IEEE MTT-V055-I02 (2007-02A) [55, 2a ed.]

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FEBRUARY 2007

VOLUME 55

NUMBER 2

IETMAB

(ISSN 0018-9480)

PART I OF TWO PARTS

PAPERS

Linear and Nonlinear Device Modeling Design of Compact Directional Couplers for UWB Applications ........ ........ ..... A. M. Abbosh and M. E. Bialkowski A Systematic State–Space Approach to Large-Signal Transistor Modeling ..... ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... M. Seelmann-Eggebert, T. Merkle, F. van Raay, R. Quay, and M. Schlechtweg Active Circuits, Semiconductor Devices, and Integrated Circuits RF Characterization of SiGe HBT Power Amplifiers Under Load Mismatch ... ......... .... A. Keerti and A.-V. H. Pham Optimum Bias Load-Line Compensates Temperature Variation of Junction Diode’s RF Resistance .... ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... .. S. C. Bera, R. V. Singh, V. K. Garg, and S. B. Sharma 0.7–2.7-GHz 12-W Power-Amplifier MMIC Developed Using MLP Technology ....... ......... . ....... ......... I. J. Bahl A 1.9-GHz CMOS Power Amplifier Using Three-Port Asymmetric Transmission Line Transformer for a Polar Transmitter ..... ......... ........ ......... ......... ........ ......... ......... ........ .. C. Park, Y. Kim, H. Kim, and S. Hong

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Millimeter-Wave and Terahertz Technologies Terahertz Performance of Integrated Lens Antennas With a Hot-Electron Bolometer ... ......... ........ ......... ......... .. .. .... A. D. Semenov, H. Richter, H.-W. Hübers, B. Günther, A. Smirnov, K. S. Il’in, M. Siegel, and J. P. Karamarkovic

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Wireless Communication Systems A New Mode-Multiplexing LINC Architecture to Boost the Efficiency of WiMAX Up-Link Transmitters ..... ......... .. .. ........ ......... ......... ........ ......... ..... M. Helaoui, S. Boumaiza, F. M. Ghannouchi, A. B. Kouki, and A. Ghazel

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(Contents Continued on Back Cover)

(Contents Continued from Front Cover) Field Analysis and Guided Waves Effective Parameters for Metamorphic Materials and Metamaterials Through a Resonant Inverse Scattering Approach .. .. ........ ......... ......... ........ ......... ......... ........ . N. G. Alexopoulos, C. A. Kyriazidou, and H. F. Contopanagos Application of Total Least Squares to the Derivation of Closed-Form Green’s Functions for Planar Layered Media ... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ..... R. R. Boix, F. Mesa, and F. Medina Filters and Multiplexers An in situ Tunable Diode Mounting Topology for High-Power -Band Waveguide Switches .. ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ..... T. Sickel, P. Meyer, and P. W. van der Walt Novel Balanced Coupled-Line Bandpass Filters With Common-Mode Noise Suppression ...... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ C.-H. Wu, C.-H. Wang, and C. H. Chen Planar Realization of a Triple-Mode Bandpass Filter Using a Multilayer Configuration ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... . C. Lugo and J. Papapolymerou

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281 287 296

Packaging, Interconnects, MCMs, Hybrids, and Passive Circuit Elements Design of a Ten-Way Conical Transmission Line Power Combiner ...... ........ ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ... D. I. L. de Villiers, P. W. van der Walt, and P. Meyer

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Instrumentation and Measurement Techniques Five-Level Waveguide Correlation Unit for Astrophysical Polarimetric Measurements . ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... .... G. Virone, R. Tascone, M. Baralis, A. Olivieri, O. A. Peverini, and R. Orta

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Microwave Photonics Optical Summation of RF Signals ........ ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... .. .. ........ ......... .. M. Chtioui, A. Marceaux, A. Enard, F. Cariou, C. Dernazaretian, D. Carpentier, and M. Achouche A Fully Electronic System for the Time Magnification of Ultra-Wideband Signals ...... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ....... J. D. Schwartz, J. Azaña, and D. V. Plant

318 327

MEMS and Acoustic Wave Components Optimization and Implementation of Impedance-Matched True-Time-Delay Phase Shifters on Quartz Substrate ....... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ B. Lakshminarayanan and T. M. Weller

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Biological, Imaging, and Medical Applications An Exposure System for Long-Term and Large-Scale Animal Bioassay of 1.5-GHz Digital Cellular Phones .. ......... .. .. ........ ......... ......... ........ ......... ...... K. Wake, A. Mukoyama, S. Watanabe, Y. Yamanaka, T. Uno, and M. Taki

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LETTERS

Corrections to “Efficient Implementations of the Crank–Nicolson Scheme for the Finite-Difference Time-Domain Method ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ...... G. Sun and C. W. Trueman Corrections to “Limits on the Performance of RF-Over-Fiber Links and Their Impact on Device Design” ..... ......... .. .. ........ ......... ......... ........ ......... ......... ........ .... C. H. Cox III, E. I. Ackerman, G. E. Betts, and J. L. Prince Information for Authors

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IEEE MICROWAVE THEORY AND TECHNIQUES SOCIETY The Microwave Theory and Techniques Society is an organization, within the framework of the IEEE, of members with principal professional interests in the field of microwave theory and techniques. All members of the IEEE are eligible for membership in the Society upon payment of the annual Society membership fee of $14.00, plus an annual subscription fee of $20.00 per year for electronic media only or $40.00 per year for electronic and print media. For information on joining, write to the IEEE at the address below. Member copies of Transactions/Journals are for personal use only. ADMINISTRATIVE COMMITTEE J. S. KENNEY, President L. BOGLIONI D. HARVEY S. M. EL-GHAZALY J. HAUSNER M. HARRIS K. ITOH

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Digital Object Identifier 10.1109/TMTT.2007.892054

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 2, FEBRUARY 2007

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Design of Compact Directional Couplers for UWB Applications Amin M. Abbosh and Marek E. Bialkowski, Fellow, IEEE

Abstract—This paper presents a simple design method for a class of compact couplers, which offer coupling in the range of 3–10 dB over an ultra-wide frequency band from 3.1 to 10.6 GHz. The proposed couplers are formed by two elliptically shaped microstrip lines, which are broadside coupled through an elliptically shaped slot. Their design is demonstrated for a 3-, 6-, and 10-dB coupling assuming a 0.508-mm-thick Rogers RO4003C substrate. Results of simulation and measurements show that the designed devices exhibit a coupling of 3 1 dB, 6 1.4 dB and 10 1.5 dB across the 3.1–10.6-GHz band. This ultra-wideband coupling is accompanied by isolation and return loss in the order of 20 dB or better. The manufactured devices including microstrip ports occupy an area of 25 mm 15 mm. Index Terms—Compact ultra-wideband (UWB) couplers, coupled circuits, directional couplers, planar coupler design.

I. INTRODUCTION

B

ROADBAND microwave directional couplers are a very important category of passive microwave circuits. They are used to combine or divide signals with appropriate phase of 90 , and are commonly used in microwave subsystems such as balanced mixers, modulators, and antenna beam-forming networks [1]. In addition, they are essential for developing the cost-effective measurement equipment [2]–[4]. Our particular interest in these devices is with respect to developing an ultra-wideband (UWB) microwave imaging system for breast cancer detection [5], [6]. In these and many other applications, the required couplers are often required to be accomplished in planar (stripline or microstrip) technology. In order to achieve their broadband operation, the approach of coupled transmission lines can be employed. The inherent feature of this approach is that matching and directivity is perfect, and independent of frequency, at least under ideal conditions. However, the challenge is to obtain a tight coupling in the range of 3–6 dB. Using coupled microstrip lines, the tight coupling can be accomplished using the Lange [7] or tandem coupler configurations [8]–[10]. However, they require wire crossovers, which

Manuscript received March 29, 2006; revised June 14, 2006. This work was supported by the Australian Research Council under Grant DP0449996 and Grant DP0450118. The authors are with the School of Information Technology and Electrical Engineering, The University of Queensland, St. Lucia, Qld. 4072, Australia (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.2006.889150

is inconvenient from the manufacturing point-of-view. In addition, the Lange coupler features narrow strips, which create additional manufacturing problems due to the requirement for strict etching tolerances. In turn, the broadband tandem coupler may require wiggles or serpentines to equalize even- and odd-mode phase velocities [9], [11] when realized in microstrip technology. In order to avoid these problems, the slot-coupling approach involving a double-sided substrate, which was first proposed by Tanaka et al. [12], can be applied to realizing a tight coupling. The structure is formed by two microstrip lines separated by a rectangular slot in the common ground plane. Its design formulas were given in [13]. When one aims only at the design of a 3-dB coupler, an alternative is the microstrip-slotline approach, which was described by de Ronde [11]. In contrast to Tanaka et al., the de Ronde’s approach preserves the one-layer microstrip format of the coupler at an expense of etching both sides of a ceramic substrate. One side of this coupler is formed by two parallel connected microstrip lines, while the other one includes a straight slotline with two circular terminating slots. In addition, de Ronde suggested the use of a capacitive disc below the slotline to enhance broadband performance. A very important feature of this coupler is a multioctave operation and a very compact size. By introducing modifications to the original de Ronde’s design, Garcia [14] demonstrated an alternative configuration of a compact planar 3-dB coupler operating, similarly as de Ronde’s device, over the 4 : 1 bandwidth. In his design, Garcia avoided the circular terminating slots and the capacitive disc. Instead, he enlarged the size of a slot below the microstrip layer. This could be the key to achieving UWB performance. By neglecting the capacitive disc beneath the slot, which appeared in the original de Ronde’s configuration, Schiek [15], and then Hoffmann and Siegl [16], produced the design rules for the microstrip-slot 3-dB coupler. However, for the simplified configurations, their designs were not as broadband as offered by de Ronde and Garcia. In this paper, we describe a class of compact planar couplers, which are capable of providing coupling between 3–10 dB over an ultra-wide frequency band. In order to find initial dimensions of these devices, simple design equations similar to the ones described in [13] are applied. Final dimensions are obtained with the use of full-wave electromagnetic analysis software package such as Ansoft’s High Frequency Structure Simulator (HFSS). The validity of the presented designs is confirmed experimentally.

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slot by their rectangular equivalents, as shown in Fig. 1(b). In this case, the rectangular microstrip width and length are and , respectively. The rectangular slot width and length are denoted by and . For the equivalent rectangular shaped microstrips and slot, the analysis and design procedure is similar to the one described in cou[13]. Assuming that the coupler is required to have pling, the even and odd mode characteristic impedances are calculated using (1) and (2) as follows: (1) (2)

Fig. 1. (a) Layout of the proposed wideband coupler including microstrip ports. (b) Equivalent configuration used to work out initial dimensions. (c) Electric field lines for odd- and even-mode excitation.

II. DESIGN The configuration of a compact coupler, which is capable of providing a tight coupling over an ultra-wide frequency band, is shown in Fig. 1(a). In concept, it is similar to the one of Tanaka et al. [12]. The differences concern the shaping of the broadside coupled strips and the slot. They seem to be the key factors behind the UWB performance. The coupler consists of three conductor layers interleaved by two dielectrics. The top conductor layer includes ports 1 and 2. The bottom conductor layer is similar to the top layer, but the ports here are ports 3 and 4. Note that ports 3 and 4 are on opposite sides of the substrate compared to ports 1 and 2, but that this is not a limitation in many applications. The two layers are coupled via a slot, which is made in the conductor supporting the top and bottom dielectrics. As observed in Fig. 1(a), the two microstrip conductors and the slot are of an elliptical shape. The curved microstrip lines are included to make connections to subminiature A (SMA) ports. By assuming that the curved microstrip lines are shortened to zero length, the structure features , double symmetry with respect to the horizontal plane . For in which the slot is located, and the vertical plane the purpose of analysis and design of this coupler, it is sufficient to utilize only the horizontal symmetry plane. In this case, an even-odd mode approach with respect to ports 1 and 3 can be applied to analyze this circuit [17]. The initial analysis and design procedure can be simplified by approximating the two elliptical conducting patches and the

where is the characteristic impedance of the microstrip ports of the coupler. and the coupling factor is Assuming that and can be calculated 3, 6, or 10 dB, the values of from (1) and (2) and are given as follows: 120.5 and 20.7 for dB, 86.7 and 28.8 for dB, and 69.4 dB. and 36.0 for Before commencing the design, we consider the operation of this coupler for the odd and even modes. When the odd mode is excited, the slot can be replaced by a perfect electric conductor. The resulting upper part of the equivalent coupler shown in Fig. 1(b) becomes a microstrip line whose charac. The width realizing can be teristic impedance is determined using standard design equations for a microstrip transmission line [17]. Alternatively, the static formulas described in this paper can be used. From Fig. 1(c), one can see that, in the odd mode, the electric field concentrates mostly in the parallel-plate region formed by the patch and ground plane. A fringe effect, also observed in Fig. 1(c), is less pronounced as becomes large in comparison with the for small substrate thickness . A different wave propagation condition occurs under the even-mode wave excitation. For this mode, the magnetic conductor replaces the slot in the ground plane. Its presence pushes an electric field (launched from the microstrip port) outside the parallel-plate region. This is because the magnetic conductor forming the lower plate does not allow the electric field to be perpendicular to its surface. As a result, the even-mode wave travels in two antipodal slot regions outside the parallel-plate region, as shown in Fig. 1(c). In order to enable a smooth launch of the even-mode wave from the microstrip port to the two antipodal slotlines, the transition formed by the elliptically shaped patches and the ground slot, as shown in Fig. 1(a), is required. and of the equivalent rectangular The dimensions shaped coupler [see Fig. 1(b)], providing the required evenand odd-mode characteristic impedances, are determined using a static approach similar to the one presented in [13]. By using and are given by (3) and (4) as follows: this approach, (3) (4)

ABBOSH AND BIALKOWSKI: DESIGN OF COMPACT DIRECTIONAL COUPLERS FOR UWB APPLICATIONS

where

is the first kind elliptical integral and . Following [13], the parameters and culated using (5) and (6) as follows:

are cal-

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TABLE I VALUES OF DESIGN PARAMETERS IN MILLIMETERS

(5) (6) is the width of the where is the thickness of the substrate, is the width of the top and bottom microstrip patches, and slot of Fig. 1(b). Using the analysis in [18], the ratio of elliptical functions appearing in (3) and (4) can be approximated by the following:

for for

(7) and for given values The synthesis task of determining and is accomplished by solving (3)–(7) using the of Gauss–Newton iteration method. The last step of the design procedure concerns the determiis chosen to be nation of the coupler’s length. Here, , where is the effective wavelength for the microstrip line and can be calculated using standard formulas such as those presented in [17]. Formulas (3)–(7) enable calculations of the equivalent parameters of the rectangular shaped coupler of Fig. 1(b). The next step is to work out the dimensions of the elliptically shaped counter part. Due to compact size, where the dimension is equal or less than a quarter of the effective wavelength, one can expect a similar performance when the rectangular and elliptically shaped couplers occupy an approximately equal area. Using this equivalence principle and assuming that the mean algebraic length of the elliptically shaped coupler is equal to its rectan, then the gular counterpart such that width of the microstrip and the width of the slot for the elliptically shaped coupler can be obtained using (8) and (9) as follows: (8) (9) are adjusted by iteratively The final dimensions , , and running the finite-element method design and analysis package Ansoft HFSSv9.2. In order to test the coupler experimentally, its ports need to be connected to SMA coaxial connectors. To minimize possible reflections, curved microstrip lines, as shown in Fig. 1(a), can be used. Our simulations have revealed that for high-quality impedance match, the radius of these curved lines should not be less than twice the width of the microstrip line.

Fig. 2. Simulated performance of the designed 3-dB directional coupler.

III. RESULTS The validity of the presented design method is tested in examples of 3-, 6-, and 10-dB directional couplers aimed for operation in the 3–10-GHz frequency band. For this band, the center frequency of operation is 6.5 GHz. A Rogers RO4003C substrate featuring a dielectric constant of 3.38 and a loss tangent of 0.0027, 0.508-mm thickness, plus 17- m-thick conductive coating is selected for the couplers development. , , and Using the proposed method, the dimensions are determined and are shown in Table I. One can find that the obtained values are not too far off from the ones calculated using mm using [13] or mm using (3)–(9). First, mm, mm, and mm. (4), mm (for mm), mm, Therefore, and mm. The return loss, coupling, and isolation of the designed couplers are first verified using HFSS. Fig. 2 shows the simulated amplitudes of the scattering parameters for the designed 3-dB coupler. These are followed by results of the phase difference between the two output ports, as shown in Fig. 3. It is clear that the designed coupler features UWB characteristics. The coupling is 3 0.8 dB for the 3.1–10.6 GHz band. The isolation and return loss are better than 28 and 22 dB, respectively, for the band. In Fig. 3, it is observed that the phase difference between ports 2 and 3 is 90 1 over the band. This

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Fig. 3. Simulated phase characteristic.

Fig. 5. Simulated performance of the designed 10-dB directional coupler.

Fig. 6. Manufactured 3-dB coupler. Fig. 4. Simulated performance of the designed 6-dB directional coupler.

result together with the magnitude results shown in Fig. 2 indicates that the coupler operates as a backward wave quadrature coupler [17]. Figs. 4 and 5 show the simulated amplitudes of the scattering parameters for the designed 6- and 10-dB couplers. It can be seen that, for the 6- and 10-dB couplers, the best result for the coupling is obtained for frequencies around the center frequency. The gradual deviation from the specified value of coupling then occurs. In general, the three couplers feature quite a good UWB performance despite only being formed by a one-quarter-wave section of (nonuniformed) coupled lines. The directional couplers are then manufactured and tested using a vector network analyzer. The photograph of the one of the manufactured 3-dB couplers is shown in Fig. 6. The overall dimensions of the coupler including bent microstrip lines are 25 mm 15 mm, indicating that the device is of a very compact size. The manufactured 6- and 10-dB couplers have the same size.

The measured results are presented in Figs. 7–9. As observed in Figs. 7–9, all of the manufactured couplers show UWB behavior with coupling 3 0.8, 6 1.4, and 10 1.5 dB for the 3-, 6-, and 10-dB couplers, respectively, across the 3.1–10.6-GHz band. The isolation is better than 23, 20, and 19 dB, while the return loss is better than 21, 18, and 19 dB for the 3-, 6-, and 10-dB couplers, respectively. As observed from the presented data in Figs. 7–9, the operation of the 3-dB coupler seems to be best and is superior over the one of Garcia [14], which showed the 3 dB 1-dB bandwidth from 4.5 to 8 GHz and the isolation of around 20 dB. The manufactured 6- and 10-dB couplers exhibit some insertion losses, which are not observed in the simulated results. These can be due to conduction and dielectric losses, the difficulty of manual aligning the two microstrip layers forming this type of coupler, and coaxial connectors. The 6- and 10-dB couplers have a smaller width than the 3-dB coupler and as such they are more sensitive to aligning errors. However, in general, the agreement between the simulated and measured results can be considered as very good.

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In some applications, one may wish to house the designed couplers in enclosures. In this case, it is important to assess the effect of shielding. Here, this problem was investigated only via computer simulations. Only brief comments concerning the results of these simulations are reported. The produced simulation results revealed that a metal cover with a height of 0.5 cm below and above the three investigated couplers did not adversely affect their performance, as the electrical characteristics were very similar to those shown in Figs. 2–5. Only small adverse effects of the enclosure were observed when the shielding height above and below the coupler structure was reduced to 0.25 cm. IV. CONCLUSION

Fig. 7. Measured performance of the manufactured 3-dB coupler.

A simple method has been proposed for the design of compact directional couplers for UWB applications. The proposed devices are formed by a multilayer microstrip structure with broadside slot coupling. The coupling is controlled by elliptical shapes of microstrip conductors and a coupling slot. The design method has been demonstrated for the case of 3-, 6-, and 10-dB coupling. The couplers have been manufactured and experimentally tested. They have shown UWB behavior across the band from 3.1 to 10.6 GHz. Due to compact size and good electrical performance, they should be of considerable interest to the designers of UWB components. Our particular aim is to use them in a UWB microwave imaging instrumentation [4]–[6]. ACKNOWLEDGMENT The authors acknowledge the assistance of D. Bill, K. Bialkowski, and S. Padhi, all with the University of Queensland, Brisbane, Australia, in the manufacturing of the couplers. REFERENCES

Fig. 8. Measured performance of the manufactured 6-dB coupler.

Fig. 9. Measured performance of the manufactured 10-dB coupler.

[1] R. Mongia, I. Bahl, and P. Bhartia, RF and Microwave Coupled-Line Circuits. Norwood, MA: Artech House, 1999. [2] G. F. Engen, “An improved circuit for implementing the six-port technique of microwave measurements,” IEEE Trans. Microw. Theory Tech., vol. MTT-25, no. 12, pp. 1080–1083, Dec. 1977. [3] M. E. Bialkowski and A. P. Dimitrios, “A step-frequency six-port network analyser with a real-time display,” AEU Int. J. Electron. Commun., vol. 47, no. 3, pp. 193–197, 1993. [4] M. K. Choi, M. Zhao, S. C. Hagness, and D. W. van der Weide, “Compact mixer-based 1–12 GHz reflectometer,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 11, pp. 781–783, Nov. 2005. [5] M. E. Bialkowski, W. C. Khor, and S. Crozier, “A planar microwave imaging system with step-frequency synthesized pulse using different calibration methods,” Microw. Opt. Technol. Lett., vol. 48, no. 3, pp. 511–516, Mar. 2006. [6] X. Li, E. J. Bond, B. D. van Veen, and S. Hagness, “An overview of ultra-wideband microwave imaging via space-time beamforming for early-stage breast-cancer detection,” IEEE Antennas Propag. Mag., vol. 47, no. 1, pp. 19–34, Feb. 2005. [7] J. Lange, “Interdigitated stripline quadrature hybrid,” IEEE Trans. Microw. Theory Tech., vol. MTT-17, no. 12, pp. 1150–1151, Dec. 1969. [8] J. P. Shelton, J. Wolfe, and R. C. Wagoner, “Tandem couplers and phase shifters for multioctave bandwidth,” Microwaves, pp. 14–19, Apr. 1965. [9] S. Uysal and A. H. Aghvami, “Synthesis and design of wideband symmetrical nonuniform couplers for MIC applications,” in IEEE MTT-S Int. Microw. Symp. Dig., 1988, pp. 587–590. [10] J.-H. Cho, H.-Y. Hwang, and S.-W. Yun, “A design of wideband 3-dB coupler with -section microstrip tandem structure,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 2, pp. 113–115, Feb. 2005.

