Page 234 - Electrical Safety of Low Voltage Systems

Safety Against Static Electricity and Residual V oltages 217

t v(t)
RC 63.2% RI
2RC 86.4% RI
3RC 95.0% RI

TABLE 13.1 Evaluation of the
Earth Potential as a Function of
the Time Constant RC

then becomes
dv(t)
I = i C (t) = C (13.4)
dt
By assuming again v(0 ) = 0, we solve the above differential equation
+
for v(t) and obtain
I
v(t) = t (13.5)
C
In this case, the voltage to earth linearly increases without having
the upper limit of RI. Therefore, when the magnitude of the potential
difference exceeds the dielectric strength of the air, a discharge of
energy occurs in the form of heat (i.e., a spark). The charge process
restarts again and repeats itself.
The graph of v(t) as a function of time is qualitatively shown in
Fig. 13.3 for both the above cases.
The energy being stored in the charged object as a function of time
is given by

1 2 1 2 2 t 2
E = Cv(t) = CR I 1 − e − RC (13.6)
2 2
If the process of accumulation of charges lasts at least three times
the time constant (i.e., t Q > 3RC), the exponential function in Eq. (13.6)

FIGURE 13.3 Graph of v(t) as a function of time.