Page 128 - Circuit Analysis II with MATLAB Applications

Chapter 4 The Laplace Transformation

­ d ½ f d – st
L ----- ft ¾ = sF s – f0 = ³ ----- ft e dt
®
¯ dt ¿ 0 dt
Taking the limit of both sides by letting s o f , we get

T d
lim > sF s f0 – @ = lim lim ³ ----- ft e – st dt
s o f s o f T o f H dt
H o 0

Interchanging the limiting process, we get

T d
lim > sF s f0 @ = lim ³ ----- ft lim e – st dt
–
s o f T o f H dt s o f
H o 0
and since
lim e – st = 0
s o f
the above expression reduces to
lim > sF s f0 – @ = 0
s o f
or
lim sF s = f 0
s o f
11. Final Value Theorem

The final value theorem states that the final value f f of the time function f t can be found from
its Laplace transform multiplied by s, then, letting s o 0 . That is,

lim ft = lim sF s = f f (4.33)
t o f s o 0
Proof:

From the time domain differentiation property,

d
----- ft œ sF s – f0
dt
or

­ d ½ f d – st
L ----- ft ¾ dt = sF s – f0 = ³ ----- ft e dt
®
dt
¯ ¿ 0
Taking the limit of both sides by letting s o 0 , we get

4-10 Circuit Analysis II with MATLAB Applications

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