2824 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 11, NOVEMBER 2011

Directional Coupler Compensation WithOptimally Positioned Capacitances

Johannes Mller, Member, IEEE, Minh N. Pham, and Arne F. Jacob, Fellow, IEEE

AbstractAn accurate design synthesis for the phase velocitycompensation in coupled line microstrip couplers by means ofparallel capacitances is presented. In contrast to previous ap-proaches, an a priori arbitrary placement of the capacitancesalong the coupled line structure is considered. By optimizing thesepositions, the directivity-bandwidth performance is significantlyimproved. Cases with two and three capacitances are treatedextensively. The findings are used to generalize the compensationscheme to any number of capacitances. Throughout the analysis,the parasitic even-mode capacitance is taken into account using arealistic model. Several design examples are presented. Simulationresults are confirmed by measurements. They compare favorablywith those reported previously.

Index TermsCapacitive compensation, coupled lines, direc-tional coupler, inhomogeneous media, phase velocity compensa-tion.

I. INTRODUCTION

C OUPLED line structures are widely used in microwavecircuits, such as Marchand baluns, matching networks,combiners, filters, and directional couplers. If realized in aninhomogeneous medium, e.g., using microstrips, the effectivepermittivities, and thus, the phase velocities of the even andodd mode differ. This leads to performance degradation, suchas output port imbalance in case of Marchand baluns, spuriouspassbands in filters, and poor directivity, as well as port mis-match in directional couplers.

Methods for compensating these phase velocity differenceshave been investigated in the literature since the early 1970s.One can distinguish between two categories. One aims at effec-tively equalizing the different phase velocities along the coupledlines. Here, the most common approach is certainly to wigglethe adjacent edges of the coupling slot [1], [2]. This extends theeffective path length of the odd-mode current and thereby com-pensates for its higher phase velocity. Alternatively, the effectiveeven- and odd-mode permittivities can be equalized. This canbe achieved by means of dielectric overlays [3], [4], anisotropicsubstrates [5], quasi-suspended substrate arrangement [6], or

Manuscript received March 08, 2011; revised July 15, 2011; accepted July29, 2011. Date of publication September 26, 2011; date of current versionNovember 16, 2011. This work was supported by the Deutsche Forschungsge-meinschaft (DFG).

The authors are with the Institut fr Hochfrequenztechnik, TechnischeUniversitt HamburgHarburg, 21073 Hamburg, Germany (e-mail: jo-hannes.mueller@tuhh.de).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMTT.2011.2165961

apertures in the ground plane [7], [8]. Stepped-impedance ap-proaches [9], [10] rely on internal reflections to achieve portisolation.

The second category relies on reactive components connectedto the coupled lines. The use of inductive elements in series tothe ports was originally proposed in [11]. Design equations werederived in [12] and further developed in [13]. Shunt inductanceswere recently considered in [14]. The achievable compensationwith all these inductive methods is, however, relatively narrow-band. Much more common is to place a capacitance between thecoupled (microstrip) lines at each end of the coupler. Ideally,this affects only the odd mode and slows it down. This methodwas first presented by Schaller [15], who also introduced anapproximate design equation for the compensating capacitance[11]. Kajfez used a similar formulation and was first to accountfor the effect of a parasitic even mode capacitance [16]. Thusfar, however, the analytical formulation was approximate andvalid for loose coupling only. Dydyk improved this method byderiving an exact design equation for both the odd-mode capac-itance and the modified odd-mode impedance of the coupledlines [17]. The latter is necessary since the capacitances effec-tively change the characteristic impedance of the odd mode. Thepresence of a parasitic even-mode capacitance, however, wasnot taken into account in [17]. In all methods, the capacitancesconnect the lines at the end of the coupler. Dydyk also derivedformulas for a single capacitance at the center of the coupler[18]. In this case, however, the compensation is more narrow-band.

In this study, we derive design equations for lumped ca-pacitances that are placed at arbitrary positions, but stillsymmetrically, along the coupler, thereby maintaining itstwofold symmetry. This simplifies the synthesis and ensuresquadrature. The parasitic even-mode capacitance is taken intoaccount as sketched in Fig. 1. In contrast to previous work, itis assumed to depend on the odd-mode capacitance instead ofbeing constant, as this is more realistic. This paper is organizedas follows. Section II describes the model and outlines theapproach. In Section III, we analyze in detail the compensationby means of two capacitances and determine their optimumposition. Section IV extends the rigorous design to the caseof three, and generalizing the previous findings, an arbitrarynumber of capacitances. Finally, measurements results arereported.

