Signals and Systems - Electrical Engineering

3.5 Analysis of LTI Systems 217

In fact, for any real poles=−α,α >0, of multiplicitym≥1, we have that

L−^1

[

N(s)

(s+α)m

]

=

∑m

k= 1

Aktk−^1 e−αtu(t)

whereN(s)is a polynomial of degree less or equal tom−1. Clearly, for any value ofα >0 and any
orderm≥1, the above inverse will tend to zero astincreases. The rate at which these terms go to zero
depends on how close the pole(s) is (are) to thejaxis: The farther away, the faster the term goes to
zero. Likewise, complex conjugate pairs of poles with a negative real part also give terms that go to
zero ast→∞, independent of their order. For complex conjugate pairs of poless1,2=−α±j 0 of
orderm≥1, we have

L−^1

[

N(s)

((s+α)^2 +^20 )m

]

=

∑m

k= 1

2 |Ak|tk−^1 e−αtcos( 0 t+∠(Ak))u(t)

where againN(s)is a polynomial of degree less or equal to 2m−1. Due to the decaying exponentials
this type of term will go to zero astgoes to infinity.

Simple complex conjugate poles and a simple real pole at the origin of thes-plane cause a steady-state
response. Indeed, if the pole ofY(s)iss=0 we know that its inverse transform is of the formAu(t),
and if the poles are complex conjugates±j 0 the corresponding inverse transform is a sinusoid—
neither of which is transient.However, multiple poles on the j-axis, or any poles in the right-hands-plane
will give inverses that grow as t→∞. This statement is clear for the poles in the right-hands-plane. For
double- or higher-order poles in thejaxis their inverse transform is of the form

L−^1

[

N(s)

(s^2 +^20 )m

]

=

∑m

k= 1

2 |Ak|tm−^1 cos( 0 +∠(Ak))u(t)

which will continuously grow astincreases.

In summary, when solving differential equations—with or without initial conditions—we have

n The steady-state component of the complete solution is given by the inverse Laplace transforms of the

partial fraction expansion terms ofY(s)that have simple poles (real or complex conjugate pairs) in the

j-axis.

n The transient response is given by the inverse transform of the partial fraction expansion terms with poles

in the left-hands-plane, independent of whether the poles are simple or multiple, real or complex.

n Multiple poles in thejaxis and poles in the right-hands-plane give terms that will increase astincreases.

nExample 3.22

Consider a second-order(N= 2 )differential equation,

d^2 y(t)

dt^2

+ 3

dy(t)

dt

+ 2 y(t)=x(t)