652 
@= |3=- 
This expression is plotted in Fig. 4 as a function of t/A for a ratio 
e/r = 1/10 (that is, amplifier time constant ten times that of the pulse). 
It is seen that, while the response curve has the same general form as the 
applied signal, it falls increasingly below the true curve as the time in- 
creases » 
If one integrates Eqe (11) to infinite time, one finds that the total 
area under the response curve is zero. Hence the curve must go negative 
and remain so for a considerable interval, this interval increasing as the 
time constant A increases. The result of zero total area is to be expected 
quite generally on the basis of the superposition theorem derivation carried 
out above. It was shown that the response area is determined primarily by 
the limiting value of the step-response characteristic at infinite time, 
which is zero for conventional amplifiers and circuits involving capacitative 
couplings (as already noted, piezoelectric or simple capacitance=type gauges 
have equivalent circuits of this form). 
The error in area measurements for an exponential pulse can be calcu- 
lated from the integral of Eqe (11). If the time t' to which the integration 
is carried is not greater than one-tenth the time constant A of the circuit, 
the fractional error dA/A in the area A(t') is approximately given by 
US 
(12) == abss, $5" 1 10 
If this error is to be less than 5 percent, tne amplifier time constant must 
be at least twenty times that of the pulse. Equation (12), although approxi- 
mate, is very useful as a guide in circuit design. 
It must be remembered that the analysis leading to Eq. (12) is based on 
the assumption that the low-frequency response is that of a single R-C cir- 
cuit. If low-frequency compensation is provided in the amplifier, the re- 
Sponse can be improved over a limited range of frequencies in the cuteff 
regions This can be expressed in terms of an equivalent time constant, based 
EER the time constant \ of the circuit is much longer than the time 
constant © of the signal, the correction term can be approximated by a series 
expansion, The first-order correction, neglecting the square and higher 
powers of (t/A), is then 
R(t) ¥ S(a)e7*/® : = (1 = 2) (4)] 
