Coplanar wave guides-CPW
A coplanar line is a structure in which all the conductors supporting wave propagation are located on the same plane, i.e. generally the top of a dielectric substrate. There exist two main types of coplanar lines: the first, called coplanar waveguide (CPW), that we will study here, is composed of a median metallic strip separated by two narrow slits from a infinite ground plane, as may be seen on the figure below.
The characteristic dimensions of a CPW are the central strip width and the width of the slots . The structure is obviously symmetrical along a vertical plane running in the middle of the central strip.
The other coplanar line, called a coplanar slot (CPS) is the complementary of that topology, consisting of two strips running side by side.
Quasi-static analysis by conformal mappings
A CPW can be quasi-statically analysed by the use of conformal mappings. Briefly speaking, it consists in transforming the geometry of the PCB into another conformation, whose properties make the computations straightforward. The interested reader can consult the pp. 886 – 910 of  which has a correct coverage of both the theoretical and applied methods. The French reader interested in the mathematical arcanes involved is referred to the second chapter of  (which may be out of print nowadays), for an extensive review of all the theoretical framework. The following analysis is mainly borrowed from , pp. 375 et seq. with additions from .
The CPW of negligible thickness located on top of an infinitely deep substrate, as shown on the left of the figure below, can be mapped into a parallel plate capacitor filled with dielectric using the conformal function: