Analysis of L-L transmission line metamaterials with Coupled inductances

Here we solve the transmission-line equations for a metamaterial having shunt and series inductances electromagnetically coupled together. We derive and analyze the anomalous wave properties that such a metamaterial may have, envisioning its realization both as a transmission-line metamaterial and as a bulk complex material whose electromagnetic relations may be read in terms of the present theory. The dependence of the wave impedances on the coupling factor between the inductive elements is derived in closed form, showing an anomalous distinct behavior between the incident and reflected waves.