Signals and Systems - Electrical Engineering

190 C H A P T E R 3: The Laplace Transform

Remarks

n The derivative property for a signal x(t)defined for all t is

∫∞

−∞

dx(t)

dt

e−stdt=sX(s)

This can be seen by computing the derivative of the inverse Laplace transform with respect to t, assuming

that the integral and the derivative can be interchanged. Using Equation (3.3):

dx(t)

dt

=

1

2 πj

σ∫+j∞

σ−j∞

X(s)

dest

dt

ds

=

1

2 πj

σ∫+j∞

σ−j∞

(sX(s))estds

or that sX(s)is the Laplace transform of the derivative of x(t). Thus, the two-sided transform does not

include initial conditions. The above result can be generalized to any order of the derivative as

L[dNx(t)/dtN]=sNX(s)

n Application of the linearity and the derivative properties of the Laplace transform makes solving differential

equations an algebraic problem.

nExample 3.8

Find the impulse response of an RL circuit in series with a voltage sourcevs(t)(see Figure 3.11).

The currenti(t)is the output and the input is the voltage sourcevs(t).

Solution

To find the impulse response of the RL circuit we letvs(t)=δ(t)and set the initial current in the

inductor to zero. According to Kirchhoff’s voltage law,

vs(t)=L

di(t)

dt

+Ri(t) i( 0 −)= 0

FIGURE 3.11

Impulse responsei(t)of an RL circuit with input

vs(t).

+

−

vs(t)

i(t)

i(t)

R

L

t