Name: checking stability with the routh table
Uploaded: 2024-11-21T14:54:10.000Z
Duration: 1 min 15 s
Channel: JoVE

정전을 방지하기 위해 안정성이 중요한 전력망을 생각해 보십시오. Routh-Hurwitz 기준을 사용하여 다양한 부하 조건 또는 결함에서 시스템의 안정성을 평가할 수 있습니다.
폐루프 전달 함수가 있다고 가정하겠습니다. Routh 테이블을 만들기 위해 행은 가장 높은 s의 거듭제곱으로 레이블이 지정됩니다.
첫 번째 행은 분모의 다른 모든 계수로 가로로 채워지며 가장 높은 거듭제곱부터 시작합니다. 두 번째 행은 다음으로 높은 거듭제곱을 따르며 건너뛴 계수를 나열합니다.
후속 항목은 이전 행의 음의 행렬식을 통해 계산되며, 이를 바로 위의 첫 번째 열 항목으로 나눕니다.
시스템의 경우 Routh 테이블 행이 계산됩니다. 양의 상수는 행을 독립적으로 확장/축소합니다.
Routh-Hurwitz 기준은 오른쪽 반평면에 있는 다항식의 근 수가 Routh 테이블의 첫 번째 열에 있는 부호 변경 수와 같으며 이는 불안정한 시스템을 나타냅니다.
모든 극점이 s-평면의 왼쪽 절반에 있으면 시스템이 안정적이며, 이는 Routh 테이블의 첫 번째 열에서 부호 변경이 없는 것과 같습니다.

로스-허위츠 기준 i

정전을 방지하기 위해 안정성이 필수적인 전력망을 생각해 보세요. 로스-허위츠 기준은 다양한 부하 조건이나 오류에서 시스템 안정성을 평가하는 데 유용한 도구입니다. 로스-허위츠 기준은 폐쇄 루프 전달 함수를 분석하여 시스템이 안정적으로 유지되는지 여부를 판단하는 데 도움이 됩니다.

로스-허위츠 기준을 적용하기 위해 Routh 표가 구성됩니다. 표의 행은 복소 주파수 변수 s의 거듭제곱으로 레이블이 지정되며 가장 높은 거듭제곱부터 시작합니다. 변수 s는 일반적으로 다음과 같은 형태의 복소수를 나타냅니다.

<img alt="Equation1" src="/files/ftp_upload/16116/16116_PT%20Equation_1.svg" style="height: 14px;">

여기서 σ는 실수 부분이고 jω는 허수 부분입니다. 첫 번째 행은 가장 높은 거듭제곱부터 시작하여 분모 다항식의 다른 모든 계수로 수평으로 채워집니다. 두 번째 행은 첫 번째 행에서 건너뛴 계수로 채워집니다.

로스 표의 후속 항목은 이전 행의 행렬식을 사용하여 계산됩니다. 구체적으로, 각 항목은 위에 있는 두 항목의 음수 행렬식을 취하고 바로 위에 있는 열의 첫 번째 항목으로 나누어 결정합니다. 이 과정은 전체 표가 채워질 때까지 계속됩니다.

예를 들어 로스 표 행이 계산되는 시스템을 고려해 보겠습니다. 필요한 경우 각 행은 계산을 단순화하기 위해 양수 상수로 독립적으로 조정됩니다. 로스-허위츠 기준에 따르면 오른쪽 절반 s 평면의 다항식 근의 수는 로스 표의 첫 번째 열의 부호 변경 수에 해당합니다. 이러한 부호 변경은 불안정성을 나타냅니다.

반대로 모든 극이 왼쪽 절반 s 평면에 있는 경우 시스템은 안정적이며, 즉 로스 허위츠 표의 첫 번째 열에 부호 변경이 없습니다. 신호 변화가 발생하지 않도록 보장하면 시스템이 다양한 작동 조건에서 안정적으로 유지될 것임을 확인할 수 있습니다.

로스-허위츠 기준을 이해하고 적용하는 것은 전력망과 같은 복잡한 시스템의 안정성을 유지하는 데 중요합니다. 모든 극이 왼쪽 반 s 평면에 있는지 확인함으로써 엔지니어는 시스템이 안정적으로 작동하여 정전과 같은 문제를 방지하고 지속적인 전력 공급을 보장할 수 있습니다. 이 방법은 안정성 분석에 대한 체계적인 접근 방식을 제공하여 시스템의 잠재적 불안정성을 식별하고 완화합니다.

로스-허위츠 기준 I

정전을 방지하기 위해 안정성이 중요한 전력망을 생각해 보십시오. Routh-Hurwitz 기준을 사용하면 시스템의 안정성이 ...

