Analysis and Synthesis of Double-Sided Parallel-Strip Transitions [PDF]

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 58, NO. 2, FEBRUARY 2010

Analysis and Synthesis of Double-Sided Parallel-Strip Transitions Pedro Luis Carro, Student Member, IEEE, and Jesus de Mingo, Member, IEEE

Abstract—Antenna feeders, mixers, and filters made in doublesided parallel-strip technology usually must be adapted to unbalanced lines like the microstrip structure, needing transitions from asymmetric to symmetric waveguides (baluns). In this paper, we propose a new method for the evaluation of a generic tapered balun based on a conformal-mapping technique and an integral equation. This method, along with the use of an optimization technique such as genetic algorithms, allows for quick evaluation of the return losses of any tapered balun and the synthesizing of specific shapes to achieve desired responses in terms of return losses or impedance values. Index Terms—Baluns, conformal mapping, parallel strips (PSs), passive circuits, ultra-wideband (UWB) technology.

I. INTRODUCTION

P

RINTED parallel-strip (PS) lines, as a balanced type of transmission lines, offer an interesting alternative to other printed transmission lines, such as the coplanar stripline, or to other unbalanced transmission structures such as the common microstrip (MS) line. Although PS topology received important attention more than 40 years ago, other transmission lines like the MS geometry were considered more interesting for millimetric applications. This was because of MS geometry’s outstanding features, including reasonable bandwidth, easy integration with active circuits, compact dimensions, and cheap manufacturing. There has been a growing interest in PS lines in recent years since they have many of the beneficial properties of MS lines. Printed PS lines are naturally balanced, without a ground plane, making them suitable for designing both passive and active microwave circuits. Additionally, they are used in antenna designs when almost omnidirectional radiation patterns are required, avoiding the possible complexities of MS antenna designs in order to achieve those patterns. We have to pay attention to the design process of millimeter integrated circuits with balanced and unbalanced devices since this situation usually implies transitions between both types of circuits called baluns. This may include impedance-matching capabilities, which are required, for example, when a PS antenna is connected to an MS line. Manuscript received November 04, 2008; revised September 22, 2009 and November 01, 2009. First published January 22, 2010; current version published February 12, 2010. This work was supported by the Spanish Government under Project TEC2008-06684-C03-02, MCI, and FEDER, the Gobierno de Aragón for WALQA Technology Park Project, and European IST Project EuWB. The authors are with the Department of Electronics and Communication Engineering, Universidad de Zaragoza, Zaragoza 50018, Spain (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2009.2038454

As far as double-sided PS and MS lines are concerned, the first balun geometry was proposed by Climer [1]. Some authors have used this balun for years in order to measure some antennas, couplers, feeding networks, and other balanced circuits. As recent examples of such circuits, printed filters and couplers are designed in [2], a bandpass filter is developed in [3] by inserting a ground plane between the strips, and [4] shows a diplexer based on a similar configuration. In spite of being extensively used in practical devices, there is not an analytical study of this balun in order to characterize its electromagnetic response (namely, the -matrix). In fact, Climer [1] analyzed the problem by estimating the even and odd voltages on the strips, computing the scattering matrix corresponding to two profiles (linear and exponential profiles). In addition, that study only focuses on perfect matching, which occurs when both the MS and PS lines have a characteristic impedance of 50 . Circular or more complex profiles were analyzed by means of an electromagnetic simulator (such as IE3D ) [5]. Alor High Frequency Structure Simulator (HFSS), though this procedure provides accurate results, it does not offer a clear physical interpretation (in terms of impedance variations) and it can be computationally expensive if a specific response is required. This paper introduces a novel fast semianalytical method used for computing the return losses of any PS to MS tapered balun. This approach is based on three different mathematical tools. First, we review the analysis method applied to any tapered transmission line geometry. This approach requires the knowledge of the characteristic impedance, which is, afterwards, worked out under the hypothesis of a quasi-TEM regime operation by applying a conformal mapping to an asymmetric double-sided printed transmission line. Finally, the new closedform formula is combined with the taper analysis method, introducing some examples of the analysis and synthesis of baluns. In terms of the analysis, return losses are computed by means of an integral approximation. The synthesis is carried out by an optimization procedure [using a genetic algorithm (GA)] that takes the analysis method into account, which provides an effective cost function for any desired electromagnetic response in terms of return losses. II. ANALYSIS OF MS TO PS TAPERED BALUNS The MS to double-sided PS tapered transition geometry consists of a (finite) ground plane, which is gradually converted into a strip, as in Fig. 1. The double-sided parallel line is achieved when the final strip is exactly identical in width to the nonground-plane strip. According to the results presented in [1], the electromagnetic performances of this transition depend on the taper applied in the gradual ground plane conversion.

