Name: root locus angle and gain rules
Uploaded: 2024-11-21T14:54:13.000Z
Duration: 1 min 5 s
Channel: JoVE

The root locus method helps analyze higher-order systems without requiring the factorization of the denominator of the transfer function.
A pole is identified when the characteristic polynomial in the transfer function&#39;s denominator equals zero.
The criteria for a point to be on the root locus are that the total angle contribution from all zeros minus the total angle contribution from all poles, is equal to an odd multiple of 180 degrees.
The gain at any point on the root locus is determined by dividing the product of the pole lengths by the product of the zero lengths.
Consider a transfer function for a unity feedback system. The angle at a specific point is the algebraic sum of angles of vectors drawn from the system&#39;s zeros and poles of the transfer function to that point.
Since the angle equals an odd multiple of 180 degrees, the point is on the root locus.
The gain can be computed by dividing the pole lengths product by the zero lengths product.

eigenschappen van de wortelplaats

De wortelplaatsmethode is een onschatbaar hulpmiddel voor het analyseren van systemen van hogere orde zonder dat de noemer van de overdrachtsfunctie hoeft te worden ontbonden. Een pool van het systeem wordt geïdentificeerd wanneer de karakteristieke polynoom in de noemer van de overdrachtsfunctie gelijk is aan nul.

Om te bepalen of een punt op de wortelplaats ligt, omvat het criterium de som van de hoeken die door alle polen en nullen aan dat punt worden bijgedragen. Deze som moet specifiek een oneven veelvoud van 180 graden zijn. De winst op elk punt op de wortelplaats wordt gevonden door het product van de lengtes van de polen tot het punt te delen door het product van de lengtes van de nullen tot het punt.

Voor een eenheidsfeedback-systeem kan de overdrachtsfunctie met deze methode worden geanalyseerd. De hoek op een specifiek punt op de wortelplaats wordt berekend door de hoeken van de nullen en polen van het systeem tot dat punt op te tellen. Om te verifiëren of een punt deel uitmaakt van de wortelplaats, moet deze som gelijk zijn aan een oneven veelvoud van 180 graden.

Zodra een punt bevestigd is op de wortelplaats, kan de versterking op dat punt bepaald worden door de afstanden van de polen en nullen van het systeem tot het punt te vergelijken. Dit houdt in dat het product van de afstanden van elke pool tot het punt berekend wordt en gedeeld wordt door het product van de afstanden van elke nul tot het punt.

Deze methode is met name nuttig bij het ontwerp en de analyse van besturingssystemen, waardoor ingenieurs kunnen voorspellen hoe veranderingen in systeemparameters de stabiliteit en respons beïnvloeden. Door de wortelplaats te begrijpen, kunnen ingenieurs systemen ontwerpen die de gewenste prestatiekenmerken behouden, waardoor stabiliteit wordt gegarandeerd in een reeks bedrijfsomstandigheden.

Samenvattend biedt de wortelplaatsmethode een systematische aanpak voor het analyseren van systemen van hogere orde door zich te richten op de hoeken en afstanden van polen en nullen tot een bepaald punt. Deze techniek helpt de stabiliteit en prestaties van een systeem te bevestigen bij wisselende versterkingen, waardoor het een essentieel hulpmiddel is bij het ontwerpen en analyseren van besturingssystemen.

Eigenschappen van de wortelplaats

The root locus method helps analyze higher-order systems without requiring the factorization of the denominator of the transfer function.
A pole is ...

Education

JoVE Core

Electrical Engineering

Elektrotechniek

Unit 1

Root-Locus Method

Root-locus-methode

KEY TERMS AND CONCEPTS

BELANGRIJKE TERMEN EN CONCEPTEN

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 from all poles and zeros to that point. The total must be an odd multiple of 180 degrees.

Once a point is confirmed on the root locus, the gain at that point can be found from distances. Multiply the lengths from each pole to the point. Then divide by the product of the lengths from each zero to the point. This gives the gain for that location on the root locus.

These rules are useful in control system design and analysis. They help engineers predict how changes in system parameters affect stability and response. By using root locus, engineers can design systems that keep desired performance across a range of operating conditions. In this way, the method offers a systematic way to study higher-order systems through angles and distances from poles and zeros.

Root Locus Angle and Gain Rules

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 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 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 uses angle criteria and pole-zero distances to check points, find gain, and study stability in unity feedback control systems.

Root Locus Angle and Gain Rules | 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 Angle and Gain Rules

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