Name: root locus breakaway and stability points
Uploaded: 2024-11-21T14:54:13.000Z
Duration: 1 min 19 s
Channel: JoVE

근좌는 일반적으로 시스템 극점이 실수 평면에서 복소 평면으로 전환될 때 발산합니다.
이탈 지점과 길입 지점은 궤적이 실제 축을 떠나고 다시 결합하는 위치를 나타냅니다. 루트 궤적 분기는 실수 축과 180/n 각도를 형성합니다.
이득은 실축의 개방 루프 극 사이의 분리 지점에서 피크를 이루고, 최소 이득은 두 0 사이의 침입 지점에서 발생합니다.
게인이 증가하면 일부 시스템 극이 오른쪽 절반 평면으로 밀려날 수 있으며, 이는 잠재적인 불안정성을 나타냅니다. jω축 교차는 안정적인 시스템 작동과 불안정한 시스템 작동 사이의 경계를 표시합니다.
루트 궤적 분석에는 특정 점을 찾고 관련 이득을 계산하는 작업이 포함됩니다.
특정 감쇠비 선을 교차할 때 근 궤적의 정확한 좌표를 알기 위해 선을 따라 여러 테스트 지점을 선택하고 각도 합계를 평가합니다.
근 궤적은 총 각도의 합이 180도의 홀수 배수와 같은 곳에 존재합니다.
해당 특정 지점에서의 이득은 극 길이의 곱을 해당 지점까지의 길이가 0인 곱으로 나누어 계산됩니다.

루트 로커스 플로팅 및 교정

루트 로커스는 종종 시스템 극이 실수 축에서 복소 평면으로 이동함에 따라 발산합니다. 이 전환의 핵심 지점은 루트 로커스가 실수 축을 떠나 다시 들어오는 지점을 나타내는 분리 및 분리 지점입니다. 루트 로커스의 가지는 실수 축과 180/n도 각도를 형성하는데, 여기서 n는 분리 또는 분리 지점의 가지 수입니다.

최대 이득은 실수 축의 오픈 루프 극 사이의 이탈 지점에서 발생하는 반면, 최소 이득은 두 개의 0 사이의 이탈 지점에서 관찰됩니다. 이득이 증가함에 따라 일부 시스템 극은 오른쪽 반평면으로 밀려나 잠재적 불안정성을 나타낼 수 있습니다. jω-축의 교차점은 안정된 작업과 불안정한 작업의 경계를 그립니다.

근궤적 분석은 이러한 중요한 지점을 식별하고 해당 이득을 계산하는 것을 포함합니다. 특정 감쇠비 선을 교차할 때 근궤적의 정확한 좌표를 결정하기 위해 이 선을 따라 다양한 테스트 지점을 선택하고 시스템의 극과 영점에서 이러한 지점까지의 각도의 합을 평가합니다. 근궤적은 총 각도가 180도의 홀수 배수인 곳에 존재합니다.

루트 로커스의 한 지점이 확인되면 해당 지점의 이득을 계산할 수 있습니다. 이는 극에서 지점까지의 거리의 곱을 0에서 지점까지의 거리의 곱으로 나누어서 수행합니다. 이 접근 방식은 시스템의 극이 이득을 변경하면서 어떻게 이동하는지 확인하고 안정성과 과도 응답을 평가하는 데 도움이 됩니다.

요약하자면, 근궤적 분석은 이득이 변함에 따라 시스템 극의 동작을 시각화하고 이해하기 위한 포괄적인 방법을 제공합니다. 엔지니어는 이탈 및 이탈 지점을 조사하고 특정 지점에서 이득을 계산하여 안정성과 원하는 성능 특성을 보장하는 제어 시스템을 설계하고 미세 조정할 수 있습니다. 이 방법은 시스템 극이 어떻게 움직이는지 예측하고 견고한 제어 시스템 설계를 보장하는 데 필수적입니다.

근좌는 일반적으로 시스템 극점이 실수 평면에서 복소 평면으로 전환될 때 발산합니다.
이탈 지점과 길들이기 지점은 궤적이 어디에 있는지 알려줍니다 ...

Education

교육

JoVE Core

Electrical Engineering

전기공학

Unit 1

제1단원

Root-Locus Method

근궤적법

KEY TERMS AND CONCEPTS

주요 용어 및 개념

Plotting and Calibrating the Root Locus

root locus breakaway and stability points

Root locus analysis tracks how system poles move as gain changes. It shows when poles leave the real axis, move into the complex plane, and affect stability. For high school students studying control systems, the key ideas are breakaway points, break-in points, and jω-axis crossings.

