What Is the Root Locus Asymptote & Breakaway Calculator?
Root locus plots closed-loop pole migrations in the complex s-plane as feedback gain K scales from zero to infinity.
How Does the Root Locus Asymptote & Breakaway Calculator Work?
Roots begin at open-loop poles and travel toward open-loop zeros or out to infinity along asymptotes centered at sigma_a.
Root Locus Asymptote & Breakaway Calculator Formula & Variables
The core mathematical equation utilized by this calculator is expressed as:
sigma_a = rac{sum ext{Poles} - sum ext{Zeros}}{n - m}, quad phi_k = rac{(2k + 1) 180^circ}{n - m}
Evans root locus centroid and asymptote departure angle equations.
How to Use the Root Locus Asymptote & Breakaway Calculator
- Input real-axis pole locations and optional zero location.
Step-by-Step Example Calculation
Type-1 Third-Order System
Input Values:
p1:0
p2:-3
p3:-8
z1:0
Worked Steps: Asymptote centroid sigma_a = -3.67, radiating angles = [60°, 180°, 300°], Breakaway sigma_b = -1.28.
Understanding Your Result
Reports asymptote center of mass sigma_a, departure angles, and breakaway points.
Factors That Affect the Result
- Relative degree (number of poles minus zeros) determines the number of radiating asymptotes.
When Should You Use This Calculator?
- Designing lead/lag compensators, tuning proportional gains, and assessing transient damping.
Assumptions & Limitations
- Applies to negative unity feedback SISO control systems.
Frequently Asked Questions
Calculation Accuracy & Reference Note
Standard Evans root locus geometric formulation.