Page 93 - Modern Control Systems

Section 2.5 The Transfer Function of Linear Systems 67
where Yi(s) is the partial fraction expansion of the natural response, Y 2(s) is the par-
tial fraction expansion of the terms involving factors of q(s), and Y 3(s) is the partial
fraction expansion of terms involving factors of d(s).
Taking the inverse Laplace transform yields
y(t) = yi(t) + yz(t) + y 3(0-
The transient response consists of y\(t) + ^(O, and the steady-state response is ^(f).

EXAMPLE 2.2 Solution of a differential equation
Consider a system represented by the differential equation
2
d y dy
-g H- 4-£ + 3y = 2K0,
dt
dy
where the initial conditions are y(0) = 1,-(0) = 0, and r(t) = 1, t ^ 0.
dt
The Laplace transform yields
2
[s Y(s) - sy(0)] + 4[sY(s) - y(0)] + 3Y(s) = 2R(s).
Since R(s) = \/s and y(0) = 1, we obtain
s + 4
Y(s) 2 + 2
s + 4s + 3 s(s + 4s + 3)'
2
where q(s) = s + 4s + 3 = (s + l)(s + 3) = Qis the characteristic equation, and
d(s) = s. Then the partial fraction expansion yields

3/2 -1/2 -1 1/3 2/3
Y(s) = + + + -^-= Yt(s) + Y 2(s) + Y 3(s).
s + 1 s + 3 s + 1 s + 3

Hence, the response is
2
l 3t 3
y(t) = -e~ - -e~ + -le~' + ie" '
_2 2
3
and the steady-state response is
>onse is
2
lim y\ 0 = =
t—x» 3 "

EXAMPLE 2.3 Transfer function of an op-amp circuit
The operational amplifier (op-amp) belongs to an important class of analog inte-
grated circuits commonly used as building blocks in the implementation of control
systems and in many other important applications. Op-amps are active elements
(that is, they have external power sources) with a high gain when operating in their
linear regions. A model of an ideal op-amp is shown in Figure 2.14.