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Modeling of Realistic Rectangular -Coaxial Lines Milan Lukic´, Student Member, IEEE, Sébastien Rondineau, Member, IEEE, Zoya Popovic´, Fellow, IEEE, and Dejan S. Filipovic´, Member, IEEE

Abstract—A comprehensive study of small inhomogeneous multilayered rectangular coaxial lines (RCLs) with irregular cross sections is presented in this paper. An accurate and efficient quasi-analytical technique based on numerical implementation of simply and doubly connected Schwarz–Christoffel conformal mapping is utilized for modeling. Misaligned and offset layers, under–over cutting due to etching, nonuniform dielectric support, and other nonidealities due to fabrication of RCLs are studied and their effects on characteristic impedance, attenuation, bandwidth, and powerhandling capacity are discussed. The validity of obtained results is verified with suitable published data, analytical models, and/or finite-element simulations. Index Terms—Attenuation, characteristic impedance, cutoff frequency, power-handling capacity, rectangular coaxial line (RCL), Schwartz–Christoffel mapping.

I. INTRODUCTION N RECENT years, there has been an increased interest in manufacturable low-loss high-density TEM lines for millimeter-wave passive circuits [1]–[4]. A number of applications across the microwave spectrum can benefit from such lines. For example, Alessandri et al. [5] demonstrated a high-power low-loss low-weight compact high-performance rectangular coaxial line (RCL) (or recta-coax) beam-forming networks in the lower microwave range for communication satellites. Latest advances in microfabrication techniques make feasible realization of highly integrated assemblies of miniature TEM structures. Several low-loss recta-coax lines with heights from 50 to 400 m have been built using different types of surface -band filter [7] micromachining. A -band hybrid [6] and a are representative examples of components manufactured using the process described in [2]. A copper/polymer/air RCL that can be fabricated with a different multilayer photolithographic process [1] is shown in Fig. 1. Early interest in RCLs was centered around the theoretical approaches for determining their characteristic impedance . The most common technique is conformal mapping (CM) [8]–[11]. Bowman [8] computed the capacitance per unit length of the square coaxial line (SCL) and his findings have been widely used by others for validating different, mainly numerical techniques. Another CM solution for a more general class of concentric rectangular conductors is derived in [9].

I

Manuscript received October 27, 2005; revised December 31, 2005. This work was supported by the U.S. Army Research Laboratory under Contract W911QX-04-C-0097, by the Defense Advanced Research Projects Agency–Microsystems Technology Office under the 3-D Micro Electromagnetic Radio Frequency Systems Program, and by BAE Systems under Subcontract 1165974. The authors are with the Department of Electrical and Computer Engineering, University of Colorado at Boulder, Boulder CO 80309-0425 USA (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2006.872792

Fig. 1. Sketch of a copper/polymer/air RCL composed of five planar layers. The horizontal walls of the outer conductor are assembled from layers 1 and 5, while the vertical walls are put together from layers 2–4. The inner conductor is a part of the third layer and it is supported by dielectric slabs firmly placed h 250 m and w h between layers 2 and 3. For a 50- SCL, w 100 m. Dielectric support is 15-m thick, and 100-m long, and periodicity is varied from 300 to 700 m.

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Chen [10] applied the same method to various L- and U-shaped bends and developed approximate formulas for both symmetrical and eccentric RCLs, valid only for low-impedance lines (small gap between conductors). Numerical inversion of the Schwarz-Christoffel conformal mapping (SCCM) is proposed in [11] where the accuracy of several approximate formulas of both symmetrical and eccentric RCLs are assessed. for In [12], approximate expressions for the conductor loss of an RCL are derived using Wheeler’s incremental inductance rule [13]. Other methods utilized for analysis of RCL include the orthonormal block analysis [14], finite differences [15], [16], finite-element method (FEM) [17], and various other numerical techniques [18]–[22]. In this paper, we demonstrate a quasi-analytical modeling approach based on two numerical implementations of the SCCM technique for analysis of miniature recta-coax lines. A three-dimensional (3-D) view of such a structure is shown in Fig. 1. The line is fabricated with five layers with a 2 : 1 maximum height/ width aspect ratio. The inner conductor is supported by integrated dielectric straps. The processing details are beyond the scope of this paper and can be found in [1]. The conformal-map-band micro-RCLs is performed ping analysis for realistic for frequencies up to -band, however, the presented results are obtained at 26 GHz. Ansoft HFSS was used for two-dimensional (2-D) and 3-D mode analysis, and for the validation of the SCCM results. This paper is organized as follows. • Section II briefly describes the simply and doubly connected SCCM techniques.

