Angle of Arrival and Angle of Departure in Root Locus with solved Examples

Rule 8 of drawing Root Locus
Angle of Departure & Angle of Arrival

1. Angle of Departure at complex poles
The angle by which the branches of Root locus departs from complex pole is called angle of Departure.
It is denoted by 𝜱d
𝜱d = 180 - 𝜱
𝜱 = 𝛴𝜱P−𝛴𝜱 𝒁

𝛴𝜱P = Contributions by the angles made by remaining open loop poles at the pole at which 𝜱d to be calculated".

𝛴𝜱Z= Contributions by the angles made by remaining open loop zeros at the pole at which 𝜱d to be calculated"

Procedure to determine Angle of Departure
To calculate 𝛴𝜱P join all remaining poles to complex pole under consideration.

Add all angles subtended by phasor joining poles to pole under consideration.

Similarly join all zeros to pole under consideration and add all angles to determine 𝛴𝜱Z

Use 𝜱 = 𝛴𝜱P−𝛴𝜱 𝒁

Calculate angle of departure by formula, 𝜱d = 180 - 𝜱

Rule 8 : Example 1 : 𝐺(𝑠)𝐻(𝑠)=(𝑲(𝒔+𝟐))/(𝒔(𝑠+𝟒)(𝒔𝟐+𝟐𝒔+𝟐))
C𝐚𝐥𝐜𝐮𝐥𝐚𝐭𝐞 𝐚𝐧𝐠𝐥𝐞𝐬 𝐨𝐟 𝐝𝐞𝐩𝐚𝐫𝐭𝐮𝐫𝐞 𝐚𝐭 𝐜𝐨𝐦𝐩𝐥𝐞𝐱 𝐜𝐨𝐧𝐣𝐮𝐠𝐚𝐭𝐞 𝐩𝐨𝐥𝐞𝐬.

2.Angle of Arrival at complex zeros
The angle by which the branches of Root locus arrives at complex zero is called angle of arrival.
It is denoted by 𝜱a
𝜱a = 180 + 𝜱
𝜱 = 𝛴"𝜱P"−𝛴"𝜱" 𝒁

𝛴𝜱P = Contributions by the angles made by remaining open loop poles at the zero at which 𝜱a to be calculated".

𝛴𝜱Z= Contributions by the angles made by remaining open loop zeros at the zero at which 𝜱a to be calculated"

Rule 8 : Example 1:𝐺(𝑠)𝐻(𝑠)=(𝑲 (𝒔𝟐+𝟒𝒔+𝟐𝟎))/𝒔(𝑠+𝟑)
C𝐚𝐥𝐜𝐮𝐥𝐚𝐭𝐞 𝐚𝐧𝐠𝐥𝐞𝐬 𝐨𝐟 𝐚𝐫𝐫𝐢𝐯𝐚𝐥 𝐚𝐭 𝐜𝐨𝐦𝐩𝐥𝐞𝐱 𝐜𝐨𝐧𝐣𝐮𝐠𝐚𝐭𝐞 𝐳𝐞𝐫𝐨𝐬.