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7.5

FIRST-ORDER CIRCUITS

[CHAP. 7

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.

Fig. 7-7

Ri þ L

di ¼ V0 dt

ið0þ Þ ¼ 0

for t > 0;

ð8Þ

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

7.6

for t > 0

ð9Þ

for t > 0

ð10Þ

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

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