Frequency Design of Lossless Passive Electronic Filters: A State-Space Formulation of the Direct Synthesis Approach

This paper deals with the frequency design of lossless passive electronic filters under magnitude constraints. With the huge increase in design complexity for mobile applications, new systematic and efficient methods are required. This paper focuses on the direct synthesis approach, an historical design approach that has not been recently updated. It consists in directly synthesizing the $LC$ values of a pre-specified circuit until the spectral mask is satisfied. While beneficial in practice, this approach typically leads to an important computational time and requires an initial guess to reduce it. Based on recent developments of the System and Control community, that led to efficient methods for system design, the direct synthesis approach is revisited. To achieve this, the port-Hamiltonian Differential Algebraic Equation (pHDAE) representation, that particularly fits the design problem, is introduced. A synthesis method is then developed, leading to solve an optimization problem of moderate complexity. For particular cases, this complexity happens to be remarkably low. Based on this observation, a second method reveals how to obtain such complexity for the more general case, using an original combination between the pHDAE and the LFT representations. Finally, a numerical example shows the validity and illustrates the benefits of this work.