II. THEORY

A. Image Parameter Approach

The approach is based on the so-called image method, a clas-sical network description used for the synthesis of filters and

0018-9480/$26.00 2011 IEEE

MLLER et al.: DIRECTIONAL COUPLER COMPENSATION WITH OPTIMALLY POSITIONED CAPACITANCES 2825

Fig. 1. Symmetric coupled microstrip lines with compensating capacitances at arbitrary positions with . Parasitic capacitances are taken into account. Symmetry lines: and .

Fig. 2. Image method representation of a: (a) general and (b) symmetric two-port.

matching networks. In this method, circuit analysis is performedfrom a wave viewpoint, much as in transmission line theory[19]. Recently, we used this theory to effectively characterizesymmetrical couplers [20]. For the sake of completeness, thetheory is briefly reviewed in the following. For a deeper under-standing, the reader is referred to [19].

Any reciprocal two-port network can be represented by itsso-called input and output image impedances, and ,and an image propagation function , as shown in Fig. 2(a).These parameters can be calculated, e.g., from its -ma-trix

(1a)

(1b)

(1c)

with being the image attenuation in neper and beingthe image phase in radian. In the case of symmetric two-ports,the equations reduce to

(2a)

(2b)

with the lower case letters representing the -matrix en-tries of half the structure, as indicated in Fig. 2(b). The imageimpedance seen at the vertical symmetry plane may differfrom the input image impedance. Due to the symmetry, however,no reflection occurs at this interface.

For a transmission line, is its characteristic impedanceand is its (complex) electrical length. As coupled transmis-sion lines can be decomposed into a pair of independent lineswith respect to their eigenmodes, the set of image parametersis, in turn, equivalent to the corresponding wave quantities. Forsymmetric cross sections, the decomposition yields the well-known even/odd-mode representation. In the following, sub-scripts (even) and (odd) or, more generally, will be usedin this context.

B. Ideal Coupler Conditions

A directional coupler is considered to be ideal if its ports arematched and if two pairs of ports are isolated from each other.In case of a backward coupler, these are P1P3 and P2P4,following the port-notation from Fig. 1. In lossless symmetricfour-ports, as are solely considered in this paper, both proper-ties hold simultaneously [21], [22]. An ideal coupler fulfills twonecessary conditions: the propagation and the impedance con-dition. For a parallel line coupler, for instance, the former meansthat even and odd mode have equal electrical length (i.e., imagephase)

(3)

In the case of homogeneous media, (3) is always fulfilled, in-dependently of frequency. Otherwise, a compensating structureyielding (3) at the design frequency is needed.

According to the second condition, the coupler impedance

(4)

has to match the system (or reference) impedance (e.g.,50 ). Here, and are the even- and odd-mode(image) impedances, respectively.

A third design condition follows from the nominal couplingfactor

(5)

which corresponds to the magnitude of the scattering parameterat frequency were the overall electrical length of the cou-

pler is . In practice, (3) might not be fulfilled. is thensometimes replaced by [6].

C. Coupler Compensation

The coupled line model with twofold symmetry is depictedin Fig. 1. Exemplarily, three capacitances are shown. Here,

is the capacitance desired for compen-sation, whereas represents the parasitic even-modecapacitance. Thus, the odd mode sees both in parallel. Thecorresponding circuit for mode is depicted in Fig. 3(a).The symmetry lines are preserved in this arrangement.Consequently, the image impedance seen at the port(s) can be

2826 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 11, NOVEMBER 2011

Fig. 3. (a) Mode schematic of coupled lines with three symmetrically placedcapacitances. (b) Corresponding image parameter representation. (c) Equivalentideal transmission line.

calculated by considering just one half of the circuit. As thestructure considered in this work is assumed to be lossless,it has a real image impedance and an imaginary propagationfunction .