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• Section III gives analysis results for an SCL (characteristic impedance, attenuation, TEM mode bandwidth) in terms of various fabrication parameters. • Section IV analyzes the effects of the inhomogeneous dielectric support on the line performance. • Section V discusses the power-handling capacity, the performance of a 250- m-high RCL and several issues associated with numerical implementation. II. MODELING Riemann’s mapping theorem from 1851 [23] provides the foundation for CM, a technique which transforms geometries by preserving local angles. Since 1923 when it was first applied to electrostatics, [24], CM has been employed for solving various boundary value problems in electromagnetics [25], [26]. The most commonly used transformations belong to the Schwarz–Christoffel family [27]–[31]. These transformations remove the discontinuities in boundary conditions at sharp conductor corners. Section II-A briefly presents this formulation applied to realistic micro RCLs. A. Schwartz–Christoffel Mapping to Simply and Doubly Connected Polygonal Domain A simply connected planar domain can be defined as the interior of a planar closed line that does not contain any holes. When this enclosed domain contains exactly one hole, the domain is said to be doubly connected. 1) Notations and Basic Equations: 2-D real vectors can be represented as complex numbers . The gradient vector operator in the complex plane becomes a complex scalar operator

(1) where

and

are the partial derivatives w.r.t.

and . The

, where denotes Laplacian operator becomes the complex conjugate. As a consequence, and by considering of the plane properties described in [23], under a CM , the gradient and the Laplacian are given by and

tions [26]. In the case of symmetrical structures, it is sufficient and simpler to map a half, a quarter, or even an eighth of the geometry. This allows the use of a simply connected polygonal domain mapping function given by

(4) where represents the Jacobi elliptic function of argu[32], is a complex constant, and are ment the vertices and associated counter-clockwise interior -normalized angles of the RCL part to be mapped, as shown in Fig. 2(a). When there is no symmetry, the problem becomes more complicated and a doubly connected polygonal mapping function is needed as follows:

(5) with the auxiliary functions

,

, and , where is the internal to external radii ratio defined in Fig. 2(b) and is a comand are the vertices plex constant. and interior -normalized angles of the outer and inner conductors of the RCL, respectively [see Fig. 2(b)]. Equation (4) maps the field distribution of a simply connected domain into the field between two parallel plates without , where is the potential fringing at the edges: difference between the two parallel plates separated by a distance [see Fig. 2(a)]. Equation (5) maps the field distribution of a doubly connected domain into that of a concentric circular , where is the coaxial line (CCL): potential difference between the conductors [see Fig. 2(b)]. contains all information about the field vector Note that at the complex position . B. RCL Parameters 1) Capacitance: Since the potential distribution is Laplacian conservative through CM, the capacitance is conserved as well. In the simply connected and doubly connected cases, i.e., (4) and (5), the capacitance per unit length is given by

(2)

where is the derivative of . Applying this to quasi-static and satisfy electromagnetic fields, the electric potentials the Laplace equation. The electric field distributions, in both the and its image one , are then related using original plane the gradient transformation through the mapping function and

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(3)

This formulation is more general than those previously reported (limited to capacitance calculation) and is well suited for the quasi-static case because the derivatives are readily available as analytic expressions. 2) Application to RCLs: As shown in Figs. 1 and 2, an RCL is composed of two conductors, both with polygonal cross sec-

for the simply connected case for the doubly connected case

(6)

where is the parallel plate’s width, and is the uniform dielectric permittivity inside the coaxial line. 2) Characteristic Impedance: The TEM transmission-line is found from the capacitance per characteristic impedance , where . unit length 3) Attenuation: Line attenuation is commonly computed using the Wheeler incremental inductance (WII) rule [13]

