Lecture 12: Examples of Root Locus Plots. Dr. Kalyana Veluvolu. Lecture 12: Examples of Root Locus Plots Dr. Kalyana Veluvolu

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1 ROOT-LOCUS ANALYSIS Example: Given that G( ) ( + )( + ) Dr. alyana Veluvolu Sketch the root locu of 1 + G() and compute the value of that will yield a dominant econd order behavior with a damping ratio, ζ.7. We have n 3 and m. Open loop zero: none Open loop pole:,, Rule 1: The loci tart from at the OL pole Rule : The loci end at at 3 infinite zero at Rule 3: The number of loci i 3, a n 3 Rule : Root loci are ymmetrical with repect to real axi 1 Dr. alyana Veluvolu Rule 5: There are (3 ) 3 aymptote. The angle of aymptote are given by: ( j + 1) π θ j ; j,1, 3 π 3π 5π,, i.e. θ j, 18, 3 ( ) Rule :The point of interection of the aymptote i OL_pole OL_zero c n m ( ) ( ). 3 Rule 7: Identify branche of loci on the real axi. Rule 9: Croing the imaginary axi. 8 The C.E. i aymptote Dr. alyana Veluvolu o 3 1 branche of loci on real axi Y ( ) 3 R( ) When 8 in the 1 row we get an all zero row. The Auxiliary Equation i + 8 ± j Rule 1: The breakaway point i the olution of ( 8 ) d d + + (3 1 8) Dr. alyana Veluvolu d d Solving,.85 i the valid breakaway point for >.

2 The Root-Locu i plotted a hown: 3 b a plane j 1 Dr. alyana Veluvolu Thi pole and it conjugate yield ζ.7 C B 3 b a -.85 A 1 θ j -plane Dr. alyana Veluvolu Stable 8 Untable 1 o With ζ.7, we have θ co ζ 5.. One of the pole that will give the required ζ i given by the interection of the root locu with the traight line that ha angle θ w.r.t. the negative real axi. 5 If the root-locu i ketched to cale, the value of that will yield the required ζ can be computed from the meaured value of A, B and C, i.e. 1 A B C Dr. alyana Veluvolu Example: Contruct the Root-Locu for a ytem with open-loop tranfer function ( + 3) G( ) H ( ) ( + 5)( + )( + + ) That i, n 5 and m 1. Open loop zero: 3 Open loop pole:, 5,, 1± Rule 1:The loci tart from at the OL pole Rule :The loci end at at the OL zero and infinite zero at Rule 3:The number of loci i 5, a n 5 Rule :Root loci are ymmetrical with repect to real axi 7 Dr. alyana Veluvolu Rule 5:There are (5 1) aymptote. The angle of aymptote are given by: ( j + 1) π θ j ; j,1,,3. π 3π 5π 7π,,, i.e. θ j 5, 135, 5 ( 135 ), 315 (5 ) Rule :The point of interection of the aymptote i OL_pole OL_zero c n m ( ) ( 3)

3 Rule 7: Identify branche of loci on the real axi. Dr. alyana Veluvolu p 1 Dr. alyana Veluvolu By conjugate ymmetry, angle of departure from the complex pole at ( 1 ) i θ d 3.7 p 3 p 5 p z p Rule 8:The angle of departure from the complex pole at ( 1 + ) i given by: θ d 18 + tan tan ( 1) + tan + tan + 9 { [ ] } ] 9 Rule 9:The point of interection of root locu with the imaginary axi uing the Routh-Hurwitz Criterion. The Characteritic Equation i ( + 5)( + )( + + ) + ( + 3) ( + ) + 3 Dr. alyana Veluvolu The Auxiliary Equation i Dr. alyana Veluvolu ( + ) ( +.77) (5..1) 3 1 For tability, > < > < 35 3 > > [ 5..1(35) ] + 3(35) > ±.33 The root loci interect the imaginary axi at ±.33 Rule 1: The breakaway point i the olution of 5 3 ( ) d d ( + 3) d ( ) ( ) + d ( + 3) ( + 3) i.e. < < 35 Solving, -5.5 i the breakaway point. Subtitute 35 in the 1 row give an all zero row. 11 1

4 Alternatively, plot v Dr. alyana Veluvolu Dr. alyana Veluvolu From the information obtained, the root-locu i ketched a hown: At 5.5, the breakaway point, i maximum max not a Multiplying Factor k Dr. alyana Veluvolu If i not a multiplying factor, ome modification of the Characteritic Equation i required for contructing the root loci. Example Conider the ytem The Characteritic Equation for the ytem i 1(1 + k) 1+ Sketch the root locu w.r.t. k. From the root locu, compute the value of k that will give a damping ratio ζ k Rewrite it a Define 1k, then the above equation become Dr. alyana Veluvolu Since k i not a multiplying factor, we modify the C.E. uch that k appear a a multiplying factor. The C.E. i k Open loop pole: p 1, ± j. Open loop zero: z 1 1

5 The root locu can be eaily ketched a given below With ζ.77, we have θ co ζ 5 1 o Dr. alyana Veluvolu From the root-locu, the value of that will yield the required ζ can be computed from the meaured value of A, B and C, i.e. B C A But 1k, hence k Α 5 j j3 j C j Β j3 j 17 Information Obtainable from the Root-Locu Dr. alyana Veluvolu (1) The tability condition for any value of, or other ytem parameter againt which the root-locu i plotted. () The limit of for which the ytem i table. (3) The effect of variation of on the performance potential of the ytem. () The actual characteritic equation for any value of. (5) Evaluation of performance criteria uch a undamped and damped natural frequencie, damping ratio and the tranient repone exponential term. 18