Page 123 - Circuit Analysis II with MATLAB Applications
P. 123
Properties of the Laplace Transform
Using integration by parts where
d
d
³ vu = uv – ³ u v (4.17)
we let du = f ' t and v = e – st . Then, u = f t , dv = – se – st , and thus
+
s
^
Lf ' t ` = ft e – st f ³ f f t e – st dt = lim ft e – st a + sF s
0 0 a o f 0
= lim e > – sa fa f0 – @ + sF s = 0 f 0 – + sF s
a o f
The time differentiation property can be extended to show that
d 2 2
-------- ft s Fs – sf 0 f ' 0 – (4.18)
dt 2
d 3
2
3
-------- ft s Fs – s f0 sf ' 0 – f '' 0 – (4.19)
dt 3
and in general
d n n n – 1 n2 n1
–
–
-------- ft s Fs – s f0 – s f ' 0 – } – f 0 (4.20)
dt n
To prove (4.18), we let
d
gt = f ' t = ----- ft
dt
and as we found above,
^ =
L g ' t ` sL g t – g0
`
^
Then,
^ =
L f '' t ` sL f ' t – f ' 0 = ssL f t @ f0 – @ f ' 0 –
>
^
>
`
2
= s Fs – sf 0 f ' 0 –
Relations (4.19) and (4.20) can be proved by similar procedures.
We must remember that the terms f 0 f ' 0 f '' 0 , and so on, represent the initial conditions.
times,
Therefore, when all initial conditions are zero, and we differentiate a time function f t n
s
this corresponds to Fs multiplied by to the nth power.
Circuit Analysis II with MATLAB Applications 4-5
Orchard Publications
