Modern Control Engineering

Section 7–1 / Introduction 401

X Y

t

Inputx(t)=Xsinvt

Outputy(t)=Ysin (vt+f)

Figure 7–2
Input and output
sinusoidal signals.

x y

G(s)

K

Figure 7–3 Ts+^1

First-order system.

whereY=X@G(jv)@. We see that a stable, linear, time-invariant system subjected to a

sinusoidal input will, at steady state, have a sinusoidal output of the same frequency as

the input. But the amplitude and phase of the output will, in general, be different from

those of the input. In fact, the amplitude of the output is given by the product of that of

the input and @G(jv)@, while the phase angle differs from that of the input by the amount

An example of input and output sinusoidal signals is shown in Figure 7–2.

On the basis of this, we obtain this important result: For sinusoidal inputs,

Hence, the steady-state response characteristics of a system to a sinusoidal input can be

obtained directly from

The function G(jv)is called the sinusoidal transfer function. It is the ratio of Y(jv)

toX(jv),is a complex quantity, and can be represented by the magnitude and phase

angle with frequency as a parameter. The sinusoidal transfer function of any linear system

is obtained by substituting jvforsin the transfer function of the system.

As already mentioned in Chapter 6, a positive phase angle is called phase lead, and a neg-

ative phase angle is called phase lag. A network that has phase-lead characteristics is called

a lead network, while a network that has phase-lag characteristics is called a lag network.

EXAMPLE 7–1 Consider the system shown in Figure 7–3. The transfer function G(s)is

For the sinusoidal input x(t)=Xsinvt, the steady-state output yss(t)can be found as follows:

SubstitutingjvforsinG(s)yields

G(jv)=

K

jTv+ 1

G(s)=

K

Ts+ 1

Y(jv)

X(jv)

=G(jv)

/G(jv)=n

Y(jv)

X(jv)

=

phase shift of the output sinusoid with respect

to the input sinusoid

@G(jv)@ =^2

Y(jv)

X(jv)

(^2) =amplitude ratio of the output sinuisoid to the

input sinusoid

f=/G(jv).