Root Loci for Positive-Feedback Systems

A Hartley oscillator is a positive feedback system that sustains oscillations by feeding the output back to the input in phase, reinforcing the signal. This phase-aligned feedback ensures the output continuously reinforces itself, maintaining steady oscillation without requiring external input to sustain the cycle.

In positive feedback systems, the root locus exists on the real axis to the left of an even number of finite open-loop poles and/or zeros, spanning the entire positive extension of the real axis. Negative feedback systems place the root locus to the right of an odd number of poles and zeros, making this distribution a fundamental distinguishing feature between the two system types.

Positive feedback systems can be interpreted as negative feedback systems with inverted feedback signals. This mathematical equivalence allows engineers to apply familiar negative feedback analysis techniques to positive feedback configurations by accounting for the signal inversion and shifting the negative gain rightward past the pick-off point.

The root locus in positive feedback systems includes all s-plane points where the system transfer function angle equals 360 degrees. This angle criterion determines which points in the complex plane represent valid closed-loop pole locations as system gain varies, establishing the fundamental geometric constraint for locus construction.

As the root locus approaches infinity, it follows asymptotes—straight lines with varying slopes but identical real-axis intercepts. These asymptotes guide the paths of the loci as they extend toward infinity, providing a framework for sketching the complete root locus diagram and predicting system behavior at high gain values.

The root locus branches and symmetry rules in positive feedback systems mirror those in negative feedback systems. Both follow identical construction principles regarding symmetry about the real axis, ensuring consistent analytical methods across feedback configurations and simplifying the sketching process for engineers.

To accurately sketch the root locus, 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's poles and zeros, and plotting and calibrating the root locus provides a visual representation of stability and response characteristics as gain varies.