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Fig. 3. Geometry cross section and parameters involved in the Schwarz– Christoffel transform ( -plane).

Fig. 1. Geometry of the printed balun transition with a general profile in the ground plane.

Fig. 2. Tapered transmission line parameters.

previously studied and evaluated [14], [15]. Hence, this analysis equation is potentially useful for computing the geometric taper to reach either optimum or specific return losses in a frequency band. The normalized impedance corresponding to the structure seems to be the main drawback in applying the previous formulation. Although the characteristic impedance of several printed transmission lines (including MS and coplanar strip lines and double-sided parallel transmission lines) has been reported in [16]–[19], to the authors’ knowledge, the case of asymmetrical double-sided parallel transmission lines has not yet been published in the literature. In Section III, we approximate such impedance in order to complete the analysis of the transition, applying a conformal-mapping technique. III. ASYMMETRICAL DOUBLE-SIDED TRANSMISSION LINE CHARACTERISTIC IMPEDANCE BY CONFORMAL MAPPING

From the physical point of view, this taper sets the characteristic impedance at any position along the balun. Therefore, the reflected power depends on this variation in . The analysis of nonuniform transmission lines has been studied by many authors, either in the frequency domain [6]–[10] or time domain [11]–[13]. In the case of tapered transmission lines (Fig. 2), the reflection coefficient at any point, , is governed by the nonlinear differential equation [7] (1) where is the propagation constant and is the normalized impedance, which is a function of the distance along the taper. If the condition is assumed and ohmic losses and dispersion are considered negligible in the transmission line , the reflection coefficient at the input is expressible in the form (2) where is the total transition length, is the effective propagation constant, is the frequency, is the speed of light, and is the effective dielectric constant (assumed to remain unchanged as a first approximation). If the PS to MS geometry is analyzed as a particular case of a nonuniform transmission line in quasi-TEM operation, (1) and (2) can be used for computing , or equivalently, , where is the angular frequency. Additionally, some methods relying on adaptive techniques to accomplish desired responses, as well as optimum solutions under certain circumstances, have been

Consider the asymmetrical double-sided strip line of Fig. 3. The dielectric sheet is assumed to be infinitely wide, and the strips are assume to have negligible thickness. The structure has been closed by an infinite ground plane in order to carefully apply the conformal-mapping technique, following a similar process to the one employed in an MS line analysis [20]. in width when The geometry resembles an MS line of , whereas if , the PS line is obtained. The objective is to yield simple closed-form formulas for obtaining the characteristic impedance given a transmission line that has a cross section like that in Fig. 3. Conformal-mapping techniques provide us with a powerful mathematical method to accomplish this task, supposing a quasi-TEM hypothesis. This implies this procedure is only valid up to a few gigahertz (for typical substrates), but this range still includes most wireless applications. Actually, we compute the capacitance per unit length when is determined, we the conformal method is applied. Once can calculate the characteristic impedance and the effective dielectric constant. This technique has been successfully applied to MS and double-sided PS lines, as well as to other complex transmission line geometries of more recent interest. Most cases have focused on symmetrical structures, although some studies deal with asymmetrical structures and finite ground planes [21], [22]. Conformal mapping is applied in [21] to a strip line with two infinite ground planes, introducing an asymmetry respect to the vertical axis, whereas the structures under study in [22] consist of finite ground planes placed in one single face above

