Magnitude and Angle Condition of Root Locus

The angle condition is used for checking whether any point lie on the root locus or not and also the validity of root locus shape.

Consider the characteristic equation (in case of a negative unity feedback system)

1+G(s)H(s)=0

Where G(s)H(s) is the open loop transfer function (OLTF) of the system.

Then,

G(s)H(s) = -1

Hence, we get magnitude condition as |G(s)H(s)| = 1

and the angle condition is ∠ G(s)H(s) = ±180⁰

or ∠ G(s)H(s) = ±[2q+1]180⁰ , where, q=0,1,2,...,integers

Angle condition may be stated as, for a point to lie on root locus, the angle evaluated at that point, using OLTF, must be an odd multiple of ±180⁰ .

Magnitude condition is used for finding gain at any point on the root locus.

Root Locus Magnitude and Angle Criterion Example

Q. Check whether s=-1+j √3 lies on the root locus or not. If it does so then find the gain K at this point. The OLTF of the system is