General Synthesis for Microwave Filters With Frequency Linear Dependent Couplings Using Circuit Approach

This article extends the circuit-based approach to synthesizing coupling matrices (CMs), allowing for the inclusion of frequency-dependent couplings (FDCs). First, it provides a comprehensive study of the inherent properties of FDCs, revisiting the minimum path rule and revealing that extending from frequency-invariant couplings (FICs) to FDC amplifies nonphysical traits, such as the generation of second-order zeros in local circuits. When FDC is used as input or output (I/O) coupling, it inevitably introduces right-half-plane (RHP) roots into the overall network. However, these nonphysical traits can be managed during physical circuit construction, allowing FDC circuits to effectively guide microwave filter design. Second, within the CM framework, the article proposes an analytical method for synthesizing FDC-coupled networks, including I/O couplings. A novel and straightforward synthesis approach based on circuit element extraction is introduced, eliminating the need for intermediate steps involving frequency-invariant matrices or extracted pole (EP) circuits. Furthermore, an analytical method for the de-normalization of directly coupled FDC filters is presented, facilitating dimensional synthesis. The versatility and effectiveness of the proposed method are demonstrated through various synthesis examples, including detailed physical dimension calculations and validation against measured data.