Investigation of metamaterial with extended constitutive relationships by using transmission line circuit theory

An equivalent circuit representation of the metamaterials with extended constitutive relationships which are coupling between the electric field and the magnetic field are derived. Those are analogized in coupled transmission lines and reciprocity, passivity, and reactivity are investigated by using this equivalent circuit. We obtain analytically that these metamaterial are non-reciprocal in general, but are reciprocal when there are opposite coupling coefficients between the electric field and the magnetic field. And they are lossless when the coupling coefficients are imaginary. To confirm the validity of our results, the frequency response S₂₁ = -S₁₂ is calculated using the proposed equivalent circuit.