(7)

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Fig. 2. Mapping of a simply connected polygonal domain onto: (a) a rectangle and a doubly connected polygonal domain onto: (b) an annulus. When a symmetry line is present in a doubly connected polygonal domain, such as the dashed horizontal line in (a), the domain can be split by symmetry into two simply connected domains. Either of these can then be mapped onto a rectangle. However, when no axial symmetry is present, as in (b), the mapping onto an annulus is used.

where represents the change in the characteristic impedance when conductor walls are receded by half of the skin depth. This rule is derived for a thick metal with small curvature. Note that the RCL in Fig. 1 meets the first assumption, but the second one is not valid at sharp corners. However, the WII rule has been widely used for many transmission lines with similar features, e.g., [33], and excellent agreement with measurements has been reported. III. ANALYSIS OF FABRICATION-INDUCED IMPERFECTIONS The formulation from Section II is utilized for the characterization of miniature recta-coax lines. In the case of a simply connected domain, the mapping function given by (4) is solved with the Schwarz–Christoffel toolbox integrated in MATLAB [27], [28]. For doubly connected domains, the evaluation of the mapping function is implemented in a FORTRAN 90 code based on the doubly connected Schwarz–Christoffel library [34]. The computed mapping parameters typically converge with accuracy better than 10 . A baseline geometry for this study is that of an SCL with a characteristic impedance of 50 . However, a lower loss 65line is also considered since it can be fabricated within the 2 : 1 height/width aspect ratio dictated by the fabrication process [1]. Table I summarizes the main properties of 50- and 65- SCL, , attenuation , specifically their characteristic impedance

TABLE I CHARACTERISTICS OF SCL WITH IDEAL CROSS SECTION (w = 250 m)

and first higher order mode cutoff frequency . Note that while is determined only by the ratio , the attenuation is dependent on the outer conductor width , operating frequency , and conductivity : . The subscript “0” denotes parameters of an SCL with an ideal cross section. When the line has imperfections, such as in Fig. 3, the characteristic impedance , attenuation , and first higher order mode cutoff frequency have the same dependence on , , and . Due to , , and , which this, normalized parameters do not depend on , , and , are used throughout this paper. The nonidealities caused by fabrication are discussed as follows: • vertically offset inner conductor due to tolerances in layer thickness [see Fig. 3(a)]; • horizontally offset layer due to mask misalignment [see Fig. 3(b)]; • trapezoidal cross section due to under/over etching [see Fig. 3(c) and (d)].

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Fig. 3. Cross section of a line with: (a) vertically offset inner conductor, (b) misaligned layers, (c) conductors of trapezoidal shape where the bottom sides of the conductors are kept constant, and (d) conductors of trapezoidal shape where the areas of the conductors are kept constant.

A. Inner Conductor Offset The relative vertical offset of the center conductor is defined as with reference to Fig. 3(a). Fig. 4 shows CM results compared to the 3-D FEM for normalized attenuation, characteristic impedance, and first higher order mode cutoff frequency as a function of relative offset. The cutoff frequency is calculated using 3-D FEM eigenanalysis of a cavity with length and the same cross section as the studied line. is extracted from the lowest resonant frequency of the cavity as . A summary of the results for 50and 65- SCL for 10% relative offset is given in Table II. Note that a 2-D FEM is also used for the mode analysis of various line irregularities. The two techniques are validated against Gruner’s results for SCL [35], as shown in Table III. The agreement is better than 0.05% with 2-D and 0.2% with 3-D FEM. From obtained results, we can conclude that this fabrication imperfection will produce minimal variations in the line performance. B. Layer Misalignment The relative horizontal misalignment of the center layer is dewith reference to Fig. 3(b). Fig. 5 shows CM fined as results compared to the 3-D FEM for normalized attenuation,