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TABLE I COMPLEX POINTS MAPPED BY INVERSE TRANSFORM

Fig. 4. Schwarz–Christoffel mappings represented geometrically. By the direct transform, structures (a) and (b) are equivalent in terms of potential.

a complex plane into a plane , and the second applies direct mapping to transform the complex plane to the plane . In this last plane, the geometry is a simple symmetrical parallel transmission line, but it has a finite substrate. The transformed geometry has an identical capacitance according to the properties of conformal mapping. Unlike the initial structure, the equivalent capacitance is evaluated using the plane capacitor approximation. A. Application of the Inverse Schwarz–Christoffel Transform

the substrate (in rectangular or polar coordinates) with symmetry respect to the same axis. The nonuniform PS line comprises only one single infinite ground plane, and different strip widths are the responsibility of the horizontal asymmetry. As a result, the complex planes used in the conformal mapping contain both finite and asymmetry features. It requires different mapping points and consequently leads to other integral transforms that need to be solved. The total capacitance of the asymmetrical line is obtained ignoring the fringing fields in the corners approximately by

The mathematical analysis begins with the definition of the points involved in the Schwarz–Christoffel transform (Table I). Mapping the -plane into the -plane is evaluated according to such a transformation. This relation is shown in (8) as follows:

(3)

(8)

is the double-sided strip capacitance, defined by a where line of height , and strip widths , and is the MS capacitance, defined by a line of height and a . strip width The analysis of the capacitance per unit length is straightforward in the case of MS geometry due to the widespread use of this transmission line, and consequently, the conformal-mapping study has been already reported. By defining the parameters [20]

defined by the finite complex points 1 and 1, and where correspond to , respectively. The analytical evaluation leads to (9) Taking into account the mapping points (Table I),constants and constrain the final inverse transform as (10)

(4)

(5)

where . and are computed applying this transThe values of formation to the strip corners. These values must fulfill the following equations:

(6)

(11)

and

the MS capacitance is expressed as

where

(12)

is the complete elliptical integral (7)

The most difficult part of the analysis is the estimation of the capacitance of the double-sided structure , which is computed by the Schwarz–Christoffel transform [23], defined in the context of complex analysis. This evaluation requires two mappings (see Fig. 4). The first is an inverse transformation from

which finally lead to (13) These points will be used in the second part of the analysis in order to achieve a finite substrate structure by means of a second mapping.

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The following equations relate the parameters of the geometry of interest with those of the final structure:

TABLE II COMPLEX POINTS MAPPED BY DIRECT TRANSFORM

(19) (20) (21)

B. Application of the Direct Schwarz–Christoffel Transform The direct transform must be applied at this point of the analysis keeping in mind the desired final cross section, which is a complete finite structure. Consequently, the previous computed points (in the -plane) are mapped into new -plane points, as in Table II. The Schwarz–Christoffel transform definition leads to the complex integral

The parameter is the main value required to quantify the capacitance. By using some algebra manipulations on (19)–(21), it holds that (22) In order to simplify (22), we introduce (23) which leads to

(14)

(24) Using this relation, the equivalent height is

where it holds that integration result (15)–(17),

. Using a fundamental (25) Finally, the capacitance is expressed as (26) (15)

C. Closed-Form Impedance Formula The impedance and effective dielectric constant of the whole structure depend on the total capacitance. We find this using (3), (4), and (26)

where

(27) (16)

where and are described in (4) and (5). Finally, the effective permittivity constant is obtained though the quotient (28)

(17)

and the final expression for the asymmetric double-sided line characteristic impedance is (29)

this integral is evaluated to obtain an explicit expression. The constants and are computed by means of the mapping conditions. Additionally, we define the parameter as (18)

This closed-form formula has been compared to the numerical result that comes from the analysis of a simple geometry, computed by means of a full-wave simulation. In order to perform this comparison, we analyzed a single cell (see Fig. 5) of l-mm length. The asymmetric PS line has been simulated

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Fig. 5. Simple transmission line geometry for impedance identification.

