Root Locus Asymptotes

Topics: Feedback Control Systems

The angle of the asymptotes of the root locus based on the degree of the numerator $d_n$ and the degree of the denominator $d_d$.
Starting from the open-loop transfer function $L$ in the following form

$$L = \beta \frac{\prod_{i=1}^{d_n}(s-z_i)}{\prod_{i=1}^{d_d}(s-p_i)}$$

The number of asymptotes in the root locus is $\# \text{ asymptotes}=d_d - d_n$. The asymptotes angles can be determined based on one of two rules.

$\beta k > 0$

$$\text{Angle of Asymptotes} = \frac{\pm 180 \degree}{d_d - d_n}, \; \frac{\pm 540 \degree}{d_d - d_n}, \; \frac{\pm 900 \degree}{d_d - d_n},\; \dots$$

$\beta k < 0$

$$\text{Angle of Asymptotes} = 0, \; \frac{\pm 360 \degree}{d_d - d_n}, \; \frac{\pm 720 \degree}{d_d - d_n}, \; \frac{\pm 1080 \degree}{d_d - d_n},\; \dots$$