Fig. 4. Computed results for normalized  , Z , and f for 50- and 65- lines with cross section shown in Fig. 3(a) versus relative offset of the inner conductor. Results for  and Z are validated with 3-D FEM and 2-D FEM simulations, respectively, where  is extracted from the simulated S -parameter using the equation jS j = e , where l is the length of the simulated line section. Small deviations (less than 0.2%) of 3-D FEM results for  from the SCCM results are within the numerical accuracy of 3-D FEM.

characteristic impedance, and first higher order mode cutoff frequency as a function of relative offset. In this case, the results

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TABLE II RELATIVE CHANGES OF Z ,  , AND f FOR LINE WITH VERTICALLY h =w = 10%) OFFSET INNER CONDUCTOR (jh

0 j

TABLE III NORMALIZED CUTOFF WAVELENGTH FOR THE TE

MODE OF AN SCL

Fig. 6. Simulation results for normalized  , Z , and f for symmetrical ( =  = ) trapezoidal 50- and 65- lines where bottom conductor widths are kept constant [see Fig. 3(c)]. FEM simulation results obtained with HFSS are given for validation. Shown variations of  are significantly larger than for other three geometries of Fig. 3, thus small discrepancies between FEM and SCCM results are not as apparent as in Figs. 4, 5, and 7.

Fig. 5. Computed results for normalized  , Z , and f for 50- and 65- lines with cross section shown in Fig. 3(b) versus relative misalignment among the layers. FEM simulations obtained with HFSS are given for validation. Deviations of FEM results for  from the SCCM results are smaller than 0.4%, while the corresponding characteristic impedances are virtually indistinguishable. TABLE IV RELATIVE CHANGES OF Z ,  , AND MISALIGNED LAYERS (2d=w

f

FOR

LINE WITH

= 10%)

for the 50- and 65- lines are nearly identical, thus, the latter and of the are omitted for clarity. Note that the effects on misalignment of layers 2 and 4 are independent. A summary of the results for 10% misalignment is given in Table IV. As seen, the horizontally misaligned layers will introduce small changes in the line performance and, for most practical cases (1%–5%), these can be ignored. C. Under/Over Etching The two lines with trapezoidal conductor cross sections (model of under/over cutting due to etching) and relevant parameters are shown in Fig. 3(c) and (d). In the first case, widths of the bottom sides of the conductors ( , ) are kept constant, while in the second case, their surface areas remained unchanged. Normalized , , and for symmetrical trapezoidal 50- and 65- lines are shown in and for a trapezoidal Figs. 6 and 7. The variations of 65- line for both geometries are noticeably smaller than for a

Fig. 7. Simulation results for normalized  , Z , and f for symmetrical ( =  = ) trapezoidal 50- and 65- lines where conductor areas are kept constant [see Fig. 3(d)]. FEM simulations obtained with HFSS are shown for validation. Variations of  , for the studied range of angle  values, are much smaller than for the previous case in which bottom conductor widths were kept constant. Consequently, small discrepancies between the FEM and SCCM results for  (less than 0.4%) are clearly observable while the results for Z remain virtually indistinguishable.

corresponding 50- line. The summarized results for from Table V show that the case where the conductor areas are kept constant is much more tolerant to under/over etching. The practical importance of these results is that known statistics of the cross-sectional dimensions can be used in design so that and are maintained throughout the nominal values for the structure. As before, the FEM mesh-dependent differences may be noticed, while the corresponding characteristic for impedances are virtually indistinguishable. and for trapezoidal line Contour plots of normalized [see Fig. 3(c)] are shown in Figs. 8 and 9. For small values of and are twice larger for angle , the variations of both than for the asymmetrical the symmetrical case , ), indicating that the effects of left- and case ( right-hand-side slants are independent.

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TABLE V RELATIVE CHANGES OF Z ,  , AND f FOR LINE WITH TRAPEZOIDAL CONDUCTOR CROSS SECTIONS ( =  =  = 10 )

Fig. 9. Contour plot of normalized  for trapezoidal 50- line of Fig. 3(c) as function of angles  and  .

