Design and synthesis of lossy microwave filters

The design of microwave filters starts from the derivation of a defined lowpass prototype network. A general lossy synthesis method is given which can 1) derive the reflection function from the transfer function when the unitary condition is not satisfied; 2) find the expressions for the complex admittance parameters and 3) synthesize the lossy coupling matrix (CM) with prescribed loss distributions. Two special cases are discussed for solving the refection function from a prescribed transfer function. An alternative approach to cope with loss is studied. In a transversal array, some resonators can be replaced by their low-Q alternatives to reduce the manufacture cost as well as the cavity size. The exact values for the dissipations of resonators or couplings can be determined analytically or by methods of gradient based optimizations. A method of CM synthesis with non-ideal load is given which can be used in designing diplexers or multiplexers. Filter networks matching to complex load impedances can be found by renormalizing reference impedances. An iteration method is introduced which can deal with frequency variant load and can deliver the required reflection zeros. A method for the synthesis of directional filters is presented which can be used for designing combiners. While each section of directional filters provides a 1st order response, more complex filter characteristics can be realized by cascading those single sections. By proper transformations, directional filter networks can be realized using normal resonators and couplings. An example utilizing coaxial resonator is given. A method for the analysis of 2-D lumped element networks is presented. The method is based on the general telegrapher’s equations of multi-wire transmission lines. A 2-D lumped element network is equivalent to a combination of sub-networks which support single mode propagations. The method can be applied to the analysis of metamaterials and can be used for the design of waffle-iron filters.