using Zeland IE3D, based on the method of moments (MoM). The impedance was estimated for several strip-width values and compared to the analytical results, setting the parameters of the mm, ). In addition, the value FR4 material ( of involved in the MS capacitance was set to with as the substrate height. The numerical impedance value is obtained by means of the return losses (30) where (31) Thus, can be computed from the scattering value , and consequently, the impedance value. As Fig. 6(a) and (b) shows, the estimation in the impedance is very accurate. The maximum error between the closed form and the numerical simulation is about 5 , which is a relative error of about 5%. IV. APPLICATION TO THE ANALYSIS AND SYNTHESIS OF TRANSITIONS As presented in Section II, the transition may be analyzed using an integral approach following (2). However, the practical use of that equation requires the knowledge of the impedance at any point of the transition, which was solved in Section III by means of a conformal-mapping method. Next, both equations must be combined in order to analyze the geometry according to the following steps. , which define Step 1) Given the geometry, and the material parameters, compute using (29). Step 2) For every frequency , compute , using as the mean value of (28) and evaluate using (2). is performed to simplify the approach, The averaging of which leads to slight deviations from the real response. Nevertheless, solving the nonlinear differential equation instead of the . In adintegral equation itself can include the effects of dition, different geometry functions can bring similar results in return losses because the same impedance values (Fig. 3) may be obtained using different combinations of and . On the other hand, the synthesis problem, starting from a required return loss, deals with estimating the values of the geometry functions . This problem

Fig. 6. Comparison of closed-form formula versus full-wave simulations (IE3D) Only one semispace region is plotted, as symmetry with respect to the diagonal holds. (a) Values obtained by means of the proposed method (analytical). (b) Difference between the analytical formula prediction and the full-wave simulation results. (a) Numerical value. (b) Numerical error.

is usually more interesting than the analysis process, as it has an inherent design application. We show here two examples of synthesis where the analysis algorithm is needed: the design of a balun with optimum return losses in a specific frequency band and the synthesis of a Klopfenstein balun [7]. Besides, as in many synthesis algorithms, the use of optimization techniques will be necessary. In this paper, a GA approach is used, although any optimization technique is directly applicable. In addition, as differential geometry functions provide better results in terms of return losses, a spline interpolation method is used for defining the geometry, decreasing the number of variare specified by ables. Thus, sets of two vectors (32) and (33)

CARRO AND DE MINGO: ANALYSIS AND SYNTHESIS OF DOUBLE-SIDED PS TRANSITIONS

These vectors allow for the evaluation of complete geometry functions. The cubic spline method assigns a cubic polynomial to each subinterval

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50–100 . The cost function chosen as

was consequently

(39) (34) satisfying some conditions

(35) (36) (37) Due to these constrains, the geometry function is continuous and differentiable. The constants involved in the polynomials are computed by the GA in order to achieve an optimum according to some cost function. In this study, the set of vectors delineating the geometry functions are established following these rules. 1) The number of points is set to nine in the ground plane layer and two in the top because, in this layer, the geometry function variation is not very wide. , the geometry functions take electrical 2) For values, constrained by the electrical boundary conditions. , where These are denotes the strip widths that lead to and , respectively. values are obtained by sampling uni3) The rest of the . formly in the interval

The convergence toward the optimum [see Fig. 7(a)] is fast, and the GA reaches the optimum in about 80 generations. The optimized geometry functions [see Fig. 7(a)] evolve towards the PS line having several local maxima and minima in the ground plane. In order to evaluate the integral method accuracy, the optimized balun was simulated using the electromagnetic code HFSS based on the finite-element method. The comparison using both approaches [see Fig. 7(b)] shows a reasonably similarity in the target band with return losses below approximately 30 dB in such frequencies. B. Klopfenstein Transition The optimum impedance taper of a fixed length matching section has been analyzed by several authors. Using the integral formulation, Klopfenstein [7] showed the optimum taper in the sense that the minimum reflection coefficient for a passband fulfills

(40) where the function

is defined as

A. Minimum In-Band Return Loss Transition

(41)