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[11] F. C. de Ronde, “A new class of microstrip directional couplers,” in IEEE MTT-S Int. Microw. Symp. Dig., May 1970, pp. 184–189. [12] T. Tanaka, K. Kusoda, and M. Aikawa, “Slot-coupled directional couplers on a both-sided substrate MIC and their applications,” Electron. Commun. Jpn., vol. 72, no. 3, pt. 2, 1989. [13] M.-F. Wong, V. F. Hanna, O. Picon, and H. Baudrand, “Analysis and design of slot-coupled directional couplers between double-sided substrate microstrip lines,” IEEE Trans. Microw. Theory Tech., vol. 29, no. 12, pp. 2123–2129, Dec. 1991. [14] J. A. Garcia, “A wideband quadrature hybrid coupler,” IEEE Trans. Microw. Theory Tech., vol. MTT-19, no. 7, pp. 660–661, Jul. 1971. [15] B. Schiek, “Hybrid branchline couplers—Useful new class of directional couplers,” IEEE Trans. Microw. Theory Tech., vol. MTT-22, no. 10, pp. 804–869, Oct. 1974. [16] R. K. Hoffmann and J. Siegl, “Microstrip-slot coupler design—Part I and II,” IEEE Trans. Microw. Theory Tech., vol. MTT-30, no. 8, pp. 1205–1216, Aug. 1982. [17] D. Pozar, Microwave Engineering, 3rd ed. New York: Wiley, 2005. [18] W. Hillberg, “From approximation to exact relations for characteristic impedances,” IEEE Trans. Microw. Theory Tech., vol. MTT-17, no. 5, pp. 259–265, May 1969. Amin M. Abbosh was born in Mosul, Iraq. He received the M.Sc. degree in communication systems and Ph.D. degree in microwave engineering from Mosul University, Mosul, Iraq, in 1991 and 1996, respectively. Until 2003, he was Head of the Information Engineering Department, Mosul University. In 2004, he joined the Centre for Wireless Monitoring and Applications, Griffith University, as a Post-Doctoral Research Fellow. He is currently a Research Fellow with the School of Information Technology and Electrical Engineering, The University of Queensland, St. Lucia, Queensland, Australia. His research interests include antennas, radio wave propagation, microwave devices, and design of UWB wireless systems.

Marek E. Bialkowski (SM’88–F’03) was born in Sochaczew, Poland. He received the M.Eng.Sc. degree in applied mathematics and Ph.D. degree in electrical engineering from the Warsaw University of Technology, Warsaw, Poland, in 1974 and 1979, respectively, and the D.Sc. Eng. (Higher Doctorate) degree in computer science and electrical engineering from The University of Queensland, St. Lucia, Queensland, Australia, in 2000. He has held teaching and research appointments with universities in Poland, Ireland, Australia, U.K., Canada, Singapore, Hong Kong, and Switzerland. He is currently a Professor with the School of Information Technology and Electrical Engineering, The University of Queensland. He has authored or coauthored over 450 technical papers, several book chapters, and one book. His research interests include antennas for mobile cellular and satellite communications, signal-processing techniques for smart antennas, low-profile antennas for reception of satellite broadcast TV programs, near-field/far-field antenna measurements, electromagnetic modeling of waveguide feeds and transitions, conventional and spatial power-combining techniques, six-port vector network analyzers, and medical and industrial applications of microwaves.

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A Systematic State–Space Approach to Large-Signal Transistor Modeling Matthias Seelmann-Eggebert, Thomas Merkle, Friedbert van Raay, Rüdiger Quay, Member, IEEE, and Michael Schlechtweg

Abstract—A state–space approach to large-signal (LS) modeling of high-speed transistors is presented and used as a general framework for various model descriptions of the dispersive features frequently observed for HEMTs at low frequency. Ensuring unrestricted LS–small-signal (SS) model compatibility, the approach allows to construct LS models from multibias SS -parameter measurements. A general transformation between state–space models is derived, which are equivalent in the SS limit, but nonequivalent under LS stimuli. This transformation has the potential to compensate deviations observed by comparing model predictions with LS measurements and to find an optimum state linear LS model without any change of the SS behavior. Index Terms—GaN HEMT, large-signal (LS) transistor modeling, low-frequency (LF) dispersion, state–space approach.

I. INTRODUCTION

L

ARGE-SIGNAL (LS) compact modeling of HEMTs has been a field of high research activity for over two decades. An LS model has to reproduce the complex dynamic of the transistor, which results from the subtle interplay of nonlinearity and memory effects. In the frequency domain, memory effects manifest themselves as dispersion, i.e., as a frequency dependence of the transfer function of the transistor two-port. To be positioned between the physics-based [1], [2] and black box [3], [4] approaches, physics-oriented semiempirical models are compact and enable efficient simulation. A variety of approaches can be found in the literature [5], [6] (and references therein), which differ by model topology and parameterization of the descriptive functions. For transistors not yet developed to perfection such as GaN HEMTs, the accuracy of semiempirical models often limits the overall reliability of nonlinear circuit simulation or is bounded to a bias region. The objective of this paper is to relate compact transistor modeling with the concepts of linear control theory [7] and, by an natural extension to nonlinear dispersion effects, exploit these in view of possible model improvements. II. COMPACT HEMT MODELING Most compact LS models build the transistor from a nonlinear intrinsic part by embedding it into a linear parasitic netManuscript received May 18, 2006; revised October 2, 2006. This work was supported by the Federal Ministry of Defense (BMVg) and by the Federal Ministry of Education and Research (BMBF). M. Seelmann-Eggebert, F. van Raay, R. Quay, and M. Schlechtweg are with the Fraunhofer Institute of Applied Solid State Physics, D-79108 Freiburg, Germany (e-mail: [email protected]). T. Merkle is with the CSIRO ICT Centre, Epping, N.S.W. 1710, Australia (e-mail: [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.2006.889154

work. The intrinsic part contributes nonlinear resistive and displacement currents. In the simplest picture, these currents are attributed to drain and gate current sources and gate charges, which are immediate functions of intrinsic voltages and, hence, independent of the system history [6], [8]. In a more realistic physical picture, both fixed and free charge carriers, which contribute to the terminal currents, result from a complex (and strongly bias dependent) interaction of transport, generation, and recombination processes, and a simple description by voltage-controlled charge and current sources cannot completely account for the frequency dependence of the device response. Therefore, it is common practice to add delay mechanisms for charge [9], current [10], or voltage [11] in order to model magnitude and phase of all four -parameters correctly. Nevertheless, the concept that the intrinsic charge and current sources are immediate functions of two intrinsic voltages is retained. If the existence of such state functions is taken for granted, then there result integrability conditions, which impose restrictions on the small-signal (SS) parameters [8]. A general compatibility of the small-signal equivalent (SSE) circuit and the LS model is not achieved anymore [11]. For GaN HEMTs, the integrability conditions are no longer satisfied on a global scale. Hence, current and charge functions give only an approximative reproduction of the -parameters, and the degree of deviation depends on the bias. Furthermore, HEMT devices often show the behavior that their dc characteristics differ from the resistive current state function obtained by integration of the -parameters at high frequencies (HFs) in the gigahertz range. In the Root model [12, p. 289], the existence of an additional state function is postulated to describe the HF current. The transition from this HF current state function to the dc characteristics is accomplished via a low-pass function. The dc-HF deviation is attributed to the influence of fixed charges [13], which modify the electric field and, as a consequence, affect the control voltages. Being influenced by the bias, these charges are thought to be trapped by imperfections located either at the surfaces between the electrodes or the substrate buffer adjacent to the channel [14]. Since their capture time constants are in the order of some microseconds, the traps cause dispersive features at frequencies below 10 MHz [low-frequency dispersion (LFD)]. However, -parameter systems for characterization of HF transistors have a low-frequency (LF) limit of typical 45 MHz, where the LFD features are absent. As a further LFD mechanism for power devices, we have to consider self-heating and the related thermal memory effects [12]. Parker and Rathmell [15] have illustrated a description of LFD by three state variables, which are determined by first-order differential equations.

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An LS model is frequently constructed from multibias SS -parameter data taken over a wide range of frequencies. For model construction, the -parameter data set has to be complete, i.e., -parameter have to be measured on a narrow voltage grid covering the complete accessible bias range or at least a large region of interest (ROI). Throughout this paper, two different models, which reproduce the same complete -parameter data set and the dc characteristics, are said to have the same SS signature. Often a model is verified by comparison of its predictions with LS load–pull and power-sweep measurements. In the semiempirical model approaches, the SS signature leads to a unique model since the state functions are completely determined by path integration of the SS parameters. Hence, there remain no degrees of freedom for fine tuning the LS behavior of the model and a tradeoff between the match of the LS data and the global match of the SS data may have to be met. However, this predictive dependence of LS data on SS data is a consequence of the chosen state representation and is relaxed for models containing more or different state variables. The goal of this paper is to tie together the semiempirical modeling approach for transistors, particularly HEMTs, and the state–space approach for modeling dynamic systems. In electronics, the state–space approach has been used since the 1960s [16], when it was applied to network analysis. Recently, the state–space approach was discussed for artificial neural network (ANN) modeling of nonlinear microwave circuits [4]. A modified state–space approach to behavioral circuit modeling was introduced by Root et al. [17]. In Section III, a general scheme for state–space modeling of transistors is presented. In Section IV, we will develop a general method of constructing a pool of LS models from a complete set of -parameters. In Section V, the proposed model construction method is applied to the standard SS field-effect transistor (FET) equivalent circuit. In Section VI, various one- and two-mode model approaches for LFD are discussed and compared with experimental power-sweep data for a GaN transistor.

The state equation (1) is the basis for nonlinear systems theory and bears a rich variety of phenomena such as limit cycles, erratic, or even chaotic behavior [18]. The output vector depends not only on the system input, but also on the system state, as described by the output equation (2) In the canonical form (1) and (2), the functions and depend neither on the output entities, nor on the derivatives of the input entities. A complete description of the system requires and output state funcknowledge of the system state function . A system description by the canonical form (1) and (2) tion is called a state–space model [18] or macro model [19]. It bears a huge amount of information, which in numerical form has to function tables in dimenbe provided by sions. A state–space model can be directly implemented in commercially available nonlinear circuit simulators. For any given of the system stimulus, the state equations are waveform solved in the time domain in a straightforward manner [7] to . In the frequency domain, (1) obtain the output response can be solved in an iterative manner by harmonic-balance algorithms. The state equation system is capable to model nonlinear systems in depth, in particular, combined effects of nonlinearity, short-term, and long-term memory. Under equilibrium conditions, we solve (1) to obtain the equilibrium state function (3) which, by definition, satisfies (4) For slight perturbations from equilibrium, a second-order Taylor expansion of the state function with respect to the state vector yields

III. NONLINEAR STATE VARIABLE APPROACH (5) A. Systems Equations and State Functions We consider a system with the input variables and the output variables . For the transistor two-port , we may ) as the input vector choose the port voltages as the output vector. In and the respective currents addition, the system has internal degrees of freedom (“states” or “modes”). At a given time, the system resides at a distinct system state, characterized by the state variable vector ns . The number of independent state variables is called the order of the system. In generalization of the linear approach [7], the state variable vector evolves driven by a first-order differential equation

(1)

The ation matrix

-matrix

of the linear term we call the relax-

(6) -tensor The the relaxation curvature tensor

of the quadratic term we call

(7) According to (4), the zeroth-order term in (5) vanishes and the residual term is of third order in . In the following, we assume that there exists an eigenvalue decomposition of the relaxation matrices, and all eigenvalues as the inverse are nonzero. The delay matrix of the relaxation matrix exists and can be written in terms of an

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eigenvector matrix and a diagonal matrix with diagonal elements resembling the delay times of the system modes

(8) Under equilibrium conditions, the output state function yields (9) For system states not too far from equilibrium, we obtain by Taylor expansion of (2) up to second order with respect to the state vector

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Even for a state linear system, all state functions may depend on the input vector in a highly nonlinear way. For frequency-domain-based simulators, RSL systems have the particular advantage that the state equations for a (periodic) time and can be solved in a dependence of single step. Hence, an RSL type of model facilitates direct extraction of parameters from LS measurements without involved harmonic-balance calculations. B. Equivalent State Transformations

(11)

In the common situation, the state variables of the transistor are not directly measured, but are only indirectly observed by their effect on the output. This fact has the interesting implication that a state–space model is not unique, but there is always a number of exactly equivalent model representations and related state functions. Two representations are equivalent if any arbitrary stimulus of the two-port device produces identical voltage and current responses. The class of mutually equivalent model representations is characterized by the group of transformations under which the nonlinear system eqautions (1) and (2) are invariant. We consider an invertible otherwise arbitrary nonlinear state transformation

(12)

(17)

describes the bias deThe equilibrium output function pendence of the output vector under equilibrium conditions. Please note that all matrices and tensors in (5)–(12), as well as the equilibrium state and output functions, are unique functions of the input vector only. If the curvature term of the state function (5) or the output function (10) can be neglected, we say the system is relaxation state linear (RSL) or output state linear (OSL), respectively. If we consider only system states close to equilibrium, we can truncate the Taylor series of both system functions (5) and (10) after the linear term and denote the system as state linearized. A state linear system obeys the simplified set of system equations

Invariance means that the canonical form of the system equations, when expressed in terms of the transformed state variables, is retained as follows:

(10) where the output matrix are defined by

and the output curvature tensor

(13) (14) For the state linear relaxation equations (13), a time-domain solution is given by the series (15) For moderate variations with time, the output current shows a short time memory effect and the dc current is modified by a (generalized) displacement current obtained by truncating the series (15) after the second term

(16)

(18) (19) The new state functions and the associated functions of the state-linearized limit can be expressed by the original functions, the state transform, and its Jacobian matrix, as listed in Table I. The transformation rules for the system functions (Table I: R3 and R4) and the equilibrium state function (Table I: R5) are verified by inserting (17) into (1), (2), and (4). Taylor expansion of the new system functions according to (5) and (10) with respect to the new state variables about the transformed equilibrium state yields the transformation rules for the output and relaxation matrices. The new relaxation matrix is given by a similarity transform of the original one. Hence, the delay times are independent of the state representation. The equilibrium output function is also not affected. The invariance of the system equations with respect to state transforms has the important consequence that no unique state and can be obtained by experiment unless the functions state variables are accessible directly by independent measurement. In general, we have to exclude this option and rely solely on measurements of the system response . Hence, we have the freedom of choosing a distinct model from a pool of large-signal equivalent (LSE) representations, e.g., under the aspect of simplicity. The preference will be an RSL representation, which, in general, still will have a state nonlinear output function.

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TABLE I EQUIVALENT STATE TRANSFORMS AND TRANSFORMATION RULES FOR ASSOCIATED STATE FUNCTIONS

C. Solution of Decoupled State Equations

IV. LS MACROMODEL SYNTHESIS

A particular simple case is a system with decoupled states, i.e., each vector component of the system state function depends . only on the respective state variable, i.e., In this case, the system can be possibly transformed into an RSL with a diagonal Jacosystem by a state transform bian. For each component, such a transform requires, according to Table I: R3

(20)

Here, we discuss the bias-dependent SS frequency response of a nonlinear system with inner state variables and analyze the prospect of macromodel synthesis [19] from such an SS signature. A. Bias-Dependent SS -Parameters Under SS conditions, the system is state linear by definition and, therefore, we cannot expect to get from an SS signature more than a state linearized truncation of the complete state–space model. The system excitation can be expressed by the input vector (24)

The solution of this equation is The state vectors have the form

(25) (21) and depends on the bias point . It is obvious that for an arbitrary state function, a state linearization can only be achieved in a local environment of a given bias point. Under the aspect of simplicity and speed, a model, which is globally RSL and diagonal, will always be the preferred choice from the pool of LSE representations. In this case, the time-scale transformation (22)

Insertion of (24) and (25) into (14) yields

(26) where is a residual term of second order with respect to the input and state variables. The gradient operation is performed matrix. row wise with respect to and results in an Evidently, in the SS limit, the current has the form (27)

leads to an explicit analytical solution to the system equations (23) which, by insertion into (2), yields the output response for any arbitrary time-dependent input signal.

where the SS part resulting from (13)

depends on the time response

(28)

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Neglecting higher order terms, we use the eigenvalue decomposition (8) of the delay matrix and solve (28) by Fourier transformation

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If two modes have identical delay times, a unique mode assignment is no longer possible. Apparent violations of the factorization conditions may occur in this degenerate case. C. Factor Reconstruction

(29) Hence, in the frequency domain, SS output and input are related by the transfer function of the system

(30) which, in the case of the transistor, corresponds to the bias-dependent admittance matrix. In (30), we introduced the input matrix (31) In terms of the input, output, and relaxation matrix, the transfer function has the form well known from linear control theory (see, e.g., [7, p. 35]). In the considered nonlinear case, the input matrix is given as a gradient of an equilibrium state function. B. Mode Matrices and Factorization With use of (8), the transfer function can also be written in the so-called pole-residue form [19]

We now investigate the feasibility to retrieve the original state linear equations (13) and (14) from a complete -parameter data set (from which the admittance matrix data can be calculated). The goal is to identify the number of system modes, their equilibrium state function, the delay matrix, and the output matrix, all of which may be bias dependent. To achieve this goal, we will have to pass through the following five steps. Step 1) Determination of mode matrices and delays. Step 2) Preparation of input and output factors. Step 3) Finding a valid eigenvector matrix. Step 4) Setting up a special state representation. Step 5) Tailoring the state representation. We suppose—by appropriate analysis of the frequency dependence of the admittance matrix—we have determined a and respective mode matrices unique set of delay times [Step 1)]. The goal is to determine the state functions on the basis of (33). For this purpose, we have to construct output factors and input factors from the mode matrices according to and (35) in Step 2). For each mode , we select a column a row for which is in the entire ROI, and obtain the input factors (37)

(32) where we have introduced the frequency-independent mode matrices

One of the four factor elements of each mode matrix can be set to an arbitrary value, hence, we can generate a special set with . With this smooth bias dependence by choosing choice, we readily obtain all remaining output factors (38)

(33) (34) Equation (32) means that the admittance matrix can be prematrix terms (modes) each of which is sented as a sum of characteristic for a specific state variable with a respective delay describes the system at time. The transit mode matrix are real valued infinite frequency. The mode matrices if all delay times are real valued. By inspection of (33), we see satisfy the that the mode matrix coefficients factorization conditions (35) In the case of a two-port device, the factorization conditions (35) mean that the determinant of each mode matrix has to vanish ) (for (36)

D. Valid Eigenvector Matrices and State Functions Unfortunately, the reconstruction of the state equations is not straightforward since the definition (35) of the mode factors results in matrices and , which are not unique. The undetermined norm of the eigenvectors in the matrix allows for multiplication with a diagonal matrix . The constructed mode factors are related to the input matrix of (31) and the output matrix (11) according to

(39) (40) and input and output matrices apparently can be determined . only within a class of invertible matrix transformations However, the variation of the eigenvectors with bias matters has to satisfy (31). Solving (39) since the output matrix and and inserting (31), we get the and (40) for

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following relation between the bias dependent set of factors and and the state functions:

integrability equation (44), i.e.,

(46) (41) where

.

(42) E. Manifold of SSE Models (43) For a given set of coefficients , (41) constitutes a condition for the otherwise unrestricted nonsingular matrix function . If is a solution of (41), it is a valid eigenvector , and we obtain the state functions matrix from (43), from (42), and from (41) as a special state representation of the state linearized are system [Step 4)]. The equilibrium state functions determined from (as path independent result) by path integration. For the state–space reconstruction, we first need to find a valid eigenvector matrix [Step 3)] as given by (41), which forms a set equations for unknown matrix elements of . We point out two approaches of finding a special solution of (41). A direct algebraic solution is possible if there are at least as many state variables as input variables. In this case, matrix elements and solve the linear we may zero equation (41) for the remaining matrix elements. Surprisingly, there is a free choice of the equilibrium state functions unless the factorizing coefficients are ill conditioned. For the other approach, we differentiate (41) with respect to the input variables and find that an equilibrium state function satisfies that satisfies (31) exists if and only if the matrix the integrability conditions

The whole manifold of state-linear models showing the same SS signature is obtained as the manifold of solutions for (41) or (44) spanned by the variety of consistent boundary conditions. The following operation will enable us to generate this manifold from a single initial model reproducing the SS signature. First, we employ a nonlinear state transform (17) and obtain a new representation, which is exactly equivalent, but nonlinear in the state variables. Next, we state linearize the new representation and obtain a new state-linear model with exactly the same SS signature. From Table I, we see that such SSE state transformations can be expressed in terms of a bias-dependent matrix , which transforms the input, output, and relaxation matrices. However, since the equilibrium state function is retained cannot be chosen comupon state linearization, the matrix in (41), it has pletely arbitrarily, but similar to the matrix to obey a subtle integrability condition. According to Table I: R6–R11, the transformation with the Jacobian of the state transform leaves the mode matrices unaffected. Also by construction (owing to Table I: R5), the state transform maintains the integrability condition (41) and (44). Vice versa, any solution of (41) can be traced to a transof the special representation. Due to (39), any form of (41) is related to the functions other solution of the special representation by

(47) (44) Equation (44) represents a quasi-linear system of partial difinput variferential equations (PDEs) (with respect to the matrix with a manifold of soluables) for the tions. The second solution approach is to choose the matrix to be diagonal. Upon temporary suppression of the state variable index, for each state variable, (41) reads , i.e., the row vector defines . For a two-port, we can the direction of the gradient of to obtain the equivalent equation eliminate (45) where The PDE (45) defines the contour lines of constant . The general invariance with respect to nonlinear state transforms leaves the freedom to assign a value of the state variable to each contour line of the equilibrium state function. Alternatively to (41), we may solve the

and are nonsingular in the en(given the matrices into a function of the state tire ROI). We turn the function variables by assigning to each bias point the respective values . By path integration given by the equilibrium function of the transformation matrix,

(48) and with use of transformation rule Table I: R5, we have estabbetween the old and lished the functional relation new state representation. For an arbitrary matrix , the integral still depends on the path, but for a proper choice in accordance with Table I: R2, it becomes path independent. Yet a valid SSE transformation matrix can be obtained from an arand an arbitrary convention bitrary nonsingular matrix of the path integration by performing (48) to establish the LSE state transform and subsequently carrying out the gradient operation on the obtained new equilibrium function.

