Routh Hurwitz Criterion Guide
What This Calculator Does
The Routh Hurwitz criterion checks whether a polynomial has roots in the right half of the complex plane. In control work, those roots usually mean an unstable closed loop. This calculator builds the Routh array from coefficients entered in descending powers. It then reads the first column and counts sign changes. Each sign change represents one right half plane root, when the table is regular. The tool also flags zero pivots and full zero rows. These special cases need careful interpretation, so the report includes notes for review.
Why Stability Testing Matters
A controller can look correct from gains alone, yet still oscillate or diverge. The characteristic equation explains that behavior. If every closed loop pole lies in the left half plane, the response decays after a disturbance. If any pole lies on the right side, the response grows. The Routh method gives this warning without solving every root. That makes it useful for quick design checks, coursework, and parameter studies.
How Inputs Affect Results
Enter all coefficients, including zeros for missing powers. For example, s^4 + 3s^2 + 2 becomes 1, 0, 3, 0, 2. A missing zero changes the order and gives a wrong array. The leading coefficient is normalized when needed because multiplying the full polynomial by a negative constant does not move roots. The tolerance setting controls when tiny values are treated as zero. The epsilon setting estimates a zero pivot as a very small positive number.
Using The Report
After submission, review the status panel first. Then check the Routh table row by row. The first column is the main stability indicator. Download the CSV when you need spreadsheet analysis. Download the PDF when you need a compact record. Use the example table to compare known patterns before testing a new equation. This calculator supports learning and early engineering review. It does not replace simulation, root solving, or expert validation for safety critical systems.
Practical Tips
Always compare the table with the original coefficient list. Keep units outside the polynomial. Avoid rounded coefficients during final checks. Test one change at a time, especially when tuning gain. Record the tolerance used, so later reports remain repeatable and clear.