The design process of a transition ( mm in length) with minimum return losses in the range between two specified frequencies can be formulated mathematically as a geometry optimization find

such that: (38)

where is the reflection coefficient in frequencies from to . If we want to solve this optimization problem with great accuracy, this requires using a full-wave simulator, which supposes time and resources. This time generally increases when dealing with optimization processes, as they usually require the evaluation of several iterations, and may be reduced using the proposed integral analysis method. In order to solve the problem by a GA, some random geometry functions are built using a natural spline interpolation scheme on the genes of the GA. Afterwards, the analysis algorithm is applied so that the score of each chromosome (transition) is obtained. The optimization process evolves for some in this study), until a requirement is fulgenerations ( filled or it is stopped. The method has been applied successfully to a transition of 25 mm in length in the range from 3.1 to 4.6 GHz. In addition, it was required in the balun matches

being the modified Bessel function, being the value , and defining the passband fulfilling the condition being the bound of the reflection coefficient in the passband. Following the work of Klopfenstein, it is possible to formulate an optimization problem based on impedance synthesis rather than the reflection coefficient as in the previous example. In this context, the utility of the proposed impedance formula is very clear. Mathematically, the optimization problem is as follows: find

such that: (42)

where would be the actual impedance value presented is the theoretby the taper section at the point and ical impedance value the taper must have, computed by means of (39). The optimization problem presented above is intended to force the Klopfenstein impedance taper. The GA uses this equation based on the quadratic error as a cost function. When the GA reaches the convergence point [see Fig. 7(c)], the geometry functions are as close as possible to the Klopfenstein values.

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Fig. 7. Synthesis profile results and electromagnetic performance comparison between full-wave simulations and the proposed technique. (a) Minimum in-band optimization convergence and optimum geometric profile. (b) Minimum in-band optimum frequency response comparison between the integral analysis and a full-wave analysis. (c) Klopfenstein balun synthesis optimization convergence and Klopfenstein geometric profile. (d) Klopfenstein scattering parameters obtained by the proposed method and a full-wave simulator. TABLE III OPTIMUM VALUES GEOMETRY FUNCTIONS

Fig. 8. Comparison between the required impedance and the synthesized impedance by means of a GA.

The obtained impedances (Fig. 8) are near the required values calculated from the Klopfenstein equations. The largest differences lay at the beginning and at the end of the transition,

namely due to the discontinuities required by the Klopfenstein method in contrast to the continuity imposed by the cubic-spline geometry functions. In addition, the full-wave numerical results agree reasonably well compared to our proposed method, as presented in Fig. 7(d). In both synthesis examples, the whole optimization process took about 3 min, whereas a single full-wave transition took about 40 min. Regarding the optimization results, Table III

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Fig. 9. Synthesized test beds and electromagnetic performance comparison between full-wave simulations and the experimental results. (a) Back-to-back Klopfenstein balun. (b) Minimum in-band back-to back test bed. (c) Experimental and simulated Klopfenstein balun scattering parameters. (d) Experimental and simulated wideband balun scattering parameters.

summarizes the obtained values after 200 generations, pointing out the difference between the transition geometry functions. V. EXPERIMENTAL VALIDATION In order to validate the analysis and synthesis method, two test samples corresponding to the presented designs have been fabricated on an FR4 substrate with dielectric constant 4.6. Measurements have been carried out in a back-to-back configuration because of the balanced circuit nature. Since it is not straightforward to characterize by measurements a single balun, we have also performed an electromagnetic simulation of the samples to compare with experimental data, which are obtained by means of the vector network analyzer ANRITSU 37247D. Fig. 9 presents the manufactured boards according to Table III and their simulated and measured scattering parameters. The Klopfenstein design [see Fig. 9(a)] constrains the bandpass ripple to a constant value. The back-to-back measurement [see Fig. 9(c)] shows that the maximum bandpass ripple is about 14 dB, pointing out a high probability of fulfilling the Klopfenstein condition. In addition, it provides a remarkable bandwidth considering the common criteria of dB

