MEASUREMENT OF PRESSURES 201 



This result is plotted in Fig. 5.24 as a function of the reduced time t/d 

 for various values of the ratio d/r. The response for the limiting case 

 that ^ ^ GO is an exponential rise to the final value Pm as given by 

 R{t) = P^(l — e~^/'"), with time constant r determined by the circuit. 

 It is evident from Fig. 5.24 that unless the value of r is much smaller 

 than the time constant 6 of the applied signal there will be considerable 

 distortion of the true curve (represented by d/r -^ ^ ) and loss of indi- 

 cated peak value. It is easily shown that the indicated peak height 

 always lies on the true curve, and the time of rise to this value must be 

 small compared to the time constant 6. For example, if a pulse of 

 duration ^ = 50 /isec. corresponding to the shock wave 3 feet from a 

 }/2 pound charge is to be recorded with a loss of less than 2 per cent in 

 peak value, the value of d/r must exceed 300 and r must be less than 

 0.17 /isec. 



Expressed in terms of frequency response, the requirement on r 

 means that the simple RC stage should have 70 per cent of its midband 

 response at a frequency /c = ^/^irRC = 950 kc./sec. For larger charges 

 the demands will be less severe, and in the case of service weapons 

 (300-500 pounds of explosive) a high frequency response flat to 100 

 kc./sec. is usually adequate. The analysis presented here is of course 

 oversimplified and in practice multistage compensated amplifiers must 

 be considered. Nevertheless, the analysis does show the nature and 

 order of magnitude of the requirements which any amplifier must 

 satisfy. 



The low frequency requirements on amplifier circuits can be made 

 evident in a similar manner. As already discussed in sections 5.7 and 

 5.8, the input impedance of an indicating circuit used with a piezoelectric 

 gauge must be high in order to prevent loss of pressure-developed 

 charge, the exact requirements depending of course on the transient. 

 A similar problem which must be considered in a-c amplifiers is the loss 

 of low-frequency response as a result of capacitative interstage cou- 

 plings. Either of these problems can be analyzed in terms of a series 

 resistance-capacity circuit, the input voltage being applied across the 

 combination of resistance R and capacitance C in series, and the output 

 voltage developed across the resistance. The relative response in this 

 case is given by 



F{j.)= ^"^^ 



1 + jo^'K 



where the time constant X = RC is the time required for the response to 

 an applied step voltage to fall to 1/e of its original value. 