Fig. 8. Contour plot of normalized Z for trapezoidal 50- line of Fig. 3(c) as function of angles  and  .

IV. INNER CONDUCTOR SUPPORT In Section III, homogeneous air-filled irregular cross-sectional SCLs were discussed. However, in practice, the inner conductor must be supported somehow. Here, a thin dielectric layer suspended between the vertical walls (Fig. 1) is used for the support [1]. Note that a dielectric with a dielectric constant greater than unity increases the line attenuation even when there is no energy loss in the dielectric itself. This is due to the fact that the dielectric decreases , and from [33], the attenuation increases. For a given transmitted power, requires a higher current, hence the loss in the a lower conductors increases. The attenuation constant of the line with is obtained as (e.g., both conductor and dielectric losses , where is the attenuation constant [33]) due to dielectric losses. Below we discuss both continuous and periodic dielectric supports.

Fig. 10. Comparison of the results for normalized  and Z versus normalized support height for an SCL with continuous dielectric supports of the inner conductor obtained by SCCM (solid lines for 50- cable and a dashed line for 65- cable) and 3-D FEM (dots).

line, and m for a 65- line, penetration depth of m, and dielectric conthe supports into the sidewalls . It can be seen that the contribution of to stant is very significant even for moderate values of . For this reason, the line of Fig. 1 is built using the periodic support studied in Section IV-B.

A. Continuous Support

B. Periodic Support

and versus normalized support Results for normalized height for a 50- SCL with continuous dielectric supports of the are inner conductor for different values of loss tangent shown in Fig. 10. Also shown in this figure are results for normalized for a 65- SCL. However, the results for normalized for a 65- line are omitted for clarity since they are almost identical as those for a 50- line. An excellent agreement with 3-D FEM results can be observed. The geometrical dimensions and physical parameters of the lines are: outer conductor width m, inner conductor width m for a 50-

The variation of for a 50- SCL with periodic dielectric supports of the inner conductor (see Fig. 1) for different is shown in Fig. 11. The height values of loss tangent m and m, and length of the supports are respectively, and the other geometrical and physical parameters is are the same as for the case of continuous support. Here, plotted as a function of the separation between consecutive supnormalized with respect to their length . It can be ports seen that the losses decrease monotonically with the increase of . The results are obtained by cascading the matrices

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Fig. 11. Comparison of the results for normalized  for an SCL with periodic dielectric supports of the inner conductor obtained by SCCM (solid lines) and 3-D FEM (dots). The SCCM results are obtained by cascading the ABCD matrices of line sections with and without dielectric supports.  = is plotted as a function of the supports periodicity L normalized with respect to their length L .

of line sections with and without dielectric supports and are validated with 3-D FEM simulations. The effective loss tangent of the line section with dielectric support is calculated by the for[36], where mula is the effective dielectric constant of the line section calculated using 2-D FEM. The small discrepancies between the two techniques are likely due to the numerical issues associated with the FEM and the inability of the SCCM to accurately account for the field effects at the transitions between homogeneous and inhomogeneous RCL sections. V. DISCUSSION A. Power-Handling Capacity From (4) and (5), it is clear that the electric-field distribution, given by (3), has a singularity at each exterior vertex of the ideal inner conductor. The electric field cannot be obtained at these perfectly sharp corners, however, perfect edges do not exist in practice. These edges are slightly chamfered and the field is computed at the middle of the chamfer. It is found that the ratio of chamfer to inner conductor dimensions of 10 is the smallest that gives converging results. This technique of chamfering the inner conductor to deal with finite field strength leads to the computation of the maximum transmitted power before air breakdown. The algorithm for computing the power handling capacity of a CCL can be found in [37]. A comparison of the normalized power and attenfor circular (CCLs), square (SCLs), uation as a function of and RCLs, for given outer conductor size, is shown in Fig. 12. The RCL studied here has square outer conductor of width equal to that for the studied SCL, and the inner conductor of height and width between and . As seen, the optimal characteristic impedance for maximum power handling for the three coaxial lines is approximately 30, 22, and 44 , respectively. The optimal characteristic impedance for minimum attenuation of an SCL is approximately 74.9 , just slightly

