Relation Between Bode Plot and Root Locus

Topics: Feedback Control Systems

The root locus imaginary crossovers, i.e., the points where the root locus intersects the imaginary axis are crucial points in the study of stability. This is because they represent the points of transition between the stable region (LHP) and unstable region (RHP), and vice versa.

These points can be read directly on the Bode Plot because of the following key ideas:

• the imaginary crossovers have the form $s=j\omega$
• Bode plot shows the response of the system to a pure sinusoid, i.e., magnitude and phase of $L(j \omega)$
• Magnitude and Angle Conditions present the conditions for a point to be on the root locus

So, by combining the above, we can find the points on the root locus that intersect the imaginary axis by using the Bode magnitude and phase plots.

From the angle condition

$$\angle L(j \omega) = -180 \degree \pm 360 \degree$$

meaning that the imaginary crossovers correspond to the points on the Bode phase plot where the phase is equal to $-180 \degree \pm 360 \degree$.

The corresponding loop gain $k$ is found from the Bode magnitude plot

$$k = \frac{1}{|L(j \omega)|}$$