Analytical closed-form solutions for different configurations of parallel-coupled microstrip lines

Closed-form solutions are presented for the even- and odd-mode impedances of the parallel-coupled microstrip lines with different configurations. The first set of equations is for the conventional-coupled microstrip lines. The second set is for the parallel-coupled microstrip lines with a slotted ground plane and the third set is for the parallel-coupled microstrip lines with a slotted ground plane and a floating-potential conductor added to cover most of the slot in the ground plane. The solution is achieved using the conformal mapping technique, which also considers the effect of any fringe capacitor that has an impact on the impedance of the coupled lines. To validate the accuracy of the presented equations, a comparison is made with the available empirical solution (only for the conventional-coupled microstrip lines), a numerical solution (using Galerkin´s technique) and the full-wave electromagnetic solution using the software HFSS. The result of the comparison indicates the high accuracy of the presented equations over a wide range of design parameters. The difference between the calculated values using the derived equations and those from the full-wave electromagnetic analysis is <5% for most of the cases. To validate the derived equations experimentally, they are used to design a 3 dB coupler. The result of measurements on the manufactured coupler confirms the accuracy of the presented equations.