Name: root locus rules for oscillator feedback
Uploaded: 2024-11-21T14:54:13.000Z
Duration: 1 min 23 s
Channel: JoVE

A Hartley oscillator, a positive feedback system, maintains oscillation by feeding the output back to the input in phase, reinforcing the signal.
Positive feedback systems can be interpreted as negative feedback systems with inverted feedback signals.
The root locus in these systems includes all s-plane points where the system transfer function angle equals 360 degrees.
These systems&#39; branches and symmetry rules mirror those in negative feedback systems.
However, the root locus exists on the real axis to the left of an even number of finite open-loop poles and/or zeros.
As the locus approaches infinity, it appears as straight lines, similar to asymptotes, with different slopes but identical intercepts on the real axis.
Negative-feedback systems with negative gain also follow these principles, forming an equivalent positive-feedback system by shifting the negative gain rightward past the pick-off point.
The root locus, left of an even count of real, finite open-loop poles and/or zeros, spans the entire positive extension of the real axis at designated intervals.
The intersections and angles are then determined to complete the sketch of the root locus.

root loci for positive-feedback systems

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 negative feedback systems. However, a key difference is that the root locus for positive feedback exists on the real axis to the left of an even number of finite open-loop poles and/or zeros.
As the root locus approaches infinity, it follows asymptotes, which are straight lines with varying slopes but identical real-axis intercepts. These asymptotes guide the paths of the loci as they extend to infinity.
In systems with negative feedback and negative gain, the root locus principles remain consistent. By shifting the negative gain rightward past the pick-off point, the system effectively behaves like a positive feedback system. Consequently, the root locus lies to the left of an even number of real, finite open-loop poles or zeros and spans the entire positive extension of the real axis at specified intervals.
To accurately sketch the root locus, one must determine the intersections and angles at which the loci depart from or approach the real axis. These points and angles are calculated based on the system&#39;s poles and zeros. The resulting root locus diagram provides a visual representation of the system&#39;s stability and response characteristics as the gain varies.
In summary, the Hartley oscillator exemplifies how positive feedback systems sustain oscillations through phase-aligned feedback. The root locus method, adapted for positive feedback, involves constructing loci to the left of an even number of open-loop poles or zeros on the real axis and extending towards infinity along asymptotes. This analysis is crucial for understanding and designing stable oscillatory systems and for predicting their behavior under different gain conditions.

Root Loci for Positive-Feedback Systems

A Hartley oscillator, a positive feedback system, maintains oscillation by feeding the output back to the input in phase, reinforcing the signal.
Positive ...

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Electrical Engineering

Unit 1

Root-Locus Method

KEY TERMS AND CONCEPTS

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 construction rules are similar to those used for negative feedback systems.

A key difference is the real-axis rule. For positive feedback systems, the root locus appears on the real axis to the left of an even number of finite open-loop poles and/or zeros. As the locus moves toward infinity, it follows asymptotes. These are straight lines with different slopes but the same real-axis intercept.

The same idea can apply when a system has negative feedback and negative gain. If the negative gain is shifted rightward past the pick-off point, the system behaves like a positive feedback system. The root locus then lies to the left of an even number of real, finite open-loop poles or zeros and spans the positive part of the real axis at the required intervals.

To sketch the root locus correctly, the departure and arrival points on the real axis must be found. Their angles are determined from the system poles and zeros. The finished root locus diagram shows how stability and response change as gain varies. The Hartley oscillator is a useful example of how positive feedback and root locus analysis work together in stable oscillatory design.

Root Locus Rules for Oscillator Feedback

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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 positive feedback, the Hartley oscillator, and root locus rules work together to show stability and oscillation as gain changes in control systems.

Root Locus Rules for Oscillator Feedback | 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 Rules for Oscillator Feedback

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