MEASUREMENT OF PRESSURES 



187 



characterized by series inductance L/d per unit length and capacitance 

 C/d per unit length, d being the length of the cable. The input and 

 output networks evidently reduce to shunt capacitances at low fre- 

 quencies and approximate resistances for sufficiently high frequency of 

 applied voltage. The steady state terminal voltage Vt for an applied 

 voltage V = VoS^^, where p = jco = j2Trf, and/ = frequency, is found by 

 standard methods^^ to be 



(5.13a) 



Vt = CoVo 



pH{p)e ^< 



Cp 



1 - J(p)e-2«' 



oCp 



In this equation, Ro is the surge impedance given by Ro = ^L/C, 

 and the quantity CoVo represents the charge developed by the equiv- 



/W Q CABLE 



Fig. 5.18 Cable termination network for piezoelectric gauge. 



alent piezoelectric gauge. The capacitance Co can be neglected in 

 comparison with the cable capacitance C, and with this approximation 

 the functions H{p) and J{p), which depend on the parameters of the 

 cable and terminal networks, are given by 



(5.13b) 



H{p) = 

 J{p) = 



2Rc 



[1 + {R2 + Ro)C2p] [1 + {Ri + Ro)Cip] 



[1 + (i^2 - Ro)C2p] [1 + {Ri - Ro)Cip] 

 [1 + (R2 + Ro)C2p] [1 + {Ri + Ro)C,p] 



The equation (5.13a) and the defining relations (5.13b) are in con- 

 venient form for calculating transient response characteristics, for the 

 reason that the exponential e-^o^p represents the phase shift corre- 

 sponding to the transit time RoC of a signal along the cable. An ex- 

 pansion of Eq. (5.13a) in powers of e~^^p therefore corresponds term- 

 by-term to the original signal and successive reflections. The transient 

 effect of these terms can be obtained by use of Laplace transform theory 

 or other operational methods. It is often desirable to analyze per- 



17 See, for example, the report by R. H. Cole (20) , in which the material of this 

 section is discussed in detail. 



