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Understanding the Root Locus Method: Breakaway Points and Angle of Departure in Control Systems

This section explores the Root Locus Method in control systems, focusing on determining breakaway points on the real axis and calculating the angles of departure and arrival. It explains that breakaway points indicate where the locus leaves the real axis, occurring at points of multiplicity of roots. The example illustrates how to graphically find the breakaway point between poles at -2 and -4, leading to the conclusion that the locus breaks away at s = -3. Additionally, it covers the analytical methods for calculating angles related to poles and zeros with visual aids.

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Understanding the Root Locus Method: Breakaway Points and Angle of Departure in Control Systems

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  1. Modern Control SystemEKT 308 Root Locus Method (contd…)

  2. Root Locus (contd…) Step 5: Determine the break away point on the real axis (if any). The locus breakaway from the real axis occurs where there is a multiplicity of roots. At the breakaway point the angle of the tangent to the locus does not change for small change in s (fig 1). Fig 1: Breakaway point

  3. Procedure for finding breakaway point Graphical method.

  4. Poles: -2, -4 Zeros: none. Breakaway point is expected between -4 and -2 Fig 2: Poles (no zeros) Let us plot p(s) between s=-4 to s=-2. Fig 3: p(s) versus s

  5. Find the maximum point in the ps-s curve. It occurs at s = -3. So the locus breaks away at s=-3 as shown in figure 3. Fig 3: Breakaway at s=-3. b) Analytical method

  6. Step 6: Determine the angle of departure from pole and of arrival at zero The angle of locus departure from a pole = Difference between net angle due to all other poles and zeros, and the criterion angle Similar formula for calculating angle of arrival at zero. Example. Poles are zeros are shown in figure 4.

  7. In order to find angle of departure at, say complex pole -p1, place a test s1 at infinitesimal distance from -p1. From angle criterion, we get Step 7: Complete the sketch Complete all the sections of the locus not covered in the previous six steps.

  8. Figure 4: Angle of departure

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