Fig. 12. Normalized power and attenuation as a function of Z for circular (CCL), square (SCL), and RCL for given outer conductor size. As seen, the optimal Z for maximum power handling for CCL and SCL is approximately 30 and 22 , respectively, while the corresponding values of Z for minimum attenuation are approximately 76.6 and 74.9 . The results are obtained using the SCCM. Analytical data for CCL are not shown for clarity (there is no visible difference with the computed ones).

lower than that for a CCL, which is approximately 76.6 . Notice that the power levels for an RCL are normalized with respect to the maximum power of an SCL, while for the other two geometries (circular and square), power levels are normalized with their own maximum values. Likewise, the attenuation for an RCL is normalized with respect to the minimum attenuation of an SCL, while for the other two geometries, the attenuation is normalized with their own minimum values. As clearly depicted in the same figure, the normalized power of an RCL is lower than that of an SCL with the same characteristic impedance. Observe also that the normalized attenuation of an RCL is higher than for an SCL with the same characteristic impedance. Normalized powers of 50- and 65- CCLs are 85.5% and 67.4%, respectively, while the corresponding values for SCLs are 89.7% and 80.4%. Thus, the ratio of normalized powers of 65- and 50- CCLs is 78.8%, while this power ratio for SCLs has a significantly larger value of 89.6%. B. Performance of a 250- m-High RCL Specific applications may require either the height or the width of the line to be restricted to a certain value. In such cases, the desired line dimensions that amount to the best performance are needed. Assuming that the heights of outer and inner conductors ( and ) are fixed to the values they have in the 50SCL from Section III, i.e., m and m, we will compute the line characteristics for widths ( and ) as the design parameters. The results of this study for , , normalized with respect to the corresponding values and for the baseline 50- SCL are plotted in Fig. 13. First higher , except for the line order mode for the studied lines is and for which that mode is with . As expected, the lines with wider outer conductor have significantly reduced bandwidth for the TEM mode operation. and , the range For . The of characteristic impedances is four dashed vertical lines in Fig. 13, connecting 50- points on curves with corresponding points on curves, have

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is less affected by the fabrication imperfections. Geometrical irregularities caused by a photolithographic fabrication process are thoroughly studied. The effects of the fabrication incurred deformations of the line cross section, such as misaligned and offset layers, under–over cutting due to the etching and nonuni, attenform dielectric support, on characteristic impedance , and TEM mode bandwidth have been presented. uation Ansoft HFSS has been used for the higher order mode analysis, and for the validation of the SCCM results. A discussion on power-handling capacity and the RCL characterization for the fixed height has also been presented. ACKNOWLEDGMENT Fig. 13. Normalized  , Z , and f for RCL versus inner conductor width w for different outer conductor widths w . The heights of outer and inner conductors (h and h ) are kept constant. The four dashed vertical lines connect 50-

points on Z curves with corresponding points on  curves. For the line with w = 1:2h and w > 1:6h , the first higher order mode is TE , while for all other lines studied here, this mode is TE .

abscissas and , respectively. With the exception of the line with and , (attenuation the studied lines have attenuation lower than of the 50- SCL). For example, the four 50- RCLs discussed and , respectively. above have C. Some Numerical Issues Numerical implementation of a simply and doubly connected SCCM is discussed in details in [27]–[29]. One of the numerical issues encountered in this study is ill conditioning due to crowding. This occurs whenever the target mapping region has areas that are relatively long and thin. We have addressed this phenomenon by utilizing mapping based on a cross-ratios of the Delaunay triangulation (CRDT) algorithm [29]. For homogeneous air-filled RCLs, the SCCM (to eight accurate digits) is typically more than ten times faster than 2-D FEM and approximately 50 times faster than (the less accurate) 3-D FEM. For the RCLs with dielectric support, effective dielectric and its derivative with respect to the support diconstant need to be known for the SCCM electric constant computation of line parameters. These can be obtained using, for example, 2-D FEM and the data can be fitted to maintain the time and memory savings of SCCM. The efficiency of SCCM compared to the 3-D FEM is even more evident in the case of the line with periodic dielectric support where the line sections with matrices. Adand without supports are cascaded using ditional computational time for cascading line sections is negligible while a 3-D FEM simulation of the entire structure takes much longer than 3-D or 2-D FEM simulations of individual homogeneous sections. VI. SUMMARY An efficient quasi-analytical technique based on numerical implementation of simply and doubly connected SCCM is used for modeling of small inhomogeneous multilayered RCLs. A lower loss alternative to the 50- recta-coax, specifically a 65line, was also investigated, and it is shown that its performance

