Modern Control Engineering

Section 7–8 / Closed-Loop Frequency Response of Unity-Feedback Systems 483

RC+– G

0.25 dB

0.5 dB

1 dB

2 dB

3 dB

4 dB

5 dB

6 dB

9 dB

• 18 dB

–12 dB

• 6 dB
  
  
  • 5 dB
  
  
  • 4 dB
  
  
  
  


• 3 dB


• 2 dB


• 1 dB


• 0.5 dB


• 0.25 dB


• 0.1 dB

0.1 dB

0 dB

12 dB

120

° 150

°

-^180

°

-^150

°

-^120

°

-^90

°

-^60

°

-^30

°

-^20

°

-^10

°

-^5

°

-^2

°

90 °

60 °

30 °

20 °

10 °

5 °

2 °

0 °

• 2 °
  
  
  • 5 °
    
    
    • 10 °
      
      
      • 20 °
      
      
      
      
    
    
    
    
  
  
  
  


• 30 °


• 60 °

36

32

28

24

20

16

12

8

4

0

• 16


• 12
  
  
  • 8
  
  
  • 4
    
    
    • 240 ° – 210 ° – 180 ° – 150 ° – 120 ° – 90 ° – 60 ° – 30 ° 0 °
      GH
    
    
    
    
  
  
  
  

|GH

| in dB

Figure 7–84
Nichols chart.

phase characteristics of the closed-loop transfer function at the same time. The Nichols

chart is shown in Figure 7–84, for phase angles between 0° and –240°.

Note that the critical point (–1+j0point)is mapped to the Nichols chart as the

point(0dB,–180°). The Nichols chart contains curves of constant closed-loop magni-

tude and phase angle. The designer can graphically determine the phase margin, gain

margin, resonant peak magnitude, resonant frequency, and bandwidth of the closed-

loop system from the plot of the open-loop locus,G(jv).

The Nichols chart is symmetric about the –180° axis. The MandNloci repeat for

every 360°, and there is symmetry at every 180° interval. The Mloci are centered about

the critical point (0dB,–180°). The Nichols chart is useful for determining the frequency

response of the closed loop from that of the open loop. If the open-loop frequency-re-

sponse curve is superimposed on the Nichols chart, the intersections of the open-loop

frequency-response curve G(jv)and the MandNloci give the values of the magni-

tudeMand phase angle aof the closed-loop frequency response at each frequency

point. If the G(jv)locus does not intersect the M=Mrlocus, but is tangent to it, then

the resonant peak value of Mof the closed-loop frequency response is given by Mr.The

resonant frequency is given by the frequency at the point of tangency.

As an example, consider the unity-feedback system with the following open-loop

transfer function:

To find the closed-loop frequency response by use of the Nichols chart, the G(jv)locus

is constructed in the log-magnitude-versus-phase plane by use of MATLAB or from

G(jv)=

K

s(s+1)(0.5s+1)

, K= 1