The compensated coupler should behave ideallyat least atone frequency. Its image parameters should thus match those ofan ideal uncompensated coupler, as depicted in Fig. 3(c). Thismeans that in (3)(5) and are now the even/odd-mode image parameters of the whole structure.

From this argumentation, it follows that the image parameterapproach is applicable to any symmetric four-port (e.g., also tobranch-line couplers). Depending on the desired functionalitythough, conditions other than (3)(5) may apply.

D. Design Parameters

Let there be capacitances in total, on each half of thesymmetric coupler. They are denoted byand define coupled line sections with impedance and elec-trical length . For odd, the center capaci-tance has to be split into two, one on each side. This yields

forfor

(6)

and

forfor

(7)

The following notations will be used throughout this paper. Thetotal electrical length of the coupled lines is

(8)

The even/odd mode inhomogeneity is characterized through

(9)

with the effective permittivities . The electrical lengths of thetwo modes are thus related through

(10)

Finally, let

with (11)

denote the fractional lengths of the individual sections.

E. Synthesis Procedure

For a desired nominal coupling , a given inhomogeneity ,and a reference impedance , the ideal image parameters forthe two modes follow from the coupler conditions (3)(5). Com-monly, the image phase of both modes is set to

. That way, the compensation and nominal coupling occurat the same frequency. The compensated multisection couplercan now be characterized by means of its image parametersby cascading the various -matrices. The goal is to findvalues for the capacitances , characteristic impedances ,and electrical lengths such that the resulting overall imageparameters satisfy conditions (3)(5).

F. Simplifications

Obviously the parameter space is very large. Some of theseparameters, however, are interdependent, such as the electricallengths of odd and even mode through (10). Also, as alreadymentioned, the even- and odd-mode capacitances are assumedto be related. The idea behind this is that, if the odd-mode capac-itance is changed during the design process, might changeas well, depending on which geometrical or electrical parame-ters are tuned. This functional dependency must be found beforethe synthesis, e.g., via simulation. In any case, is not an in-dependent design variable here and we have

(12)

To further reduce the parameter space, the different coupledline sections are assumed to have equal cross section and thusimpedance

(13)

MLLER et al.: DIRECTIONAL COUPLER COMPENSATION WITH OPTIMALLY POSITIONED CAPACITANCES 2827

For a certain arrangement of , one can now find oneor more solutions for the remaining independent quantities,namely, the impedances , the capacitances , and theoverall electrical length . In the following sections, couplercompensation is formulated analytically for different numbers

of capacitances and the solutions are discussed.

III. COMPENSATION WITH TWO CAPACITANCES

As a first generalization of known designs, the case of twocapacitances with variable positioning is considered.

A. Derivation

For the fractional position of the capacitance, we set. Thus, for , the capacitances are placed

at the outer ends of the coupler, and for , they merge atthe center.

To obtain a frequency-independent solution, we consider thesusceptances

(14)

instead of the corresponding capacitances. For each mode, the-matrix of, for example, the left sub-circuit, is calculated

by multiplying the -matrices of its three parts

(15)

For the lossless case and with

(16)

the -matrix entries for mode become

(17a)

(17b)

(17c)

(17d)

Inserting (17a)(17d) into (2b) yields the even- and odd-modeelectrical lengths. Applying the propagation condition (3) leadsto a quadratic equation, which can be solved for the auxiliaryvariables . then depend on parameters , , and the stillunknown

(18)

Next, (17a)(17d) is inserted into (2a) and the remaining con-ditions (4) and (5) are applied. This yields, after some manip-ulation, the following expressions for the even- and odd-modeimpedances of the coupled line sections:

(19)

It should be mentioned that (19) was obtained using the negativesquare root in (18) as this leads to , the physicallymeaningful solution for microstrip lines.

In the next step, (18) and (19) are inserted into (16). Thisyields expressions for and as a function of , , , ,and . Using these expressions in (12), one obtains an equationthat can be solved numerically for , with following from(10). Here, represents an additional degree of freedom, whichwill be used for optimization.

In the case is constant, as in [16], is calculated first,following from (10). Knowing , and for a given value of pa-rameter , are obtained from (18), from (19), and finally,

from (16).The two extreme cases and lead to the following

simplified formulas.

(20)

(21)

(22)

(23)

Again, (or ) has to be calculated first, as explained above.

