585 
15 
distortionless transmission system. It may be noted that the phase of the 
original signal has been altered by subtracting a constant time, LVLC. Since 
VLC = 1/v, where v is the phase velocity, 1VLC represents a time lag which is 
independent of the frequency. It is the time required for a sinusoidal com- 
ponent of the signal, as in a Fourier analysis, to traverse the length of the 
cable. If a nonsinusoidal voltage is impressed on the line, all the harmonic 
components will be subjected to the same amplitude and phase changes during 
the transmission. Thus a replica of the impressed voltage will arrive at the 
receiving end. This addition of the harmonic components is possible since 
all the differential equations are linear. 
The important question that remains is the following. What magni- 
tude of error is introduced by having the terminations Z, and Z, not equal 
to Z? The following analysis will give criteria for the length of line and 
the size of terminating impedance that will keep the distortion below a spec- 
ified amount. 
Substituting the values of r,, r,, and LVLC = 1CVL/C = |CZ,=C,Z, 
into Equation [4], we obtain 
E\2Z 2, 
ee [7] 
Z,(Z, + Zp) coswZ,C, +(Z, + 2Z,Z,)j sinwZ,C, 
E, 
C, = 1C is the distributed capacitance of the whole cable. 
In this problem the pressure pulse gives rise to the generator 
voltage E,«?"* of the piezoelectric gage. The generator has an internal im- 
pedance 
il 
jw, 
2,= 
where C, is the capacitance of the piezoelectric pickup element. If gq is the 
charge generated by the gage then 
q=E,C, 
and therefore 
ne 92,2, 
= 4" "(J -—— [8] 
1 2 Zeer 
Z\C,2, at ia cos wZ,C, + (Z, Cat aa jsinwZ,C, 
It is customary to terminate the cable with a capacitance to con- 
trol the output voltage. Therefore let 
ae | 
G2 = Fut, 
where C, is the terminating capacitance. Then Equation [8] becomes 
