On the Functional Relation Between Quality Factor and Fractional Bandwidth

TL;DR

This paper explores the relationship between quality factor definitions and fractional bandwidth in resonant systems, revealing that only certain definitions maintain proportionality in first-order systems and highlighting limitations in higher-order systems.

Contribution

It compares various quality factor definitions and clarifies their proportionality to fractional bandwidth in different order systems, providing insights for antenna and resonator design.

Findings

Only the impedance differentiation-based Q is proportional in first-order systems.

Classical Q is not proportional to fractional bandwidth.

Higher-order systems break the proportionality of impedance-based Q.

Abstract

The functional relation between the fractional band-width and the quality factor of a radiating system is investigated in this note. Several widely used definitions of the quality factor are compared on two examples of RLC circuits that serve as a simplified model of a single resonant antenna tuned to its resonance. It is demonstrated that for a first-order system, only the quality factor based on differentiation of input impedance has unique proportionality to the fractional bandwidth, whereas e.g. the classical definition of the quality factor, i. e. the ratio of the stored energy to the lost energy per one cycle, is not uniquely proportional to the fractional bandwidth. In addition, it is shown that for higher-order systems the quality factor based on differentiation of the input impedance ceases to be uniquely related to the fractional bandwidth.