Electric Circuit Analysis

Some Theorems on Laplace Transforms    13.21

Figure 13.7-2 shows the two components of

response in dotted curves.

13.7.2 Frequency-Shifting Theorem

If  v (t)  = f (t)  u (t)  has  a  Laplace  transform 

V (s),  then,  v

d 

(t)  =

v t e

s t

o

( )

  has  a  Laplace 

transform 

V s V s s

d o

( ) ( ).= −

This theorem follows from the defining equation

for Laplace transforms.

V s

v t e dt v t e e dt v t e

d d

st

s t

st

s s t

o o

( ) ( ) ( ) ( )

( )

= = ==

−

∞

−

∞

− −

∞

− − −

∫ ∫

0 0 0

∫∫

= −dt V s s

o

( )

13.7.3 Time-Differentiation Theorem

If v (t) = f (t) u (t) has a Laplace transform V (s), then, v

d 

(t) =

dv t

dt

( )

 has a Laplace transform 

V s sV s v

d

( ) ( ) ( ).= −

−

0  Note that  ...