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Related Concept Videos

Plotting and Calibrating the Root Locus01:19

Plotting and Calibrating the Root Locus

378
Root loci often diverge as system poles shift from the real axis to the complex plane. Key points in this transition are the breakaway and break-in points, indicating where the root locus leaves and reenters the real axis. The branches of the root locus form an angle of 180/n degrees with the real axis, where n is the number of branches at a breakaway or break-in point.
The maximum gain occurs at the breakaway points between open-loop poles on the real axis, while the minimum gain is...
378
Construction of Root Locus01:15

Construction of Root Locus

345
The construction of a root locus involves several key steps to analyze and visualize the behavior of a system's poles with varying gain. The number of branches in the root locus equals the number of closed-loop poles and is symmetrical about the real axis.
For positive gain values, the root locus exists on the real axis to the left of an odd number of finite open-loop poles or zeros. The root locus starts at the open-loop poles and traces the paths of the closed-loop poles as the gain...
345
Time and frequency -Domain Interpretation of Phase-lead Control01:24

Time and frequency -Domain Interpretation of Phase-lead Control

398
Phase-lead controllers are commonly used in various control systems to enhance response speed and stability. Adjusting the brightness on a television screen offers a practical example of phase-lead control. When contrast is enhanced, a phase-lead controller is employed. Mathematically, phase-lead control is identified when the first parameter is smaller than the second.
The design of phase-lead control involves the strategic placement of poles and zeros to balance steady-state error and system...
398
Root Loci for Positive-Feedback Systems01:23

Root Loci for Positive-Feedback Systems

284
The Hartley oscillator is a positive feedback system that sustains oscillations by feeding the output back to the input in phase, thereby reinforcing the signal. Positive feedback systems can be viewed as negative feedback systems with inverted feedback signals. In these systems, the root locus encompasses all points on the s-plane where the angle of the system transfer function equals 360 degrees.
The construction rules for the root locus in positive feedback systems are similar to those in...
284
Properties of the Root Locus01:05

Properties of the Root Locus

263
The root locus method is an invaluable tool for analyzing higher-order systems without needing to factor the denominator of the transfer function. A pole of the system is identified when the characteristic polynomial in the transfer function's denominator equals zero.
To determine if a point lies on the root locus, the criterion involves the sum of angles contributed by all poles and zeros to that point. Specifically, this sum must be an odd multiple of 180 degrees. The gain at any point on...
263
Time and frequency -Domain Interpretation of Phase-lag Control01:21

Time and frequency -Domain Interpretation of Phase-lag Control

364
Phase-lag controllers are widely used in control systems to improve stability and reduce steady-state errors. A dimmer switch controlling the brightness of a light bulb serves as a practical example of phase-lag control, gradually adjusting the bulb's brightness. Mathematically, phase-lag control or low-pass filtering is represented when the factor 'a' is less than 1.
Phase-lag controllers do not place a pole at zero, but instead influence the steady-state error by amplifying any...
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Locating fixed points in the phase plane.

Yanhua Zhang1, Yeyin Zhao1, Lizhu Chen2

  • 1Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079, China.

Physical Review. E
|December 25, 2019
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Summary
This summary is machine-generated.

This study introduces a novel method to precisely identify critical points in finite-size scaling. By analyzing the width of scaled observables, researchers can determine phase transition temperatures and scaling exponent ratios for various physical systems.

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Area of Science:

  • Statistical Mechanics
  • Condensed Matter Physics
  • Computational Physics

Background:

  • Critical points are fundamental in understanding phase transitions.
  • Finite-size scaling is a key technique for analyzing critical phenomena.
  • Characterizing fixed points requires precise determination of scaling exponents.

Purpose of the Study:

  • To develop a robust method for quantifying critical points in finite-size scaling.
  • To precisely determine phase transition temperatures and scaling exponent ratios.
  • To distinguish between critical points, first-order phase transitions, and crossover regions.

Main Methods:

  • Defining the width of scaled observables for varying system sizes at specific temperatures and scaling exponent ratios.
  • Identifying the fixed point by locating the minimum of this width.
  • Applying the method to the three-dimensional three-state Potts model.

Main Results:

  • The minimum of the scaled observable width accurately reveals the fixed point's position.
  • The method successfully determines phase transition temperatures and scaling exponent ratios.
  • The determined ratio effectively classifies the nature of the fixed point.

Conclusions:

  • The proposed method offers a more precise and effective approach than conventional techniques for analyzing critical phenomena.
  • This technique has potential applications in high-energy physics, such as the Beam Energy Scan at the Relativistic Heavy Ion Collider.