Propagation Constant Measurement of Microwave Networks With Symmetric/Asymmetric Reflections

Abstract

We proposed a Line-Line (LL) method for complex propagation constant $\gamma $ determination of uniform transmission lines or uniform lines loaded by samples having symmetric or asymmetric reflections using calibration-free scattering (S-) parameter measurements. To achieve our goal, we first derived pseudo wave-cascading-matrix (WCM) presentation of a network modeling a uniform line with asymmetric reflections ( $S_{11} \ne S_{22}$ ). Then, we proved that such a network can be represented by the product of three matrices: two matrices for impedance transitions and one matrix for phase and amplitude changes. While the first impedance transition matrix is a $2\times 2$ square matrix with unity values at diagonal entries and two different reflection coefficient values at off-diagonal entries, the second one is the inverse of the first one. Besides, the phase-amplitude change matrix is a diagonal matrix with diagonal entries corresponding to phase and amplitude changes. Next, using such representation of the pseudo WCM, we implemented the LL method to determine $\gamma $ of a network with asymmetric (or symmetric) reflections by a simple trace operation (summation of the diagonal terms in a square matrix) without resorting to ascertaining the reflection coefficient corresponding to an impedance transition. The method was validated by $\gamma $ measurements of an empty waveguide section with length 7.70 mm, the same section loaded by a polyethylene sample, and another waveguide section with length 10.16 mm loaded by bianisotropic metamaterial slabs in C-shape form.