How to Sketch the Root Locus

The root locus involves sketching the passage of a system and how variations change within the system over time. Project goals can be estimated by simulating the final product by graphing the root locus using a conventional x- and y-axis graph with plots. A root locus will show on a graph how a system changes over time. This has many applications, including business and mechanics. Root locus graphs can show the process of a control system, even something as simple as a gatekeeper opening and closing a gate could be graphed with the root locus equation.

Things You'll Need

  • Graph paper
  • Ruler
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Instructions

    • 1

      Draw an x- and y-axis on a piece of graph paper. Across the center of the page from left to right, draw a straight line, following one of the graph paper lines. Draw another line down the center from the top to the bottom of the page. Notate both of these planes as "0" on the x- and y-axis. On the x-axis, write in the negative points on the graph from the middle to the left of the page. The first one should be -5; the second, -10; followed by -15 and so on.

    • 2

      Mark the open-loop poles on the graph with an "X" on the x-axis. The poles are found in the bottom portion of the equation, since we want to zero out the equation, you must use negative numbers to achieve zero as the denominator. For instance, if the bottom part of the equation is (s+4)(s+6)(s+11), you will mark the plots on the x-axis that equal -4, -6 and -11. The "X" spots on the graph are your poles.

    • 3

      Draw a line from the end of the negative side of the x-axis to the first number. In the example from Step 2, you will have a solid line from the edge of the paper to -11. The line theoretically continues infinitely in the negative direction.

    • 4

      Skip over the gap between the last equation and the second equation and draw a line between the first two plots on the graph. For example, draw a line between -6 and -4 on the x-axis. Find the distance between -6 and -4, in this case the distance is 2.

    • 5

      Draw a line from the second pole, in this example, -6, to the number 2 (obtained from Step 4) in the graph. Do this on either side of the x-axis.

    • 6

      Determine the breakaway point by using the equation -G(s)H(s)/ds = 0.

    • 7

      Draw the breakaway point on the graph by using the slope from the answer you received from Step 6.

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