# Chapter7 - CHAPTER 7 7.1 COMPENSATOR DESIGN USING ROOT...

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CHAPTER 7 COMPENSATOR DESIGN USING ROOT LOCUS PLOTS 7.1 (a) – s A = – 2 ; f A = 60º, 180º, 300º ; intersection with j w -axis at ± j 5.042; angle of departure f p from – 1 + j 4 = 40º. (b) – s A = – 1.25 ; f A = 45º, 135º, 225º, 315º ; intersection with j w -axis at ± j 1. 1; angle of departure f p from – 1 + j 1 = – 71.6º; multiple roots at s = – 2.3. (c) Angle of arrival f z at – 3 + j 4 = 77.5º ; multiple roots at s = – 0.45. (d) – s A = 1.33 ; f A = 60º; 180º, 300º; intersection with j w -axis at ± j 5 ; angle of departure f p from 2 + j 1 = – 63.43º; multiple roots at s = – 1, – 1.67. (e) – s A = – 1.5 ; f A = 90º, 270º (f) – s A = – 5.5 ; f A = 90º, 270º; multiple roots at s = – 2.31, – 5.18. (g) – s A = 0.5 ; f A = 90º, 270º; intersection with j w -axis at ± j 2.92. 7.2 (i) – s A = – 2; f A = 60º, 180º, 300º; intersection with j w -axis at ± j 10 ; angle of departure f p from – 3 + j 1 = – 71.5º. Two root loci break away from s = – 1.1835 at ± 90º. Two root loci approach s = – 2.8165 at ± 90º j w j ÷10 s –3+ 1 j –2 (i)

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SOLUTION MANUAL 55 (ii) Intersection with j w -axis at ± j 23 ; angle of departure f p from – 3 + j 3 = – 60º. Three root loci approach the point s = – 2, and then break away in directions 120º apart. j w j w j ÷3 s s –2 (ii) (iii) –9 –4 –3+ 60° j 2 ÷3 (iii) – s A = – 4 ; f A = 90º, 270º The characteristic equation has three roots at s = – 3; the three root loci originating from open-loop poles approach this point and then breakaway. Tangents to the three loci breaking away from s = – 3 are 120º apart. (iv) – s A = – 1 ; f A = 45º, 135º, – 45º, – 135º; intersection with the j w axis at ± j 1.58; angle of departure f p from – 1 + j 2 = – 90º There are two roots at s = – 1, two roots at s = – 1 + j 1.22, and two roots at s = – 1 – j 1.22. The break away directions are shown in the figure. (v) There are four roots at s = – 1. The break away directions are shown in the figure.
56 CONTROL SYSTEMS: PRINCIPLES AND DESIGN j w j w s s –1+ 1 j –1+ 2 j –2 (v) (iv) 45° 45° 7.3 s A = – 1 ; f A = 60º, 180º, 300º; intersection with the j w -axis at ± j 2 ; multiple roots at s = – 0.42. The z = 0.5 loci passes through the origin and makes an angle of q = cos –1 z = 60º with the negative real axis. The point s d = – 0.33 + j 0.58 on this line satisfies the angle criterion. By magnitude criterion, the value of K at this point is found to be 1.04. Using this value of K , the third pole is found at s = – 2.33. Therefore, M ( s ) = 104 033 0 0 233) . ( . .58)( . .58)( . sj s ++ +- + 7.4 s A = – 3 ; f A = 45º, 135º, 225º, 315º; intersection with the j w -axis at ± j 3.25; departure angle f p from – 4 + j 4 = 225º; multiple roots at s = – 1.5. At the intersection of the z = 0.707 line with the root locus, the value of K , by magnitude criterion is 130. The remaining pair of complex roots for K = 130 can be approximately located graphically. It turns out that real part of complex pair away from j w -axis is approximately four times as large as that of the pair near j w -axis. Therefore, the transient response term due to the pair away from the j w -axis will decay much more rapidly than the transient response term due to the pair near j w -axis.

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## This note was uploaded on 04/08/2012 for the course MT 423 taught by Professor M.lee during the Spring '12 term at National Taiwan University.

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Chapter7 - CHAPTER 7 7.1 COMPENSATOR DESIGN USING ROOT...

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