Application of Belevitch Theorem for Pole-Zero Analysis of Microwave Filters With Transmission Lines and Lumped Elements

This paper presents the application of Belevitch theorem for pole-zero analysis of microwave filters synthesized with transmission lines and lumped elements. The scattering (S) matrix determinant (Δ) based on the Belevitch theorem, aptly called Belevitch determinant, comprises poles and zeros that are separated in different half-plane regions. Using the Belevitch determinant, the poles and zeros of filter transfer functions can be determined separately with certainty, e.g., by applying the contour integration method based on argument principle. Note that the contour integration can be evaluated numerically without requiring complicated overall analytical expressions. The proposed method is able to solve the poles and zeros for filters synthesized with noncommensurate transmission lines and lumped elements, where the transform method and the eigenvalue approach are inapplicable. Several applications are discussed to demonstrate the use of Belevitch theorem and the contour integration method to determine the poles and zeros of various microwave filters on the complex plane.