Unitary Condition, Group Delay and Quality Factor in Microwave Filters

In this paper, the unitary condition of scattering parameters for lossless and lossy filters is investigated. The primary objective is to establish a mathematical framework that explains the limitations observed in filter amplitude response sharpness at passband edges during practical measurements. It is achieved by analytically evaluating the unitary condition using a closed form coupling matrix model. Various numerical examples are presented to demonstrate the impact of loss on the unitary condition and filter sharpness in practical scenarios. Moreover, we propose a mathematical form of the T term based on the extension to the unitary condition, serving as a gauge to quantify the rounded passband edges in filter response. This T term is related to filter design parameters and the resonators' quality factor, enabling a deeper understanding of filter performance in real-world conditions. The developed theory implication is discussed in analyzing the measurement results of three different filter in different frequency bands.