Chapter 10 - Wiley · 2 Chapter 10: Frequency Response Techniques Figure 10.1 The HP 35670A Dynamic Signal Analyzer obtains frequency response data from a physical system. The displayed

©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e

Chapter 10: Frequency Response Techniques1

Chapter 10

Frequency Response Techniques



Figure 10.1The HP 35670A Dynamic Signal Analyzer obtainsfrequency responsedata from a physicalsystem. Thedisplayed data can be used to analyze, design, or determine a mathematical modelfor the system.

Courtesy of Hewlett-Packard.



Figure 10.2Sinusoidal frequencyresponse:a. system;b. transfer function;c. input and outputwaveforms



Figure 10.3System withsinusoidal input



Figure 10.4Frequency responseplots for G(s) =1/(s + 2):separate magnitudeand phase



Figure 10.5Frequency responseplots for G(s)= 1/(s + 2): polar plot



Figure 10.6Bode plots of(s + a):a. magnitude plot;b. phase plot.



Table 10.1Asymptotic and actual normalized and scaledfrequency response data for (s + a)



Figure 10.7Asymptotic and actual normalized and scaled magnitude response of(s + a)



Figure 10.8Asymptotic and actual normalized and scaled phase response of (s + a)



Figure 10.9Normalized and scaledBode plots fora. G(s) = s;b. G(s) = 1/s;c. G(s) = (s + a);d. G(s) = 1/(s + a)



Figure 10.10Closed-loop unityfeedback system



Figure 10.11Bode log-magnitudeplot for Example 10.2:a. components;b. composite



Table 10.2Bode magnitude plot: slope contribution from each pole and zero in Example 10.2



Table 10.3Bode phase plot: slope contribution from each pole and zero in Example 10.2



Figure 10.12Bode phaseplot for Example 10.2:a. components;b. composite



ζω22 +s

Figure 10.13Bode asymptotesfor normalizedand scaled G(s) =

a. magnitude;b. phase



Table 10.4Data for normalized and scaled log-magnitude and phase plots for(s2 + 2ζωns + ωn

2).Mag = 20 log(M/ωn

2) (table continues)

Freq.ω/ωn

Mag(dB)

ζ = 0.1

Phase(deg)

ζ = 0.1

Mag(dB)

ζ = 0.2

Phase(deg)

ζ = 0.2

Mag(dB)

ζ = 0.3

Phase(deg)

ζ = 0.3



Table 10.4 (continued)

Freq.ω/ωn

Mag(dB)

ζ = 0.5

Phase(deg)

ζ = 0.5

Mag(dB)

ζ = 0.7

Phase(deg)

ζ = 0.7

Mag(dB)ζ = 1

Phase(deg)ζ = 1



Figure 10.14Normalized and scaledlog-magnitude response for



Figure 10.15Scaled phase response for



Table 10.5Data for normalized and scaled log-magnitude and phase plots for1/(s2 + 2ζωns + ωn

2).Mag = 20 log(M/ωn

2) (table continues)

Freq.ω/ωn

Mag(dB)

ζ = 0.1

Phase(deg)

ζ = 0.1

Mag(dB)

ζ = 0.2

Phase(deg)

ζ = 0.2

Mag(dB)

ζ = 0.3

Phase(deg)

ζ = 0.3



Table 10.5 (continued)

Freq.ω/ωn

Mag(dB)

ζ = 0.5

Phase(deg)

ζ = 0.5

Mag(dB)

ζ = 0.7

Phase(deg)

ζ = 0.7

Mag(dB)ζ = 1

Phase(deg)ζ = 1



Figure 10.16Normalized and scaled log magnitude response for



Figure 10.17Scaled phase response for



Figure 10.18Bode magnitudeplot for G(s) =(s + 3)/[(s + 2)(s2 + 2s + 25)]:a. components;b. composite



Table 10.6Magnitude diagram slopes for Example 10.3



Table 10.7Phase diagram slopes for Example 10.3



Figure 10.19Bode phase plot forG(s) = (s + 3)/[(s +2)(s2 + 2s + 25)]:a. components;b. composite



Figure 10.20Closed-loop controlsystem



Figure 10.21Mapping contour Athrough function F(s)to contour B



Figure 10.22Examples of contourmapping



Figure 10.23Vector representationof mapping



Figure 10.24Contour enclosingright half-planeto determine stability



Figure 10.25Mapping examples:a. contour doesnot enclose closed-loop poles;b. contour doesenclose closed-loop poles



Figure 10.26a. Turbine andgenerator;b. block diagram of speed control system forExample 10.4



Figure 10.27Vector evaluation ofthe Nyquist diagramfor Example 10.4:a. vectors on contour at low frequency;b. vectors on contour around infinity;c. Nyquist diagram



Figure 10.28Detouring around open-loop poles:a. poles on contour;b. detour right;c. detour left



Figure 10.29a. Contour for Example 10.5;b. Nyquist diagram for Example 10.5



Figure 10.30DemonstratingNyquist stability:a. system;b. contour;c. Nyquist diagram



Figure 10.31a. Contour for Example 10.6;b. Nyquist diagram



Figure 10.32a. Contour and root locus of system thatis stable for small gain and unstable for large gain;b. Nyquist diagram



Figure 10.33a. Contour and root locus of system that is unstable for small gain and stable for large gain;b. Nyquist diagram



Figure 10.34a. Portion of contour to be mapped for Example 10.7;b. Nyquist diagram of mapping of positive imaginary axis



Figure 10.35Nyquistdiagramshowing gainand phasemargins



Figure 10.36Bodelog-magnitudeand phase diagramsfor the system of Example 10.9



Figure 10.37Gain and phasemargins on the Bodediagrams



Figure 10.38Second-order closed-loopsystem



Figure 10.39Representative log-magnitude plot ofEq. (10.51)



Figure 10.40Closed-loop frequency percentovershoot for a two-pole system



Figure 10.41Normalized bandwidth vs. damping ratiofor:a. settling time;b. peak time;c. rise time



Figure 10.42Constant M circles



Figure 10.43Constant N circles



Figure 10.44Nyquist diagramfor Example 10.11 and constant M and N circles



Figure 10.45Closed-loopfrequency responsefor Example 10.11



Figure 10.46Nichols chart



Figure 10.47Nichols chart with frequency response forG(s) = K/[s(s + 1)(s + 2)] superimposed. Values for K = 1 and K = 3.16 are shown.



Figure 10.48Phase margin vs.damping ratio



Figure 10.49Open-loop gain vs. open-loop phase angle for –3 dB closed-loop gain



Figure 10.50a. Block diagram(figure continues)

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Chapter 10 - Wiley · 2 Chapter 10: Frequency Response Techniques Figure 10.1 The HP 35670A Dynamic Signal Analyzer obtains frequency response data from a physical system. The displayed