---
title: "Routh-Hurwitz Criterion | Electrical Circuits II"
description: "Routh-Hurwitz Criterion tests LTI stability from a characteristic polynomial without solving roots, which is handy for state equations in Circuits II."
canonical: "https://fiveable.me/electrical-circuits-systems-ii/key-terms/routh-hurwitz-criterion"
type: "key-term"
subject: "Electrical Circuits and Systems II"
unit: "Unit 12"
---

# Routh-Hurwitz Criterion | Electrical Circuits II

## Definition

The Routh-Hurwitz Criterion is a stability test for an LTI system's characteristic polynomial. In Electrical Circuits and Systems II, you use it to tell whether the system's state response stays stable without finding every root.

## What It Is

The Routh-Hurwitz Criterion is a shortcut for checking whether an Electrical Circuits and Systems II system is stable by looking at its characteristic polynomial, not by solving every root directly. For a linear time-invariant system, stability means the natural response dies out over time, which happens when all the poles have negative real parts.

Instead of factoring a high-order polynomial, you build a Routh array from the polynomial coefficients. The first two rows come from alternating coefficients, and the rest of the table is filled using a specific determinant-based pattern. Once the array is built, the sign pattern in the first column tells you how many roots lie in the right half-plane.

That is the big payoff: you do not need to compute messy roots to know whether the system is stable. If the first-column entries are all the same sign, there are no sign changes, so there are no right-half-plane poles. If the signs change, each sign change means one unstable root pair or one unstable root depending on the polynomial structure.

In this course, the criterion shows up when you are working with state equations, state matrices, or closed-loop characteristic equations. You might get a matrix A, form the characteristic polynomial from det(sI - A), and then use Routh-Hurwitz to check whether the state trajectory settles down. That makes it a practical bridge between algebra and system behavior.

There are a few special cases to know. If a row becomes all zeros, that usually points to symmetric root patterns or repeated imaginary-axis roots, and you have to use an auxiliary polynomial to keep going. If the first coefficient is not positive or the polynomial is not written in descending powers, the table can get misleading fast. The method is mechanical, but only if the polynomial is set up cleanly first.

## Why It Matters

Routh-Hurwitz matters because Electrical Circuits and Systems II is full of systems where you care less about the exact root values and more about whether the response behaves. A circuit or control system can have a neat-looking transfer function and still be unstable if one pole slips into the right half-plane. This criterion gives you a fast way to catch that before you waste time on full root solving.

It also connects directly to state space work. When you write a system in state space form, the eigenvalues of the state matrix determine the natural response. Routh-Hurwitz lets you test the characteristic polynomial of that matrix and decide whether the states decay, oscillate, or blow up.

That makes it a useful check in design problems too. If you are adjusting feedback, changing parameters, or comparing a few candidate systems, you can use the first-column sign pattern as a quick stability screen. In other words, it is not just a math trick, it is a decision tool for circuits and control models.

## Connections

### Characteristic Polynomial

Routh-Hurwitz starts with the characteristic polynomial, because that polynomial contains the poles or eigenvalues that decide stability. In practice, you either get it from a transfer function denominator or from det(sI - A) in state space. If you set up the polynomial wrong, the whole Routh table will give you the wrong stability answer.

### [State Matrix](/electrical-circuits-systems-ii/key-terms/state-matrix)

The state matrix is where this criterion often shows up in Circuits II. Its eigenvalues are the system poles, so checking the characteristic polynomial of the state matrix tells you whether the state response decays or grows. Routh-Hurwitz gives you a way to inspect that polynomial without solving for every eigenvalue explicitly.

### [Lyapunov Stability](/electrical-circuits-systems-ii/key-terms/lyapunov-stability)

Both Routh-Hurwitz and Lyapunov Stability are about whether a system settles down, but they approach it differently. Routh-Hurwitz works through the characteristic polynomial, while Lyapunov methods work by finding an energy-like function or matrix condition. They are useful together because they give two different stability viewpoints.

### [state feedback](/electrical-circuits-systems-ii/key-terms/state-feedback)

state feedback changes the closed-loop characteristic equation, which means it can move poles into a stable region. After you design feedback, you can use Routh-Hurwitz to check whether the new polynomial has all the right sign conditions. That makes it a quick verification step after controller design.

## On the AP Exam

A quiz or problem-set question usually gives you a characteristic polynomial and asks whether the circuit or state-space system is stable. You build the Routh array, check the first column, and count sign changes to identify unstable poles. If a row of zeros appears, you need the auxiliary polynomial step, not a guess.

You might also see this paired with a state matrix problem, where you first find det(sI - A), then use Routh-Hurwitz instead of solving the eigenvalues directly. The main skill is staying organized with the coefficients and catching sign errors early. One wrong entry can change the stability conclusion completely, so neat setup matters as much as the arithmetic.

## Routh-Hurwitz Criterion vs Pole-Zero Plot

A pole-zero plot shows pole locations directly on the complex plane, so you can visually judge stability. Routh-Hurwitz reaches the same kind of conclusion from the polynomial coefficients without graphing or solving for all poles. The plot is visual, while the criterion is algebraic.

## Key Takeaways

- Routh-Hurwitz Criterion checks stability from the characteristic polynomial, so you do not have to solve for every root.
- In Electrical Circuits and Systems II, it is most useful for state equations, transfer-function denominators, and closed-loop characteristic equations.
- The first-column signs of the Routh array tell you how many roots are in the right half-plane, which means how many unstable modes you have.
- A row of all zeros is a special case that needs an auxiliary polynomial, not just a continued table.
- If the polynomial is set up incorrectly, the Routh array can look valid but still lead you to the wrong stability answer.

## FAQs

### What is Routh-Hurwitz Criterion in Electrical Circuits and Systems II?

It is a method for checking whether a linear time-invariant system is stable by looking at the coefficients of its characteristic polynomial. In Circuits II, that usually means testing whether the poles or state-matrix eigenvalues stay in the left half-plane. You use the Routh array to do that without solving the full polynomial.

### How do you know if a system is stable using Routh-Hurwitz?

Build the Routh array and inspect the first column. If all the first-column entries have the same sign, there are no sign changes and the system is stable. Each sign change indicates a root in the right half-plane, which means an unstable mode.

### What does a row of zeros mean in the Routh table?

A row of zeros means the polynomial has a special symmetry or repeated imaginary-axis roots, so the basic table cannot finish the job by itself. You use an auxiliary polynomial formed from the row above to continue. This is a common place to lose points because it is not a normal arithmetic step.

### Is Routh-Hurwitz the same as a pole-zero plot?

No. A pole-zero plot shows where the poles are on the complex plane, so you can judge stability by their location. Routh-Hurwitz gives the same stability information from the polynomial coefficients, which is useful when you do not want to calculate or sketch every pole.

## Related Study Guides

- [12.3 Solution of state equations](/electrical-circuits-systems-ii/unit-12/solution-state-equations/study-guide/JoAovZpaK2VvS0FU)

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