Wolfram Root Locus Calculator

The Wolfram Root Locus Calculator plots and analyzes root locus diagrams for control systems. This tool helps engineers and students visualize how the roots of a characteristic equation move in the complex plane as a system parameter varies.

What is Root Locus?

Root locus is a graphical method used in control theory to analyze the stability and transient response of linear time-invariant systems. It shows how the roots of the characteristic equation of a system move in the complex plane as a parameter (typically gain) varies.

Key characteristics of root locus plots include:

Starting points (poles of the open-loop transfer function)
Ending points (zeros of the open-loop transfer function)
Asymptotes (lines that the root locus approaches as gain increases)
Break-in points (where the root locus crosses the real axis)

The root locus helps engineers determine the optimal gain for a control system to achieve desired performance characteristics such as stability, settling time, and overshoot.

How to Use the Calculator

To use the Wolfram Root Locus Calculator:

Enter the numerator coefficients of your transfer function
Enter the denominator coefficients of your transfer function
Click "Calculate" to generate the root locus plot
Interpret the resulting diagram to analyze your system's stability

Note: The calculator assumes a proper transfer function (degree of numerator ≤ degree of denominator). For improper functions, you may need to rewrite the transfer function in proper form.

Formula Used

The root locus is determined by solving the characteristic equation:

1 + K * G(s) * H(s) = 0

Where:

K is the gain parameter
G(s) is the forward transfer function
H(s) is the feedback transfer function

The root locus plot shows the paths that the roots of the characteristic equation take as K varies from 0 to ∞.

Worked Example

Consider a system with transfer function:

G(s) = (s + 2)/(s² + 5s + 6)

Using the calculator with numerator coefficients [1, 2] and denominator coefficients [1, 5, 6], we generate the root locus plot.

The resulting diagram shows:

Two branches starting from the poles at s = -2 and s = -3
One branch ending at the zero at s = -2
Asymptotes at ±60° angles

This analysis helps determine the optimal gain for stable operation of the system.

FAQ

What is the difference between root locus and Bode plot?: Root locus shows how roots move in the complex plane as gain varies, while Bode plots show frequency response characteristics of a system.
How do I interpret the root locus plot?: The plot shows the paths that system poles take as gain changes. Stable systems have all roots in the left half-plane.
What does it mean if the root locus crosses the imaginary axis?: This indicates the system becomes unstable for certain gain values, as roots move into the right half-plane.
Can I use this calculator for discrete-time systems?: This calculator is designed for continuous-time systems. For discrete-time systems, you would need a different approach.
How accurate are the results?: The calculator provides accurate root locus plots based on the transfer function you provide, using standard control theory methods.