On the Angles of Root Loci Branches that Arrive or Depart from Breakaway Points

In most textbooks on principles of automatic control, very limited content is given on the angles of arrival or departure from breakaway points of root loci. Sometimes, for simplicity of description, the angles of arrival or departure from breakaway points are given by $180^{\circ}/ l$ apart, where l denotes the number of branches of root loci that meet at the breakaway point; however, no detailed proof is given for this result. In this paper, detailed proof is provided for the angles apart of the root loci branches from the breakaway point. It is proved that, the result is valid with strictly positive real open loop transfer function in both cases of unit negative feedback and unit positive feedback. The proof is based on the property of angles of departure or arrival and the angle condition of the root locus itself. Several examples are given to support the proposed proof. The proposed results can be lectured to students for better understanding and more precise plot of root locus.