Education

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JoVE Core

Electrical Engineering

전기공학

Unit 1

제1단원

Transient and Steady-state Response Analysis

과도 및 정상 상태 응답 분석

transient-and-steady-state-response-analysis

KEY TERMS AND CONCEPTS

주요 용어 및 개념

Routh-Hurwitz Criterion I

checking stability with the routh table

Checking stability with the Routh table is a useful part of analyzing a closed-loop transfer function. It helps determine whether a system will remain stable when load conditions change or when a fault occurs. In a power grid, this kind of check is important because instability can lead to blackouts.

The Routh-Hurwitz criterion uses a table built from the denominator polynomial of the transfer function. The rows of the Routh table are labeled by powers of the complex variable s, starting with the highest power. Here, s is a complex number with a real part, σ, and an imaginary part, jω.

The first row is filled with every other coefficient from the denominator polynomial. It starts with the coefficient from the highest power term. The second row then uses the coefficients that were skipped in the first row.

The remaining entries are found from the rows above them. Each new value is calculated from the negative determinant of the two entries above it, divided by the first entry in the column directly above. If needed, each row can be scaled by a positive constant to make the calculations simpler.

Once the table is complete, the first column gives the stability result. The number of sign changes in that column matches the number of roots in the right-half s-plane. Sign changes show instability. If there are no sign changes, then all poles are in the left-half s-plane and the system is stable. This makes the Routh-Hurwitz criterion a systematic way to test whether a system can operate reliably under different conditions.

Checking Stability with the Routh Table

transient and steady-state response

Control System Test Signals and Stability

Control system test signals help measure how a system reacts to change, stability, and accuracy. A ramp function is useful when a system changes gradually. A step function is used when a system faces a sudden disturbance. An impulse function is the best test signal for a shock input. These signals are used to study two main performance features: transient response and steady-state response. Transient response is the short-term change from the initial state to the final state. It shows how fast...

first order systems

RC System Responses to Step Inputs

RC circuits and other first-order systems show how a simple input can shape system response over time. These systems are often studied through their transfer function, which links the input and output under zero initial conditions. That relationship makes it easier to analyze how the system behaves when the input changes. A unit-step input gives a response that starts at zero and rises toward one. The Laplace transform of the step input is applied to the transfer function, then the result is...

second order systems i

Servo System Response in Second-Order Models

A servo system is a second-order model that links an input position to an output position. It uses a proportional controller and load elements to keep the output aligned with the input. This behavior can be written as a second-order differential equation. When the equation is transformed with the Laplace transform under zero initial conditions, it becomes a transfer function. The transfer function shows how the input is converted into the output. It can also be rewritten as a closed-loop...

second order systems ii

Step Response of Second-Order Systems

Second-order systems react to a unit-step input in different ways depending on the damping ratio. When the damping ratio, zeta, is between 0 and 1, the system is underdamped. In this case, inverse Laplace methods give an output that oscillates with a decreasing amplitude. The error signal, which is the difference between the input and output, also shows damped oscillations before it goes to zero at steady state. When zeta equals 0, the system is undamped. The response keeps oscillating forever...

stability

System Response and Stability

System response and stability describe how a linear time-invariant (LTI) system reacts to a change in input. The time response has two parts: transient response and steady-state response. The transient response is the first reaction and it fades to zero over time. The steady-state response remains after the transient effects have disappeared. An LTI system is stable when it gives a bounded output for a bounded input. For these systems, stability depends on the roots of the characteristic...

routh-hurwitz criterion i

Checking stability with the Routh table is a useful part of analyzing a closed-loop transfer function. It helps determine whether a system will remain stable when load conditions change or when a fault occurs. In a power grid, this kind of check is important because instability can lead to blackouts. The Routh-Hurwitz criterion uses a table built from the denominator polynomial of the transfer function. The rows of the Routh table are labeled by powers of the complex variable s, starting with...

routh-hurwitz criterion ii

Routh Table Special Cases

Routh table special cases can make stability analysis harder when using the Routh-Hurwitz criterion. Two situations need extra care. One is a zero in the first column. The other is an entire row of zeros. When a single zero appears in the first column, division by zero would occur. The usual fix is to replace that zero with a very small number, epsilon (ε). The stability test then continues by assuming a sign for ε. If ε is positive, any sign change in the first column means the system is...

key terms and concepts

Learn how the Routh-Hurwitz criterion uses a Routh table to test system stability by tracking sign changes and pole locations in the s-plane.

Checking Stability with the Routh Table | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

Checking Stability with the Routh Table

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