Lagrangian formulation of a linear microstrip resonator: theory and experiment

A Lagrangian approach is used to formulate the electromagnetic scattering properties of a linear microstrip resonator. The resonator design includes a center microstrip separated from the source and output loads by dielectric gaps. The gaps of the resonator are represented by capacitively coupled π-networks whereas the Lagrangian is formulated in the discrete limit such that the equations of motion transform into two boundary conditions, relating the waves on the two sides of the gap in terms of two lumped capacitors. The effective capacitors expressed in the boundary conditions are actually the capacitances experienced by the even and odd excitation modes of the system, respectively. The calculations, therefore take into account the characteristic impedance, effective dielectric constant, conductivity of the metal strip and the ground plane, and dielectric loss tangent of the dielectric material. In addition, the two capacitor values representing the gaps are included in a consistent manner with the above parameters by the application of the Lagrangian formulation. Calculated and measured reflection coefficients of linear microstrip resonators are compared, and general agreement is found between theory and experiments