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

The root locus construction involves determining the number of branches that equals the number of closed-loop poles. The root locus is symmetrical about the real axis.
The root locus exists on the part of the real axis where the sum of the total number of finite open-loop poles and zeros on the left is odd.
Originating at open-loop poles, the locus traces closed-loop system poles with increasing gain and ends at open-loop zeros, where system poles stabilize with rising gain.
A function that approaches infinity as s approaches infinity has a pole at infinity. If it approaches zero, it has a zero at infinity.
As the root locus moves towards infinity, it approaches certain straight lines known as asymptotes. For a given system, the real-axis intercept and angle are calculated. As the gain increases, the angles start repeating themselves.
Their equation, based on real-axis intercept and angle, reveals the loci&#39;s path from poles to infinity.
The locus originates from the open-loop poles and terminates at the open-loop zeros.

constructie van de wortellocus

Het construeren van een root locus omvat verschillende belangrijke stappen om het gedrag van de polen van een systeem met variërende gain te analyseren en visualiseren. Het aantal takken in de root locus is gelijk aan het aantal closed-loop polen en is symmetrisch ten opzichte van de reële as.
Voor positieve gainwaarden bevindt de root locus zich op de reële as links van een oneven aantal eindige open-loop polen of nullen. De root locus begint bij de open-loop polen en volgt de paden van de gesloten-loop polen naarmate de gain toeneemt. Hij eindigt bij de open-loop nullen, waar de systeempolen stabiliseren naarmate de gain blijft stijgen.
Als een functie oneindig nadert terwijl s oneindig nadert, heeft deze een pool op oneindig. Aan de andere kant, als een functie nul nadert, heeft deze een nul op oneindig. Naarmate de root locus zich uitstrekt naar oneindig, volgt deze specifieke asymptoten. De vergelijkingen van deze asymptoten worden bepaald door het snijpunt van de reële as en hun hoeken, die de paden aangeven die de loci nemen van de polen naar oneindig.
Om deze paden voor een gegeven systeem te berekenen, bepaalt men de intercept van de reële as en de hoeken van de asymptoten. De hoeken worden afgeleid op basis van het aantal polen en nullen en beginnen zich te herhalen naarmate de winst toeneemt. De formule voor de hoek van asymptoten wordt gegeven door,
<img alt="Equation1" src="/files/ftp_upload/16122/16122_PT_Equation_001.svg" style="height: 30px;">
waarbij k een geheel getal is, n het aantal polen en m het aantal nullen.
De complete root locus plot begint bij de open-loop polen en eindigt bij de open-loop nullen. Deze plot houdt zich aan de regels die de root locus methode beheersen, waardoor de stabiliteit en respons van het systeem grondig geanalyseerd en voorspeld kunnen worden.
Door deze stappen te volgen, kunnen engineers de root locus-methode effectief gebruiken om controlesystemen te ontwerpen en af ​​te stemmen, en zo de gewenste prestaties en stabiliteit te garanderen in verschillende bedrijfsomstandigheden. Deze aanpak biedt een duidelijke visualisatie van hoe de systeempolen migreren met toenemende versterking, wat helpt bij het robuuste ontwerp van controlestrategieën.

Constructie van de wortellocus

The root locus construction involves determining the number of branches that equals the number of closed-loop poles. The root locus is symmetrical about ...

Education

JoVE Core

Electrical Engineering

Elektrotechniek

Unit 1

Root-Locus Method

Root-locus-methode

KEY TERMS AND CONCEPTS

BELANGRIJKE TERMEN EN CONCEPTEN

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

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root-locus method

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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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