Online Root Locus Calculator

Root locus analysis is a fundamental tool in control system design. This calculator helps you plot and analyze root locus diagrams for linear time-invariant systems.

What is Root Locus?

Root locus is a graphical method used to analyze the stability and transient response of linear time-invariant control systems. It plots the paths of the system's closed-loop poles as a parameter (typically the gain) varies.

The root locus diagram provides valuable insights into system behavior, including:

System stability boundaries
Dominant pole locations
System response characteristics
Optimal gain selection

Root locus analysis is essential for engineers designing control systems in various applications, from aerospace to industrial automation.

How to Use the Calculator

Using the online root locus calculator is straightforward:

Enter the numerator coefficients of your transfer function
Enter the denominator coefficients of your transfer function
Click "Calculate" to generate the root locus plot
Analyze the resulting diagram and stability information

The calculator will display the root locus plot along with key stability information and critical points.

Formula

The root locus plot is generated 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 is the set of values of s that satisfy the characteristic equation for some value of K.

Example Calculation

Consider a system with transfer function:

G(s) = s + 1 / s² + 2s + 2

Using the calculator, we can plot the root locus for this system. The resulting diagram will show the paths of the closed-loop poles as the gain K varies from 0 to ∞.

Interpreting Results

When analyzing the root locus plot, look for:

Locations of closed-loop poles
Stability boundaries (where poles cross the imaginary axis)
Dominant pole pairs that determine system response
Break-in and break-away points where root locus paths change direction

The root locus plot helps engineers select appropriate gain values to achieve desired system performance while maintaining stability.

FAQ

What is the difference between root locus and Nyquist plot?

Root locus plots the paths of closed-loop poles as gain varies, while Nyquist plots the open-loop transfer function in the complex plane. Both are used for stability analysis but provide different perspectives on system behavior.

How do I determine system stability from the root locus plot?

A system is stable if all closed-loop poles lie in the left half of the complex plane. The root locus plot shows where poles cross the imaginary axis, indicating potential instability.

What are break-in and break-away points?

Break-in points are where root locus paths enter the real axis, and break-away points are where they leave. These points indicate changes in system behavior as gain varies.