Question

A system has two zeros and four poles. Then two asymptotes in the root loci plane move towards infinity along:

• A. \( \pm 60^\circ \)
• B. \( \pm 90^\circ \)
• C. \( \pm 45^\circ \)
• D. \( \pm 30^\circ \)

Hint:

The angles of root locus asymptotes are calculated using the formula \( \theta_k = \frac{(2k + 1) 180^\circ}{P - Z} \). This helps in determining system stability.

Answer 1

The Correct Option is B

Step 1: The number of root locus asymptotes is given by: \[ N_a = P - Z \] where \( P \) is the number of poles and \( Z \) is the number of zeros. 
Step 2: Given: - \( P = 4 \) (number of poles) - \( Z = 2 \) (number of zeros) \[ N_a = 4 - 2 = 2 \] 
Step 3: The angles of asymptotes are determined using the formula: \[ \theta_k = \frac{(2k + 1) 180^\circ}{N_a}, \quad k = 0, 1, \dots, (N_a - 1) \] 
Step 4: Substituting \( N_a = 2 \): \[ \theta_0 = \frac{(2 \times 0 + 1) 180^\circ}{2} = 90^\circ \] \[ \theta_1 = \frac{(2 \times 1 + 1) 180^\circ}{2} = -90^\circ \] 
Step 5: Thus, the two asymptotes move towards infinity along \( \pm 90^\circ \).