Name: root locus paths and asymptotes
Uploaded: 2024-11-21T14:54:13.000Z
Duration: 1 min 15 s
Channel: JoVE

근 궤적 구성에는 폐쇄 루프 극의 수와 동일한 분기의 수를 결정하는 작업이 포함됩니다. 루트 궤적은 실제 축에 대해 대칭입니다.
근궤적(root locus)은 실수 축의 일부에 존재하며, 왼쪽의 유한한 개루프 극점과 0의 총 개수의 합이 홀수입니다.
개루프 극에서 시작된 궤적은 이득이 증가함에 따라 폐루프 시스템 극을 추적하고 개방 루프 0에서 끝나며, 여기서 시스템 극은 이득이 상승함에 따라 안정화됩니다.
s가 무한대에 가까워짐에 따라 무한대에 접근하는 함수는 무한대에서 극점을 갖습니다. 0에 접근하면 무한대에서 0이 됩니다.
근 궤적이 무한대를 향해 이동함에 따라 점근선으로 알려진 특정 직선에 접근합니다. 주어진 시스템에 대해 실제 축 절편과 각도가 계산됩니다. 게인이 증가함에 따라 각도가 반복되기 시작합니다.
실축 절편과 각도를 기반으로 하는 그들의 방정식은 극에서 무한대까지의 loci의 경로를 보여줍니다.
궤적은 개루프 극점에서 시작하여 개루프 0에서 끝납니다.

루트 로커스의 구축

루트 로커스의 구성에는 다양한 이득을 가진 시스템 극의 동작을 분석하고 시각화하기 위한 몇 가지 핵심 단계가 포함됩니다. 루트 로커스의 분기 수는 닫힌 루프 극의 수와 같으며 실수 축에 대해 대칭입니다.

양의 이득 값의 경우 근궤적은 홀수의 유한한 개방 루프 극 또는 영점의 왼쪽에 있는 실수 축에 존재합니다. 근궤적은 개방 루프 극점에서 시작하여 이득이 증가함에 따라 폐쇄 루프 극점의 경로를 추적합니다. 이득이 계속 증가함에 따라 시스템 극점이 안정화되는 개방 루프 영점에서 끝납니다.

함수가 s가 무한대에 가까워질 때 무한대에 가까워지면, 무한대에 극이 있습니다. 반대로 함수가 0에 가까워지면 무한대에 영이 있습니다. 근궤적이 무한대로 확장됨에 따라 특정 점근선을 따릅니다. 이러한 점근선 방정식은 실수 축 절편과 각도에 의해 결정되며, 이는 궤적이 극에서 무한대로 이동하는 경로를 나타냅니다.

주어진 시스템에 대한 이러한 경로를 계산하려면 실수 축 절편과 점근선의 각도를 결정합니다. 각도는 극과 영점의 수에 따라 도출되며 이득이 증가함에 따라 반복되기 시작합니다. 점근선 각도에 대한 공식은 다음과 같습니다.

<img alt="Equation1" src="/files/ftp_upload/16122/16122_PT_Equation_001.svg" style="height: 30px;">

여기서 k는 정수이고, n는 극의 개수이고, m는 영점의 개수입니다.

완전한 근궤적 플롯은 오픈 루프 극에서 시작하여 오픈 루프 제로에서 끝납니다. 이 플롯은 근궤적 방법을 지배하는 규칙을 준수하여 시스템의 안정성과 응답을 철저히 분석하고 예측할 수 있도록 합니다.

이러한 단계를 따르면 엔지니어는 루트 로커스 방법을 효과적으로 사용하여 제어 시스템을 설계하고 조정하여 다양한 작동 조건에서 원하는 성능과 안정성을 보장할 수 있습니다. 이 접근 방식은 시스템 폴이 이득이 증가함에 따라 어떻게 이동하는지 명확하게 시각화하여 제어 전략의 견고한 설계에 도움이 됩니다.

근 궤적 구성에는 폐쇄 루프 극의 수와 동일한 분기의 수를 결정하는 작업이 포함됩니다. 근 궤적은 ...

Education

교육

JoVE Core

Electrical Engineering

전기공학

Unit 1

제1단원

Root-Locus Method

근궤적법

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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 zeros. The paths begin at the open-loop poles and move as gain increases.

As gain continues to rise, the root locus ends at the open-loop zeros. At those points, the system poles settle. If a function goes to infinity as s approaches infinity, it has a pole at infinity. If a function goes to zero, it has a zero at infinity.

When the root locus extends toward infinity, it follows asymptotes. An asymptote is a line that shows the direction of a branch far from the poles and zeros. The asymptotes are found from the real-axis intercept and their angles. The angle formula depends on the number of poles, n, the number of zeros, m, and an integer k. These angles repeat as gain increases.

A complete root locus plot starts at the open-loop poles and ends at the open-loop zeros. It follows the rules of the root locus method so stability and response can be studied and predicted. Engineers use this plot to design and tune control systems for the desired performance across different operating conditions. The result is a clear picture of how system poles migrate with increasing gain.

Root Locus Paths and Asymptotes

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 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 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...

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 branches move from open-loop poles to zeros, use asymptotes, and show how gain affects stability in control systems.

Root Locus Paths and Asymptotes | 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 Paths and Asymptotes

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