Root Locus Angle of Departure

When open-loop complex poles or zeros exist, the root locus does not start or end along the real axis. The direction in which a branch leaves a complex pole (or arrives at a complex zero) is precisely defined by the angle of departure (or angle of arrival). These angles are critical for sketching the locus near complex singularities and are routinely tested in GATE numerical problems.

Without computing these angles, you cannot correctly sketch the root locus when complex poles or zeros are present. The angle of departure determines whether the locus moves toward or away from the right half plane immediately after leaving a complex pole, giving an early indication of stability behavior at low gains.

Core Concept Explanation

The angle of departure is derived directly from the angle condition of the root locus. For any point on the locus, the net angle contribution from all poles and zeros must equal (2q+1) x 180 degrees. When the test point is placed infinitesimally close to a complex pole p1, all other poles and zeros contribute known angles to the test point. The remaining unknown angle is exactly the departure angle from p1.

Let sum_phi_p be the sum of angles from all poles other than p1 to p1, and sum_phi_z be the sum of angles from all zeros to p1. Then the angle of departure from p1 is: theta_d = 180 + sum_phi_z - sum_phi_p. This ensures the total angle condition of 180 degrees is satisfied at the test point.

Similarly, the angle of arrival at a complex zero z1 is: theta_a = 180 - sum_phi_z_other + sum_phi_p, where sum_phi_z_other is the angle contribution from all other zeros to z1 and sum_phi_p is the total pole angle contribution to z1. The angle of arrival tells you the direction from which the locus approaches the zero as K goes to infinity.

Mathematical Expression

The angle of an individual contribution from a pole pi to a test point s0 is the phase angle of the vector drawn from pi to s0, measured as the angle of (s0 - pi) in the complex plane. If s0 = sigma_0 + j*omega_0 and pi = sigma_i + j*omega_i, then the angle is: phi_i = arctan[(omega_0 - omega_i) / (sigma_0 - sigma_i)]. For a test point at complex pole p1, you substitute s0 = p1 and compute angles from all other singularities.

The final formula for departure angle from complex pole p1 is: theta_d = (2q+1) x 180 - sum of angles from all other OL poles to p1 + sum of angles from all OL zeros to p1. Take q = 0 for the principal value, giving theta_d = 180 - sum(pole angles) + sum(zero angles).

Practical Understanding

The angle of departure is most important in systems with lightly damped or undamped open-loop poles (poles near or on the imaginary axis). In such cases, a small change in the departure angle can determine whether the locus immediately moves into the RHP (unstable region) or into the LHP (stable region) for small K. A departure angle pointing toward the RHP is a warning that even small gains could cause instability.

Solved Numerical Example

Consider G(s) = K / [(s+1-j2)(s+1+j2)(s+3)]. The complex poles are at p1 = -1+j2 (conjugate pair) and a real pole at p3 = -3. Find the angle of departure from p1 = -1+j2. No zeros are present.

Given:
Poles: p1 = -1+j2, p2 = -1-j2, p3 = -3. No zeros.
Find: angle of departure from p1 = -1+j2

Why this formula applies:
Angle condition at test point = p1 gives departure angle

Formula:
theta_d = 180 - (phi_p2 + phi_p3) [no zeros]

Substitution:
Vector from p2 to p1: p1 - p2 = (-1+j2) - (-1-j2) = j4
Angle phi_p2 = angle of j4 = 90 degrees

Vector from p3 to p1: p1 - p3 = (-1+j2) - (-3) = 2+j2
Angle phi_p3 = arctan(2/2) = arctan(1) = 45 degrees

Calculation:
theta_d = 180 - (90 + 45) = 180 - 135 = 45 degrees

Final Answer:
Angle of departure from p1 = -1+j2 is 45 degrees (measured from positive real axis)

Exam Tip: Always measure angles of vectors from the other poles/zeros TO the departure pole, not the other way. A reversed direction flips the sign and gives a completely wrong answer. This sign reversal is the most common mistake in GATE problems on angle of departure.

Mechanism: Vector Angle Calculation

• Draw vectors FROM each other pole and zero TO the pole at which departure is needed.
• Compute the angle (in degrees) of each such vector using arctan(imaginary/real) taking care of the quadrant.
• Apply the formula: theta_d = 180 - sum(other pole angles to p1) + sum(zero angles to p1).
• For angle of arrival at a complex zero, the formula reverses: theta_a = 180 + sum(pole angles to z1) - sum(other zero angles to z1).
• The conjugate pole always departs at the negative of the departure angle of p1, consistent with real-axis symmetry of the locus.

Quick Revision

• Angle of departure applies when open-loop complex poles exist. It is the initial direction of the locus branch leaving that pole.
• Formula: theta_d = 180 - sum(angles from other poles to p1) + sum(angles from zeros to p1).
• Angle of arrival at complex zero: theta_a = 180 + sum(pole angles to z1) - sum(other zero angles to z1).
• Vectors are always drawn FROM other singularities TO the departure/arrival point.
• Conjugate poles always have departure angles that are negatives of each other.
• Exam trap: Reversing vector direction (drawing from p1 to others) gives wrong angles. Always draw toward p1.