How to find breakaway points in root locus

In summary, for an exam question, the speaker needs to sketch the root locus of a system with the given transfer function. They attempt to calculate the number of poles and zeroes, the centroid, and the angles of the asymptotes, but struggle with finding the breakaway points. A helpful resource for generating Bode diagrams is suggested.
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MattH150197
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Homework Statement


For an exam question i need to be able to sketch the root locus of a system, for example the following: g(S) = 200(S+3) / ((S+2)(S+4)(S+6)(S+8)(S+10)

The Attempt at a Solution


So i counted number of poles and zeroes and calculated no. of asymtodes: p-z = 4 and calculated the centroid by: (sum of poles - sum of zeroes)/ p-z = -27/4. Then i got the angle of each asymtodes to be: -45, 45, 135 and 225 however it is the next step of actually finding the breakaway points i can't find how to calculate, would appreciate any help you can offer. Thanks
 
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1. How do I determine the breakaway points in a root locus plot?

The breakaway points in a root locus plot can be found by setting the derivative of the open-loop transfer function with respect to the gain equal to zero. This will result in a quadratic equation that can be solved to find the breakaway points.

2. What is the significance of identifying breakaway points in a root locus plot?

The breakaway points indicate the points on the root locus where the system transitions from stable to unstable or vice versa. They are important for understanding the stability and performance of a control system.

3. Can I use graphical methods to find breakaway points in a root locus plot?

Yes, graphical methods such as the angle criterion and magnitude criterion can also be used to find the breakaway points in a root locus plot. These methods involve drawing lines on the root locus plot and finding the intersection points to determine the breakaway points.

4. How do I interpret the breakaway points in a root locus plot?

The location and number of breakaway points can provide insight into the stability and performance of the control system. For example, a single breakaway point at a low gain indicates good stability, while multiple breakaway points at high gains may indicate potential instability.

5. Are there any software tools available to help find breakaway points in a root locus plot?

Yes, there are many software tools, such as MATLAB and Scilab, that have built-in functions for generating and analyzing root locus plots. These tools can automatically identify and plot the breakaway points, making it easier for engineers and scientists to analyze the stability of control systems.

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