B. Some Limitations

The inhomogeneity depends on the geometry of the coupledline structure and the dielectric property of the medium. In thecase of microstrip lines, for example, . It increases withthickness and permittivity of the substrate, as well as withthe coupling factor , and may reach values of up to .Other transmission line structures, such as broadside coupledlines on suspended substrate, may exhibit inhomogeneities of

, which vary in a much wider range [23].Following the previous derivation, compensation is theoret-

ically possible for any . The type of inhomgeneity imposes,however, some constraints on the compensating capacitances.From (21) or (23), it follows:

Type 1: (24)

Type 2: (25)

2828 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 11, NOVEMBER 2011

Fig. 4. Compensating odd-mode susceptance versus for 1020-dB cou-pling with and .

Fig. 5. Normalized even- and odd-mode impedance versus for 1020-dBcoupling. and .

Thus, the upper (lower) limit of for Type 1 Type 2is determined by the coupling factor through

. Weaker (stronger) coupling therefore relaxesthe constraints on the capacitance ratio for Type 1 Type 2 .This makes compensation easier.

In the remainder of this work, only edge-coupled microstriplines are considered. In this case, the required capacitance valuesincrease with , as will be shown in Section III-C. Dependingon the farication technique, large capacitance values might bedifficult to realize. In the following analysis, we choose

and , which are typical values for coupledmicrostrip lines. For reasons of comparability, these values arekept throughout this paper.

C. Analysis

To illustrate the above, several examples are studied. Com-pensation is performed at . Fig. 4 shows the compen-sating odd-mode susceptance versus its position for differentcoupling levels. The value of the susceptance grows stronglywith coupling and decreases when the position approaches thecenter of the coupler (increasing ). Fig. 5 depicts the normal-ized even- and odd-mode impedances. They may significantlydiffer from the image impedances. In Fig. 6, the scattering pa-rameters of a 15-dB coupler are shown for different values ofand for the uncompensated case as well. They are plotted versusthe normalized frequency.

Independently of , perfect compensationis achieved at , as intended. The broadband behavior, how-ever, is about 20 dB better for compared to or

. Maximum coupling occurs at about for bothand the uncompensated case, while it is shifted to lower

Fig. 6. Scattering parameters for the uncompensated case and for , , and .

(higher) frequencies for . The response can beexplained by inspecting the resulting image parameters. Fig. 7depicts the frequency dependence of the even- and odd-modeimage impedances of the resulting coupler impedance normal-ized to the 50- reference impedance and of the electrical lengthdifference . For the uncompensated case,the impedances are constant and fulfill conditions (4) and (5) in-dependently of frequency. The electrical length difference, how-ever, is around 10 at and is roughly proportionalto . This strongly degrades the isolation, whereas the inputreflection, which depends more on the impedance level, remainsbelow 40 dB. In the compensated cases, the image impedancesand the electrical length difference meet the design values at .In general, though, they are dispersive. While the image imped-ances increase (decrease) with frequency for ,they exhibit almost no dispersion for . Also showsmuch less dispersion for , compared to the cases with

and , which display similar dispersive behavior thistime.

D. Ideal Capacitance Position

The image parameter dispersion depends strongly on , aswas demonstrated in Section III-C. The next step is to find theposition of the capacitance yielding maximum bandwidth. Tothis end, the solution space is further examined and evaluatedusing a figure-of-merit , which is defined as the minimumdirectivity within a certain fractional bandwidth (FBW)

Directivity dB (26)

MLLER et al.: DIRECTIONAL COUPLER COMPENSATION WITH OPTIMALLY POSITIONED CAPACITANCES 2829

Fig. 7. Image impedances and (top), coupler impedance nor-malized to (middle), and the electrical length difference (bottom) forthe uncompensated case, as well as for , , and .

Fig. 8. Minimum directivity over an FBW of 100% as a function of and thenominal coupling.

In this study, FBW is set to 100%. Fig. 8 depicts the contourplot of in the -space for a typical microstripcoupler. The optimum position is at about for strongcoupling and increases to values around for weakercoupling. A minimum directivity in excess of dB isthen achievable with a 10-dB coupler. If, on the other hand, thecapacitances are placed at the ends or at the center of the coupler,the minimum directivity in the band drops below dBand dB, respectively.