to define the operative band, which is verified from 1.7 GHz up to 9.4 GHz. This transition can be used in practice for feeding antenna structures in ultra-wideband (UWB) systems. The wideband transition [see Fig. 9(b)] measurements, shown in Fig. 9(d), agree with the proposed target bandwidth. The goal was to design a transition from 3.1 to 4.6 GHz, and the experimental results are excellent at the required frequency band. The return losses are below 25 dB and achieve extremely low values at 4 GHz ( 40 dB). The conformal-mapping technique can be validated indirectly (at least up to 6 GHz) according to the experimental data and its agreement with our proposal. Slight differences can be explained from the fact that ohmic losses have not been taken into account in our approach. In this case, the method can be applied for designing baluns at any required frequency band. VI. CONCLUSION In this paper, we have proposed a new method based on conformal mapping allowing to evaluate and optimize the return losses of a tapered MS to double-sided PS balun with impedance-matching capabilities. The Swartz–Christoffel

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transform applied to this electromagnetic structure led to impedance and effective dielectric constant closed-form formulas, offering a new approach to analyze and optimize the tapered balun in terms of impedance variations. These expressions have been verified by comparing their predictions to full-wave computation results, showing very small differences. In addition, two case studies have demonstrated the validity of this method, which has been applied in order to design structures with minimum return losses in a specific frequency band and approximate the Klopfenstein geometric shape. The synthesis procedure combines a closed-form impedance formula and an optimization technique (in this case, GAs) providing some advantages: fast convergence toward the optimum solution, small computation cost compared to a full-wave numerical approach, and a better understanding about the electromagnetic balun behavior from the impedance variation point of view. The synthesized proposals have been fabricated and measured using a vector network analyzer, ensuring the validity of the algorithm and confirming indirectly the conformal-mapping approach.

[15] M. Kobayashi and N. Sawada, “Analysis and synthesis of tapered microstrip transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 40, no. 8, pp. 1642–1646, Aug. 1992. [16] D. Park, “Planar transmission lines,” IEEE Trans. Microw. Theory Tech., vol. MTT-3, no. 3, pp. 8–12, Apr. 1955. [17] H. A. Wheeler, “Transmission-line properties of parallel wide strips by a conformal-mapping approximation,” IEEE Trans. Microw. Theory Tech., vol. MTT-12, no. 3, pp. 280–289, May 1964. [18] H. A. Wheeler, “Transmission-line properties of parallel strips separated by a dielectric sheet,” IEEE Trans. Microw. Theory Tech., vol. MTT-13, no. 2, pp. 172–185, Mar. 1965. [19] H. A. Wheeler, “Transmission-line properties of a strip line between parallel planes,” IEEE Trans. Microw. Theory Tech., vol. MTT-26, no. 11, pp. 866–876, Nov. 1978. [20] C. Nguyen, Analysis Methods for RF, Microwave, and Millimeter-Wave Planar Transmission Line Structures. New York: Wiley, 2000. [21] J. S. Rao and B. N. Das, “Analysis of asymmetric stripline by conformal mapping,” IEEE Trans. Microw. Theory Tech., vol. MTT-27, no. 4, pp. 299–303, Apr. 1979. [22] M. Duyar, V. Akan, E. Yazgan, and M. Bayrak, “Analyses of elliptical coplanar coupled waveguides and coplanar coupled waveguides with finite ground width,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 4, pp. 1388–1395, Jun. 2006. [23] G. D. Zill, A First Course in Complex Analysis With Applications. Bloomfield, NJ: Jones and Bartlett, 2003.

