numerical methods - 17.25

The output function can be converted to a function of time using the transform tables, as shown below.

Vo( t) =

–1

[ Vo( s) ] =

–1

1.5s + 1.5

–1.5

–1

1.5s + 1.5

–1

–1.5

L

L

-----------------------

+ -----------

=

L

-----------------------

+ L

-----------

s

s

2

+ 1

1

s + 1

s

2

+ 1

s + 1

Vo( t)

–1

–1

–t

=

1.5L

-------------

+ 1.5L

-------------

– 1.5e

s2 + 1

s2 + 1

Vo( t)

=

1.5 cos t + 1.5 sin t – 1.5e–t

2

2

1.5

–t

Vo( t)

=

1.5

+ 1.5

sin

t + atan ------

– 1.5e

1.5

Vo( t)

π

–t

=

2.121 sin

t +

--

– 1.5e

4

Figure 17.29 A circuit example (continued)

17.7 ADVANCED TOPICS

17.7.1 Input Functions

In some cases a system input function is comprised of many different functions, as shown in Figure 17.30. The step function can be used to switch function on and off to create a piecewise function. This is easily converted to the s-domain using the e-to-the-s functions.

numerical methods - 17.26

f(t)

5

t

seconds 0 1 3 4

f( t) = 5tu( t) – 5( t – 1) u( t – 1) – 5( t – 3) u( t – 3) + 5( t – 4) u( t – 4)

f( s) =	5			5e–s			5e–3s			5e–4s
	----		–	---------		–	------------		+	------------
	s	2		s	2		s	2		s	2

Figure 17.30 Switching on and off function parts

17.7.2 Initial and Final Value Theorems

The initial and final values an output function can be calculated using the theorems shown in Figure 17.31.

x( t →

∞ )

=

lim [ sx( s) ]

Final value theorem

s → 0

x( t →

∞ ) =

lim

1s

=

lim

1

=

1

1

---------------------------------

-------------------------

------------------------------------ =

--

s → 0

( s2 + 3s + 2) s

s → 0

s2 + 3s + 2

( 0) 2 + 3( 0) + 2

2

Initial value theorem

x( t →

0)

=

lim [ sx( s) ]

s →

∞

x( t →

0)

=

lim

1( s)

=

1

1

= 0

---------------------------------

--------------------------------------------

= ---

s → ∞

( s2 + 3s + 2) s

( ( ∞ ) 2 + 3( ∞ ) + 2)

∞

Figure 17.31 Final and initial values theorems