IV. COMPENSATION WITH CAPACITANCES

The number of capacitances is now increased, the place-ment still ensuring symmetry, as described in Section II-D. Ex-emplarily, we consider compensation with three capacitances,and to explore the limits, with an arbitrary number . To re-strict the number of unknowns, we assume , as

Fig. 9. Minimum directivity as a function of and susceptance .

Fig. 10. Scattering parameters and versus the number of capacitances for dB, , and .

in Section II-F. As the derivation is similar to the case with twocapacitances only, distinct aspects are mentioned here. In thissection, we consider exclusively 15-dB couplers.

For , the position of the two outer capacitances is adesign variable and has to be optimized. One-half of the sym-metric structure is depicted in Fig. 3. The -matrix (15)has to be multiplied with the one of half the center shunt suscep-tance . As the position of is fixed, there isonly one additional degree of freedom, namely, its value. Eachfractional position spans a solution space for the possible com-binations of and , each of which yielding a set of valuesfor and . The minimum directivity is shown as a con-tour plot versus and in Fig. 9. The highest valuedB is found for . In the cases and ,is thus optimum if the coupler is subdivided into approximatelyequal symmetric T-sections, each consisting of a shunt capaci-tance between two identical coupled line sections. To simplifythe design in case of capacitances, this finding is general-ized. The coupler is thus subdivided into identical T-sections.All capacitances are equal . This might not be quiteoptimal, but reduces the parameter space and also simplifies thecoupler design in practice. Indeed, the T-sections can be speci-fied from (22) and (23). For this, has to be replaced by .Fig. 10 shows the scattering parameters of such a structure forincreasing . For 16 and more capacitances, the isolation ex-ceeds 80 dB over the whole frequency range. The minimumdirectivity and the resulting odd-mode susceptance are de-picted in Fig. 11 as a function of . For , is alreadylarger than 50 dB. It should be mentioned, however, that for in-

2830 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 11, NOVEMBER 2011

Fig. 11. Minimum directivity and the odd-mode susceptance as a func-tion of .

Fig. 12. Geometry of the interdigital capacitances.

TABLE IGEOMETRIC PARAMETERS OF THE FABRICATED COUPLERS

creasing , the lumped compensation resembles more and morea distributed one. For , one has uniformly coupled lineswith equal even- and odd-mode velocities. This justifies in someway the simplifying assumptions made above.

V. MEASUREMENTS

Several 15-dB couplers were designed and fabricated onRogers Ro4003c substrate (thickness mm, )for a center frequency of 2 GHz. The measurements wereperformed using a test fixture and a four-port vector networkanalyzer (Rhode & Schwarz ZVA50). The measurement resultswere deembedded up to the transition between the singleand the coupled lines using the multiline TRL-calibrationsoftware StatistiCAL from NIST [24]. This way, we wereable to characterize the fabricated couplers by means of theirimage parameters, allowing for straightforward redesigns withoptimized results [20]. The capacitances were realized asinterdigital capacitors, as depicted in Fig. 12. The geometricparameters for the various fabricated couplers are listed inTable I. Couplers with and capacitors having

Fig. 13. Coupler layout with two capacitances at . Measured scatteringparameters and extracted image parameters versus frequency.

Fig. 14. Coupler layout with two capacitances at . Measured scatteringparameters and extracted image parameters versus frequency.

or fingers were realized. Besides optimal posi-tioning, the case with and was also investigated.The measured performances are documented in Figs. 1316.The figures include the corresponding layouts.

The two capacitance cases in Figs. 13 and 14 demonstrate thebenefits of optimal positioning, which improves the minimumdirectivity from 27 to 38 dB. The image parameters reveal that,

MLLER et al.: DIRECTIONAL COUPLER COMPENSATION WITH OPTIMALLY POSITIONED CAPACITANCES 2831

Fig. 15. Coupler layout with three capacitances . Measured scat-tering parameters and extracted image parameters versus frequency.