REFERENCES [1] B. Climer, “Analysis of suspended microstrip taper baluns,” Proc. Inst. Elect. Eng., vol. 135, pt. H, pp. 65–69, Apr. 1988. [2] S. Kim and K. Chang, “Ultrawide-band transitions and new microwave components using double-sided parallel-strip lines,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 9, pp. 2148–2152, Sep. 2004. [3] C. Jian-Xin, C. H. K. Chin, and X. Quan, “Double-sided parallel-strip line with an inserted conductor plane and its applications,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 9, pp. 1899–1904, Sep. 2007. [4] Q. Xue and J. X. Chen, “Compact diplexer based on double-sided parallel-strip line,” Electron. Lett., vol. 44, no. 2, pp. 123–124, Jan. 2008. [5] P. L. Carro Ceballos and J. de Mingo Sanz, “Ultrawideband tapered balun design with boundary curve interpolation and genetic algorithms,” presented at the IEEE AP-S Int. Symp., Albuquerque, NM, Jul. 9–15, 2006. [6] H. A. Wheeler, “Transmission lines with exponential taper,” Proc. IRE, vol. 27, no. 1, pp. 65–71, Jan. 1939. [7] R. W. Klopfenstein, “A transmission line taper of improved design,” Proc. IRE, vol. 44, no. 1, pp. 31–35, Jan. 1956. [8] R. E. Collin, “The optimum tapered transmission line matching section,” Proc. IRE, vol. 44, no. 4, pp. 539–548, Apr. 1956. [9] R. P. Hecken, “A near-optimum matching section without discontinuities,” IEEE Trans. Microw. Theory Tech., vol. MTT-20, no. 11, pp. 734–739, Nov. 1972. [10] D. C. Youla, “Analysis and synthesis of arbitrarily terminated lossless nonuniform lines,” IEEE Trans. Circuit Theory., vol. 11, no. 3, pp. 363–372, Sep. 1964. [11] J. E. Schutt-Aine, “Transient analysis of nonuniform transmission lines,” IEEE Trans. Circuits Syst., vol. 39, no. 5, pp. 378–385, May 1992. [12] T. Dhaene, L. Martens, and D. De Zutter, “Transient simulation of arbitrary nonuniform interconnection structures characterized by scattering par arne ters,” IEEE Trans. Circuits Syst., vol. 39, no. 11, pp. 928–937, Nov. 1992. [13] W. Bandurski, “Simulation of single and coupled transmission lines using time-domain scattering parameters,” IEEE Trans. Circuits Syst., vol. 47, no. 8, pp. 1224–1234, Aug. 2000. [14] J. P. Mahon and R. S. Elliott, “Tapered transmission lines with a controlled ripple response,” IEEE Trans. Microw. Theory Tech., vol. 38, no. 10, pp. 1415–1420, Oct. 1990.

Pedro Luis Carro (S’06) was born in Zaragoza, Spain, on 1979. He received the Engineer of Telecommunication M.S. degree and Ph.D. degree from the Universidad de Zaragoza, Zaragoza, Spain, in 2003 and 2009, respectively. In 2002, he carried out his Master’s thesis on antennas for mobile communications with the Department of GSM and Antenna Products, Ericsson Microwave Systems AB, Göteborg, Sweden. From 2002 to 2004, he was an Electrical Engineer with the Space and Defense Department RYMSA S.A., where he was involved in the design of antennas and passive microwave devices for satellite communication systems. From 2004 to 2005, he was an RF Engineer with the Research and Development Department Telnet Redes Inteligentes, where he was involved with radio-over-fiber systems. In 2005, he joined the Universidad de Zaragoza, where he was an Associate Professor, and since 2006, an Assistant Professor with the Departamento de Ingeniería Electrónica y Comunicaciones. His research interests are in the area of mobile antenna systems, passive microwave devices, and power amplifiers.

Jesus de Mingo (M’98) was born in Barcelona, Spain, on 1965. He received the Ingeniero de Telecomunicación degree from the Universidad Politécnica de Cataluña (UPC), Barcelona, Spain, in 1991, and the Doctor Ingeniero de Telecomunicación degree from the Universidad de Zaragoza, Zaragoza, Spain, in 1997. In 1991, he joined the Antenas Microondas y Radar Group, Departamento de Teoría de la Señal y Communicationes, Universidad Politénica de Cataluña. In 1992, he was with Mier Comunicaciones S.A., where he was involved with the solid-state power amplifier design until 1993. Since 1993, he has been an Assistant Professor ,and since 2001, a Professor with the Departamento de Ingeniería Electrónica y Comunicaciones, Universidad de Zaragoza. He is a member of the Aragon Institute of Engineering Research (I3A). His research interests are in the area of linearization techniques of power amplifiers, power amplifier design, and mobile antenna systems.