Chapter 10
10.8 Evans Root Locus Construction Rule #6: Angles of Departures/Arrivals at Complex Poles/Zeros
This Rule deals with angles of departure (arrival) from/to complex poles (zeros).
Rule 6: If [latex]G(s)[/latex] has a pole p of multiplicity r, then r branches of the root locus depart from p. The angle of departure of these root loci from p are described by this equation:
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[latex]\theta_{dep}[/latex]= [latex]\frac{ \angle [G(s)(s-p)^r]_{s=p}+(2k+1) \cdot 180^{\circ}}{r}[/latex] [latex]k=0,1,2,...,r-1[/latex] Equation 10-13 |
Similarly, if [latex]G(s)[/latex] has a zero z of multiplicity r, then r branches of the root locus arrive at z. The angle of arrival of these root loci to z are described by this equation:
| [latex]\theta_{arr} = \frac{-\angle[\frac{G(s)}{(s-z)^r}]_{s=z}+(2k+1) \cdot 180^{\circ}}{r} k=0,1,2,...,r-1[/latex]
Equation 10-14 |
See examples in the next section for illustration of this rule.