Breakaway points are locations where the root locus leaves the real axis. Break-in points are locations where it returns to the real axis. At either type of point, the branches meet the real axis at an angle of 180/n degrees, where n is the number of branches at that point.

Gain also changes at these points. The maximum gain occurs at breakaway points between open-loop poles on the real axis. The minimum gain occurs at break-in points between two zeros. As gain increases, some poles may move into the right half-plane, which can signal instability.

The jω-axis crossings mark the boundary between stable and unstable operation. Root locus analysis uses these crossings to show how far the system can move before stability changes. It also helps find the corresponding gain at a chosen point on the locus.

To locate a point on a damping ratio line, several test points are checked along that line. The angles from the system poles and zeros to each point are added. The point lies on the root locus when the total angle is an odd multiple of 180 degrees.

Once a point is confirmed, the gain can be calculated from distances. The product of the distances from the poles to the point is divided by the product of the distances from the zeros to the point. This method helps predict how poles migrate with changing gain and supports control system design with the desired stability and transient response.

Root Locus Breakaway and Stability Points

root-locus-breakaway-and-stability-points

control system problem

Root Locus and Closed-Loop Stability

Root locus and closed-loop stability are key ideas in control systems. A basic thermostat is an open-loop system, so its transfer function poles affect the response but do not set stability. When feedback is added, the system becomes closed-loop, like an advanced thermostat that changes heating based on room temperature. In a closed-loop system, stability depends on the new poles of the closed-loop transfer function. If those poles move into the unstable region, the system can start to...

vector representation of complex numbers

Complex Numbers as Vectors in Polar Form

Complex numbers can be shown as vectors, not just as points in Cartesian coordinates. In polar form, a vector is described by its magnitude, or length, and its angle. This makes complex numbers easier to compare by size and direction. The same idea also helps describe complex functions. When a complex number goes into a function, the output is another complex number. The function can be viewed as having a zero point, and the vector picture starts from that point. For a function written as a...

root-locus method

Cruise Control Stability with Root Locus

Root locus helps show how cruise control stays stable as conditions change. In a car, cruise control keeps a set speed by adjusting the gas pedal automatically. It keeps checking the car’s speed and makes small pedal changes to hold that target speed. This system can be drawn as a block diagram, and its transfer function gives the math model. The model is useful when the car goes uphill, downhill, or faces strong wind resistance. To find the system poles for different pedal forces, the...

properties of the root locus

Root Locus Angle and Gain Rules

Root locus angle and gain rules help analyze higher-order control systems. The method works without factoring the denominator of the transfer function. It is especially useful for unity feedback systems, where the transfer function can be studied by tracking how poles and zeros shape the response. A pole is a point where the characteristic polynomial in the denominator of the transfer function equals zero. To check whether a point lies on the root locus, use the angle criterion. Add the angles...

construction of root locus

Root Locus Paths and Asymptotes

Root locus paths and asymptotes show how closed-loop poles move as gain changes. This method helps students track system stability by following the pole locations on a graph. It also shows where the poles start, where they may end, and how they travel in between. The number of root locus branches equals the number of closed-loop poles. The plot is symmetric about the real axis. For positive gain, the root locus lies on the real axis to the left of an odd number of finite open-loop poles or...

plotting and calibrating the root locus

Root locus analysis tracks how system poles move as gain changes. It shows when poles leave the real axis, move into the complex plane, and affect stability. For high school students studying control systems, the key ideas are breakaway points, break-in points, and jω-axis crossings. Breakaway points are locations where the root locus leaves the real axis. Break-in points are locations where it returns to the real axis. At either type of point, the branches meet the real axis at an angle of...

root loci for positive-feedback systems

Root Locus Rules for Oscillator Feedback

Root locus rules help describe how oscillator feedback systems behave as gain changes. The Hartley oscillator is one example of a positive feedback system. It keeps oscillating by feeding the output back to the input in phase, which reinforces the signal. Positive feedback systems can also be treated as negative feedback systems with the feedback signal inverted. In root locus analysis, the locus is the set of points on the s-plane where the transfer function angle equals 360 degrees. The...

key terms and concepts

Learn how root locus breakaway and break-in points, jω-axis crossings, and gain calculations help predict pole motion, stability, and response in control systems.

Root Locus Breakaway and Stability Points | 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

Root Locus Breakaway and Stability Points

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