Root Locus - Sketching Root Locus

Sketching Root Locus

• Mark open-loop poles and zeros
• Mark real axis portion to the left of an odd number of poles and zeros
• Find asymptotes

Let P be the number of poles and Z be the number of zeros:

number of asymptotes

The asymptotes intersect the real axis at and depart at angle given by:

where is the sum of all the locations of the poles, and is the sum of all the locations of the explicit zeros.

• Phase condition on test point to find angle of departure
• Compute breakaway/break-in points

The breakaway points are located at the roots of the following equation:

Once you solve for z, the real roots give you the breakaway/reentry points. Complex roots correspond to a lack of breakaway/reentry.

The break-away (break-in) points are obtained by solving a polynomial equation