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## Rule 11: Departure (arrival) angle

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****’ ’ Total angle contributed by “other” singularities ’ Rule 11:Departure (arrival) angle s0 p1 s0 is on the RL if - + + =(2k+1) -’ +’ +’ Angle of departure from p1 6.5 Root-locus: Other Rules & Examples**Rule 11 (Cont’d)**For the CRL: For the RL: Angle of departure from a pole =(2k+1)π+total angle contributed by “other” singularities Angle of arrival at a zero =(2k+1)π-total angle contributed by “other” singularities 6.5 Root-locus: Other Rules & Examples**Example**n=4; m=1 # of branches =max(4,1)=4 Starting at 4 poles 1 end at the zero 3 end at infinity 6.5 Root-locus: Other Rules & Examples**Example (cont’d)**3 asymptotes at angles: (2k+1)π/3=180o,60o intersecting at: {-3-2-0-0+(-1)}/(4-1)=-2 Real axis loci: Between -2 & 1 & to the left of -3 6.5 Root-locus: Other Rules & Examples**Example (cont’d)**jω-axis crossing: R-H test yields Single-pole crossing for K=6 Two-pole crossing for K=0 (start) & K=50 at s=j11= j 3.31 6.5 Root-locus: Other Rules & Examples**Departure angle from +j1:**(2k+1)180o +135 -90 -tan-1(1/2) -tan-1(1/3) =180o 6.5 Root-locus: Other Rules & Examples**yields**BA points? 3s4+6s3-6s2-14s-11=0 Roots are: 1.6677 -2.2888 -0.6894j0.6966 None on the RL 6.5 Root-locus: Other Rules & Examples**COMBINE!**6.5 Root-locus: Other Rules & Examples**Check by MATLAB**rlocus([1 -1],[1 5 7 5 6]),grid,figure(1) 6.5 Root-locus: Other Rules & Examples**Gain for a root to be at s0**Rule 12:Calibration 6.5 Root-locus: Other Rules & Examples**z1**z0 |z0-z1| Note: 6.5 Root-locus: Other Rules & Examples**Apply to example by MATLAB**rlocfind([1 -1],[1 5 7 5 6]), grid,figure(1) 6.5 Root-locus: Other Rules & Examples**Examples**6.5 Root-locus: Other Rules & Examples**Introduce a zero in LHP**rlocus([1 1],[1 0 0]) 6.5 Root-locus: Other Rules & Examples**Introduce a 2nd pole in LHP**6.5 Root-locus: Other Rules & Examples**Introduce a zero to 2nd order system**6.5 Root-locus: Other Rules & Examples**Introduce a pole**6.5 Root-locus: Other Rules & Examples**COMPARE**6.5 Root-locus: Other Rules & Examples**End of Chapter**Restart section Next chapter Restart chapter i The End General index End show 6.5 Root-locus: Other Rules & Examples