

 advantages of transducers (sensitivity and wide frequency range) are to be 

 exploited to the full. 



The equivalent circuit technique will be used again on pages 492, 495, 

 where it will be shown that the mechanical and electrical properties of an 



476 



