Attenuation Characteristics of Conventional, Micromachined, and Superconducting Coplanar Waveguides

Abstract

CLOSED FORM EQUATIONS FOR CONVENTIONAL CPW ATTENUATION CONSTANTCoplanar waveguide (CPW) circuits fabricated using standard photolithography process on copper-clad soft plastic-like sheets have a conductor thickness t several times the skin depth . Typically this thickness t is on the order of 17 m, which is the result of rolling a half ounce of copper over an area of one square feet [1]. Hence, for deriving an expression for the CPW attenuation constant, the conductors are considered to be electrically very thick and with conductivity the same as that of bulk copper. For the case of thick conductors a simple closed form expression for CPW attenuation constant is derived in [2] using conformal mapping technique.In monolithic microwave integrated circuits the conductor patterns are formed by evaporating or by sputtering precious metal such as gold on the surface of a dielectric substrate. These deposition processes yield conducting films with thickness on the order of a micron. The thickness is then built up to about 2.5 m by an electroplating process. The conductor thickness t in this case is said to be thin and in some cases may be less than or equal to the skin depth at the operating frequency. When t -, the fields penetrate into the conductors and the conductivity inside the conductors can be accounted for by introducing a complex dielectric constant. The CPW attenuation constant for this case is determined using accurate full wave analysis, such as the modematching method [3]. The disadvantage of the mode-matching method is that it requires significant amount of computer resources to carry out the computations. However, closed form expressions that approximate the results of the mode-matching method are also available [4].Besides thickness the cross section of the CPW conductors in MMICs may deviate from the ideal rectangular shape. The shape is strongly dependent on the fabrication process involved. In most cases the shape would be a trapezoid with 60°or 70°angles. The CPW attenuation constant for this case is derived by the modified matched asymptotic expansion technique [5], [6]. In this technique an approximate current density in first assumed on the CPW conductors and an equation for the CPW attenuation constant is derived. This equation involves a contour integral. Instead of evaluating this integral out to the edge of the conductors where the fields are singular, the limits of the integral are taken at some distance just before the edge. This short length before the edge is defined as the stopping distance [6]. Once is determined for a given edge shape as a function of t/ , a closed form expression is derived for the CPW conductor loss in [6]. In the subsections that follow closed form equations based on the foregoing numerical methods as well as measurementbased design equations are presented, and the corresponding computed attenuation constant is compared with measured data.