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F. Consequences and Construction Strategies It is interesting to note that, for a transistor two-port, there always exists a diagonal state-linear model, which is consistent with its SS signature. If there are at least two state variables, we can apply the algebraic solution approach to solve (41) and choose the port voltages (or combinations of them) as equilibrium state functions. We then use the obtained eigenvector maas matrix in (48) to construct an equivalent trix state transform by path integration of the equilibrium state function. By construction, the new state representation has a diagonal relaxation matrix. Hence, for a well-conditioned system , there always exists a transformation, which diagonalizes the relaxation matrix for all bias conditions, yet this state representation is not unique. The set of all state-linear and diagonal models can also be obtained directly by solving the contour (45) or (46) for a diagonal state representation. As discussed in Section IV-D for the contour degrees of lines of the equilibrium state, there still remain freedom for nonlinear expansions and contractions along the diagonal state axes associated with a spectrum of respective state functions. In the state-linear limit, all these diagonal state representations are mutually equivalent. However, state linearity is lost upon any LSE state transformation (17) and there is a clear difference between state–space representations under LS excitation when quadratic state terms become important [see (5) and (10)]. This difference can be used for an individual nonlinear readjustment of the scale for each state variable to maintain state linearity for LS conditions. The considered SSE transforms have the potential to prepare an optimum state-linear model. However, in general, they will not be able to prepare an optimum RSL model since this possibly possesses a state-nonlinear output function. Please note that any state model leaves the free choice of setting up the linear scales for the state variables. Gain and offset constants of each individual scale remain to be defined, e.g., by assigning appropriate values to the equilibrium state function at convenient bias points. V. LS APPROACH FOR THE HEMT STANDARD EQUIVALENT CIRCUIT

Fig. 1. Equivalent circuit of the HEMT as composed: (a) by lumped elements and (b) by block elements.

effect of a drain capacity is emulated by the dispersion of the output conductance. The combined effects of the transit time of the current source and the delay of the control voltage are . Since each approximated by a mode with the delay time component in (49) is of the form (50), in Step 1) (with ), we directly obtain the mode matrix decomposition (32) (51)

(52)

In the following, we construct a global LS extension for the SS standard equivalent circuit of the FET (Fig. 1), which consists of bias-dependent lumped elements. The intrinsic admittance matrix can be written as

A factorization [Step 2)] of the mode matrices is given by

(49) (53) Each of the subunits in Fig. 1(b) can be assigned to a distinct system mode and be described by three constants, i.e.: 1) an LF conductivity ; 2) an HF conductivity ; and 3) a time constant . We can use the identity (50) where to eliminate all capacitors from the LS model. By assuming a very small value for , the

(54) In the Step 3), we would have to select eight nonzero elements of the 4 4 eigenvector matrix and set up the linear equations for the algebraic solution of (41). However, the simple form of facilitates direct integration and yields the unity matrix as an eigenvector matrix. The output matrix is given by (53) and

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the equilibrium state functions are simple combinations of the port voltages [Step 4)]

in the For each LF mode, we have to add an extra row in the output matrix input matrix and an extra column

(55)

(61)

The state variables correspond to delayed voltages with the diagonal delay matrix

(62) For an LS model, the input matrix has to equal the gradient of the equilibrium function according to (31) (63)

(56) If the system description is correct, the mode matrices should satisfy the condition

(57) Insertion of the state functions (53) and (55)–(57) into (13) and (14) yields an LS model, which is consistent with the SSE circuit of Fig. 1 over the entire bias range. No additional assumptions have to be made about the conductances, which define the system by (52) and (53) and contain the gate charge functions, implicitly. VI. LFD LFD leads to a situation where the extrapolated LF limit of the admittance matrix deviates from the dc admittance and (57) is apparently violated. This deviation has to be attributed to additional system modes with large time constants so that the respective dispersion is not observed in the investigated frequency range with the lower limit . A. Separation of the LFD Modes

and from the state-linear approach (30), we obtain the following SS relation between the (measured) LFD matrix and (sought) state functions: (64) The respective state-linear output equations are

(65) If the input signal contains only (harmonic) frequency components well above the LFD frequency , then the state variable is nearly constant even in LS operation (“frozen state”). If (and only if) we have an RSL representation, then according to (13), the state variable is close to the time average (66)

B. Single-Mode LFD Models It is common practice to approximate the LFD by a single mode (dispersion model of Root [12], [20]) with the properties

It is convenient to introduce an LFD matrix as the difference (67) i.e., (58) (68) The dispersion matrix may be composed of several components, which, in principle, can be resolved by -parameter measurements at LF. However, if such information is not available, we may assume that there is only a single dispersion mode or that the dispersion modes are degenerate. Since the delay constant of LFD is not known exactly, commonly an estimated voltage-independent value is used. If (and only if) there is no coupling between HF and LF dispersive modes, then there are two separate sets of state equations (13) for the LF and HF modes and the output matrix, i.e., (59) (60)

(69) The Root model postulates the existence of a primitive state and that the function for the drain current at the LF limit LFD matrix satisfies corresponding integrability conditions. It can be shown that even if (67) is not satisfied globally, there still exists a state function that matches the LFD matrix exactly intersecting the along a given straight or bent line -bias plane. However, deviations of the voltage trajectories from such a line may be considerable and pose a problem for global LS models. We now discuss two extensions of the Root model, which are globally compatible with the LFD matrix. If there is only and a single mode, we begin with

SEELMANN-EGGEBERT et al.: SYSTEMATIC STATE–SPACE APPROACH TO LS TRANSISTOR MODELING

and solve the PDE (46) for a single bias-depen. With any regular solution dent transformation element in the ROI) of this equation, we have the output ( matrix

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and obtain the equilibrium function of the LFD state by path integration

For the dispersion models discussed above, assumptions about the output matrix were made and the equilibrium state function was derived in turn. We now investigate the opposite option of choosing an equilibrium state function first and finding a suitable output matrix. It is a surprising fact that there is a free choice for the equilibrium functions under the premises that: 1) there are two dispersion modes and 2) their equilibrium functions are invertible. Under these conditions, (64) can always be inverted as follows:

(71)

(78)

(70)

This equation looks similar to the path integration used for the ), but now by construction yields a Root model (with path-independent result. In practice, construction of the dispersion equilibrium state function on the basis of the PDEs (45) or (46) turns out to be difficult since complications arise in regions where all elements of the dispersion mode matrix are close to zero.

The simplest choice for the equilibrium functions, which complies with these requirements, is (79) and results in a voltage lag model [21] with (80) (81)

C. Double-Mode LFD Models These problems can be circumvented by another extension of the simple Root model, which is based on the assumption that there are two LFD modes with the same or similar delay constants. For the state-linearized system, we make the approach (72)

The assumption of two LFD modes is plausible in view of distinct gate and drain lag effects [14]. These effects are thought to arise from charges at deep traps, which are located on the drain side and the source side of the gate and which are influenced by the bias conditions. D. Analytic Comparison of LFD Models

(73) defines a preferential where the unit vector axis in the bias plane. This double current state (DCS) approach yields the LFD mode matrix (74)

It is interesting to compare the analytical expressions obtained with the voltage lag model (80), (81) and the Root model (68), (69). Under the assumption that (63) is satisfied, both models will generate identical -parameter data. Under LS conditions (such as load–pull experiments) with negligible LF input signal components, according to (66) the Root model predicts (82)

The basic idea of the DCS approach is to use the identity whereas the two-mode voltage lag model yields (75) and to choose a direction for the path integral where the second term vanishes. For the special case , i.e., , we have (76) (77) The first equation is readily integrated [with the freedom to choose an additive function ] and yields the first summand of the second equation, which then can be resolved for the second state function. Resubstitution into the first equation yields the first state function. The same procedure works for any direction of the unit vector by path integration along the direction orthogonal to .

(83) The predictions of the two models clearly differ as soon as has to be taken into the nonlinearity of the function account. Finally, we exemplify the effect of the equivalent state transform (17) with the Root model. By application of (17) and neglect of higher order terms, we have the most general state-linear one-mode LFD model for the special LFD matrix (67) satisfying the integrability constraint. With use of Table I: R7, it yields the LS response (84)

The scalar function is arbitrary, but has to be either globally monotonic increasing or decreasing.

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E. Power-Sweep Simulations In the following, we want to exemplify the effect and usefulness of SSE transforms by comparison of model prediction with experimental data for a transistor in LS operation. A GaN power HEMT with eight fingers and a total gatewidth of 480 m was stimulated at 2 GHz under class AB conditions V V , and the microwave power consumed at an ohmic output load of 50 was measured in dependence of the input power. Simultaneous measurements of the components of the input and output voltage waves up to the ninth harmonic order facilitated the reconstruction of the current and voltage trajectories in the time domain. A rhomboidal-shaped region was filled by the voltage trajectories implying that a suitable LS model V V and should at least cover the voltage range V V. The influence of LFD phenomena on the performance of this transistor was shown by dc-pulse measurements, which showed a pronounced variation of the output characteristics with the quiescent bias. To built a state-of-the-art LS HEMT model, a complete -parameter set was measured in the frequency range from 300 MHz to 20 GHz for a bias list covering the ROI. The model obtained from this SS database comprises the elements of the parasitic network, the voltage-dependent charge, and delay functions and current functions for the dc input and output currents. Output conductance and transconductance as extrapolated from the -parameter measurements clearly did not form the complete differential (67), as required for the Root model, since above the knee voltage) the transfer characteristics (at any consistently showed an SS gain, which was consistently smaller, and an output conductance, which was substantially larger than obtained from the dc current. On the other hand, the experimental output current trajectory was found to be qualitatively was rescaled by a reproduced if the dc current function . The optimum varies with the input power, factor load, and operation bias. For the considered 5- load agreement of the output power was achieved at SS levels by setting . In order to study the effect of the different LFD approaches on power sweep data, we will discuss four models is a constant. None of these under the assumption that models are intended to give a global description of the LFD phenomena. The first model disregards LFD. The second model (model Ro) is a simple one-mode Root model [see (67)–(69)] with as the equilibrium function. This LFD implementation predicts a uniform reduction of the drain current when the LS excitation is changed from equilibrium (dc) to frozen state (HF). A third model is obtained from the second by a specific nonlinear state transformation (model RoM) for which a change in LS output current is expected according to (84). The fourth model (model VLag) is a two-mode LFD model according to (79)–(81). The second row of the output matrix has been intentionally chosen to equal the gradient of the equilibrium function of model Ro. This way it is accomplished that the models Ro, RoM, and Vlag have the same global SS signature and dc characteristics. However, the to current response on a voltage pulse from will be quite different for the four model situations. While the

2

Fig. 2. Power-sweep characteristics for a 8 60 GaN HEMT in class AB operation in comparison with predictions from four different LFD models. (a) Output power. (b) Gain. (c) PAE versus input power.

models Ro and RoM lead to a uniform current decrease, the voltage lag model introduces bead-shaped structures in the output characteristics. All models and modes have the same s. delay time Fig. 2 shows a comparison of the experimental power-sweep characteristics and the model predictions. For very low input power ( 1 dBm) output power, gain, and power-added efficiency (PAE) coincide by construction for the models Ro, RoM, and VLag, whereas gain and output power is overestimated if LFD is neglected. As compression starts, the models differ appreciably. While the Root model (in this specific version) underestimates output power and PAE, the VLag model gradually approaches the too optimistic predictions under LFD neglect. For

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the model RoM, there is a good correspondence of predicted and measured output power and PAE. This is not surprising since the (used to create RoM specific state transform from Ro) was found by trial and error with the goal to reproduce the compression of the output power without any distortion of the SS signature. This example shows that higher order terms (5) and (10) in the state variables are normally important, but can be possibly compensated for by an appropriate choice of the state representation.

VII. CONCLUSION In this paper, we have presented a general and consistent state-variable approach for LS modeling of transistors, which facilitates the treatment of nonlinearity and memory effects. As shown in Section VI, a complete -parameter data set does not suffice to retrieve all possible output variations and off-equilibrium state configurations accessible under LS excitation; however, it allows for the construction of a state-linear LS model consisting of a relaxation matrix, an output matrix, an equilibrium state function, and an equilibrium output function. The key entities for model construction are the mode matrices. They satisfy factorization relations with factors, which yield trial input and output matrices. Bias-dependent similarity transformations are employed to obtain appropriate input matrices, which satisfy integrability conditions and provide the equilibrium state functions. In summary, a complete -parameter data set gives insight into the dimension and the relaxation times of the state space and devises state functions in a state-linearized limit. From an initial model representation, a large pool of models, which conform with a given SS signature, can be generated by SSE state transforms. An LS database is necessary to prepare from this pool a state-linear or an RSL model or to construct higher order state terms of the state functions. LFD has been described consistently by states relaxing at LFs. Single- and double-mode approaches have been presented for the description of LFD phenomena and are compared with experimental power-sweep data for a GaN transistor.

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[6] M. Iwamoto, D. E. Root, and J. Wood, “Device modelling for III–V semiconductors,” in IEEE CSIC Dig., 2004, pp. 279–282. [7] T. Glad and L. Ljung, Control Theory: Multivariable and Nonlinear Methods. New York: Taylor & Francis, 2000. [8] P. Jansen and D. Schreurs, “Consistent small-signal and large signal extraction techniques for heterojunction FET’s,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 1, pp. 87–93, Jan. 1995. [9] A. Werthof, F. van Raay, and G. Kompa, “Direct nonlinear power MESFET parameter extraction and consistent modelling,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1993, vol. 4, pp. 645–648. [10] M. C. Foisy, P. E. Jeroma, and H. H. Martin, “Large-signal relaxation time model for HEMTs and MESFETs,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1992, pp. 251–254. [11] I. Schmale and G. Kompa, “A symmetric non quasi-static large-signal FET model with a truly consistent analytic determination from DCand S -parameter data,” in Proc. 29th Eur. Mircrow. Conf., Munich, Germany, Oct. 1999, vol. 1, pp. 258–261. [12] R. Anholt, Electrical and Thermal Characterization of MESFETs, HEMTs and HBTs. Boston, MA: Artech House, 1995. [13] S. C. F. Lam, P. C. Canfield, and D. J. Allstot, “Modeling of frequency and temperature effects in GaAs MESFETs,” IEEE J. Solid-State Circuits, vol. 25, no. 2, pp. 299–306, Feb. 1990. [14] S. C. Binari, P. B. Klein, and T. E. Kazior, “Trapping effects in GaN and SiC microwave FETs,” Proc. IEEE, vol. 90, no. 6, pp. 1048–1058, Jun. 2002. [15] A. E. Parker and J. G. Rathmell, “Bias and frequency dependence of FET characteristics,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 2, pp. 588–592, Feb. 2003. [16] E. S. Kuh and R. A. Rohrer, “The state-variable approach to network analysis,” Proc. IEEE, vol. 53, no. 7, pp. 672–685, Jul. 1965. [17] D. E. Root, J. Wood, N. Tufillaro, D. Schreurs, and A. Pekker, “Systematic behavioral modeling of nonlinear microwave/RF circuits in the time domain using techniques from nonlinear dynamical systems,” in Proc. IEEE Int. Behavioral Modeling and Simulation Workshop, Oct. 2002, pp. 71–74. [18] H. K. Khalil, Nonlinear Systems. New York: Macmillan, 1992. [19] A. Stelzer, R. Neumayer, F. Haslinger, and R. Weigel, “On the synthesis of equivalent circuit models for multiports characterized by frequency-dependent parameters,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2002, vol. 43, pp. 721–724. [20] D. E. Root, “Measurement based active device modelling for circuit simulation,” in 23rd Eur. Adv. Microw. Microwave Conf., Devices, Characterization, Modeling Workshop, 1993, pp. 191–193. [21] P. A. Traverso, M. Pagani, F. Palomba, F. Scappaviva, G. Vannini, F. Filicori, A. Raffo, and A. Santarelli, “Improvement of PHEMT intermodulation prediction through the accurate modelling of low-frequency dispersion effects,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2005, pp. 465–468.

REFERENCES [1] C. M. Snowden, “Nonlinear modelling of power FETs and HBTs,” Int. J. Microw. Millimeter-Wave Comput.-Aided Design, vol. 6, pp. 219–234, Jul. 1996. [2] W. Batty, C. E. Christoffersen, A. J. Pank, S. Davi, C. M. Snowden, and M. B. Steer, “Electro-thermal CAD of power devices and circuits with fully physical time-dependent compact thermal modelling of complex nonlinear 3-D systems,” IEEE Trans. Compon. Packag. Technol., vol. 30, no. 3, pp. 566–590, Sep. 2001. [3] D. Schreurs, J. Verspecht, E. Vandamme, N. Vellas, C. Gaquiere, M. Germain, and G. Borghs, “ANN model for AlGaN/GaN HEMTs constructed from near-optimal-load large-signal measurements,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2003, pp. 447–450. [4] J. Xu, M. C. E. Yagoub, R. Ding, and Q. J. Zhang, “Neural based dynamic modeling of nonlinear microwave circuits,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2002, pp. 1101–1104. [5] A. Alabadelah, T. Fernandez, A. Mediavilla, B. Nauwelaers, A. Santarelli, D. Schreurs, A. Tazon, and P. A. Traverso, “Nonlinear models of microwave power devices and circuits,” in Proc. 12th GaAs Symp., Oct. 2004, pp. 191–193.

Matthias Seelmann-Eggebert received the Diploma degree and Ph.D. degree in physics from the University of Tübingen, Tübingen, Germany, in 1980 and 1986, respectively. From 1980 to 1996, he was involved with research and development related to infrared detectors based on HgCdTe and developed electrochemical and surface physical methods for the characterization of compound semiconductor surfaces. From 1990 to 1991, he was a Visiting Scientist with Stanford University. From 1997 to 2000, he was engaged in the growth of chemical vapor deposition (CVD) diamond. Since 2001 he has been a member of the Department of High-Frequency Electronics, Fraunhofer Institute of Applied Solid State Physics (IAF), Freiburg, Germany, where he is concerned with the preparation and development of simulation models for active and passive III–V devices.

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Thomas Merkle received the Dipl.-Ing. Degree in electrical engineering from the University of Stuttgart, Stuttgart, Germany, in 1999. His doctoral thesis concerned the investigation of intermodulation distortion in HEMTs, which has been submitted to the University of Ilmenau. Upon the completion of the M.Sc. degree, he joined the Fraunhofer Institute of Applied Solid State Physics (IAF), Freiburg, Germany, where he was involved with the field of nonlinear characterization and modeling of GaAs and GaN HEMTs, as well as the design of monolithic microwave integrated circuits (MMICs) from 30 to 110 GHz. Since 2005, he has been with the CSIRO ICT Centre, Epping, N.S.W., Australia, where he is involved with the design of active integrated antennas from 60 to 100 GHz. His general research interests comprise devices and systems with wanted and unwanted nonlinear behavior at HFs, especially problems in model identification, parameter extraction, and design methods.

Friedbert van Raay received the M.Sc. degree in electrical engineering from the Technical University of Aachen, Aachen, Germany, in 1984, and the Ph.D. degree from the University of Kassel, Kassel, Germany, in 1990. From 1992 to 1995, he was with SICAN GmbH, Hannover, Germany, where he was involved with RF system development and measurement techniques. In 1995, he returned to the University of Kassel, as a Senior Engineer, where he supervised the Microwave Group, Institute of High Frequency Engineering. In November 2001, he joined the Fraunhofer Institute of Applied Solid-State Physics (IAF), Freiburg, Germany, as a Supervisor of the Device Modeling Group. His current research interests are development and characterization of high-speed digital and high-power millimeter-wave GaAs and GaN devices and circuits.

Rüdiger Quay (M’01) was born in Cologne, Germany, in 1971. He received the Diplom degree in physics from Rheinisch-Westfälische Technische Hochschule (RWTH) Aachen, Aachen, Germany, in 1997, and the the Doctoral degree in technical sciences (with honors) from the Technische Universität Wien, Vienna, Austria, in 2001. He currently is a Senior Research Engineer with the Fraunhofer Institute of Applied Solid-State Physics (IAF), Freiburg, Germany. He has authored or coauthored over 50 refereed publications. His scientific interests include RF semiconductor device and MMIC development, heterostructure device modeling and simulation, advanced measurement techniques, and circuit and reliability issues.

Michael Schlechtweg received the Dipl.-Ing. degree in electrical engineering from the Technical University Darmstadt, Darmstadt, Germany, in 1982, and the Dr.-Ing. degree from the University of Kassel, Kassel, Germany, in 1989. Since 1989, he has been with the Fraunhofer Institute for Applied Solid-State Physics (IAF), Freiburg, Germany, where he was initially involved with the design and characterization of microwave and millimeter-wave integrated circuits and with nonlinear characterization and modeling of active RF devices. Since 1996, he has been Head of the High-Frequency Devices and Circuits Department, where he has focused on the design and the characterization of devices and integrated circuits based on III–V compound semiconductors (GaAs, InP, and GaN) for HF applications. He has authored or coauthored over 150 scientific publications. He holds two patents. Dr. Schlechtweg was the recipient of the 1993 Fraunhofer Prize and the 1998 European Microwave Prize.

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RF Characterization of SiGe HBT Power Amplifiers Under Load Mismatch Arvind Keerti, Student Member, IEEE, and Anh-Vu H. Pham, Senior Member, IEEE

Abstract—We present the RF characterization of silicon–germanium heterojunction bipolar transistor (SiGe HBT) power amplifiers (PA) under load mismatched conditions. Experimental results demonstrate a strong dependence of a PA’s RF performance on the phase of mismatched antenna loads. For a load mismatch of voltage standing-wave ratio (VSWR) of 10 : 1, the 1-dB compressed RF output power ( 1 dB ), transducer gain ), output third-order intercept point, and power-added effi( ciency differ by 7.5 dBm, 8.1 dB, 8.3 dBm, and 15%, respectively, between the optimal and worst phase conditions, for an SiGe HBT PA biased at CC = 3 3 V, collector current CE = 400 mA, and frequency of 1.88 GHz. At the optimal phase condition up to VSWR of 10 : 1, the SiGe HBT PA maintains its linearity, RF output power, gain, and efficiency close to that at a VSWR of 1 : 1. At all the nonoptimal phases, the deterioration in the RF performance increases with the magnitude of load mismatches. The nonlinear characteristic of a PA under load mismatches is due to amplitude and phase-distortion mechanisms. Index Terms—Antenna, distortion, linearity, load, mismatch, phase, power amplifier (PA), SiGe HBT, voltage standing-wave ratio (VSWR), wireless.

