Modern Control Engineering

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

response, the damped natural frequency in the transient response is somewhere

between the gain crossover frequency and phase crossover frequency.

3.The resonant peak frequency vrand the damped natural frequency vdfor the step

transient response are very close to each other for lightly damped systems.

The three relationships just listed are useful for correlating the step transient re-

sponse with the frequency response of higher-order systems, provided that they can be

approximated by the standard second-order system or a pair of complex-conjugate

closed-loop poles. If a higher-order system satisfies this condition, a set of time-domain

specifications may be translated into frequency-domain specifications. This simplifies

greatly the design work or compensation work of higher-order systems.

In addition to the phase margin, gain margin, resonant peak Mr, and resonant fre-

quency vr, there are other frequency-domain quantities commonly used in performance

specifications. They are the cutoff frequency, bandwidth, and the cutoff rate. These will

be defined in what follows.

Cutoff Frequency and Bandwidth. Referring to Figure 7–76, the frequency vbat

which the magnitude of the closed-loop frequency response is 3 dB below its zero-fre-

quency value is called the cutoff frequency. Thus

For systems in which

The closed-loop system filters out the signal components whose frequencies are greater

than the cutoff frequency and transmits those signal components with frequencies lower

than the cutoff frequency.

The frequency range 0vvbin which the magnitude of is greater

than–3dB is called the bandwidthof the system. The bandwidth indicates the frequency

where the gain starts to fall off from its low-frequency value. Thus, the bandwidth indicates

how well the system will track an input sinusoid. Note that for a given vn, the rise time in-

creases with increasing damping ratio z. On the other hand, the bandwidth decreases with

the increase in z. Therefore, the rise time and the bandwidth are inversely proportional to

each other.

C(jv)R(jv)

2

C(jv)

R(jv)

(^2) 6- 3 dB, for v 7 vb

@C(j0)R(j0)@ = 0 dB,

2

C(jv)

R(jv)

2 6 2

C(j0)

R(j0)

(^2) - 3 dB, forv 7 vb
dB
0

• 3

Bandwidth

vb

v in log scale

Figure 7–76

Plot of a closed-loop

frequency response

curve showing cutoff

frequency vband

bandwidth.

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