The authors would like to thank G. Potvin and D. Fontaine, both with BAE Systems, Nashua, NH, Dr. C. Nichols and the Rohm and Haas Company 3-D Micro Electromagnetic Radio Frequency Systems (MERFS) team, Blacksburg, VA, Dr. J. Evans, Defense Advanced Research Projects Agency (DARPA)–Microsystems Technology Office (MTO), Arlington, VA, E. Adler, Army Research Laboratory (ARL), Adelphi, MD, and K. Vanhille, University of Colorado at Boulder, for useful discussions and support. REFERENCES [1] Rohm & Haas Electronic Materials LLC, “Coaxial waveguide microstructures and methods of formation thereof,” U.S. Patent 7012489 B2, Mar. 2006. [2] E. R. Brown, A. L. Cohen, C. A. Bang, M. S. Lockard, G. W. Byrne, N. M. Vendelli, D. S. McPherson, and G. Zhang, “Characteristics of microfabricated rectangular coax in the Ka band,” Microw. Opt. Technol. Lett., vol. 40, p. 365, Mar. 2004. [3] I. H. Jeong, S. H. Shin, J. H. Go, J. S. Lee, C. M. Nam, D. W. Kim, and Y. S. Kwon, “High performance air-gap transmission lines and inductors for millimeter-wave applications,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2850–2855, Dec. 2002. [4] J.-B. Yoon, B.-I. Kim, Y.-S. Choi, and E. Yoon, “3-D construction of monolithic passive components for RF and microwave IC’s using thick-metal surface micromachining technology,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 1, pp. 279–288, Jan. 2003. [5] F. Alessandri, M. Mongiardo, and R. Sorrentino, “Computer-aided design of beam forming networks for modern satellite antennas,” IEEE Trans. Microw. Theory Tech., vol. 40, no. 6, pp. 1117–1127, Jun. 1992. [6] J. R. Reid and R. T. Webster, “A 60 GHz branch line coupler fabricated using integrated rectangular coaxial lines,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2004, pp. 441–444. [7] R. T. Chen, E. R. Brown, and C. A. Bang, “A compact low-loss Ka-band filter using 3-dimensional micromachined integrated coax,” in 17th IEEE Int. Microelectromech. Syst. Conf., Jan. 2004, pp. 801–804. [8] F. Bowman, Introduction to Elliptic Functions. New York: Dover, 1961. [9] H. J. Riblet, “The exact dimensions of a family of rectangular coaxial lines with given impedance,” IEEE Trans. Microw. Theory Tech., vol. MTT-20, no. 8, pp. 538–541, Aug. 1972. [10] T. S. Chen, “Determination of the capacitance, inductance and characteristic impedance of rectangular lines,” IEEE Trans. Microw. Theory Tech, vol. MTT-8, no. 9, pp. 510–519, Sep. 1960. [11] E. Costamagna and A. Fanni, “Analysis of rectangular coaxial structures by numerical inversion of the Schwarz–Christoffel transformation,” IEEE Trans. Magn., vol. 28, no. 3, pp. 1454–1457, Mar. 1992. [12] K. H. Lau, “Technical memorandum: Loss calculations for rectangular coaxial lines,” Proc. Inst. Elect. Eng.—Microw., Antennas, Propag., vol. 135, no. 3, pp. 207–209, Jun. 1988. [13] H. A. Wheeler, “Formulas for the skin effect,” Proc. IRE, vol. 30, no. 9, pp. 412–424, Sep. 1942. [14] O. R. Cruzan and R. V. Garver, “Characteristic impedance of rectangular coaxial transmission lines,” IEEE Trans. Microw. Theory Tech., vol. MTT-20, no. 9, pp. 488–495, Sep. 1964.