I. INTRODUCTION ODERN MOBILE wireless standards like code division multiple access (CDMA)-IS95 and wideband code division multiple access (W-CDMA) use nonconstant envelope modulation schemes, and impose a stringent requirement on the linearity of a power amplifier (PA) in a wireless system. SiGe HBT-based technologies have become a viable solution for high-frequency wireless PAs and transceivers because of their high linearity, high current gain, low noise, and compatibility with BiCMOS processes [1]–[3]. RF linearity characteristics of SiGe HBTs are analyzed in detail using Volterra series [4] and charge control theory [5], whereas the effect of load and source impedances on phase distortion and linearity is analyzed in [6]. The optimization of the input and output matching network for linearity and efficiency in HBT PAs is further discussed in [6]. However, the linearity of SiGe HBT PAs under load mismatches has not been studied.

M

Manuscript received February 10, 2006; revised August 17, 2006. This work was supported in part by the University of California at Davis and by the University of California MICRO. A. Keerti was with the Electrical and Computer Engineering Department, University of California at Davis, Davis, CA 95616 USA. He is now with Qualcomm Inc., Campbell, CA 95008 USA (e-mail: [email protected]). A.-V. H. Pham is with the Electrical and Computer Engineering Department, University of California at Davis, Davis, CA 95616 USA (e-mail: pham@ece. ucdavis.edu). 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.2006.889326

A wireless device encounters impedance mismatches at the antenna due to signal reflections from the surrounding objects. RF power is reflected back into a PA due to the antenna mismatches and degrades the PA’s linearity, RF output power, and efficiency [7], [8]. Traditionally, an isolator is placed between the PA and an antenna to protect the output power transistor from large voltage/current swings and improve the PA linearity under load mismatches. However, isolators are bulky and expensive for handheld devices, as well as reduce the transmitted power level. Hence, the goal is to develop isolatorless modules for linear PA applications. Previous work on load mismatches has focused on ruggedness of power transistors and control circuitry to protect the transistor at large impedance mismatches [9], [10]. SiGe HBTs are found to be rugged enough to handle large voltage excursions caused by load mismatches [11], [12]. Circuit techniques have been developed to improve the linearity of a PA under antenna mismatches using drive-level adjustment [7], [13]. However, lowering the drive level reduces the output power necessary for a communication link. Tuning circuits are placed between the transceiver and an antenna so that the effective impedance seen towards the antenna at the output of a transmitter is the matched load (50 ), even when the actual antenna impedance is mismatched. These impedance tuners are simple or T networks with their elements tuned digitally, based on genetic algorithms to achieve high-speed impedance tuning [14]–[17]. In order to maximize the performance of a PA under antenna mismatches, it is imperative to understand the RF output power, linearity, transducer gain, power-added efficiency (PAE), and breakdown voltage of a PA terminated with a mismatched load. This paper presents a systematic analysis to understand the effect of load impedance mismatches, on RF output power, linearity, transducer gain, PAE, and breakdown voltage of an SiGe HBT PA. At each load voltage standing-wave ratio (VSWR), as the phase of the load reflection coefficient at the output of a PA varies from 0 to 360 , the linearity, RF output power, gain, and efficiency swing from the optimal values to worst values. A performance identical to that of the matched condition is achieved at the optimal phase condition for load mismatches as large as 10 : 1. At all other phase conditions, the RF performance degrades rapidly. The degradation in the RF performance at the nonoptimal phases increases with a rise in the magnitude of the VSWR. This phase dependence of PA performance under impedance mismatches is pronounced at different collector currents. A detailed investigation shows that the amplitude and phase-distortion mechanisms cause nonlinearities in a PA under load mismatches. Section II describes the dependence of a PA’s RF parameters on load impedance. Ruggedness of SiGe HBTs is further

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is the -parameters of the PA output matching netwhere work, which is passive and reciprocal. The load line of a transistor changes depending on the magniat the collector [7]. tude and phase of reflection coefficient Thus, the SiGe HBT is no longer terminated by the optimum required to achieve opload-line matched impedance timal RF performance—maximum output power, PAE, and linearity [19].

TABLE I REFLECTED POWER AND TRANSMISSION LOSS AT THE ANTENNA

C. SiGe HBT Breakdown Voltage Ruggedness of a transistor is a concern in PAs since large voltages appear at the collector under load mismatches and can result in damage to the HBTs. The maximum peak collector voltage is expressed as (2) Fig. 1. Block diagram of a PA output stage.

discussed in Section II. Section III shows the measured RF performance of an SiGe HBT PA under load mismatches. Section IV gives detailed analysis to illustrate the cause of RF performance variation and its relation to the load-line impedance. Time-domain analysis is presented to determine the nonlinear mechanism under impedance mismatches. Finally, the measured results of a commercially available SiGe PA under load mismatches are outlined in Section V. In this paper, the term “VSWR” always refers to “load VSWR.” II. MISMATCH AT ANTENNA A. Antenna Impedance The input impedance of an antenna in a wireless communication device is affected by objects in its vicinity. Coupling of an antenna’s radiated field with these surrounding objects changes the antenna input impedance from its nominal value of 50 [18]. This causes an impedance mismatch between an antenna and the transmission line (or matching networks) at its input. Due to this impedance mismatch, RF power is reflected back into a transmitter. As shown in Table I, the reflected power is as high as 36% of the transmitted power and the transmission loss through an antenna is 2 dB when its input impedance is mismatched by a magnitude of VSWR of 4 : 1.

where is the dc-bias voltage. At high collector voltages, the electric field is high and avalanche breakdown occurs when the base current becomes negative [20]. If the impedance of a bias circuit driving the RF transistor is high (a current source applied at the base), the collector–base breakdown current is fed back into the base–emitter junction and the base–emitter voltage increases. The collector current increases and breakdown occurs due to . thermal runaway. This occurs at the conventional If the impedance of a bias circuit driving the RF transistor is low (a voltage source applied at the base), the collector–base breakdown current is initially shunted by the external base biasing resistor and breakdown occurs at a voltage higher than [12]. the In linear SiGe HBT PAs, dc bias is usually applied through high-impedance current mirror circuit and, hence, the bias . However, these transisvoltage should not exceed the tors are driven by low-impedance networks and, thus, peak RF , close to [11]. The voltage swings well above work in [11] demonstrated that the relevant breakdown voltage for an SiGe HBT in a 0.5- m SiGe BiCMOS process (similar to the one used for this work) is the voltage under emitter open or base grounded breakdown breakdown condition instead of the conventional lower open circuit condi. tion of III. SIGE HBT PA—RESULTS AND DISCUSSION A. Measurement Setup

B. PA Performance A PA is the key component determining the linearity, efficiency, and power transmitted from the transmitter chain. Due to antenna mismatches, the reflected power flows back directly into a PA and affects its performance more significantly than the losses in the antenna itself. Fig. 1 shows the output stage of a PA . When the load impedance connected to the load (antenna) (or load reflection coefficient ) alters, the effective impedance (and reflection coefficient ) seen by the collector of a transistor is changed by (1)

On-wafer load–pull measurements are performed on an SiGe m fabricated in an IBM HBT of emitter area 0.5- m SiGe BiCMOS process. Fig. 2 shows the transistor formed by connecting 96 unit cells (0.5 m 20 m 2) in parallel. A ballasting resistor of 3.86 is added in the emitter of each unit cell for thermal stability. The Maury Microwave automatic tuner system (ATS) used for on-wafer load–pull measurements is shown in Fig. 3. It consists of an SiGe HBT as the device-under-test (DUT), a tuner at the output (T2), and a tuner at the input (T1). The DUT is placed on a Cascade Microwave probe station. The input and output fixtures consist of low-loss 150- m-pitch microwave

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TABLE II LOAD-LINE MATCHED IMPEDANCE

TABLE III SIGE HBT PA PERFORMANCE AT Z

= 50

Fig. 2. Fabricated IBM SiGe HBT.

Fig. 3. Load–pull measurement setup for load mismatches.

probes and cables. Initially, the output fixture with the output tuner T2 (tuned for maximum output power) serves as the output matching network, and the input fixture with the input tuner T1 (tuned for maximum gain) serves as the input matching netat the work, thereby constituting a PA. The load impedance output of tuner T2 is 50 . For mismatch measurements, the ATS software automatically determines the effective impedance by emulating the load impedance variations (assuming the output matching network is fixed). This effective impedance is due to the cascaded combination of the output matching network and the load. Consequently, the output tuner T2 is di, as seen rectly tuned to synthesize the effective impedance by the output of a DUT under varying load impedances. During mismatch measurements, the output fixture and tuner T2 represent the cascaded combination of the output matching network and the load. Thus, load is varied to change the magnitude and phase of and to create a specific impedance mismatch for this study. The measured -parameters of each block (tuners, fixtures, directional couplers, bias tees, and isolators) in the load–pull system are incorporated into the computer-controlled ATS software and SiGe HBT is deembedded to set the reference planes at the collector and base of a transistor. This facilitates the measurement of the RF power at node A (collector of transistor). A constant current (base current ) bias is applied to the base of an SiGe HBT through a resistive type constant current source and input bias tee, while the collector is biased at a supply voltage of 3.3 V through an output bias tee.

mA, is matched at the An SiGe HBT, biased at to achieve a maximum output to to RF output power and at the input to achieve a maximum transducer gain. This constitutes an SiGe is initially set to a 50- matched conHBT PA whose load dition. Two-tone measurements are conducted to determine the fundamental and third-order intermodulation component of the RF output power at the output of a transistor (node A). Furand ther, the load is tuned for , respectively, and two-tone of these in measurements are carried out at phases steps of 30 . RF performance of this discrete SiGe HBT PA is measured for all the above-mentioned load conditions at a freGHz, a supply voltage V, and quency and mA, recollector bias currents is same for all the collector currents spectively. Impedance and the source tuner is fixed for all measurements. Impedance changes with collector currents and the output . Table II shows the optimum loadtuner is tuned for each for different collector currents. line matched impedance B. RF Performance—

V,

mA

Table III summarizes the measured RF performance of an , GHz, in class A SiGe HBT PA, at 50- load V, and mA. The PAE is mode with , while is the maximum value. RF output reported at power at 50- load termination is slightly lower than other reported devices [21] due to a small series resistance at the base resulting in lower effective beta. The RF performance of an SiGe HBT PA, measured at GHz, V, mA, and a load mismatch magnitude of is shown in Fig. 4. , , PAE, and output thirdAs seen from this figure, the of the load order intercept point (OIP3) vary with the phase , OIP3, , and PAE reflection coefficient. The optimal occur at , which is referred as the optimal phase con, OIP3, , and PAE dition. The worst degradation in , which is referred as the worst phase condioccurs at swings from 23.5 to 16 dBm, OIP3 varies from 34.9 tion.

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Fig. 4. Measured RF output power (P ), OIP3, G , and PAE of an SiGe = 3:3 V, I = 400 mA. HBT PA at VSWR = 10 : 1, f = 1:88 GHz, V

), OIP3, G , and PAE of an SiGe Fig. 6. Measured RF output power (P HBT PA with magnitude of load mismatch, at optimal ( = 180 ), and worst = 3:3 V, I = 400 mA. ( = 0 ) phases, f = 1:88 GHz, V

Fig. 5. Measured IM3 of an SiGe HBT PA at VSWR = 10 : 1, f = = 3:3 V, I = 400 mA. 1:88 GHz, V

Fig. 7. Measured IM3 of an SiGe HBT PA at different VSWRs, at optimal = 3:3 V, ( = 180 ), and worst ( = 0 ) phases, f = 1:88 Hz, V = 400 mA. I

to 26.6 dBm, varies from 14.8 to 6.7 dB, while the PAE changes from 17.5% to 2.63%. The third-order intermodulation (IM3) distortion ratio over for the same bias conthe RF output power at dition is shown in Fig. 5. IM3 varies with for all the output and lowest at power levels. Highest IM3 occurs at , where linearity is the best. IM3 differs by 18 dBc between the optimal and worst phase conditions at for output power levels up to 20 dBm. , Fig. 6 shows the measured RF output power transducer gain, PAE, and OIP3 of an SiGe HBT PA with varying VSWR at the worst phase condition and the optimal phase condition, respectively. At the nonoptimal phases of , the RF output power , transducer gain, PAE, and OIP3 drop rapidly as the magnitude of increases. However, at the optimal phase condition, these four performance parameters have values close to that of the 50- load termina(optimal phase)—OIP3 (worst tion. Define phase); (optimal phase)— (worst (optimal phase)— (worst phase); phase);

(optimal phase)—PAE (worst phase). , , , and increase as VSWR rises. IM3 over the RF output power for different VSWRs, at the optimal and worst phase condition, is shown in Fig. 7. For all at all the the output power levels, the IM3 increases with other than the optimal phase, where it is close to phases of the 50- matched condition. C. RF Performance as Function of The RF performance of an SiGe HBT PA as a function of is shown in Fig. 8(a) and (b). Under load mismatches, varies with phases of RF performance of a PA at each load reflection coefficient, similar to the class A bias condition mA . The optimum performance and the worst degradation occur at and , respectively, and mA, as compared to for and , respectively, for mA. This is because changes the optimum load-line matched impedance with collector currents, becoming more inductive as is mA decreased. As collector current varies from

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Fig. 9. Measured IM3 of an SiGe HBT PA as a function of I

10 : 1, optimal and worst phases, f = 1:88 GHz, V

at VSWR =

= 3:3 V.

(a)

(b)

Fig. 10. Smith chart showing load–pull contours, and the VSWR circles and Z circles for an SiGe HBT PA at load mismatches of VSWR = 4 : 1; 10 : 1; and all phases.

Fig. 8. (a) Measured OIP3 and G of an SiGe HBT PA with magnitude of = 3:3 V. (b) Measured mismatch as a function of I at f = 1:88 GHz, V P and PAE of an SiGe HBT PA with magnitude of mismatch as a function at f = 1:88 GHz, V = 3:3 V. of I

Finally, during measurements, the SiGe HBT did not break , as down and survived load mismatches up to evident from the work in [11].

to whereas

Measurement results in Section III demonstrate the effect of load mismatches on the RF performance of an SiGe HBT PA. System-level analysis is provided here to understand the effect of load changes on the PA’s RF output power, transducer gain, PAE, and linearity. Further, time-domain analysis is used to investigate the linearity behavior under load mismatches.

IV. PERFORMANCE ANALYSIS mA, increases from 14.5% to 27.5%, decreases by 8 dB at a load VSWR of 10 : 1. and decrease by 1 and 1.6 dBm, respecchanges from 400 to 200 mA at the same load tively, as is decreased from 200 to 100 mA, VSWR. However, as both and increase by 2.5 dBm. at Fig. 9 shows the IM3 distortion ratio with varying for an SiGe HBT PA. It is seen that the for all the absolute value of IM3 decreases with increase in phases of at all the RF output power levels. Difference between the optimal and worst phase IM3 is 18 dBc for mA, and this difference further decreases with decrease in , exhibiting a smaller difference at mA. As the collector current reduces, the SiGe HBT approaches towards class B mode, where it is highly nonlinear and the effect of terminating load impedance on linearity is less as compared to that at higher collector currents

A. Load-Line Impedance and Load–Pull Contours The measured impedance seen by the collector of a transistor under matched and mismatched load is plotted on a Smith chart in Fig. 10. Fig. 10 shows the load–pull contours, circles, and VSWR circles. The load–pull contours for RF output power and OIP3 are shown in Fig. 10, whereas and the PAE contours closely overlap with the power contours. is the load-line matched impedance, determined through load–pull measurements, mA V under a for an SiGe HBT

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matched 50- load. This impedance point is the center of the oval-shaped load–pull contours, and the power (or OIP3) drops by 1 dBm for each contour from its center. As the normalized is varied through all the phase load reflection coefficient angles at VSWR of 4 : 1 and 10 : 1 (shown by VSWR circles at the center), the corresponding transformed impedance is represented by the smaller circles towards the left of the Smith chart. The inner smaller circle (denoted by ) represents at VSWR of 4 : 1 and outer smaller circle (denoted by ) at VSWR of 10 : 1, for all values of . At the represents . It is seen that since center of the smaller circles is is a small value, the circles compress towards the left edge of the Smith chart occupying a smaller area. It is observed from the load–pull power measurements that on each circle, which corresponds to the there is a optimal phase condition. is for the VSWR of 4 : 1. for the VSWR the of 10 : 1 also occurs at the same point in this case. As seen and are very close to from Fig. 10, and, hence, the SiGe HBT is capable of deliv, transducer gain, PAE, and ering the RF output power OIP3 corresponding to that of the 50- case at all the impedance points up to a VSWR of 10 : 1. For each circle, at all other values (or phases of ) , the RF output power , OIP3, gain, other than its moves away from its . The worst and PAE degrade as degradation in the RF performance for the SiGe HBT PA occurs and for at impedances a VSWR of 4 : 1 and 10 : 1, respectively. As evident from the contours in Fig. 10, these impedances points are the farthest , and collector voltage swing at (at 180 offset) from its these points is high. Hence, the power drop at these points is circles are the maximum. For larger magnitudes of , the in Fig. 10. As a spread out, as seen for at the nonoptimal phases of moves result, the impedance further away from its on each circle and collector voltage excursions are still larger. This causes the increasing , OIP3, gain, and PAE with drop in RF output power an increase in the magnitude of load mismatch. depends on the phase shift through the output matching , as determined through (1). Since the network and the load elements of the output matching network (or output tuner) are at which is the same for fixed, the phase of each load VSWR. In this case, occurs at for each load VSWR and the RF performance is optimum at this for all load VSWRs. Similarly, the worst degradaphase of for each load VSWR. As the tion occurs at the same phase of collector current decreases, becomes inductive and the load–pull contours are oriented at an axial symmetry along 30 phase of . Hence, the optimal performance is at 150 and the worst degradation a load reflection phase of is at . depends on the output matching network of Impedance a PA, and the varying load impedance. For a PA, the output matching network is not unique and it may vary from one implementation to another. If the output matching network is changed, then the phase of the load reflection coefficient at which RF performance is optimum (or worst) is relatively

shifted depending on the phase shift through a matching network. B. Linearity Time-domain analysis is performed to further analyze the linearity phenomena of an SiGe HBT PA under load mismatches. The nonlinear characteristics in a PA produce amplitude and phase distortions. If the RF input power is large, the SiGe HBT saturates and causes voltage clipping at positive and negative peaks of the modulated output waveform. This results in amplitude modulation to amplitude modulation (AM–AM) distortion—compression of the transfer characteristic as the drive level is increased. Amplitude modulation to phase modulation (AM–PM) is the change in phase of the transfer characteristic as the drive level is increased [22]. Consider a two-tone envelope signal applied to the input of an SiGe HBT given by (3) The RF output voltage at the collector is distorted and is represented as [19]

where

(4) is the peak voltage amplitude of output envelope signal. is the peak magnitude of the AM–PM phase shift occurring at the positive or negative amplitude peaks of the modulated output voltage waveform [19]. is assumed to be small, and radians. This phase shift depends on the nonlinear output and base–collector capacitance of an HBT and the matching at the collector. Analysis in [6] shows the strong impedance dependence of the phase shift on the impedance . The magnitude of each of the fundamental components, and the magnitude of each of the IM3 components of RF output voltage at the collector of an SiGe HBT are represented by (5) and (6), respectively, as follows:

(5)

(6) and are the first- and third-order distortion terms where due to AM–AM nonlinearities. The phase shift is the contribution from the AM–PM distortion. Thus, the amplitude and phase distortion affects the IM3 products significantly and increases the IM3 levels. It is difficult to measure the individual contributions of the two distortion phenomena to the IM3. Hence, the phase distortion at output of a PA is plotted to illustrate that the AM–PM is contributing to the nonlinearities under load mis. Fig. 11 shows the meamatches up to

KEERTI AND PHAM: RF CHARACTERIZATION OF SiGe HBT POWER AMPLIFIERS UNDER LOAD MISMATCH

Fig. 11. Measured phase distortion of an SiGe HBT PA at 50 , optimal ( = 180 ) and worst ( = 0 ) phases at VSWR of 4 : 1 and 10 : 1, f = 1:88 GHz, V = 3:3 V, I = 400 mA.

sured phase distortion at the output of an SiGe HBT PA as a function of the RF output power for the 50- load condition, and the optimal and worst phase conditions at a load VSWR of 4 : 1 and 10 : 1, respectively. At the optimal phase condition , the phase distortion is almost zero at up to dBm, but at the worst phase condition, there is a sigdBm for nificant amount of phase distortion of 3 at and further increases as seen for . The phase distortion is dominant at all other phases of , except at the optimal phase. This causes a reduction in fundamental RF output power and an increase in IM3 levels over the phase of . Thus, time-domain analysis quantifies that the degradation in linearity under load mismatches is due to amplitude (voltage clipping), as well as phase distortion. V. WIRELESS PA—RESULTS The effect of load mismatches is experimentally demonstrated on a commercially available SiGe PA circuit at frequency GHz and V. Input and output matching networks are a part of this PA circuit, and a tuner (T2) is used only to vary the load impedance. RF performance is measured , rather than at the collector as in the previous at the load sections. SiGe PA at a frequency of 2.4 GHz and 3.3-V supply with the voltage has similar characteristics at varying from 18 to 13 dBm; OIP3 swinging from 29 to varies by 4 dB. Fig. 12 shows the measured 22.3 dBm, and RF output power , transducer gain, PAE, and OIP3 of ) at the worst the PA with varying VSWR (magnitude of phase condition and the optimal phase condition, respectively. , and the worst The optimal performance occurs at degradation occurs at due to a different matching network than the one used in Section III. As the magnitude , of load mismatch increases, the RF output power at the worst phase condition decreases for the OIP3, and PA. However, at the optimal phase condition, the RF output , OIP3, and remains close to that of the power 50- condition up to a VSWR of 6 : 1. PAE reduces drastically with impedance mismatch at all the phases. This experiment on the PA circuit illustrates that the PA circuit’s response to

213

), OIP3, G , and PAE of an SiGe Fig. 12. Measured RF output power (P PA with magnitude of mismatch, optimal ( = 150 ), and worst ( = 30 ) = 3:3 V. phases, f = 2:4 GHz, V

0

load mismatches is analogous to that of a hybrid SiGe HBT PA analyzed in Section IV. The only small deviation is in the PAE selected for an SiGe performance. The optimum HBT used in the PA circuit, and the matching network, are the possible causes of the minor deviation in PAE performance. VI. CONCLUSION Characterization of an SiGe HBT PA under load mismatches empowers the development of circuit techniques to improve the PA RF performance, and provides insight for isolatorless linear PA modules. It has been found that the RF output power, linearity, efficiency, and transducer gain of an SiGe HBT PA is heavily dependent on the phase of load. RF performance of a PA under severe impedance mismatches (up to VSWR of 10 : 1) stays close to that at 50 , at an optimum phase of load reflection coefficient, whereas it degrades at all other phases. The deterioration in the RF performance at the nonoptimal phases increases with a rise in the magnitude of load VSWR. Time-domain analysis demonstrates that the AM–AM and AM–PM distortion mechanisms cause linearity degradation under load mismatches. ACKNOWLEDGMENT The authors would like to thank Dr. P. Zampardi, Skyworks Solutions Inc., Newbury Park, CA, for reviewing this study and providing his valuable suggestions. The authors also acknowledge Tahoe RF Semiconductor Inc., Auburn, CA, and IBM Inc., Yorktown Heights, NY, for support and fabrication. REFERENCES [1] J. D. Cressler, “SiGe HBT technology: A new contender for Si-based RF and microwave circuit applications,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 5, pp. 572–589, May 1998. [2] A. Raghavan, D. Heo, M. Maeng, A. Sutono, K. Lim, and J. Laskar, “A 2.4 GHz high efficiency SiGe HBT power amplifier with high-Q LTCC harmonic suppression filter,” in IEEE MTT-S Int. Microw. Symp. Dig., Seattle, WA, Jun. 2002, vol. 2, pp. 1019–1022. [3] P. D. Tseng, L. Zhang, G.-B. Gao, and M. F. Chang, “A 3-V monolithic SiGe HBT power amplifier for dual-mode (CDMA/AMS) cellular handset applications,” IEEE J. Solid-State Circuits, vol. 35, no. 9, pp. 1338–1344, Sep. 2000.

