ANALYTIC FORMULAS FOR CALCULATING THE QUASISTATIC PARAMETERS OF A MULTILAYER CYLINDRICAL COPLANAR STRIP LINE Adnan Gorur, 1 Mehmet Duyar, 2 and Ceyhun Karpuz 1 ¨¨ 1 Department of Electrical and Electronics Engineering Faculty of Engineering and Architecture Nigde University 51100 Nigde, Turkey 2 Department of Electrical and Electronics Engineering Aksaray Engineering Faculty Nigde University Aksaray, Turkey Recei  ed 16 March 1999 ABSTRACT: Fast, simple, and accurate analytic expressions are deri  ed for calculating the quasistatic TEM parameters of a multilayer cylindrical ( ) coplanar strip line MC CPS . The effects of the dielectric materials on the quasi-TEM characteristics are in estigated using the conformal mapping method. The deri  ed results are compared with those of a corresponding planar CPS. The cur  ature effects on the quasistatic parameters are also discussed. In addition, it has been obser  ed that the effect of the inner substrate on the quasistatic parameters is much less than the outer substrate.  1999 John Wiley & Sons, Inc. Microwave Opt Technol Lett 22: 432436, 1999. Key words: cylindrical CPS; conformal mapping; CAD-oriented formulas INTRODUCTION   To date, many authors 1  5 have extensively studied sym- Ž . metric coplanar strip lines CPSs on planar-shaped sub- strates due to the flexibility in the design of complex mi- crowave 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 discontinuity parasitics, and elimi- nating 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, the CPS supports all of the advantages of a coplanar Ž . waveguide CPW , which is complementary to CPW, and also, it is efficient in the use of wafer area. In addition, the CPS is useful in manufacturing lines with high characteristic impedances as it is easier to realize a high impedance with CPS than with CPW. Furthermore, due to its balanced con- figuration, it is also useful for balanced circuits such as mixers and for differential drivers such as those used in some high-speed digital applications. Unlike these structures, there is only a limited number of publications dealing with qua- sistatic parameters of CPS on cylindrically shaped substrates. We have reported the analytic expressions for quasi-TEM  parameters of the single-layer cylindrical CPS 7 . However, according to our knowledge, there is no study on cylindrical CPS with multilayered dielectrics, whereas, in practice, there are many circumstances in which the substrates are multi- layer, for example, in MICs, connection lines are either on or buried between dielectric layers. Here, the multilayer cylin- Ž . drical CPS MC CPS is analyzed using the conformal map- ping technique. As is well known, the quasi-TEM approxima- tion is valid only at low frequencies. However, Bedair and  Wolff 8 have shown that a simple quasistatic approximation can be used in the design of coplanar monolithic microwave Ž . integrated circuits MMICs up to 40 GHz because of their very low dispersion. Furthermore, Goverdhanam, Simons,  and Katehi 4 have shown, for conventional CPS structures, that there are no higher order modes up to the millimeter- wave frequency range. The CPS with smaller line dimensions and thinner substrates can increase the cutoff frequency to higher frequencies, thus extending the operating range. Namely, due to the fewer dispersion characteristics of CPS, the quasi-TEM approximation can be used to discuss the CPS propagation characteristics, at least up to the millimeter-wave frequency range. In this case, the characteristic impedance and effective dielectric constant may be calculated by the formulas based on the conformal mapping method because conformal mapping techniques lead to closed-form analytical solutions suitable for CAD software packages. Also, full-wave methods, such as spectral-domain analysis, the integral equa- tion method, and the finite-difference method, are mathemat- ically complex and require time-consuming numerical calcu- lations, and so they are not very attractive for the interactive CAD. Therefore, the aim of the study is to present fast, simple, and accurate CAD-oriented analytic expressions for the quasistatic parameters of the MC CPS. QUASISTATIC FORMULATION The configuration of the multilayer cylindrical CPS to be studied is shown in Figure 1, where the CPS line is sand- wiched between one top and two bottom cylindrical dielectric materials. In the following analysis, the two strips of width w are assumed to be infinitely thin and perfectly conducting, and these strips are separated by a distance of s. It is further assumed that the air  dielectric and dielectric  dielectric in- terfaces can be dealt with as though perfect magnetic walls are present in them. The three-layer cylindrical CPS shown in Figure 1 consists of three cylindrical CPSs. According to the   method of superposition of partial capacitances 8  11 , since the total capacitance C per unit length of the three-layer T cylindrical CPS is the sum of the capacitances of these CPSs, it can be written as Ž. C  C  C  C  C 1 T 0 1 2 3 where capacitance C is the free-space capacitance of the 0 CPS in the absence of all dielectrics, and capacitance C is 1 introduced by a dielectric substrate of thickness h having an 1 Ž . equivalent dielectric constant    . Similarly, the ca- r 1 r 2 Figure 1 Geometry of a multilayer cylindrical CPS MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 22, No. 6, September 20 1999 432