Introduction:

In control system analysis and design, the root locus method is a crucial tool that helps engineers to predict and analyze the transient behavior of a closed-loop system. The root locus diagram is a graphical representation of how the system’s poles change as the gain is adjusted. By understanding this concept, it becomes easier to find the appropriate values for system parameters to achieve stable, fast, and accurate response. In this article, we will walk you through the steps involved in drawing the root locus of a system.

Step 1: Determining the Open-Loop Transfer Function

The first step in drawing root locus is to determine the open-loop transfer function (OLTF) of the system. OLTF is represented by G(s)H(s), where G(s) denotes the forward path transfer function and H(s) signifies feedback path transfer function.

Step 2: Characteristic Equation

Once you have determined the OLTF, write down the characteristic equation of the closed-loop system. The characteristic equation can be represented as 1 + G(s)H(s) = 0.

Step 3: Obtain Poles and Zeros

Identify all poles and zeros from your transfer function (G(s)H(s)).

Poles: Points where the denominator is zero.

Zeros: Points where the numerator is zero.

Mark all poles as ‘X’ and zeros as ‘O’ on your graph paper.

Step 4: Determine Real Axis Segments

To determine which segments of the real axis belong to the root locus, draw imaginary lines starting from each pole or zero and extend them towards their neighboring pole or zero.

If there are an odd number of poles or zeros to its right, that entire segment belongs to the root locus, otherwise it doesn’t.

Step 5: Compute Asymptotes

To find asymptotes for the root locus plot, you need to calculate the angle of asymptotes and centroid.

Angle of asymptotes: (2n ± 1) × 180° / (No. of poles – No. of zeros)

Centroid: (Sum of real parts of poles – Sum of real parts of zeros) / (No. of poles – No. of zeros)

Step 6: Identify Breakaway and Break-in Points

Breakaway points occur when multiple branches in the root locus converge, and break-in points occur when multiple branches diverge. They typically occur on straight segments of the real axis where dK/ds = 0.

To find breakaway or break-in points:

1. Write K as a function of s.

2. Compute dK/ds.

3. Set dK/ds = 0.

4. Solve for s.

Step 7: Determine Angle of Departure and Angle of Arrival

Calculate the angle of departure for each complex conjugate pole pair and angle of arrival for each complex conjugate zero pair using the phase-angle equation.

Step 8: Draw Root Locus

Now that all necessary information has been obtained, start sketching the root locus by connecting segments, asymptotes, breakaway points, and break-in points as appropriate.

Conclusion:

Drawing root locus diagrams is essential for understanding system behavior and tuning control systems to achieve desired performance characteristics. By following these steps, engineers can analyze transient response and stability parameters to optimize their closed-loop control system design.