ROOT LOCUS DIAGRAM PLOTTING WITH EXAMPLE|NUMERICAL|CHEMICAL ENGINEERING| PROCESS CONTROL|PDC|GATE|

Introduction

This tutorial will guide you through the process of plotting root locus diagrams, a critical tool in process control and chemical engineering. Understanding root locus plots helps in analyzing system stability and response, making it a valuable skill for engineers, especially for those preparing for exams like GATE.

Step 1: Understand the Basics of Root Locus

• Definition: Root locus is a graphical method for examining how the roots of a system change with varying system parameters, particularly gain.
• Purpose: It helps in determining the stability of a system as gain changes.
• Key Components:
  • Poles: Values of 's' that make the denominator of the transfer function zero.
  • Zeros: Values of 's' that make the numerator of the transfer function zero.

Step 2: Identify the Transfer Function

• Formulate the Transfer Function: Start with the system's differential equation and derive the transfer function, usually in the form:

  [ G(s) = \frac{N(s)}{D(s)} ]

  where (N(s)) is the numerator and (D(s)) is the denominator.

• Example: For a simple system, you might have:

  [ G(s) = \frac{5}{s^2 + 2s + 5} ]

Step 3: Find Poles and Zeros

• Calculate Zeros: Solve (N(s) = 0).
• Calculate Poles: Solve (D(s) = 0).
• Practical Tip: Use the quadratic formula for second-order systems or numerical methods for higher-order systems.

Step 4: Plot the Root Locus

• Axis Setup: Draw the real and imaginary axes.
• Plot Poles and Zeros: Mark the poles with ‘X’ and zeros with ‘O’.
• Determine Branches:
  • Each pole will have a branch that indicates how it moves in the complex plane as gain increases.
• Use Software Tools: Consider tools like MATLAB or Python for complex systems, which can automate the plotting process.

Step 5: Analyze the Plot

• Stability Criteria: Check if all poles remain in the left half of the complex plane for stability.
• Gain Calculation: Use the formula for gain to determine how much gain can be applied before instability occurs.
• Common Pitfalls: Ensure that all poles and zeros are correctly plotted, and verify calculations for potential errors.

Step 6: Example Problem Walkthrough

• Example Transfer Function: Suppose you have:

  [ G(s) = \frac{4s + 8}{s^2 + 3s + 2} ]

• Identify Poles and Zeros:

  • Zeros: Set (4s + 8 = 0) → (s = -2)
  • Poles: Set (s^2 + 3s + 2 = 0) → (s = -1, -2)

• Plot: Follow the plotting steps outlined earlier.

Conclusion

Root locus diagrams are essential for assessing the stability of control systems in chemical engineering. By following these steps—understanding the basics, identifying the transfer function, finding poles and zeros, plotting the root locus, and analyzing the plot—you can effectively evaluate system behavior as gain varies. For more complex systems, consider utilizing software tools to assist in the plotting and analysis process.