0195136047.pdf

3.3 LAPLACE TRANSFORM 153

Transfer functions V 2 (s); V 2 (s); I 2 (s); I 2 (s)

V 1 (s) I 1 (s) V 1 (s) I 1 (s)

+

−

+

−

V 1 (s) V 2 (s)

I 1 (s) I 2 (s)

Input Network Output

Figure 3.3.5Network transfer func-

tions.

EXAMPLE 3.3.3

A network function is given by

H(s)=

2 (s+ 2 )

(s+ 1 )(s+ 3 )

(a) Forx(t)=δ(t), obtainy(t).

(b) Forx(t)=u(t), obtainy(t).

(c) Forx(t)=e−^4 t, obtainy(t).

(d) Express the differential equation that relatesx(t) andy(t).

Solution

(a) Forx(t)=δ(t), X(s)=1. Hence,

Y(s)=H(s)X(s)=H(s)=

2 (s+ 2 )

(s+ 1 )(s+ 3 )

=

K 1

s+ 1

+

K 2

s+ 3

where

K 1 =

2 (− 1 + 2 )

− 1 + 3

= 1

K 2 =

2 (− 3 + 2 )

− 3 + 1

= 1

Thus,

y(t)=e−t+e−^3 t

Note thatL−^1 [H(s)] yields the natural response of the system.

(b) Forx(t)=u(t), X(s)= 1 /s. Hence,

Y(s)=H(s)X(s)=H(s)

1

s

=

2 (s+ 2 )

(s+ 1 )(s+ 3 )

1

s

=

K 1

s+ 1

+

K 2

s+ 3

+

K 3

s

where

K 1 =

2 (− 1 + 2 )

(−^1 +^3 )(−^1 )

=− 1