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[4] G. Niu, Q. Liang, J. D. Cressler, C. S. Webster, and D. L. Harame, “RF linearity characteristics of SiGe HBTs,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 9, pp. 1558–1565, Sep. 2001. [5] M. Vaidyanathan, M. Iwamoto, L. E. Larson, P. S. Gudem, and P. M. Asbeck, “A theory of high-frequency distortion in bipolar transistors,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 2, pp. 448–461, Feb. 2003. [6] T. Iwai, S. Ohara, H. Yamada, Y. Yamaguchi, K. Imanishi, and K. Joshin, “High efficiency and high linearity InGaP/GaAs HBT power amplifiers: Matching techniques of source and load impedance to improve phase distortion and linearity,” IEEE Trans. Electron Devices, vol. 45, no. 6, pp. 1196–1200, Jun. 1998. [7] A. van Bezooijen, C. Chanlo, and A. H. M. van Roermund, “Adaptively preserving power amplifier linearity under antenna mismatch,” in IEEE MTT-S Int. Microw. Symp. Dig., Philadelphia, PA, Jun. 2003, vol. 3, pp. 1515–1518. [8] A. Keerti and A. Pham, “Dynamic output phase to adaptively improve the linearity of the power amplifier under antenna mismatch,” in IEEE Radio Freq. Integr. Circuits Symp., Long Beach, CA, Jun. 2005, pp. 675–678. [9] J. Pusl, S. Sridharan, P. Antognetti, D. Helms, A. Nigam, J. Griffiths, K. Louie, and M. Doherty, “SiGe power amplifiers ICs with VSWR protection for handset applications,” Microw. J., pp. 100–113, Jun. 2001. [10] M. Spirito, L. C. N. de Vreede, L. K. Nanver, S. Weber, and J. N. Burghartz, “Power amplifier PAE and ruggedness optimization by second harmonic control,” IEEE J. Solid-State Circuits, vol. 38, no. 9, pp. 1575–1583, Sep. 2003. [11] J. B. Johnson, A. J. Joseph, D. C. Sheridan, R. M. Maladi, P. O. Brandt, J. Persson, J. Andersson, A. Bjorneklett, U. Persson, F. Abasi, and L. Tilly, “Silicon–germanium BiCMOS HBT technology for wireless power amplifier applications,” IEEE J. Solid-State Circuits, vol. 39, no. 10, pp. 1605–1614, Oct. 2004. [12] A. Inoue, S. Nakatsuka, R. Hattori, and Y. Matsuda, “The maximum operating region in SiGe HBT’s for RF power amplifiers,” in IEEE MTT-S Int. Microw. Symp. Dig., Seattle, WA, Jun. 2002, pp. 1023–1026. [13] K. Yamamoto, S. Suzuki, K. Mori, T. Asada, T. Okuda, A. Inoue, T. Miura, K. Chomei, R. Hattori, M. Yamanouchi, and T. Shimura, “A 3.2-V operation single-chip dual-band AlGaAs/GaAs HBT MMIC power amplifier with active feedback circuit technique,” IEEE J. SolidState Circuits, vol. 35, no. 8, pp. 1109–1120, Aug. 2000. [14] Y. Sun and J. K. Fidler, “High speed automatic antenna tuning units,” in 9th Int. Antennas Propag. Conf., Eindhoven, The Netherlands, Apr. 1995, vol. 1, pp. 218–222. [15] M. Thompson and J. K. Fidler, “Fast antenna tuning using transponder based simulation annealing,” Electron. Lett., vol. 36, no. 7, pp. 603–604, Mar. 2000. [16] J. J. Mallorqui, A. Aguasca, A. Cardama, R. Pagés, and J. M. Haro, “Automatic self-matching network for industrial microwave heating base don conjugate gradient algorithm,” Electron. Lett., vol. 35, no. 4, pp. 311–312, Feb. 1999. [17] J. de Mingo, A. Crespo, A. Valdovinos, D. Navarro, and P. García, “A radio frequency electronically controlled impedance tuning network design and its application to antenna input impedance automatic matching system,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 2, pp. 489–497, Feb. 2004.

[18] S. R. Best, “Antenna properties and their impact on wireless system performance,” Cushcraft Corporation, Manchester, NH, 1998 [Online]. Available: http://www.cushcraft.com/comm/support/pdf/AntennaProperties-an-14998.pdf [19] S. Cripps, RF Power Amplifiers for Wireless Communications. Norwood, MA: Artech House, 1999. [20] S. Heckmann, R. Sommet, J. M. Nebus, J. C. Jacquet, D. Floriot, P. Auxemery, and R. Quere, “Characterization and modeling of bias dependent breakdown and self-heating in GaInP/GaAs power HBT to improve high power amplifier design,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2811–2819, Dec. 2002. [21] J. Deng, P. S. Gudem, L. E. Larson, and P. M. Asbeck, “A high averageefficiency SiGe HBT power amplifier for WCDMA handset applications,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 2, pp. 529–537, Feb. 2005. [22] P. B. Kenington, High-Linearity RF Amplifier Design. Norwood, MA: Artech House, 2000.

Arvind Keerti (S’05) received the B.E. degree in electronics engineering from the University of Pune, Pune, India, in 1999, and the M.S. and Ph.D. degrees in electrical engineering from the University of California at Davis, in 2004 and 2006, respectively. From 2000 to 2001, he was a Design Engineer with Motorola Inc., Noida, India, where he developed application-specific integrated circuits (ASICs) for cellular handsets. He is currently an RF Integrated Circuit (RFIC) Design Engineer with Qualcomm Inc., Campbell, CA. His research interests are RFICs for wireless communication systems.

Anh-Vu H. Pham (SM’03) received the B.E.E. (with highest honors), M.S., and Ph.D. degrees from the Georgia Institute of Technology, Atlanta, in 1995, 1997, and 1999, respectively. In 1997, he cofounded RF Solutions LLC, an RFIC company that was acquired by Anadigics in 2003. He has held faculty positions with Clemson University and the University of California at Davis, where he is currently an Associate Professor. He is also active as a consultant to industry. He coauthored approximately 60 technical papers. His research interests are in the area of RF and high-speed packaging and signal integrity and RFIC design. Dr. Pham is a member of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium (IMS) Technical Program Committee (TPC) on Power Amplifiers and Integrated Circuits. He has been the chair of the IEEE MTT-12 Microwave and Millimeter Wave Packaging and Manufacturing Technical Committee of the IEEE MTT-S. He was the recipient of the 2001 National Science Foundation CAREER Award.

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Optimum Bias Load-Line Compensates Temperature Variation of Junction Diode’s RF Resistance Subhash Chandra Bera, Member, IEEE, Raj Vir Singh, Vinesh Kumar Garg, and Sashi Bhushan Sharma, Senior Member, IEEE

Abstract—This paper presents a novel temperature-compensation technique to compensate variation of the junction diode’s RF resistance. Here we have theoretically analyzed and experimentally demonstrated the temperature sensitivity of the junction diode’s RF resistance and proposed an optimum dc-bias load line technique to minimize temperature variation of RF resistance of the Schottky barrier, p-i-n, and p-n junction diodes. The proposed optimum load line biasing technique eliminates the requirement of conventional temperature-compensation circuits with temperature sensors to achieve temperature-invariant RF performance of diode-based RF circuits such as a linearizer, attenuator, phase shifter, etc. for various RF applications where it experiences wide temperature variations. The circuit responds directly to the junction temperature of the diode; thus, there will be no compensation error due to temperature gradient and self-heating of the diodes. This technique is very simple, accurate, and suitable to implement in monolithic-microwave integrated-circuit technology. Index Terms—Attenuation, linearizer, load line, phase shifter, p-i-n diode, p-n junction diode, RF resistance, Schottky diode, temperature compensation.

I. INTRODUCTION CHOTTKY barrier diodes and p-i-n diodes are popularly used for various RF applications such as a linearizer, variable attenuator, phase shifter, etc. [1]–[11]. In these applications, diodes RF resistance determines the RF performance of the circuits. Performance of these circuit changes with temperature since RF resistance of the diode varies with temperature [13]. Conventionally, a temperature-compensation circuit with a separate temperature sensor is used to compensate the performance variation of the diode-based circuits [14]. Today’s satellite communication transponder uses a linearizer with high-power microwave amplifiers, such as traveling-wave-tube amplifiers (TWTAs) and solid-state power amplifiers (SSPAs) to improve linearity without sacrificing transmitter efficiency. A diode-based linearizer [1]–[11] is a good choice for satellite applications due to its small size and less power consumption. RF impedance of the Schottky barrier diode determines the performance of the linearizer. p-i-n diodes are also used for a linearizer [9]–[11] and to control the RF signal level in different applications such as an attenuator and a phase shifter.

S

Manuscript received May 27, 2006; revised September 22, 2006. The authors are with the Space Applications Centre, Indian Space Research Organization, Ahmedabad 380-015, Gujarat, India (e-mail: [email protected]. gov.in). 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.2006.889155

RF performance of these diode-based circuits changes with temperature. In the conventional approach, digital or analog compensation circuits with temperature sensors are used for temperature compensation [14], which lead to a higher part count and complex circuit, resulting in less reliability. We have previously presented a control circuit for temperature-invariant attenuation of the p-i-n diode-based attenuator circuit [15]. Here we have presented complete theory, simulation, and test results about the temperature dependency of the junction diode’s RF resistance and have proposed a novel optimum bias load line technology to achieve temperature-invariant RF resistance of the diodes. This paper addresses the temperature behavior of p-i-n diodes, Schottky barrier diodes, and p-n junction diodes. Mathematically and by measurements, it is shown that the proposed optimum bias load line technology is applicable to achieve temperature-invariant RF resistance of all these diodes. Here it is also shown that the same technology is applicable at small-signal operation, as well as large-signal operation of the diodes. Proposed temperature compensation of the RF performance of the diode-based circuits is achieved without using any additionally complicated circuits with a separate temperature sensor. Section II will discuss RF resistance variation of the junction diodes over the change of temperature under different types of bias condition. Section III discusses the different types of RF circuits based on RF resistance of the junction diodes. This section presents performance variation of the diode-based RF circuits over temperature under a different biasing condition. Section IV discusses the proposed optimum load line bias technique with complete theoretical discussion. Section V discusses the effectiveness of the proposed optimum bias load line technique to achieve temperature-invariant RF resistance with respect to conventional constant current bias condition. This section also presents the sensitivity of the proposed technique under variation of the diode’s parameter and supply voltage. Section VI presents the experimental and test results of the p-i-n diode and Schottky barrier diode-based RF circuits. Section VI then presents a conclusion. II. RF RESISTANCE OF JUNCTION DIODE AND THEIR TEMPERATURE DEPENDENCY Forward dc current of the junction diode is related to the forward junction voltage by the well-known equation (neglecting a term corresponding to the reverse saturation current)

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(1)

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Fig. 1. RF equivalent circuit of forward biased junction diode.

TABLE I PARAMETER VALUES OF DIODES [15]

Fig. 2. Simulated RF resistance variation with temperature at fixed current bias and fixed voltage bias.

where is the electron charge, is the Boltzman’s constant, is the barrier height in volts for the Schottky barrier diode and bandgap potential for the p-i-n and p-n junction diode, and is the ideality factor. is a constant related to the area of the diode. The RF equivalent circuit of a forward biased junction diode is the RF resistance and is the is shown in Fig. 1. Here, can be considered as capacitance of the diode. Capacitance independent of temperature at the forward bias condition. The RF resistance of the Schottky barrier diode and p-n junction diode is the dynamic resistance of the diode and can be written as [16] (2) and the RF resistance of the p-i-n diode [13] is given by (3) as follows: (3) is the width of the diode. and are where the ambipolar carrier lifetime and ambipolar carrier mobility at temperature . Thus, RF resistance of the Schottky, p-i-n, and p-n junction diode can be written by the generalized equation given by (4) as follows: (4) where is a constant independent of temperature. Table I shows the parameter’s value of different diodes. Equation (4) shows that RF resistance of junction diodes increases with the increase of temperature when the diodes are , mainbiased through constant current source, i.e., tained constant over the operating temperature range. Fig. 2 shows the simulated RF resistance (normalized to the value at 30 C) variation over the diode temperature variation from 20 C to 80 C. This figure shows that RF resistance increases with the increase of temperature when diodes are biased by constant current source, but RF resistance decreases drastically when the diodes are biased by a voltage source bias.

Thus, performance of the diode-based RF circuits, where constant voltage bias was used [7] [8], will be affected severely over the change of temperature, whereas the performance of constant current bias circuits will be less affected with a change of temperature. III. RF CIRCUITS BASED ON RF RESISTANCE OF DIODES p-i-n and Schottky barrier diodes are being used for various RF circuit applications such as linearizers, variable attenuators, phase shifters, etc. p-i-n diodes have been widely used to control RF/microwave signals, e.g., limiting, phase shifting, RF switching, attenuating, etc. p-i-n diode-based variable attenuators are especially used for gain control and RF leveling applications. They are also used for linearizer applications [9]–[11]. Conventionally, p-i-n diode attenuators have been controlled by constant current bias because they are known as current controlled devices, and the attenuation level is determined by RF resistance of the diode that depends upon the bias current [12]. RF resistance of the diode increases with the increase of temperature at a fixed current bias condition and drastically decreases at a fixed voltage bias condition, as shown in Fig. 2. Thus, the performance of the p-i-n diode-based circuits also changes with temperature under these conventional bias conditions. Schottky diodes are used for detectors, mixers, linearizers, harmonic generators, etc. RF impedance of Schottky diode determines the performance of the linearizer [1]–[11] and other circuits. Schottky diodes are especially used as a distortion generator for RF/microwave linearizer applications due to its low power consumption and compact size. In linearizer applications, RF resistance of the Schottky diode increases or decreases with the increase of the RF power level and generates amplitude and phase nonlinearity to compensate for the nonlinearity of the high-power amplifiers. The typical I–V characteristic of a Schottky diode in the presence of an RF signal is shown in Fig. 3. Two types of bias load lines for nearly constant current bias and constant voltage bias are also shown in this figure. It is clear from this figure that the RF resistance of the Schottky diode increases with the increase of the RF power level when the diodes are biased by a constant current bias condition, whereas RF resistance of the diode decreases with an increase of the RF power level when the diodes are biased with a constant voltage bias source. This property enables two types of diode-based linearizer, i.e., one is a parallel diode linearizer [4], [5] and the other is a series diode linearizer [7], [8]. As shown in

BERA et al.: OPTIMUM BIAS LOAD-LINE COMPENSATES TEMPERATURE VARIATION OF JUNCTION DIODE’S RF RESISTANCE

Fig. 3. I–V characteristic and load lines of Schottky diode in presence of RF power.

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Fig. 4. Equiresistance curve and load line.

Fig. 2, a series diode-based linearizer will be seriously affected in a temperature-varying environment since these are biased by a constant voltage source. Comparatively, there will be less performance variation of the constant current bias parallel diode linearizer. Fig. 5. Simple diode bias circuit.

IV. PROPOSED TEMPERATURE-COMPENSATION SCHEME As discussed above, whether diodes are biased by a constant voltage source or a constant current source, the RF performance changes with a variation of the temperature. Constant current source bias gives less temperature sensitive performance, but constant current source circuit itself is complex and it is difficult to accommodate to realize in the monolithic-microwave integrated-circuit (MMIC) technology. Voltage-controlled bias circuit is though simple and easy to realize, but it seriously affects the RF resistance over temperature variation and needs temperature-compensation circuits. Moreover, performance variation of a constant current biased diode circuit may also not meet the accuracy requirement of certain applications in a temperature-varying environment such as spacecraft applications, where precise gain control of the transponder is required and the circuit experiences large temperature variation. Therefore, a temperature-compensation circuit is required to obtain temperature-invariant RF performance of the diode-based circuits. There are various analog, as well as digital temperature-compensation techniques. All these techniques are based on the use of separate temperature sensors to sense the temperature and accordingly vary the diode current to obtain temperature-invariant circuit performance. Since all these scheme uses temperature sensors, theses circuits suffers due to the temperature gradient between the sensor and diode. In addition, these circuits are relatively complex and difficult to accommodate in MMIC technology. Here, we will mathematically investigate the condition for achieving temperature-invariant RF resistance of the diode and we will see whether a simple bias scheme can meet the current requirement. To achieve temperature-invariant RF resistance of the diodes, the bias network must meet the current requirement to achieve over the operating temperature a constant RF resistance of the diode. Therefore, by combining (1) and (4) and setting , (5)

where

is given by (6)

This parametric equation is the equation of ideal bias-point locus that will satisfy over the diodes temperature range. This locus is called the equiresistance curve [15] for the . required RF resistance Fig. 4 shows the plot of (5) for the diode’s typical paramV, and over eters the temperature range of 260 C to 300 C. The plot shows that the equiresistance curve is highly linear over the very wide range of temperature. Thus, the load line of a simple bias circuit, as shown in Fig. 5, can be adjusted to coincide with the equiresistance curve within the temperature range of interest and will then maintain the desired RF resistance. The open circuit voltage of the bias network will be the voltage axis intercept of the bias load line and resistance will be the reciprocal of the slope of the bias load line. Combining (4)–(6) and eliminating , another form of the equation of the equiresistance curve can be written as

(7) The equation of tangent to the equiresistance curve of (7) at and corresponding current at temperature diode voltage is given by

(8)

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Taking this tangent to be the optimum bias load line, its is voltage intercept

(9) will be the open-circuit voltage This optimum voltage of the bias network. The open-circuit voltage is seen to be practi, implying cally independent of the selected bias point that the optimum no-load output voltage of the bias circuit is the same for all values of RF resistance of the diode. and are constant and For a particular type of diode, independent of temperature. Putting typical values of these conand in (9), the optimum open-cirstants, i.e., cuit voltage becomes

This is the expression of the temperature coefficient of the RF resistance variation of the diodes. Using (15), we can estimate the temperature coefficient of the diodes RF resistance when the diode parameters shown in Table I are varied. Putting nominal values of all diode parameters from Table I and varying all theses parameters simultaneously by 10%, the worst case temperature coefficient of the diodes RF resistance becomes 0.074 %/ C. For and , expression (15) reduces to (16) when diodes are biased by the constant current source, , and C. Now, when the diodes are biased by the proposed optimum in (16) from (10), bias load line condition, putting the temperature coefficient will then be

(10) Thus, open-circuit voltage is very close to the bandgap potential of the p-i-n, p-n junction diode, and barrier height of since the last term is of the order of a Schottky barrier diode is the property of the diodes matefew millivolts. Parameter rial; it is independent of process technology and does not vary from one die to another. Thus, this temperature-compensation technology provides very stable temperature-invariant performance under different parameter variation of the diodes. V. SENSITIVITY OF THE PROPOSED TEMPERATURE-COMPENSATION SCHEME Now we will investigate the accuracy of the optimum bias load line technique over the operating range of temperature. The C, temperature coefficient of RF resistance variation, say, of the diode can be written from (4) as (11)

(17) , the temperature coefThis expression shows that at ficient of the RF resistance becomes zero. Now we will investigate the temperature coefficients of the diode’s RF resistances at other temperatures considering typical diode parameters. V and For silicon p-i-n diodes, putting K (25 C), the temperature coefficient will increase from 0 to 0.015 times when the temperature changes from 25 C to 25 C 45 C with respect to the variation in fixed current biasing condition. V and For the silicon Schottky diode, putting K (25 C), the temperature coefficient will increases from 0 to 0.010 times when the temperature changes from 25 C to 25 C 45 C with respect to the variation in fixed current biasing condition. from Fig. 5 to Putting diode current (3) and differentiating RF resistance with respect to supply voltage , we can write

From the bias circuit of Fig. 5, we can write (18) (12) Combining (11) and (12), we can write (13) Differentiating (6) with respect to temperature , (14) Putting this expression in (13), we can write

(15)

This determines the sensitivity of the RF resistance variation under a different biasing condition of the diode due to supply voltage variation. From this it is clear that, under the quasi-con), is very large stant current biasing condition (high value compared to , thus, percent variation of RF resistance will be equal to the percent variation of supply voltage. However under a constant voltage biasing condition (a very low value of is nearly equal to ), the denominator of (18) will be very small, thus, RF resistance will be highly sensitive to the supply voltage variation. In the case of the proposed optimum ( is greater than bias load line technique, (9) determines ), thus, RF resistance variation under the supply voltage variation will be better than the constant voltage bias condition, but poorer than the constant current biasing condition. Thus, it will be better to use a stable voltage supply source in the case of the optimum load line biasing scheme.

