ESTABLISHING A DC CURRENT IN AN INDUCTOR
If a dc source is suddenly applied to a series RL circuit initially at rest, as in Fig. 7-7(a), the current grows exponentially from zero to a constant value with a time constant of L=R. The preceding result is the solution of the ﬁrst-order diﬀerential equation (8) which is obtained by applying KVL around the loop. The solution follows.
Ri þ L
di ¼ V0 dt
ið0þ Þ ¼ 0
for t > 0;
Since i ¼ ih ðtÞ þ ip ðtÞ, where ih ðtÞ ¼ AeRt=L and ip ðtÞ ¼ V0 =R, we have i ¼ AeRt=L þ V0 =R The coeﬃcient A is found from ið0þ Þ ¼ A þ V0 =R ¼ 0 or A ¼ V0 =R. The current in the inductor and the voltage across it are given by (9) and (10) and plotted in Fig. 7-7(b) and (c), respectively. iðtÞ ¼ V0 =Rð1 eRt=L Þ di vðtÞ ¼ L ¼ V0 eRt=L dt
for t > 0
for t > 0
THE EXPONENTIAL FUNCTION REVISITED
The exponential decay function may be written in the form et= , where is the time constant (in s). For the RC circuit of Section 7.2, ¼ RC; while for the RL circuit of Section 7.4, ¼ L=R. The general decay function f ðtÞ ¼ Aet=
ðt > 0Þ
is plotted in Fig. 7-8, with time measured in multiples of .
It is seen that
f ðÞ ¼ Ae1 ¼ 0:368A