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Root Locus Calculator Wolfram

Reviewed by Calculator Editorial Team

Root locus analysis is a graphical technique used in control theory to analyze the stability of linear time-invariant systems. This calculator implements Wolfram's methods for generating root locus plots, which help engineers understand how system poles move as gain changes.

What is Root Locus?

The root locus is a graphical representation of the paths that the poles of a closed-loop control system take as a parameter (usually gain) is varied. It provides valuable insights into system stability and performance.

Key characteristics of root locus plots include:

  • Number of branches: Equal to the number of poles
  • Starting points: Open-loop poles
  • Ending points: Open-loop zeros
  • Asymptotes: Lines that the root locus approaches as gain increases

Root locus analysis helps engineers design control systems by identifying stable operating regions and optimal gain values.

How to Use This Calculator

To use the root locus calculator:

  1. Enter the numerator coefficients of your transfer function
  2. Enter the denominator coefficients of your transfer function
  3. Click "Calculate" to generate the root locus plot
  4. Interpret the resulting plot to analyze system stability

Note

This calculator implements Wolfram's methods for root locus analysis. For complex systems, you may need to adjust the gain range for optimal visualization.

Formula Used

Transfer Function

G(s) = (bnsn + bn-1sn-1 + ... + b0) / (amsm + am-1sm-1 + ... + a0)

The root locus is generated by solving for the roots of the characteristic equation as gain K varies from 0 to ∞.

Worked Example

Consider a system with transfer function:

G(s) = (s + 2) / (s2 + 3s + 2)

Using the calculator:

  1. Enter numerator coefficients: 1, 2
  2. Enter denominator coefficients: 1, 3, 2
  3. Click "Calculate"

The resulting plot will show the root locus for this second-order system, helping you analyze its stability characteristics.

Interpreting Results

When analyzing root locus plots:

  • Look for branches crossing the imaginary axis - these indicate potential instability
  • Check if all branches move toward the left half-plane as gain increases
  • Note the gain values where branches cross the imaginary axis

A well-designed control system should have all root locus branches in the left half-plane for all positive gain values.

FAQ

What is the difference between root locus and Nyquist plot?

Root locus shows how poles move as gain changes, while Nyquist plot shows how the frequency response changes. Both are used for stability analysis but from different perspectives.

How does root locus help with controller design?

Root locus helps select appropriate controller gains and types by showing how system poles move with different parameter values. This allows engineers to choose designs that provide desired stability and performance.

What are the limitations of root locus analysis?

Root locus assumes linear time-invariant systems and may not account for nonlinearities or time-varying parameters. It's most useful for single-input single-output systems.