FAST AND SIMPLE CAD-ORIENTED CLOSED-FORM FORMULAS FOR DOUBLE-SIDED COPLANAR STRIP LINES Ceyhun Karpuz 1 and Adnan Gorur 1 ¨¨ 1 Department of Electrical and Electronics Engineering Faculty of Engineering and Architecture Nigde University 51100 Nigde, Turkey Recei  ed 11 January 1999 ABSTRACT: Closed-form analytical expressions suitable for CAD soft- ware packages are presented for calculating the quasistatic TEM parame- ( ) ters of the double-sided coplanar strip line DS CPS . It has been shown that the present transmission line reduces to practical  alues the charac- teristic impedance of the con entional CPS. Furthermore, the effect of dielectric o erlaying on the quasistatic parameters of the DS CPS is discussed by using the present expressions deri  ed by the conformal mapping techniques. This paper’s results are compared with the results a ailable in the literature.  1999 John Wiley & Sons, Inc. Microwave Opt Technol Lett 22: 215218, 1999. Key words: CPS; double-sided CPS; dielectric o erlaying; conformal mapping INTRODUCTION Because of the recent developments in monolithic microwave Ž . integrated circuits MMICs , better understanding is required of the characteristics of various coplanar-type transmission   lines. To date, many authors 1  5 have extensively studied Ž . symmetric coplanar strip lines CPSs due to their flexibility in the design of complex microwave circuitry requiring shunt and series connections. Due to these properties as well as several advantages, such as excellent propagation with linear dispersion, less sensitivity to substrate thickness, small dis- continuity parasitics, and eliminating the need of via holes, the coplanar strip line is suitable for the emerging wireless communications industry, and for the design of low-cost uniplanar microwave circuits such as mixers, filters, antennas,  and optoelectronic devices 6 . Briefly, CPS supports all of the advantages of the conventional CPW, and also, they are efficient in the use of wafer area.  Nguyen 7 has presented the dispersive properties of the Ž . broadside-coupled CPW BC CPW using the spectral-do- main method, and has shown that a quasistatic TEM ap- proach can be used for designing practical microwave and millimeter-wave circuits employing broadside-coupled CPW because of their very low dispersion. However, to our knowl- edge, there is only a limited number of publications dealing Ž . with double-sided coplanar strip lines DS CPSs , whereas the coplanar strip line is a transmission medium suitable for Ž . application to microwave integrated circuits MICs , and to phase arrays as a feed line. In one of these publications, BC CPW directional couplers for double-sided printed antennas have been proposed, and have been analyzed by utilizing the  rectangular boundary division method 8 . Tefiku, Yamashita, and Funada have discussed a section of a broadside-coupled micro CPW which consists of two broadside-coupled strips placed near two asymmetric coplanar ground planes, as a part    of a BC CPW directional coupler 8, Fig. 2b . Lee 9 has presented the quasistatic parameters of a double-sided CPS employing the well-known moment method based on a quasi- TEM analysis. As is well known, the quasi-TEM approximation is valid   only at low frequencies. However, Bedair and Wolff 10 have shown that a simple quasistatic approximation can be used for BC CPW analysis because of their very low dispersion.  Also, Goverdhanam, Simons, and Katehi 4 have shown that the conventional CPS has excellent propagation characteris- tics with linear dispersion. Hence, the quasi-TEM approxima- tion can be used to discuss the CPS propagation characteris- tics, at least up to the millimeter-wave frequency range. In other words, the characteristic impedance and effective di- electric constant may be calculated using analytic formulas based on the conformal mapping method because conformal mapping techniques lead to closed-form analytical solutions suitable for CAD software packages. Therefore, the purpose of this study is to present the fast, simple, and accurate CAD-oriented analytic expressions for calculating the qua- sistatic parameters of the DS CPS using conformal mapping techniques. FORMULATION Figure 1 shows the cross-section view of a double-sided Ž . dielectric-overlaid DSD CPS which consists of two strips of width w with spacing s. A coplanar strip line is symmetrically printed on each side of a dielectric substrate whose thickness is h and whose relative dielectric constant is  . In addition, 1 r 1 another dielectric material whose thickness is h and whose 2 relative dielectric constant is  is covered on each of the r 2 CPSs. All conductors are assumed to be infinitely thin and perfectly conducting. It is assumed that the air  dielectric or dielectric  dielectric interfaces, where the strip conductors are located, can be dealt with as though perfect magnetic walls are present in them. It should also be noted that, as  done by Lee 9 , we have only analyzed the odd-mode param- eters between the strips for practical applications. In this case, the overall capacitance per unit length of the structure can be considered as the sum of the capacitances in the air and dielectric layer. The quasistatic parameters of the DS CPS are evaluated by assuming that an electric wall is present at the centerline A A  in the vertical direction and a mag- netic wall at the center line B B  in the horizontal direction. The total air capacitance per unit length can be considered as the sum of two components C and C , representing the a1 a2 Ž . Ž. electric field in region I and region II  III im Figure 2 a , Ž respectively. The expressions for these two components C a1 Figure 1 Cross section of double-sided coplanar strip line MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 22, No. 3, August 5 1999 215