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's poles and zeros. The resulting root locus diagram provides a visual representation of the system'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.