Fig. 16. Coupler layout with 16 equally distributed capacitances. Measuredscattering parameters and extracted image parameters versus frequency.

while the electrical length difference remains below 0.3in both cases, the image impedance becomes much flatter at

. It should be mentioned that, in both cases, the achievedminimum directivity is higher than predicted (Fig. 7). Indeed,one has dB instead of dB

and dB instead of dB. Thereasons for this can be manifold. On the one hand, the actualvalues of and are slightly smaller than assumed inthe simulation. On the other hand, instead of being lumped el-ements, as presumed in the analysis, the realized capacitanceshave finite length. Full-wave simulation with interdigital struc-tures yields the same qualitative behavior of with respect toand the number of capacitances . It is, however, very time con-suming and does not perfectly converge due to numerical noise.It should also be mentioned that, owing to the distributed natureof the realized capacitors, the case is not realizable. Thus,the fractional position of the capacitor with respect to its centercorresponds to .

Compensation with three and 16 capacitances (Figs. 15and 16) yields minimum directivities of dBand dB, respectively. While these are goodresults for broadband compensated coupler, the theoreticallyachievable performances of dB (Fig. 9) and

dB (Fig. 11) could not be reached. Sev-eral limitations cause this difference. First, the measurementuncertainty, as determined from the redundant multiline thru-re-flect-line (TRL) calibration [24], is around 45 dB, which canbe considered as the accuracy limit for the directivity mea-surement. Secondly, the in-house fabrication process exhibitsetching tolerances of around 8 m. The resulting variationsof the coupler geometry have a nonnegligible influence on thedirectivity at such low signal levels. Finally, losses may alsohave some effect.

VI. CONCLUSION

An accurate design synthesis for the phase velocity compen-sation of coupled lines by means of parallel capacitances wasinvestigated for directional-coupler applications. In contrastto previous approaches, an arbitrary number of capacitanceswas considered. Their positions were optimized with respect tobandwidth-directivity performance. Significant improvementswere demonstrated. The parasitic even-mode capacitance wastaken into account throughout the analysis. Compensatedcouplers with two, three, and 16 interdigital capacitances werefabricated and measured. Due to measurement uncertainties andfabrication tolerances, the measurements do not quite match thepredictions. Still, the achieved performance with directivities ofabout 40 dB over an FBW of % is, to the authorsknowledge, unmatched in literature. Thus, the measurementresults confirm the novel compensation approach. Ongoinginvestigations deal with the distributed nature of the interdigitalcapacitors and its inclusion in the optimization process.

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Johannes Mller (S04M10) was born inCologne, Germany, in 1977. He received the LicencedIngnierie lectrique degree from the Universitde St. Jrme, Marseille, France, in 2001, and theDipl. Ing. degree from the Technische UniversittHamburgHarburg, Hamburg, Germany, in 2005.

From 2003 to 2005, he was also with the EuropeanTechnology Center, Panasonic, Lneburg, Germany.In 2006, he joined the Institut fr Hochfrequen-ztechnik, Technische Universitt HamburgHarburg.His research interests include the design and devel-

opment of microwave and millimeter-wave circuits and components.

Minh N. Pham was born in Ho Chi Minh City,Vietnam, in 1985. He is currently working towardthe Diploma degree at the Institut fr Hochfrequen-ztechnik, Technische Universitt HamburgHarburg,Hamburg, Germany.

Since 2005, he has been with the Technische Uni-versitt HamburgHarburg. He is engaged in the de-sign and development of microwave and millimeter-wave planar circuits.

Arne F. Jacob (S79M81SM02F09) was bornin Braunschweig, Germany, in 1954. He received theDipl.-Ing. and Dr.-Ing. degrees from the TechnischeUniversitt Braunschweig, Braunschweig, Germany,in 1979 and 1986, respectively.

From 1986 to 1988, he was a Fellow with theCentre Europen pour la Recherche Nuclaire(CERN) (the European Laboratory for ParticlePhysics), Geneva, Switzerland. He then spent threeyears with the Accelerator and Fusion Research Di-vision, Lawrence Berkeley Laboratory, University of

California at Berkeley. In 1990, he joined the Institut fr Hochfrequenztechnik,Technische Universitt Braunschweig, as a Professor. Since 2004, he hasbeen a Professor wit the Technische Universitt HamburgHarburg, Hamburg,Germany. His current research interests include the design, packaging, andapplication of integrated (sub-)systems up to millimeter frequencies, and thecharacterization of complex materials.