Direct Synthesis of Complex Loaded Chebyshev Filters in a Complex Filtering Network

In this paper, a direct synthesis approach is proposed for designing a Chebyshev filter that matches the frequency variant complex loads at both the ports. The approach is based on the power wave renormalization theory and two practical assumptions: 1) the prescribed transmission zeros are stationary and 2) the reflection zeros are located along an imaginary axis. Three conditions are derived to stipulate the characteristic polynomials of the filter's responses through the renormalization of reference load impedances. These conditions are sequentially applied to ensure that the filter is physically realizable and well matched to the complex loads. It has been shown that the optimally synthesized filtering network that consists of an ideal filter cascaded with a piece of transmission line of an optimal length at each port can match the complex loads over a given frequency band with the best effort. By incorporating with an adjustment of the junction parameters, the approach provides an analytical yet flexible way to synthesize advanced microwave circuits composed of multiple filters connected together through some generalized junctions. The effectiveness of the method is demonstrated through three synthesis examples.