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Phase angle and margin

I'm trying to find the angles of departure and angles of arrival for an equation. In my current problem, I am looking for the angle of departure for

GH=10(s+z)/s^2 (also GH(jw)=10(jw+z)/(jw^2)

The angle of departure is defined as 180 degrees plus the phase angle of GH computed at the complex pole, but ignoring the contribution of that particular pole.

The s^2 is the pole portion of the equation and it represents 2 poles at the orgin (on the jw axis).

Please help me understand how to determine the angle of departure from GH.

Also, (separate from above question) the phase margin is the phase angle at cross over. For g(jw)=50/(jw(jw/5+1)), what is and how do I (explicity)find the phase margin? Thanks!

Thank you.

Solution Preview

Please see the attached file and contact me for any future questions.

Answer:

1. Angle of departure
The angle of departure/arrival is used in plotting Root-locus for a linear system. Basically, the root locus always departs from poles and arrives at zeros.
angle of departure. At each complex pole, add up the angles from the zeros to the current pole, then subtract the angles from the other poles to the current pole. In mathematical terms, for a given pole, the angle of departure is

where is the angle between the ith pole and the given pole, and is the angle between the jth zero and the given pole. and can be calculated using trigonometry. ...

Solution Summary

This solution is provided in 460 words in an attached .doc file. It describes angle of departure and the calculation for obtaining this, as well as a graph of the problem to find the solution. Root locus diagrams are also provided to find both the angle of departure and the phase margin.

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