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Fig. 8. Measured p-i-n diode I–V data and load lines to determine V

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.

Fig. 6. Schematic circuit diagram.

Fig. 9. Measured p-i-n diode attenuation variation with temperature.

A. Test Results of p-i-n Diode

Fig. 7. Diode circuit.

VI. EXPERIMENT AND TEST RESULTS Fig. 6 shows the schematic circuit diagram of the circuit that we have used for experiment. An isolated port of a 3-dB hybrid coupler is used as the output of the circuit, two diodes are connected at the direct port and coupled port of the hybrid, and is the load resistor to another port is used as the input port. drive the circuit from a voltage source . Depending upon RF resistance of the diodes, determined by the current through the diodes, some portion of the RF power will be dissipated within the diode and the rest of the power will be reflected from the diode. The reflected power from both the diodes will be comof the circuit bined at the output port of the circuit. Thus, is given by (19) is the RF impedance of the diode, i.e., parallel comwhere and capacitance . The circuit bination of RF resistance . A Lange is realized in a 20-mil alumina substrate coupler is used for the 3-dB coupler at a center frequency of 3.7 GHz and a layout is made for beam lead diodes. A photograph of the realized circuit is shown in Fig. 7. This circuit is used to demonstrate the temperature variation of RF resistance of the p-i-n diode and Schottky barrier diode.

A beam lead silicon p-i-n diode Hewlett-Packard HPND4005 is used for this test. At first, optimum voltage is determined , the circuit is kept in a experimentally. To determine thermal chamber, and the voltage and current of the diode is measured at three different chamber temperature (i.e., 10 C 25 C 60 C) for attenuation setting of 20, 15, and 10 dB at a frequency of 3.7 GHz by adjusting the current of the diodes. Fig. 8 shows the plot of these data and load lines for 10-, 15-, and 20-dB attenuation. All three load lines intersect the voltage axis at nearly 1.2 V, which is the optimum voltage . This agrees with the value obtained from (9), putting V and at the ambient temperature. Attenuation variation over the temperature range of 10 C to 60 C for attenuation setting of 5, 10, 15, and 20 dB is shown in Fig. 9. It is seen that, under optimum load line biasing condition, attenuation variation remains within 0.2 dB for all the attenuation settings over this temperature range. However, under a conventional constant-current biasing condition, peak-to-peak attenuation variation is nearly 5.0, 3.4, 2, and 0.9 dB for attenuation setting of 20, 15, 10, and 5 dB (at ambient temperature), respectively, over the temperature range of 10 C to 60 C. B. Test Results of Schottky Diode The same circuit of Fig. 7 is used with two numbers of a beam lead medium-barrier Schottky diode of type MSS-40,148-B10B of M/s Metalics for the experiment. The circuit parameter “attenuation” is measured to verify the RF resistance variation of the diode with a change of temperature.

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Fig. 10. Measured Schottky diode I–V data and load lines for different attenuation at P = 30 dBm.

0

Fig. 12. Measured Schottky diode I–V data and load lines for different attenuation at P = 20 dBm.

0

Fig. 11. Measured Schottky diode I–V data and load lines for different attenuation at P = 25 dBm.

0

Fig. 10 shows the plot of measured voltage and current data of the Schottky diodes for 4-, 8-, and 12-dB attenuation at three different temperatures of 60 C 25 C, and 10 C. These data are taken at 30-dBm RF power input to the circuit. It is clear that the I–V data of each attenuation value for different temperatures lies in a straight line. Load lines are drawn for the attenuation of 4, 8, and 12 dB. All the load lines are intersecting the voltage axis at the same point of value nearly 0.72 V, which is nearly equal to the barrier potential (0.69 V) of the used Schottky diode and matches with (9). Figs. 11 and 12 shows the plot of the similar measurements data, as in Fig. 10, but at different RF input power levels of 25 and 20 dBm, respectively. Theses figures shows that the I–V data of each attenuation value for different temperatures also lies in a straight line for both of the RF power levels, but the intercept point of the load line to the voltage axis increases with the increase of the RF power level. The voltage axis intercept, i.e., optimum open circuit voltages, are 0.80 and 1.05 V for the input power level of 25 and 20 dBm, respectively. The increase of the optimum open circuit voltage with the increase of the RF power level is for self-biasing of the Schottky barrier diode due to the rectification of the RF signal, which is not taken into account in (2). Measurement shows that with optimum load line bias, with the open circuit voltage of 0.72, 0.80, and 1.05 V corresponding to the RF input power of 30 25 and 20 dBm, respectively, the attenuation variation remains within 0.2 dB for all the attenuation level of 4, 8, and 12 dB, over the temperature range from 10 C to 60 C.

Fig. 13. Measured S 21 variation over RF power level for V

= 0:75 V.

Therefore, for temperature compensation of the Schottky barrier diode-based circuit’s RF performance over the entire RF power level, it is required to change the optimum voltage over the operating range of the RF power level. For that, the circuit requirement will be complex. Fig. 13 shows the measured temperature variation of magof the circuit of Fig. 7 for nitude and phase of V (approximately equal to a small signal optimum voltage). Measurement shows that amplitude variation remains within 0.6 dB and phase remains within 2 over the temperature range of 10 C to 60 C over the entire RF power level from 30 to 10 dBm. It is also clear that attenuation and phase variation over the temperature range is negligible at the small-signal RF level condition. p-i-n diodes are used as the linear RF resistor in most of the RF circuit applications such as the attenuator, phase shifter linearizer, etc. [10]–[14]. Thus, depending upon the operating frequency and RF power level, p-i-n diodes of proper carrier lifetime are selected so that there will be no rectification effect and, thus, no self-biasing effect, as happened in the case of the

BERA et al.: OPTIMUM BIAS LOAD-LINE COMPENSATES TEMPERATURE VARIATION OF JUNCTION DIODE’S RF RESISTANCE

Schottky barrier diode. However, at large RF power level, e.g., more than 10 dBm for the p-i-n diode of type HPND-4005 at the -band frequency range, it starts to rectify the RF signal and, due to the self-biasing effect, the requirement of optimum open-circuit voltage will also increase, as discussed in the case of a Schottky barrier diode. VII. CONCLUSION In this paper, an unexplored optimum bias load line technique has been discussed to achieve temperature-invariant RF resistance of the junction diodes. Measurement has shown that this novel bias technique leads to temperature-invariant RF resistance of the diodes without using any separate temperature sensor and compensation circuits, as used in conventional temperature-compensation circuits. It has also been shown that the proposed temperature-compensation technique is valid under a large-signal operation condition of the diodes. In this case, the circuit responds directly to the junction temperature of the diode and, thus, there will be no compensation error due to the temperature gradient and self-heating of the diodes under high-power operation. REFERENCES [1] A. Katz, R. Sudarsan, and C. Aubert, “A reflective diode linearizer for spacecraft applications,” in IEEE MTT-S Int. Microw. Symp. Dig., 1985, pp. 661–664. [2] N. Imai, T. Nojima, and T. Murase, “Novel linearizer using balanced circulators and its application to multilevel digital radio systems,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 8, pp. 1237–1243, Aug. 1989. [3] K. Yamanchi, Y. Kazuhisa, K. Mori, M. Nakayama, Y. Itoh, Y. Mitsu, and O. Ishida, “A novel series diode linearizer for mobile radio power amplifiers,” in IEEE MTT-S Int. Microw. Symp. Dig., 1996, pp. 831–834. [4] K. Yamauchi, “A microwave miniaturized linearizer using a parallel diode with a bias feed resistance,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 12, pp. 2431–2435, Dec. 1997. [5] K. Yamauchi, K. Mori, M. Nakayama, Y. Mitsui, and T. Takagi, “A microwave miniaturized linearizer using a parallel diode,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1997, pp. 1199–1202. [6] T. Yoshimasu, M. Akagi, N. Tanba, and S. Hara, “An HBT MMIC power amplifier with an integrated diode linearizer for low voltage portable phone applications,” IEEE J. Solid-State Circuits, vol. 33, no. 9, pp. 1290–1296, Sep. 1998. [7] K. Yamauchi, “A novel series diode linearizer for mobile radio power amplifiers,” in IEEE MTT-S Int. Microw. Symp. Dig., 1996, pp. 831–834. [8] C. Haskins, T. Winslow, and S. Raman, “FET diode linearizer optimization for amplifier predistortion in digital radios,” IEEE Microw. Guided Wave Lett., vol. 10, no. 1, pp. 21–23, Jan. 2000. [9] W.-M. Zhang and C. Yuen, “A broadband linearizer for -band satellite communication,” in IEEE MTT-S Int. Microw. Symp. Dig., May 1998, pp. 1203–1206. [10] S. C. Bera, P. S. Bhardhwaj, R. V. Singh, and V. K. Garg, “A diode linearizer for microwave power amplifiers,” Microw. J., vol. 46, no. 11, pp. 102–, Nov. 2003. [11] S. C. Bera, R. V. Singh, and V. K. Garg, “A compact -band linearizer for space application,” in Proc. Asia–Pacific Microw. Conf., 2004, pp. 37–38. [12] P. Sahjani and J. F. White, “PIN diode operation and design trade-offs,” Appl. Microw. Wireless, pp. 68–78, Spring 1991. [13] R. H. Caverly and G. Hiller, “Temperature effects on PIN diode forward bias resistance,” Solid State Electron., vol. 38, no. 11, pp. 1879–1885, 1995. [14] B.-J. Jang, “Voltage-controlled p-i-n diode attenuator with a temperature-compensation circuit,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 1, pp. 7–9, Jan. 2003. [15] S. C. Bera and P. S. Bharadhwaj, “Insight into p-i-n diode behavior leads to improved control circuit,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 52, no. 1, pp. 1–4, Jan. 2005. [16] S. A. Maas, Nonlinear Microwave Circuit. Norwood, MA: Artech House, 1988.

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Subhash Chandra Bera (M’05) received the B.Sc. degree in physics (with honors) from Presidency College, Calcutta, India, in 1989, and the B.Tech. and M.Tech. degrees in radio physics and electronics from the Institute of Radio Physics and Electronics, University of Calcutta, Calcutta, India, in 1992 and 1994, respectively. Since 1994, he has been with the Space Applications Centre, Indian Space Research Organization (ISRO), Ahmedabad, India, where he has been involved in many communication and navigation payload projects such as the INSAT-2, INSAT-3, INSAT-4, and GSAT series of spacecraft. His research interests include temperature behavior and compensation of microwave active circuits.

Raj Vir Singh received the B.Tech degree in electrical engineering from GB Pant University, Pantnagar, India, in 1972. Since October 1972, he has been with the working in Space Applications Centre (SAC), Indian Space Research Organization (ISRO), Ahmedabad, India, where he has been involved with the design and development of various high-reliability low-noise amplifiers, solid-state amplifiers, and communication receivers for satellite transponders. He is responsible for the design and development of communication transponders for the INSAT-3A satellite as Associate Project Director. He is currently Group Director of the Power Amplifier Group of the SATCOM Payload Technology Area, SAC, ISRO.

Vinesh Kumar Garg received the B.Tech. degree in electronics and communications from the Indian Institute of Technology (IIT), Madras, India, in 1970. In July 1971, he joined the Experimental Satellite Communication Earth Station (ESCES), Ahmedabad, India, after spending one year with the BARC Training School (14th batch). Since 1971, he has been involved in all major communications satellite projects of the Indian Space Research Organization (ISRO), Ahmedabad, India, such as the SITE, STEP, APPLE, INSAT-1, INSAT-2, INSAT-3, and now the INSAT-4 series of spacecrafts. He was Associate Project Director for INSAT-2C and INSAT-2D communication payloads. He is currently Deputy Director of the Satcom Payload Technology Area, ISRO. He guides the design and development of regenerative payloads, special payloads, and state-of-the-art technology subsystems.

Ka

Ku

Shashi Bhushan Sharma (SM’06) was born in Moradabad, India, in 1947. He received the B.E. degree in electronics and communication and M.E. degree in microwave engineering from the University of Roorkee, Roorkee, India, in 1970 and 1972, respectively, and the Ph.D. degree in microwave engineering from Gujarat University, Ahmedabad, India, in 1987. He possesses over 32 years of academic and diversified research and development experience in the design and development of antenna systems for satellite communication and remote sensing. He is currently an Outstanding Scientist with the Indian Space Research Organization (ISRO), Ahmedabad, India, and the Deputy Director of the Antenna Systems Area (ASA), Space Application Center, ISRO. He has authored or coauthored approximately 100 publications. Dr. Sharma was the recipient of the 1992 Dr. Vikram Sarabhai Research Award in the field of electronics, telematics, and automation for his outstanding contributions to the development of various types of antenna systems for ground, airborne, and spaceborne systems.

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0.7–2.7-GHz 12-W Power-Amplifier MMIC Developed Using MLP Technology Inder J. Bahl, Fellow, IEEE

Abstract—A design approach and test data for a broadband high-power amplifier monolithic microwave integrated circuit (MMIC) developed using MSAG MESFETs with multilevel-plating technology are presented. A low-loss matching design technique was used in the development of a two-stage amplifier. The UHF/ / -band amplifier has exhibited greater than 12-W power output and better than 22% power-added efficiency over the 0.7–2.7-GHz frequency range. To our knowledge, these power results represent the state-of-the-art in multioctave high-power MMIC amplifiers. Index Terms—Broadband power amplifiers, high-power amplifiers (HPAs), MESFET HPAs, monolithic-microwave integratedcircuit (MMIC) HPAs, two-octave HPAs.

I. INTRODUCTION VER THE past two decades, tremendous progress has been made in the design of narrowband monolithic-microwave integrated-circuit (MMIC) power amplifiers. Most of the products have been developed for bandwidth less than 50%. Several applications such as broadband communications and electronic warfare require multioctave high-power amplifiers (HPAs). However, to date, not much work on multioctave MMIC HPAs has been reported in the published literature. Progress in broadband MMIC power amplifiers is summarized [1]–[5] in Table I.1 2 Although microwave integrated-circuit (MIC) technology can be used to develop broadband power amplifiers, power MMIC amplifiers, in general, offer smaller size and light weight, higher gain, wider bandwidth, higher reliability, lower cost, and much better unit to unit amplitude and phase-tracking capability when manufactured in large volume. Monolithic technology is particularly beneficial to broadband HPAs due to the elimination of the parasitic effects of bond wires and discrete components used in hybrid MICs. The design of broadband MMIC HPAs poses a significant challenge because of low device impedances and thermal limitations due to low PAE. There are various circuit topologies used to realize broadband MMIC power amplifiers including traditional reactive/resistive matching, shunt or parallel resistive feedback, and traveling-wave approach. The reactive/resistive topologies are suitable for bandwidths less than one octave and require more matching elements to implement wider

O

Manuscript received June 28, 2006; revised September 20, 2006. The author is with the Integrated Products Business Unit, M/A-COM, Roanoke, VA 24019 USA (e-mail: [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.2006.889151 1TriQuint

Semiconductor, Dallas, TX. RF and Microwave Product Solutions, Lowell, MA.

2M/A-COM,

bandwidths. They also have poor gain flatness and voltage standing-wave ratio (VSWR) performance. However, when such amplifiers are designed for balanced configuration, their gain flatness and both input and output VSWR are greatly improved. The resistive feedback configuration results in poor power density per device due to an additional loss in the feedback resistor. However, this scheme is good for gain flatness, VSWR, and multioctave bandwidth performance. The traveling wave or distributed power-amplifier approach has excellent gain-bandwidth characteristics with flat gain and low VSWR, and has the capability of multioctave bandwidth; however, with limited power output and poor power-added efficiency (PAE) [1], [6], [7]. The distributed approach provides the best gain-bandwidth product, while the reactive/resistive matching method is more suitable for high power and high PAE. We selected the latter approach in the development of a low microwave frequency broadband MMIC HPA. We have successfully developed the high-power and high-efficiency MMIC amplifiers using the high-performance and highly reliable multifunction self-aligned gate (MSAG) MESFET integrated circuit (IC) process [8]–[16] being used at M/A-COM, Roanoke, VA. These amplifiers over approximately 15% bandwidth demonstrated [12], [13] in excess of 55% and 40% PAE with associated power output of 14 and 12 W at the - and -band, respectively. We also developed MMIC HPAs having output power and PAE of 5 W and 40%, and 4 W and 24% working over the frequency bands of 8–14 and 10–17 GHz, respectively. These results represent the state-of-the-art power-amplifier performance at - through -bands with the MSAG MESFET technology. Although the high-performance HPAs work over narrow bands, the MSAG technology is also suitable for broadband applications below 20 GHz and provides a low-cost solution to broadband HPAs. This paper describes the design and test results of a fully monolithic class-AB broadband 12-W power-amplifier MMIC designed to operate at a nominal power supply voltage of 10 V. The two-stage amplifier is designed to cover the frequency range from 0.7 to 2.7 GHz and have good input match. Possible applications include radar, point-to-point radio, satellite communication, and wireless communication. II. GENERAL DESIGN APPROACH We selected MSAG MESFET technology to develop broadband power amplifiers because of its high across-wafer uniformity, excellent linearity, and low-cost capabilities. One of the basic requirements for achieving high-power output and PAE on a single MMIC chip is the high across-wafer uniformity of and device cutoff frethe saturated drain–source current quency in order to combine all unit field-effect transistor (FET) cells efficiently. The high across-wafer uniformity for

0018-9480/$25.00 © 2007 IEEE

BAHL: 0.7–2.7-GHz 12-W POWER-AMPLIFIER MMIC DEVELOPED USING MLP TECHNOLOGY

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TABLE I SUMMARY OF BROADBAND MMIC POWER AMPLIFIERS. PERFORMANCE LISTED IS MINIMUM OVER THE FREQUENCY BAND. MMICS WITH GREATER THAN 1 W WERE SELECTED FOR THIS COMPARISON

efficient power combining becomes more important at higher power levels when larger device sizes are needed for broadband applications. The MSAG transistors, based on a self-aligned gate approach, have a planar channel structure attributing to better across-wafer uniformity, reproducibility, and manufac, , breakdown voltage, and turability. Typical values of are 460 mA/mm, 3 V, 20 V, and 21 GHz, respectively. Their standard deviation values are 4.8%, 9%, 8.4%, and 5%, respectively. In this process, the ion-implanted active devices use 0.4- m gates deposited by employing low-cost optical lithography, leading to higher throughput and lower cost. The MSAG process has unique features: it does not use air bridges, has polyimide scratch protection, has multilevel-plating (MLP) capability [15] for low-loss passive components, has no hydrogen poisoning susceptibility, and is very reproducible unit to unit. These features all lead to mean time to failure (MTTF) greater than 100 years at the channel temperature of 150 C, higher assembly yields with MSAG chips, and provide cost-effective solutions. The MSAG process also uses three layers of polyimide , i.e., interlevel dielectric (3- m thick), inductor crossover layer (7 m) or low-loss microstrip (10- m thick), and a scratch protection buffer layer (7- m thick) for mechanical protection of the finished circuitry. Three metal layers are metal 1 (0.5- m thick), first plated gold (4.5- m thick), and second plated gold (4.5- m thick). The MLP process enables to reduce the chip size, to lower resistive loss in passive components, and to realize broadband biasing networks. Low-capacitance metallization crossovers are achieved by a polyimide intermetal dielectric layer. Front side processing is completed by the pattering of a polyimide buffer layer. The buffer layer provides mechanical protection of the circuit structures during backside processing, dicing, and subsequent assembly operations. Finally, the wafers are thinned to its final thickness of 75 m, thru-wafer vias are etched, and the backside is metallized. A. FET Size The design of the MMIC power amplifier starts with the selection of the number of stages and unit FET sizes based on the gain, PAE/linearity, and output power requirements. The choice of the FET cell size impacts the matching networks, combining topology, chip size and electrical performance. Larger FET cell sizes reduce the chip area because fewer combiners

Fig. 1. Physical layout of the 2.0-mm FET with four vias.

are required. However, they have lower input and output impedances, which increase the impedance-matching ratio of the matching networks, increasing circuit mismatch loss (ML) and reducing bandwidth. In addition, there is reduction in the FET’s performance due to increased parasitic reactances and resistance. This latter effect is minimized with careful FET design. In narrowband MMIC amplifiers, we have demonstrated a V of FET periphery. In power density of 0.6 W/mm@ two-octave bandwidth HPAs, we have scaled back to 0.4 W/mm to determine the total FET periphery needed in the design of these power amplifiers. For example, for the 12-W two-octave band HPA, this translates to 16 2.0-mm gate periphery FETs. The physical layout of the 2.0-mm FET with four vias is shown in Fig. 1. After the selection of FET sizes, the FET’s gate periphery, the number of fingers in each FET cell, and gate-to-gate pitch are finalized. The unit gatewidth for each FET is the FET size or periphery divided by the number of fingers, and is a function of device type and frequency of operation. The gate pitch determines the area over which the FET dissipates power; larger pitch reduces FET channel temperature. The design of the output FETs is key to reducing the chip area. Keeping in mind the electrical, physical, and thermal design requirements for each design, we selected appropriate physical dimensions for each FET. The 2-mm FET has six fingers and 54- m gate–gate pitch. The small-signal gain is greater than 18 dB at 3 GHz.

