CHAP. 16]

415

THE LAPLACE TRANSFORM METHOD

The network has an equivalent impedance in the s-domain ZðsÞ ¼ 10 þ

ð5 þ 1=sÞð5 þ 1=0:5sÞ 125s2 þ 45s þ 2 ¼ 10 þ 1=s þ 1=0:5s sð10s þ 3Þ

Hence, the current is IðsÞ ¼

VðsÞ 50 sð10s þ 3Þ 4ðs þ 0:3Þ ¼ ¼ ZðsÞ s ð125s2 þ 45s þ 2Þ ðs þ 0:308Þðs þ 0:052Þ

Expanding IðsÞ in partial fractions, IðsÞ ¼

1=8 31=8 þ s þ 0:308 s þ 0:052

and

1 0:308t 31 0:052t e e þ 8 8

ðaÞ

16.20 Apply the initial- and ﬁnal-value theorems to the s-domain current of Problem 16.19.      1 s 31 s þ ¼4A ið0þ Þ ¼ lim ½sIðsÞ ¼ lim s!1 s!1 8 s þ 0:308 8 s þ 0:052      1 s 31 s ið1Þ ¼ lim½sIðsÞ ¼ lim þ ¼0 s!0 s!0 8 s þ 0:308 8 s þ 0:052 Examination of Fig. 16-18 shows that initially the total circuit resistance is R ¼ 10 þ 5ð5Þ=10 ¼ 12:5 , and thus, ið0þ Þ ¼ 50=12:5 ¼ 4 A. Then, in the steady state, both capacitors are charged to 50 V and the current is zero.

Supplementary Problems 16.21

Find the Laplace transform of each of the following functions. ðaÞ ðbÞ Ans:

ðeÞ f ðtÞ ¼ cosh !t ð f Þ f ðtÞ ¼ eat sinh !t

See Table 16-1 ! ðs þ aÞ2  !2

ðaÞðeÞ ðfÞ

16.22

ðcÞ f ðtÞ ¼ eat sin !t ðdÞ f ðtÞ ¼ sinh !t

f ðtÞ ¼ At f ðtÞ ¼ teat

Find the inverse Laplace transform of each of the following functions. ðaÞ ðbÞ ðcÞ Ans:

s ðs þ 2Þðs þ 1Þ 1 FðsÞ ¼ 2 s þ 7s þ 12 5s FðsÞ ¼ 2 s þ 3s þ 2 FðsÞ ¼

ðaÞ ðbÞ ðcÞ

2e2t  et 3t

e

e

2t

10e

3 þ 6s þ 9Þ sþ5 ðeÞ FðsÞ ¼ 2 s þ 2s þ 5 2s þ 4 ð f Þ FðsÞ ¼ 2 s þ 4s þ 13 ðdÞ

ðdÞ

4t

FðsÞ ¼

sðs2

1 1 3t 3  3e t

 te3t

ðgÞ

ðgÞ

FðsÞ ¼

ðs2

10 29 cos 2t

2s þ 4Þðs þ 5Þ

5t 4 þ 29 sin 2t  10 29 e

ðeÞ e ðcos 2t þ 2 sin 2tÞ t

 5e

ðfÞ

2e2t cos 3t

16.23

A series RL circuit, with R ¼ 10  and L ¼ 0:2 H, has a constant voltage V ¼ 50 V applied at t ¼ 0. Find the resulting current using the Laplace transform method. Ans: i ¼ 5  5e50t ðAÞ

16.24

In the series RL circuit of Fig. 16-19, the switch is in position 1 long enough to establish the steady state and is switched to position 2 at t ¼ 0. Find the current. Ans: i ¼ 5e50t ðAÞ

Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An
Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An