Drawing the root locus

I have recently (summer 2020) developed this page to help student learn how to sketch the root locus by hand. You can enter a numerator and denominator for G(s)H(s) (i.e., the loop gain) and the program will guide you through the steps to develop a sketch of the root locus by hand.

It is not the intention of this page to be used as a generic design tool, it is simply a teaching tool. As such, it is only designed to work with poles and zeros that are relatively small numbers (say between 0 and 20).

Please let me know if you find any problems with this page (contact links are at the bottom). Here is link to known problems with page.





We start with the loop gain transfer function:

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The characteristic equation (i.e., the denominator of the closed loop transfer function) is \[ 1+K \cdot G(s) \cdot H(s) = 0 \textrm{, or } 1+K \cdot \frac{N(s)}{D(s)} = 0,\] which we can rewrite as: Replace.

If we plot the roots of this equation as K varies, we obtain the root locus. A program (like MATLAB) can do this easily, but to make a sketch, by hand, of the location of the roots as K varies we need some information:
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Select rule to be explained

(The explanation of the rule applied to this loop gain is below the graph.)















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Only the root locus is shown

As you can see the locus is symmetric about the real (horizontal) axis. You can select one of the root locus rules (above) to see how it is applied for the given loop gain.

Starting and Stopping Points

Link to in depth description of rule for finding starting and stopping points (and for finding closed loop pole locations given K).

If you set K=0 (below), the starting points are displayed (i.e., the poles of the closed loop transfer function when K=0) as pink diamonds. As you increase K the closed loop poles (i.e., pink diamonds) move towards the stopping points as K→∞.

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Locus on the real axis

Link to in depth description of rule for finding where the locus exists on the real axis.

The root locus exists on the real axis to the left of an odd number of poles and zeros of the loop gain, G(s)H(s), that are on the real axis. The real pole and zero locations (i.e., those that are on the real axis) are highlighted on the diagram by pink diamonds, along with the portion of the locus that exists on the real axis that is shown by a pink line.

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Asymptotes as |s|→∞

Link to in depth description of rule for finding the intersect and asymptote of closed loop poles as K (and s) goes to infinity.

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Break-Out (and -In) points

Link to in depth description of rule for finding Break-out and Break-in points on the real axis..

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Angle of Departure

Link to in depth description of rule for finding angle of departure from complex poles of G(s)H(s).

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Angle of Arrival

Link to in depth description of rule for finding angle of arrival to complex zeros of G(s)H(s)..

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Locus Crossing the Imaginary Axis

Link to in depth description of rule for finding where the locus crosses the imaginary axis.

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Set Pole Location and Determine K

Link to in depth description of rule for finding the value of K corresponding to a desired closed loop pole location.

Use the mouse to click the desired root location on the locus.

Known Problems

As I find problems with the code, I will list them here.


References

© Copyright 2005 to 2020 Erik Cheever    This page may be freely used for educational purposes.

Erik Cheever       Department of Engineering         Swarthmore College