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Lecture 6

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McMaster University**We aren't endorsed by this school
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ENG TECH 4CT3
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Electrical Engineering
Date
Jan 10, 2025
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28
Uploaded by BailiffDangerWalrus49
Lecture 6Root Locus Plots4CT3 Fall 2024 Lec6 Dr. C. Tang1
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Root Loci of Feedback Control SystemsSuppose we can adjust the gain K. As Kincreases from 0 to infinity, what are the trajectories (loci) of the system poles? These trajectories are called Root Loci. How can we sketch them?4CT3 Fall 2024 Lec6 Dr. C. Tang2
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Recap: Transient Response & Pole Location4CT3 Fall 2024 Lec6 Dr. C. Tang3
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Closed Loop/Open Loop Transfer Functions4CT3 Fall 2024 Lec6 Dr. C. Tang4Closed loop transfer functionOpen loop transfer function = KG(s)H(s)=K is the gain of the forward path and varies from 0 to infinity.
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System Poles on Root LocusMagnitude CriterionCharacteristic equation:1 + ? · ?? ??= 0Assume K varies from 0 to ∞.If ‘s’ lies on a root locus, then? · ?? ??= −1| 𝑲 · ?𝒔 ?𝒔| = 𝟏Angle CriterionIf ‘s’ lies on a root locus, thenangle of ? · ?? ??= ±180⁰ orangle of ?𝒔 ?𝒔=±180⁰ ? · ?? ??is called the Open-Loop Transfer Function.4CT3 Fall 2024 Lec6 Dr. C. Tang5
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How to Sketch Root Loci4CT3 Fall 2024 Lec6 Dr. C. Tang6Image source: Engineering Funda
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Root Locus (RL) Sketching Rules4CT3 Fall 2024 Lec6 Dr. C. Tang7Rule 8: Angle of Departure = θ𝐷(or Arrival = θ𝐷)θ𝐷= 1800− [Σθ?𝑖− Σθ𝑧𝑖]θ𝐴= 1800− [Σθ𝑧𝑖− Σθ?𝑖](assuming more poles than zeros)(complex poles exist in pairs)(or infinity)(centroid)
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General Procedure for Plotting Root LocusStep 1: Identify poles and zeros of the open loop transfer function KG(s)H(s) and the number of branches of root lociStep 2: Identify loci on the real axisStep 3: Identify asymptotes (centroid σasym& θasym)Step 4: Identify break-away and/or break-in pointsStep 5: Identify angles of departure and arrivalStep 6: Identify j-axis crossing pointsStep 7: Sketch the root loci4CT3 Fall 2024 Lec6 Dr. C. Tang8
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Example L6-1Sketch the root loci for the following feedback control system:4CT3 Fall 2024 Lec6 Dr. C. Tang9
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Rule 1: Find poles, zeros, and number of branches4CT3 Fall 2024 Lec6 Dr. C. Tang10How many branches of root loci does the system have?𝑂??? ???? 𝑇?𝑎????? ???𝑐?𝑖?? ?? ??=?(? + 2)?2+ 2? + 10
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Rule 3: Root Loci on Real Axis4CT3 Fall 2024 Lec6 Dr. C. Tang11𝑂??? ???? 𝑇?𝑎????? ???𝑐?𝑖?? ?? ??=?(? + 2)?2+ 2? + 10
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Rule 5: Determine Asymptotes4CT3 Fall 2024 Lec6 Dr. C. Tang12asymptote
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Rule 6: Determine Break-in/away Points4CT3 Fall 2024 Lec6 Dr. C. Tang13Break-in/away pointBreak-in point
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Rule 7: Determine intersection points with j-axis4CT3 Fall 2024 Lec6 Dr. C. Tang14No j-axis crossing point
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Rule 8: Determine Angles of Departure4CT3 Fall 2024 Lec6 Dr. C. Tang15θD(Rule 2: symmetry about the real axis )
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Final Sketch: Putting everything together4CT3 Fall 2024 Lec6 Dr. C. Tang16In Matlab: rlocus(OpenLoopTransferFunction)
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Example L6-1Sketch the root loci of a feedback control system with the following open loop transfer function:?? ??=??(? + 5)(? + 10)4CT3 Win 2024 Lec6 Dr. C. Tang17
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Rule 1: No. of BranchesNo. of branches = No. of polesThere are 3 poles and no zeros:pole 1 = 0pole 2 = -5pole 3 = -10Here there are 3 root locus branches (blue, green, and red)4CT3 Win 2024 Lec6 Dr. C. Tang18
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Rule 2: Symmetry about real axisSince complex poles must exist in complex conjugates, root locus branches are symmetrical about the real (or σ) axis.4CT3 Win 2024 Lec6 Dr. C. Tang19
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Rule 3: Real-axis segmentsReal-axis segments are to the left of an odd number of real-axis finite poles and zeros.The green and blue segments are to the left of 1 poleThe red segment is to the left of 3 poles.4CT3 Win 2024 Lec6 Dr. C. Tang20
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Rule 4: Start point and end pointEach root locus branch starts at a pole and ends at a zero or infinity.4CT3 Win 2024 Lec6 Dr. C. Tang21
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Rule 5: AsymptotesNumber of asymptotes NA: NA= Np –NzCentroid of asymptotes σA:σA=σA= (0 -5 -10)/(3 –0) = -54CT3 Win 2024 Lec6 Dr. C. Tang22
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Rule 5: Asymptotes –cont’dAngles of asymptotes θA: θA=m = 0, 1, 2, … (N –1)θ1= 180⁰/3 = 60⁰θ2= 3x180⁰/3 = 180⁰θ3= 5x180⁰/3 = 300⁰4CT3 Win 2024 Lec6 Dr. C. Tang23θ1= 180⁰/3 = 60⁰
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Rule 6: Breakaway/Break-in PointsBreakaway or break-in points are where the root locus branches leave or enter the real axis.There is one breakaway point in this example.Isolate ‘K’ in the characteristic equation.4CT3 Win 2024 Lec6 Dr. C. Tang24
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Rule 6: Cont’dCharacteristic equation:1 + GH = 1 + =0s(s+5)(s+10)+ K =0k = -s(s+5)(s+10)k = -s(s2+15s+50)k = -(s3+15s2+50s)dk/ds = -(3s2+30s+50)=0s = -7.89 or -2.11The breakaway point is at σ=-2.11 and cannot be at -7.894CT3 Win 2024 Lec6 Dr. C. Tang25
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Rule 7: Imaginary Axis CrossingsThe easiest way is to use Routh table to determine the imaginary axis crossing points.4CT3 Win 2024 Lec6 Dr. C. Tang26
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Rule 7: Cont’ds31 50 0s215 K 0s1 50-K/15 0s0 KIf the s1row is all zeros, then K = 750Auxiliary eqn: 15s2+750=0s2+50=0s = +7.07 or -7.074CT3 Win 2024 Lec6 Dr. C. Tang27
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Rule 8: Angles of Departure/ArrivalAngles of departure or arrival apply to complex poles or zeros.There are none of these in this example.4CT3 Win 2024 Lec6 Dr. C. Tang28
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