406

THE LAPLACE TRANSFORM METHOD

[CHAP. 16

(a) IðsÞ ¼ 1=s and the unit-step response is 1 RLs 1 1 ¼ 2 s RLCs þ Ls þ 1 C ðs þ Þ2 þ !2d 1 t vðtÞ ¼ e sin ð!d tÞuðtÞ C!d

VðsÞ ¼

where 1 ; ¼ RC

and

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ   1 2 1 !d ¼  2RC LC

(b) IðsÞ ¼ 1 and the unit-impulse response is 1 1 C ðs þ Þ2 þ !2d 1 t vðtÞ ¼ e ½!d cos ð!d tÞ   sin ð!d tÞuðtÞ C!d

VðsÞ ¼

Solved Problems 16.1

Find the Laplace transform of eat cos !t, where a is a constant. Ð1 Applying the deﬁning equation l½ f ðtÞ ¼ 0 f ðtÞest dt to the given function, we obtain ð1 l½eat cos !t ¼ cos !teðsþaÞt dt 0 " #1 ðs þ aÞ cos !teðsþaÞt þ eðsþaÞt ! sin !t ¼ ðs þ aÞ2 þ !2 0 sþa ¼ ðs þ aÞ2 þ !2

16.2

If l½ f ðtÞ ¼ FðsÞ, show that l½eat f ðtÞ ¼ Fðs þ aÞ.

Apply this result to Problem 16.1.

Ð1

f ðtÞest dt ¼ FðsÞ. Then, ð1 ð1 l½eat f ðtÞ ¼ ½eat f ðtÞest dt ¼ f ðtÞeðsþaÞt dt ¼ Fðs þ aÞ

By deﬁnition, l½ f ðtÞ ¼

0

0

0

Applying (5) to line 6 of Table 16-1 gives l½eat cos !t ¼

sþa ðs þ aÞ2 þ !2

as determined in Problem 16.1.

16.3

Find the Laplace transform of f ðtÞ ¼ 1  eat , where a is a constant. ð1 est dt  eðsþaÞt dt 0 0 0  1 1 1 1 1 a eðsþaÞt ¼  ¼ ¼  est þ s sþa s s þ a sðs þ aÞ 0

l½1  eat  ¼

ð1

ð1  eat Þest dt ¼

ð1

ð5Þ

Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An
Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An