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B. Thermal Design Thermal modeling of semiconductor devices can be performed by using numerical techniques such as the finite-difference and finite-element analyses or by using simple analytic methods such as the Cooke model [17]. For larger FETs (gate periphery greater than 1.5 mm), our measurements using infrared (IR) and liquid-crystal techniques have shown a close agreement with the Cooke model predictions. The first step is of each FET used in to calculate the thermal resistance the designs. Based on the FET structure (gate–gate pitch, unit gatewidth, and FET size), substrate properties, and maximum channel temperature, the thermal resistance is calculated. In our design, we used a 2-mm FET having gate–gate pitch of 54 m, six fingers, and unit gatewidth of 333.3 m. The calculated thermal resistance of the FET is 35.8 C/W. The GaAs substrate thickness and thermal conductivity at room temperature are 75 m and 0.46 W/cm C, respectively. The difference in from the bottom surface (carrier) to the the temperature, top surface (channel) of the MMIC chip is calculated using , where is the net power dissipated in the device. For these calculations, the GaAs thermal conductivity is chosen based on the maximum allowed junction temperature of 150 C. The next step is to calculate the net power dissipation in the FETs under RF drive. We used an in-house program to calculate the power delivered to each FET and the power delivered out of each FET in order to calculate the net power dissipated in the devices. Based on these calculations, an average value of 0.7-W/mm power dissipation was used to calculate the value for each FET. The value for each FET used in the of broadband amplifier designs is 50.1 C. C. Load Impedance for the FET is equivalent to a parThe load impedance and a capacitor . For allel combination of a resistor and are a 1-mm FET operating at 10 V, the values of

pF

(1) (2)

D. Linear Models The design of broadband amplifiers was based on the linear MSAG FET models. For linear simulations, we used equivalent-circuit (EC) models and small-signal -parameters obtained over 0.5–20 GHz at the operating bias point. The EC model topology used is typical of most FET models in the commercial simulators. The -point of the FETs was selected for ) of the device in class-AB operation (approximately 25% order to obtain the maximum power output and PAE. The operating voltage is 10 V and the two-terminal breakdown voltage is approximately 20 V. Four sets of -parameter data, corresponding to device low gain, high gain, low current, and high current were used in the amplifier designs. E. Amplifier Design Considerations Here, a summary of design considerations, used in the design of broadband power amplifiers, is described. This includes chip

size, loss in matching networks, electromigration requirements, and stability considerations. F. Chip Size In general, the larger size for power-amplifier MMICs perform better in terms of RF parameters and thermal design. However, reducing chip area will be a significant cost-saving requirement provided that all other characteristics such as reliability, RF yield, etc., are almost the same. Reducing chip area contributes to MMIC cost reduction in two ways: larger number of chips per wafer and higher visual and functional yields. Thus, reducing the chip area of the power amplifier is an important factor in reducing their costs. Due to low-frequency and broadband operation, we selected a 40-mm chip area for the 12-W HPA MMIC as our goal to achieve better than 0.3-W/mm power density at -band and below operating at 10 V. G. Low-Loss Matching (LLM) Networks It is very desirable to lower the dissipative loss (DL) in the power amplifier’s output matching network using microstrip lines [18] in order to improve the output power and PAE performance. We have also improved the dissipation loss in the microstrip matching networks by using relatively thick conductors [15]. All microstrip line conductors are 9-mm thick, which also improve their current and power-handling capabilities. Thick conductors are realized by combining two 4.5-mm-thick conductors available in the MLP process [15]. The additional thick metallization layer available in the MLP process offers benefits in lower loss at lower microwave frequencies and in the area of dc current routing. Most straightforwardly, the designer now has the flexibility to use 4.5- or 9- m-thick conductor lines. The other benefit of the additional thick metal layer is the option to create 3-D inductors and high current spiral inductors. Previously, spiral inductors would be single layer and current limited based on the width of the thin metallization (0.5–1- m thick) underpass to get from the center of the spiral. With MLP, a spiral inductor can be fabricated with 4.5- m-thick conductors and 4.5- m-thick underpasses. This also allows the design of compact two-layer inductors with an optimum performance [19] and the realization of small chip size at lower frequencies. H. Electromigration Requirements Electromigration requirements dictate the microstrip and inductor line widths carrying dc current. We use a very conservative current density of 2.2 10 A/cm as an electromigration limit [19]–[21]. This translates to a maximum allowed current per unit linewidth of 10 mA m for our 4.5- m-thick gold conductors and 20 mA m for our 9- m-thick gold conductors. Since the maximum current expected in the output FET for this design is approximately 6 A, and we are using a busbar biasing approach, this dictates that, in the output matching network, the dual drain bias lines should be used, and the bus line must be 300- m wide. The bus line runs over metal–insulator–metal (MIM) capacitors and via-hole pads, only 4.5- m thickness is allowed for continuous lines running over RF lines. The requirement for such wide lines complicates efforts to shrink the chip

BAHL: 0.7–2.7-GHz 12-W POWER-AMPLIFIER MMIC DEVELOPED USING MLP TECHNOLOGY

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Fig. 2. Two-stage power-amplifier configuration depicting the load required at the drain of each FET stage.

area. Compact lumped elements are used in the earlier stages where power and current densities are less. I. Stability Our experience has been that, for MSAG FETs, the standard even-mode and odd-mode stability analyses are adequate to avoid microwave oscillations. However, under large-signal condition and pulsed operation, it is necessary to design a worst case -factor greater than 1 based on -paramV and 50% eters data for various bias conditions from to V and 25% to ensure the amplifier’s stability. This approximately replicates the envelope a full cycle of the input signal experiences during the large signal and pulsed condition for operation. We found that imposing a V and 25% small-signal -parameters is adequate to ensure unconditional stable operation [22], [23] under all conditions. J. Biasing Amplifier Biasing low microwave frequency HPA MMICs generally need off-chip RF chokes. In our design, we have used an asymmetric broadside coupled line transformer in the output matching network, which facilitate both biasing drain, as well as realizing broadband impedance matching. Both the gate bias and drain bias are applied using corporate feed configurations. Each FET’s gate is individually biased through a gate resistor. The drain bias is applied to each drain using low-resistance busbar topology employing the top conductor of the MIM capacitors. Both the gate-bias resistor and MIM capacitor values resistor provide robust stability. Further, exercising special care in maintaining the symmetry in the amplifier’s layout and properly selecting isolation resistors prevent odd-mode oscillations. III. HPA DESIGN Traditionally, a power amplifier can be designed based on the loadline method [24]–[27]. The design of the broadband MMIC power amplifier was based on a combination of loadline and LLM [16], [28] design methodology using small-signal FET model and load–pull data obtained at the operating bias point. In this method, the loadline technique [24] is used to optimize the circuit parameters. For example, in a two-stage and at the drain of HPA, the optimum load impedances first- and second-stage FETs, respectively, to realize maximum output power and PAE, are shown in Fig. 2. The value of and are calculated from (1) and (2). In the LLM design technique [28], both the resistive or DL and

Fig. 3. Schematic of the 12-W HPA, only one-quarter is shown. Four of them are combined in parallel.

ML for each stage are calculated and controlled as required in the design. Generally, DL and ML for the output match are kept at a minimum and ML for the interestage is minimized. The controlling factors for DL and ML for the interstage include stability criteria and electrical performance. This also helps in optimizing the FET aspect ratio. The DL is for the individual passive stage, i.e., input, interstage, output, etc., and the ML is the difference between the required device’s optimum load impedance and the transformed 50- output impedance at the drain terminal of the FET. The above method is based on the assumption that the device input impedance depends strongly on the load connected at the drain terminal rather than its large-signal parameters. Fig. 3 shows a simplified schematic of the 12-W HPA (only one-quarter of the circuit is shown). Four quarters are combined in parallel. The determination of DL and ML for each passive stage is described in [16] and [28]. For the input, the DL is given by (3) where ’s are the two-port -parameters of the input-matching is the input impedance looking into the input of network. If is the source impedance, the is given the amplifier and by where

(4)

The broadband amplifier design is carried out by considering the power gain rolloff of the FET with frequency (usually 6 dB/octave), the gain-bandwidth limitations of the input and output of the FET, and the overall amplifier stability versus frequency. The design of a broadband common-source amplifier requires a compromise between several competing requirements such as bandwidth, power output, gain, and PAE. In the matching networks, we used a 16-way reactive cluster combining matching topology. A 16-way cluster combining also provides wider bandwidth than four- or eight-way combining. Both lumped elements and distributed circuit elements were used for impedance-matching networks. In the design optimization using a loadline technique, four sets of -parameter

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Fig. 4. Input-matching network’s dissipative and total loss of the two-stage 12-W MMIC power amplifier.

data, corresponding to low gain, high gain, low current, and high current, were used. These data files represent the possible fabrication changes, allowing to realize a design more tolerant to process variations. The input stage, which has a limited gain compensation network, was designed for good input match, as well as for maximum power transfer at the high frequency end. The input matching consists of a bridged T-coil network [29] and 3-D inductors and is designed to match to 50 with a return loss better than 10 dB. Fig. 4 shows the simulated DL and the total loss (the sum of DL and ML) for the input-matching network. The higher DL at lower frequencies is for device gain slope compensation. It may be noted that the ML, the difference between the total loss and DL, over the 0.75–3-GHz range is less than 0.5 dB. This confirms that the amplifier is matched to 50 . In the multistage broadband HPAs, the design of interstage and output matching networks pose serious challenges. In the interstage, it is the impedance ratio, gain slop compensation, and the bandwidth, whereas in the output, it is the bandwidth and resistive loss that affect the output power and PAE. The interstage matching networks are designed to provide flat gain response and enough output power to succeeding stage FETs for achieving maximum output power and PAE. The interstage matching comprised of RLC lumped-based topology using thin-film resistors, single-layer and 3-D inductors, and MIM capacitors. Fig. 5 shows simulated DL and the total loss for the interstage matching network. The DL is monotonically deceasing with frequency and is adjusted to obtain the unconditional stable operation of each stage. It may be noted that the ML over the 0.75–3-GHz range is lower than 3 dB. If one can design the output match for a correct load and maintain the insertion loss less than 1 dB, one can achieve PAE in the range of 25%–35%, depending on the power level, device technology, frequency range, and bandwidth. The low gain value, small device aspect ratio, high interstage, and output match losses, unable to harmonic tune over wider bandwidths, greatly affect the PAE of broadband amplifiers. Improper second harmonic termination might also lower the PAE at some

Fig. 5. Interstage matching network’s dissipative and total loss of the two-stage 12-W MMIC power amplifier.

Fig. 6. Output matching network’s dissipative and total loss of the two-stage 12-W MMIC power amplifier.

frequencies in the amplifiers working over greater than octave bandwidth. The output matching elements were selected to provide an optimum load match with minimum possible insertion loss since the efficiency is reduced to a greater extent by a given amount of loss due to decreased power out, gain, and available dc power at the FET drain pads. The output match uses 9- m lines, asymmetric broadside coupled line transformer, busbar for biasing drain of second stage FETs, and single-layer highinductors [19]. Fig. 6 shows the dissipative and total loss for the output matching network. The DL is approximately 1.2 dB over 0.7–2.5 GHz. In this case, the ML is less than 0.3 dB over most of the band. Each stage, as well as the complete amplifier, were designed to be unconditionally stable over 3–10-V drain power supply and 25%–50% drain current. Electromagnetic (EM) simulations were used extensively during circuit optimization for closely packed passive circuit components and discontinuities. This broadband HPA designs require a bias supply from both sides. Fig. 7 shows a photograph of the 12-W broadband HPA.

BAHL: 0.7–2.7-GHz 12-W POWER-AMPLIFIER MMIC DEVELOPED USING MLP TECHNOLOGY

Fig. 7. UHF/L/S -band 12-W power amplifier. Chip size is 5.0 mm

227

2 8.0 mm.

Fig. 9. Small-signal input and output VSWR of the 12-W HPA at 25% I V = 10 V.

Fig. 8. Small-signal gain of the 12-W HPA at 25% I

,V

,

= 10 V.

IV. TEST DATA FOR BROADBAND MMIC CHIPS Several MMIC amplifier chips were assembled on goldplated Elkonite (Cu–W alloy) carriers for RF characterization after “on-wafer” pulsed power screening. The Elkonitematerial was chosen for its good thermal conductivity and good thermal expansion match to GaAs and alumina. The ICs were die attached using gold tin (AuSn) at 300 C on a pedestal in order to keep minimum bond-wire lengths between the chip and the input and output microstrip feed lines, which were printed on 15-mil alumina substrate. The test fixtures were fitted with high-performance microstrip-to-coaxial connectors having return loss greater than 20 dB up to 18 GHz. All chips were tested at V and 25% under continuous-wave (CW) conditions, and the MMIC base temperature was kept at 60 C. Typical measured small-signal gain and input and output VSWR of the two-stage UHF/ / -band power amplifier are shown in Figs. 8 and 9. Over the 0.7–2.7-GHz frequency range, the gain was greater than 20 dB, and input and output VSWR were lower than 2 : 1 and 2.2 : 1, respectively. Simulated data is also shown for comparison.

Fig. 10. Output power and PAE at V . I

= 10 V, P

= 22 dBm, and 25%

Average measured output power and PAE of the two-stage broadband power amplifier are shown in Fig. 10. At 22 dBm of input power, the power gain was greater than 19 dB over the 0.7–2.7-GHz frequency range. The output power was greater than 12 W and the PAE was better than 22%. Over 0.8–2.0 GHz, the output power and PAE were better than 14 W and 27%, respectively. Fig. 11 shows the saturated output power versus frequency at drain voltage of 6, 8, and 10 V, and the output power is approximately linear with drain voltage. A plot of output power at 1-dB gain compression versus frequency at drain voltage of 6, 8, and 10 V is show in Fig. 12. The dB power level is approximately linear with drain voltage and at 10 V it is greater than 11 W. Fig. 13 depicts saturated power at 3 C, 60 C, and 100 C of MMIC base temperature.

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Fig. 11. Saturated output power versus frequency and drain voltage at 25% I .

Fig. 14. Output power versus input power and frequency at 10 V and 25% . I

Fig. 15. PAE versus input power and frequency at 10 V and 25% I Fig. 12. 1-dB power compression point and drain voltage at 25% I

.

.

Fig. 16. Second harmonic versus frequency and input power at 10 V and 25% I . Fig. 13. Saturated output power versus frequency and temperature at 10 V and 25% I .

The output power and PAE as a function of input power for various frequencies are plotted in Figs. 14 and 15, respectively. The second and third harmonic power levels of the broadband HPA were also measured at various input power levels. Figs. 16 and 17 depict the second and third harmonic levels versus frequency, respectively. The second harmonic power levels at input power levels of up to 24 dBm are less than 20 dBc, whereas the third harmonic power levels are less than 14 dBc, over 0.7–3.0 GHz. The low second harmonic power levels are probably due to the use of coupled line transformers at the output.

Fig. 17. Third harmonic versus frequency and input power at V 25% I .

= 10 V,

BAHL: 0.7–2.7-GHz 12-W POWER-AMPLIFIER MMIC DEVELOPED USING MLP TECHNOLOGY

V. CONCLUSION A broadband 12-W power-amplifier MMIC has been developed using M/A-COM’s MSAG MESFET technology. The power amplifiers have demonstrated 12–15-W output power and good PAE performance over multioctave bandwidths. This outstanding power performance was only possible because of high across-wafer uniformity of saturated drain–source current and cutoff frequency for the MSAG process. The second harmonic levels were measured below 20 dBc, demonstrating state-of-the-art performance for multioctave bandwidth HPA. Accurate nonlinear models, predicting the output power and harmonics, are needed to improve the PAE over multioctave bandwidths. ACKNOWLEDGMENT The author wishes to acknowledge the support of M/A-COM’s layout, test, standard microwave products, and wafer-processing groups. REFERENCES [1] Y. Ayasi et al., “A monolithic GaAs 1–13 GHz traveling-wave amplifier,” IEEE Trans. Microw. Theory Tech., vol. MTT-30, no. 7, pp. 976–981, Jul. 1982. [2] J. P. Fraysse et al., “A 2 W, high efficiency, 2–8 GHz, cascode HBT MMIC power distributed amplifier,” in IEEE MTT-S Int. Microw. Symp. Dig., 2000, pp. 529–532. [3] J. J. Xu et al., “A 3–10-GHz GaN-based flip-chip integrated broadband power amplifier,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 12, pp. 2573–2578, Dec. 2000. [4] A. Sayed and G. Boeck, “Two-stage ultrawide-band 5-W power amplifier using SiC MESFET,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 7, pp. 2441–2449, Jul. 2005. [5] D. Conway, M. Fowler, and J. Redus, “New process enables wideband high-power GHz amplifiers to deliver up to 20 W,” Defense Electron., Overland Park, KS, Feb. 2006, pp. 8–11. [6] C. Duperrier, M. Campovecchio, L. Roussel, M. Lajugie, and R. Quere, “New design method of non-uniform distributed power amplifiers. Application to a single stage 1 W PHEMT MMIC,” in IEEE MTT-S Int. Microw. Symp. Dig., 2001 , vol. 2, pp. 1063–1066. [7] ——, “New design method of uniform and nonuniform power amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 12, pp. 2494–2500, Dec. 2001. [8] A. E. Geissberger, I. J. Bahl, E. L. Griffin, and R. A. Sadler, “A new refractory self-aligned gate technology for GaAs microwave power FETs and MMICs,” IEEE Trans. Electron. Devices, vol. 35, no. 5, pp. 615–622, May 1988. [9] I. J. Bahl et al., “Class-B power MMIC amplifiers with 70% poweradded efficiency,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 9, pp. 1315–1320, Sep. 1989. [10] ——, “C -band 10 W MMIC class-A amplifier manufactured using the refractory SAG process,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 12, pp. 2154–2158, Dec. 1989. [11] ——, “Multifunction SAG process for high-yield low cost GaAs microwave integrated circuits,” IEEE Trans. Microw. Theory Tech., vol. 38, no. 9, pp. 1175–1182, Sep. 1990.

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[12] W. L. Pribble and E. L. Griffin, “An ion-implanted 13 watt C -band MMIC with 60% peak power added efficiency,” in IEEE Microw. Millimeter-Wave Monolithic Circuits Symp. Dig., 1996, pp. 25–28. [13] E. L. Griffin, “X -band GaAs MMIC size reduction and integration,” in IEEE MTT-S Int. Microw. Symp. Dig., 2000, pp. 709–712. [14] I. J. Bahl, “Design of a generic 2.5 W, 60 percent bandwidth, C -band MMIC amplifier,” Microw. J., vol. 45, pp. 54–70, Aug. 2002. [15] M. Ashman and I. Bahl, “High performance wideband MSAG gain block/driver amplifier MMICs using MLP technology,” Microw. J., vol. 47, pp. 74–88, Dec. 2004. [16] I. Bahl, “MESFET process yields MMIC Ka-band PAs,” Microw. RF, vol. 44, pp. 96–112, May 2005. [17] H. F. Cooke, “Precise technique finds FET thermal resistance,” Microw. RF, pp. 85–87, Aug. 1986, correction of this paper in Microw. RF, p. 13, Feb. 1987. [18] I. J. Bahl et al., “Low loss multilayer microstrip line for monolithic microwave integrated circuits applications,” Int. J. RF Microw. Comp.Aided Eng., vol. 8, pp. 441–454, Nov. 1998. [19] I. J. Bahl, “High current capacity multilayer inductors for RF and microwave circuits,” Int. J. RF Microw. Comput.-Aided Eng., vol. 10, pp. 139–146, Mar. 2000. [20] ——, “High-Q and low-loss matching network elements for RF and microwave circuits,” IEEE Micro, vol. 1, pp. 64–73, Sep. 2000. [21] I. Bahl, Lumped Elements for RF and Microwave Circuits. Norwood, MA: Artech House, 2003. [22] R. G. Freitag et al., “Stability and improved circuit modeling considerations for high power MMIC amplifiers,” in IEEE Microw. Millimeter-Wave Monolithic Circuits Symp. Dig., 1988, pp. 125–128. [23] M. Ohtomo, “Stability analysis and numerical simulation of multidevice amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 6, pp. 983–991, Jun./Jul. 1993. [24] S. C. Cripps, RF Power Amplifiers for Wireless Communications. Norwood, MA: Artech House, 1999. [25] E. L. Griffin, “Application of loadline simulation to microwave high power amplifiers,” IEEE Micro, vol. 1, pp. 58–66, Jun. 2000. [26] K. Chang, I. Bahl, and V. Nair, RF and Microwave Circuit and Component Design for Wireless Systems. Hoboken, NJ: Wiley, 2002, ch. 11. [27] I. Bahl and P. Bhartia, Microwave Solid State Circuit Design, 2nd ed. Hoboken, NJ: Wiley, 2003, ch. 10. [28] I. J. Bahl, “Low loss matching (LLM) design technique for power amplifiers,” IEEE Micro, vol. 5, pp. 66–71, Dec. 2004. [29] R. Goyal, Ed., High-Frequency Analog Integrated Circuit Design. Hoboken, NJ: Wiley, 1995, pp. 179–180. Inder J. Bahl (M’80–SM’80–F’89) received the B.S. degree in physics from Punjab University, Punjab, India, in 1965, the M.S. degree in physics and M.S. (Tech.) degree in electronics engineering from the Birla Institute of Technology and Science, Pilani, India, in 1967 and 1969, respectively, and the Ph.D. degree in electrical engineering from the Indian Institute of Technology, Kanpur, India, in 1975. He is currently a Distinguished Fellow of Technology with M/A-COM, Roanoke, VA. He has authored or coauthored over 145 research papers and 12 books. He holds 16 patents. He is the Editor of the International Journal of RF and Microwave Computer-Aided Engineering. His interests are in the area of device modeling, high-efficiency power amplifiers, and MMIC products for commercial and military applications. Dr. Bahl is a member of the Electromagnetic Academy.

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A 1.9-GHz CMOS Power Amplifier Using Three-Port Asymmetric Transmission Line Transformer for a Polar Transmitter Changkun Park, Student Member, IEEE, Younsuk Kim, Student Member, IEEE, Haksun Kim, Member, IEEE, and Songcheol Hong, Member, IEEE

Abstract—A 1.9-GHz CMOS differential power amplifier for a polar transmitter is implemented with a 0.18- m RF CMOS process. All of the matching components, including the input and output transformers, are fully integrated. The concepts of injection locking and variable load are applied to increase the efficiency and dynamic range of the amplifier. An asymmetric three-port transmission line transformer is proposed to embody the variable load effectively. The power amplifier achieved a power-added efficiency of 40% at a maximum output power of 32 dBm. The dynamic range was 20 dB at supply voltages ranging from 0.5 to 3.3 V. The improvement of the low power efficiency was 290% at an output power of 16 dBm. Index Terms—Class-E, CMOS, dynamic range, global system for mobile communication (GSM), injection locking, polar transmitter, power amplifier, transmission line transformer, variable load.

