Modern Control Engineering

416 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response Method

f

0 °

• 90 °


• 180 °

G 1 (jv)

G 2 (jv)

v

Figure 7–13

Phase-angle

characteristics of the

systemsG 1 (s)and

G 2 (s)shown in

Figure 7–12.

The pole–zero configurations of these systems are shown in Figure 7–12. The two sinu-

soidal transfer functions have the same magnitude characteristics, but they have differ-

ent phase-angle characteristics, as shown in Figure 7–13. These two systems differ from

each other by the factor

The magnitude of the factor (1-jvT)/(1+jvT)is always unity. But the phase

angle equals –2tan–1vTand varies from 0° to –180° as vis increased from zero to infinity.

As stated earlier, for a minimum-phase system, the magnitude and phase-angle char-

acteristics are uniquely related. This means that if the magnitude curve of a system is

specified over the entire frequency range from zero to infinity, then the phase-angle

curve is uniquely determined, and vice versa. This, however, does not hold for a non-

minimum-phase system.

Nonminimum-phase situations may arise in two different ways. One is simply when

a system includes a nonminimum-phase element or elements. The other situation may

arise in the case where a minor loop is unstable.

For a minimum-phase system, the phase angle at v=qbecomes–90°(q-p),

wherepandqare the degrees of the numerator and denominator polynomials of the

transfer function, respectively. For a nonminimum-phase system, the phase angle at

v=qdiffers from –90°(q-p). In either system, the slope of the log-magnitude curve

atv=qis equal to –20(q-p)dBdecade. It is therefore possible to detect whether

the system is minimum phase by examining both the slope of the high-frequency

asymptote of the log-magnitude curve and the phase angle at v=q. If the slope of the

log-magnitude curve as vapproaches infinity is –20(q-p)dBdecade and the phase

angle at v=qis equal to –90°(q-p), then the system is minimum phase.

G(jv)=

1 - jvT

1 +jvT

jv

1

• T
  1


• T 1
  1


• T 1
  s

G 1 (s)= 11 ++TTs

1 s

jv

1

T

s

G 2 (s)= 11 +–TTs

1 s

0 0

Figure 7–12

Pole–zero

configurations of a

minimum-phase

systemG 1 (s)and

nonminimum-phase

systemG 2 (s).

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