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Milan Lukic´ (S’02) received the Dipl. Eng. degree in electrical engineering from the University of Banjaluka, Banjaluka, Bosnia and Herzegovina, in 1998, the M.S.E.E. degree from the University of Mississippi, University, in 2002, and is currently working toward the Ph.D. degree at the University of Colorado at Boulder. His research interests include multilayered rectangular waveguide dyadic Green’s functions, mode matching, conformal mapping (CM), transmission lines, and antennas. Mr. Lukic´ was the recipient of the 2002 Graduate Achievement Award presented by the University of Mississippi and the 1998 Gold Medal presented by the University of Banjaluka.

Sébastien Rondineau (M’04) received the Diplôme d’ Ingénieur en Informatique et Télécommunications degree in signal processing and telecommunications and the Ph.D. degree from the University of Rennes 1, Rennes, France, in 1999 and 2002, respectively. He is currently a Research Assistant Professor with the Microwave and Active Antenna Laboratory, Electrical and Computer Engineering Department, University of Colorado at Boulder. His research interests include the method of analytical regularization in computational electromagnetics, mode matching, conformal mapping (CM), propagation and scattering of waves, dielectric lenses, discrete lens arrays, and antennas.

Zoya Popovic´ (S’86–M’90–SM’99–F’02) received the Dipl. Ing. degree from the University of Belgrade, Serbia, Yugoslavia, in 1985, and the Ph.D. degree from the California Institute of Technology, Pasadena, in 1990. Since 1990, she has been with the University of Colorado at Boulder, where she is currently a Full Professor. She has developed five undergraduate and graduate electromagnetics and microwave laboratory courses and coauthored the textbook Introductory Electromagnetics (Prentice-Hall, 2000) for a junior-level core course for electrical and computer engineering students. Her research interests include microwave and millimeter-wave quasi-optical techniques, high-efficiency microwave circuits, smart and multibeam antenna arrays, intelligent RF front ends, RF optical techniques, batteryless sensors, and broadband antenna arrays for radio astronomy. Dr. Popovic´ was the recipient of the 1993 Microwave Prize presented by the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) for the best journal paper. She was the recipient of the 1996 URSI Isaac Koga Gold Medal. In 1997, Eta Kappa Nu students chose her as a Professor of the Year. She was the recipient of a 2000 Humboldt Research Award for Senior U.S. Scientists from the German Alexander von Humboldt Stiftung. She was also the recipient of the 2001 Hewlett-Packard (HP)/American Society for Engineering Education (ASEE) Terman Award for combined teaching and research excellence.

Dejan S. Filipovic´ (S’97–M’02) received the Dipl. Eng. degree in electrical engineering from the University of Nis, Nis, Serbia and Montenegro, in 1994, and the M.S.E.E. and Ph.D. degrees from The University of Michigan at Ann Arbor, in 1999 and 2002 respectively. From 1994 to 1997, he was a Research Assistant with the School of Electrical Engineering, University of Nis. From 1997 to 2002, he was a Graduate Student Research Assistant with The University of Michigan at Ann Arbor. He is currently an Assistant Professor with the University of Colorado at Boulder. His research interests are antenna theory and design, modeling and design of passive millimeter-wave components for future microelectromagnetic RF systems, as well as computational and applied electromagnetics. Mr. Filipovic´ was the recipient of the prestigious Nikola Tesla Award for his outstanding graduation thesis. He and his students were corecipients of the Best Paper Award presented at the IEEE Antennas and Propagation Society (AP-S)/ URSI and Antenna Application Symposium conferences.