I. INTRODUCTION OLAR transmitters are expected to be very popular owing to their many advantages over conventional Cartesian transmitters, especially in global system for mobile communications (GSM) and EDGE systems. Accordingly, their power amplifiers based on GaAs HBTs are being studied intensively. However, few studies have focused on CMOS power amplifiers for polar transmitters. Although CMOS power amplifiers are expected to be cheaper than GaAs HBT power amplifiers and easier to integrate with other circuits, they are not considered to be a useful RF power amplifier with a watt-level output power. Recently, the potential of a CMOS power amplifier was successfully demonstrated using a distributed active transformer [1]–[3]. The distributed active transformer is considered to have the potential to lend improvements to the performance of a CMOS power amplifier [4]. In this study, the concept of the distributed active transformer is used to design a CMOS power amplifier for polar transmitter applications.

P

Manuscript received July, 13, 2006; revised October, 19, 2006. This work was supported by the Korea Science and Engineering Foundation under the Engineering Research Center Program through the Intelligent Radio Engineering Center in Korea. C. Park and S. Hong are with the Department of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea (e-mail: [email protected]). Y. Kim was with the Department of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology, Daejeon 305701, Korea. He is now with the Samsung Electro-Mechanics Company Ltd., Suwon 443-803, Korea. H. Kim is with the Department of Radio-Wave Engineering, Hanbat National University, Daejeon 305-719, Korea. Digital Object Identifier 10.1109/TMTT.2006.889152

Fig. 1. Simplified block diagram of a polar transmitter.

Fig. 1 shows a block diagram of a polar transmitter. The polar transformer ( - to - ) decomposes the input of a power amplifier into two types of input signals. The first of these is an RF phase signal, which is applied to the input of a power amplifier, and the second is an envelope signal, applied as a . The supply voltage has to be applied supply voltage through a dc–dc converter or a low drop output regulator circuit in order to supply sufficient power. Polar transmitters can use switching-mode power amplifiers, as the input signals do not contain envelope information. These include class-D, class-E, and class-F amplifiers, which are nonlinear, but very efficient. A CMOS power transistor is known to be viable for switching power amplifiers rather than linear amplifiers; thus, it common to study CMOS switching power amplifiers for polar transmitters [1]–[3], [5]. There are two important specifications of power amplifiers for polar transmitters. The first is the output dynamic range. It is crucial to obtain enough dynamic range with a given supply voltage range. The second specification is related to efficiency at a low output power. In general, a power efficiency close to the maximum output power is fairly high. However, the efficiency at a low output power is very low. Therefore, a stage-convertible power amplifier using a variable load method [6] is proposed as a means of increasing the dynamic range of the power amplifiers and improving the efficiency at a low output power. An asymmetric three-port transmission line transformer is also proposed to embody the variable load method efficiently. II. LOAD IMPEDANCE TRANSFORMATION A. Power Efficiency of Class-E Power Amplifiers Polar transmitters can use switching-mode power amplifiers, as mentioned in Section I. The power efficiencies of polar transmitters are, therefore, expected to be higher than those of conventional transmitters; however, in a general switching mode, power amplifiers require a driving power greater than that for linear amplifiers. In a conventional class-E power amplifier, the

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PARK et al.: 1.9-GHz CMOS POWER AMPLIFIER USING THREE-PORT ASYMMETRIC TRANSMISSION LINE TRANSFORMER

Fig. 2. Drain efficiency and PAE of a conventional and a variable load power amplifier.

power consumption of the driver stage degrades the overall efficiency severely regardless of the high drain efficiency of the power amplifier. As the output power of a conventional class-E power ampliwith a fixed input power, as shown fier is in proportion to by (1) the gain is decreased severely at a low [7]. Although the drain efficiency maintains a high value, the power-added efficiency (PAE) of the power amplifier is degraded as the decreases. The drain efficiency and PAE of a power amplifier are expressed as follows: Drain Efficiency

(2)

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Fig. 3. Dynamic range of a power amplifier with a variable load.

power. As the load impedance of a conventional class-E power amplifier is fixed, the dynamic range of the conventional power amplifier for polar transmitters is determined by the variable . To satisfy the dynamic range of a GSM system, range of i.e., 20 dB, the minimum supply voltage must be as low as 0.33 V with a maximum supply voltage of 3.3 V. However, 0.33 V is too low to generate with a dc–dc converter. Additionally, such a low supply voltage degrades the overall efficiency of a dc–dc converter. Moreover, the phase distortion and the feed-through of the power amplifier are significant issues under a low supply voltage [8]. To solve these problems, a variable load method can be applied to the power amplifier for a polar transmitter. As the output , the output power is decreased as power is in proportion to is decreased. If the load impedance is increased at a low region, the dynamic range can be increased, as shown by

(3) dBm

Dynamic range where is the input power, is the output power, is the on resistance of the power transistors, and is the load impedance of the power amplifier. Fig. 2 shows the drain of a conventional power efficiency and PAE versus the amplifier, and a variable load power amplifier when the varies with a fixed . As the output power is in proportion to , as shown in (1), the drain efficiency and PAE versus can be determined, as shown in Fig. 2. The dotted lines denote the drain efficiency and PAE of a conventional power amplifier. If the low load impedance is increased two times in the low output power region, the drain efficiency and PAE are increased according to (2) and (3), respectively. The drain efficiency and PAE curves with variable loads appear as solid lines in Fig. 2. Although a low load impedance is required to obtain high maximum output power, the load impedance must be increased is decreased in order to increase the efficiency. The varias able load allows a high maximum output power at high power and a high efficiency at low power. B. Dynamic Range of Class-E Power Amplifiers The dynamic range of a power amplifier is the difference between the maximum output power and the minimum output

dB

dBm

(4)

is the maximum In this equation, the parameter , is the minimum , is the low load impedance for the high output power, and is the high load impedance for the low output power. As shown in (4), the dynamic range can be controlled by not only the range of , but also through the ratio between and . variable , 0.7 V for , Assuming the values of 3.3 V for and 3 for , the required load impedance ratio should be 5 for the dynamic range of a GSM system. , the load Fig. 3 explains (4). At a certain point of changes from low to high. As versus impedance decreases as increases, as shown in (1), the dynamic

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Fig. 4. Simplified schematic of the output transformer (after [1] and [2]).

range increases for a given range of . In Fig. 3, the dashed of a conventional power amplifier. The solid line shows of a power amplifier with an line in this figure shows of a abruptly varied load, and the dotted line shows power amplifier with a gradually varied load. The dotted arrow shows the dynamic range of a conventional power amplifier and the solid arrow shows the dynamic range of a power amplifier with a variable load. It is important to note that the dynamic range can be extended using the variable load method. III. PROPOSED THREE-PORT ASYMMETRIC TRANSMISSION LINE TRANSFORMER

Fig. 5. (a) Output matching network. (b) Equivalent circuit for the output matching network.

, , and in Fig. 5(b), can be used as output e.g., matching components. The following equations demonstrate the process of impedance transformation using the transmission and . line transformer and the additional capacitors, of Fig. 5 can be calculated as follows:

A. Impedance Transformation In contrast to GaAs technology, there is no via process in CMOS technology. Thus, a bond wire is needed to make an ac ground in RF CMOS circuits. By applying a differential structure, it is possible to obtain virtual grounds, thereby preventing the gain reductions that are induced by the bond wires. However, in general, the antenna and filter connected to the output of the power amplifier are single-ended components. Thus, a transformer is needed to connect the differential power amplifier to the single-ended components. In this study, a high- transmission line transformer is used as the output transformer. The transformer significantly influences the output power and efficiency. The output matching is completed with an additional metal–insulator–metal (MIM) capacitor. Two differential pairs of a power stage are used, as shown in Fig. 4. If the -factor of the transformer used in Fig. 4 is 1, as is ideal, and there is no parasitic inductance in the transmission line transformer, the load impedance of 50 is transformed into 12.5 with an impedance transforming ratio of 1 to 4 [1], [2]. However, the inductance of the transmission line transformer plays an important role in the output matching network. To verify the role of the transmission line transformer in the output matching network, Fig. 5 shows an equivalent circuit for the output matching network, which is composed of the transmission line transformer and additional MIM capacitors and . To simplify the analysis, the turn ratio of the transmission line transformer, which is used in the analysis, is 1, as shown in Fig. 5(a). Fig. 5(b) shows the equivalent circuit of the output matching network shown in Fig. 5(a). and are the self-inductances. The parasitic inductances of the transmission line transformer,

(5) The imaginary part of (5) is equal to zero at resonance with and , as described in (6) Here,

is expressed by (7)

and

are calculated by

(8)

(9) Afterwards, the imaginary part of (9) can be equal to zero using the value from Fig. 5, as described in

(10) Following this, the value

can be expressed as (11)

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Fig. 7. Simplified schematic of the proposed three-port asymmetric transmission line transformer.

an increase of increases.

,

cannot be increased effectively as

B. Implementation of a Three-Port Asymmetric Transmission Line Transformer Fig. 6. Simulation results of: (a) the capacitance load impedance R .

C and (b) the transformed

can, therefore, be expressed by

(12) is 25 , the transAssuming the -factor is 0.5 and formed load impedance versus the varied inductance of the primary part was simulated using (12). The inductance of the secondary part is assumed to be 0.5, 1.0, 1.5, and 2.0 nH in the simulation. Fig. 6 shows simulated and . As shown in Fig. 6(b), increases as increases. Thus, although the impedance transformation ratio of the ideal transformer is 1, the impedance transformation ratio of the transmission line transformer can be controlled by the parasitic component of the transformer. From the results shown in Fig. 6, the inductance of the primary part must be small in order to obtain the low load impedance for a high maximum output power. As shown in Section II, the inductance of the primary part must be large in order to obtain a high load impedance for a high dynamic range and a high efficiency at the low output power region. It is interesting to note that, for a high load impedance, the -factor of the transmission line transformer does not increase as the inductance increases. If the -factor increases with

A three-port transmission line transformer is proposed in order to obtain a low and high load impedance with a transformer, as simplified in Fig. 7. The primary part of the transformer consists of two transmission lines, which have different inductances. The inductances can be controlled by the length and width of the slab waveguides. One of these is designed for a low load impedance to obtain a high maximum output power. The other is designed for a high load impedance in order to obtain a high dynamic range and high efficiency at a low output power. The proposed three-port asymmetric transmission line transformer, therefore, provides two load impedances. The -factor of the primary high-impedance transmission line to the secondary line should not exceed that of the low-impedance line in order to obtain the two load impedances effectively. If the -factor of the high load impedance part exceeds that of the low load impedance part, the value of of the high load impedance part cannot be higher than that of of the low load impedance part. In this study, a three-port transmission line transformer is embodied, as shown in Fig. 8. The primary part for the low-power mode has spiral turns to increase the parasitic inductance of the transformer. The proposed three-port transmission line transformer is applied to the proposed power amplifier for a polar transmitter application in order to achieve a high maximum output power, high dynamic range, and high efficiency at a low output power region. The simulated loss of the transformer itself for a high power mode is approximately 1.7 dB.

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Fig. 8. Three-port asymmetric transmission line transformer. (a) Transformer for a high-power mode. (b) Transformer for a low-power mode. (c) Three-port transmission line transformer.

Fig. 9. Power amplifier with the bypass switch.

Fig. 11. Schematic of the stage-convertible power amplifier.

Fig. 10. Proposed stage-convertible power amplifier architecture.

IV. DESIGN OF THE 1.9-GHz CMOS POWER AMPLIFIER A. Proposed Stage-Convertible Power Amplifier The proposed stage-convertible power amplifier is a modified structure of a power amplifier with a bypass switch, as shown in Fig. 9 [9]. The switch is located parallel to the power stage, as shown in Fig. 9. The power stage is turned on and the switch is turned off to achieve the maximum gain and output power. For the low output power, the power stage is turned off and the switch is turned on to bypass the power stage. The output of driver stage then becomes the output of the power amplifier. In the proposed power amplifier, a low-power matching network is located parallel to the power stage. The output of the power stage and the output of the matching network are combined in a power combiner, as shown in Fig. 10. For a high-power mode, the power stage is turned on and the driver stage drives the power stage. In addition, a certain amount of driver power is coupled to the output directly through the three-port transmission line transformer. The load impedance for the power stage is low to achieve a high maximum output

power. For a low-power mode, the power stage is turned off and only power from the driver is transmitted into the output port through the low-power matching network. The load impedance of the driver stage must be high in order to obtain a high dynamic range in addition to a high efficiency in the low output power region. In the proposed stage-convertible power amplifier, high- and low-power modes of the power amplifier are selected by turning on, and off, respectively, the power stage. The power combiner and low-power matching network are implemented with the three-port transmission line transformer, as shown in Fig. 11. A schematic of the proposed power amplifier is shown in Fig. 11, which has a differential structure. The driver and power stages were designed as class-E amplifiers. The output of the power stage is connected to the one primary part of the transformer that has a low load impedance. The output of the driver stage is connected not only to the input of the power stage, but also to the other primary part of the transformer with a high load impedance. For the high-power mode, all of the stages in the power amplifier are turned on to generate a high output power. For the low-power mode, the power stage is turned off and the output power is generated only by the driver stage. The output power of the driver stage is transmitted into the output while in the low-power mode. In Fig. 11, a pair of differential power stages and a pair of differential driver stages are cross-coupled to create

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Fig. 13. Schematics of the power stages for: (a) conventional cascode structure and (b) self-biased cascode structure.

Fig. 12. Schematic of the designed power amplifier.

node “A” and node “B” in the phase. In the proposed power amplifier, an additional power stage is not needed for the lowpower mode. The driver stage works as the output stage in the low-power mode. The chip size is, therefore, reduced. A schematic of the designed power amplifier is shown in Fig. 12. The concept of a distributed active transformer is used to realize the voltage-combining technique. Two differential pairs of the power stage and the driver stage are used. The output matching network of the power stage is implemented with the three-port transformers, the drain–source capacitance of the power stage, and an additional MIM capacitor. The output matching network of the driver stage is composed of the transformers, the drain–source capacitance of the driver stage, the gate–source capacitance of the power stage, and an additional MIM capacitor. The input transformer consists of a spiral-type transformer. B. Auto-Switching Technique An auto-switching technique is used to switch to the highand low-power modes automatically with respect to the of the power amplifier, which is an amplitude signal of a polar transmitter. Therefore, an additional external control signal to change the mode of the power amplifier is not needed. As mentioned in Section II, the output power of a polar transof the power amplifier. mitter is controlled via the variable is increased. The output power is, therefore, increased as Here, both the driver stage and power stage are turned on when the supply voltage becomes high. However, the driver stage is turned on and the power stage is turned off automatically when the supply voltage is low. A cascode structure is used in each stage to realize the auto-switching technique. A conventional

Fig. 14. Chip photograph of the implemented power amplifier.

cascode structure is shown in Fig. 13(a), and a self-biased cascode structure is shown in Fig. 13(b) [10]. If the supply voltage of the self-biased cascode structure is decreased, the on resisis increased and the current through the power trantance of sistor of the self-biased cascode structure becomes smaller than that of the conventional cascode structure. In this study, the conventional cascode structure is used in the driver stage and the self-biased cascode structure is used in the power stage. The power stage is programmed to be turned on or turned off automatically according to the variable supply voltage. The contribution of the power stage to the output power is decreased and the contribution of the driver stage to the output is decreased. Thus, the high-power power is increased as region and the lowmode becomes dominant in the higher region. power mode becomes dominant in the lower V. MEASUREMENT RESULTS Fig. 14 shows the implemented power amplifier using 0.18- m RF CMOS technology. The size of the chip is 1.2 mm 1.8 mm including pads. All of the matching components, including the input and output transformers, are fully integrated. The area of the transformer is 1.0 mm 1.0 mm. The metal for the transformer is 2.34- m-thick aluminum.

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Fig. 15. Circuit configurations of: (a) conventional cascode structure, (b) selfbiased cascode structure, and (c) measurement results.

The losses of the bond-wire, input transformer, and printed circuit board interconnections are included in the amplifier’s measured performance. Fig. 15 shows the measured PAE versus , while varies from 0.5 to 3.3 V. In Fig. 15(a), conventional cascode structures are used in the power and driver of the power stage varies while of the stages, and driver stage remains fixed. In Fig. 15(a), the gate bias in the cascode transistor of the power stage is fixed at 3.3 V. If the gate bias is lower than 3.3 V, the maximum output power and PAE are degraded. In Fig. 15(b), a self-biased cascode structure is used in the power stage. The measured PAE of Fig. 15(b) is higher than that of Fig. 15(a). For the circuit configuration of Fig. 15(b), decreases. the contribution of the power stage decreases as The efficiency of the structures in Fig. 15(b) in the low-power region is, therefore, higher than those in Fig. 15(a). The measured dynamic range is not sufficient for a GSM specification, and the efficiency improvement in the low output power region is limited. To extend the dynamic range and to increase the efficiency of the driver stage also varies with of further, the power stage, as shown in Fig. 16. A measurement of the

Fig. 16. Circuit configurations of: (a) high-power mode, (b) low-power mode, (c) auto-switching technique, and (d) measurement results.

high-power mode shows that the maximum output power is 32 dBm with a PAE of 40%. Theoretically, the dynamic range of a conventional class-E amplifier is approximately 16.4 dB, varies from 0.5 to 3.3 V. However, the dynamic while range of the high-power mode is extended to nearly 20 dB, as the input power of the power stage also decreases when of the driver stage decreases. For the low-power mode, the power stage is turned off by an external signal. Due to the high load impedance of the low-power mode, the efficiency in a low output power region is increased. With the circuit configuration shown in Fig. 16(c), the auto-switching technique is realized by the self-biased cascode structure in the power stage. In the low output power region, the input power of the power stage is decreased. The contribution of the power stage is, therefore, decreased and the contribution of the driver stage that has a

PARK et al.: 1.9-GHz CMOS POWER AMPLIFIER USING THREE-PORT ASYMMETRIC TRANSMISSION LINE TRANSFORMER

Fig. 17. Measured gains of different modes.

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Fig. 19. Measured frequency response.

Fig. 20. Measured degradation of the output power with time. Fig. 18. Extension of dynamic range.

higher load impedance is increased. Thus, the power and driver stages are always turned on. As shown in Fig. 17, the mode of the power amplifier is changed automatically and smoothly . This is due to the self-biased cascode structure of with the power stage. Fig. 17 shows the measured gains of different modes. For , the gain of a high-power mode always appears a given higher than that of a low-power mode due to the fixed input power of the class-E amplifier. As indicated in the graph, the mode of the power amplifier with the auto-switching method is changed automatically and smoothly in the mode-change region. Fig. 18 shows an extension of the dynamic range. The white symbols show the theoretically calculated output power of a conventional class-E amplifier. The dynamic range of a conventional class-E amplifier is approximately 16.4 dB, while varies from 0.5 to 3.3 V. The black symbols show the output power of the circuit configuration shown in Fig. 16(c). The measured extension of dynamic range is nearly 4 dB. Fig. 19 shows the frequency response under the maximum output power conditions. The single-ended output power is higher than 31 dBm over a frequency range of 1.70–1.92 GHz. Fig. 20 shows the measured degradation of the output power with time under the maximum output power conditions. The nonlinearities of a polar transmitter are mainly due to signal delay mismatches between amplitude path and phase

path. There are also due to the AM–PM distortion of the amplifier. However, the nonlinearities can be solved using a digital predistorter in the polar transmitter. VI. CONCLUSIONS A 1.9-GHz power amplifier for a polar transmitter application has been implemented using 0.18- m RF CMOS technology. Stage-convertible power-amplifier architecture, an auto-switching technique, and a three-port asymmetric transmission line transformer have been proposed in order to improve efficiency and dynamic range. The improvement of the dynamic range is approximately 4 dB, and the total dynamic range is nearly 20 dB. The low-power efficiency improvement of the power amplifier is 290% at an output power of 16 dBm. REFERENCES [1] I. Aoki, S. D. Kee, D. B. Rutledge, and A. Hajimiri, “Distributed active transformer—A new power-combining and impedance-transformation technique,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 1, pp. 316–331, Jan. 2002. [2] ——, “Fully integrated CMOS power amplifier design using the distributed active-transformer architecture,” IEEE J. Solid-State Circuits, vol. 37, no. 3, pp. 371–383, Mar. 2002. [3] Y. Kim, C. Park, H. Kim, and S. Hong, “CMOS RF power amplifier with reconfigurable transformer,” Electron. Lett., vol. 42, no. 7, pp. 405–407, Mar. 2006. [4] S. Kim, K. Lee, J. Lee, B. Kim, S. D. Kee, I. Aoki, and D. B. Rutledge, “An optimized design of distributed active transformer,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 380–388, Jan. 2005.

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[5] A. Mazzanti, L. Larcher, R. Brama, and F. Svelto, “A 1.4 GHz–2 GHz wideband CMOS class-E power amplifier delivering 23 dBm peak with 67% PAE,” in IEEE Radio Freq. Integr. Circuits Symp. Dig., Jun. 2005, pp. 425–428. [6] A. Shirvani, “A CMOS RF power amplifier with parallel amplification for efficient power control,” IEEE J. Solid-State Circuits, vol. 37, no. 6, pp. 684–693, Jan. 2002. [7] N. O. Sokal and A. D. Sokal, “Class-E: A new class of high-efficiency tuned single-ended switching power amplifiers,” IEEE J. Solid-State Circuits, vol. SSC-10, no. 6, pp. 168–176, Jun. 1975. [8] P. Reynaert and M. S. J. Steyaert, “A 1.75-GHz polar modulated CMOS RF power amplifier for GSM-EDGE,” IEEE J. Solid-State Circuits, vol. 40, no. 12, pp. 2598–2608, Dec. 2005. [9] J. Staudinger, “Stage bypassing in multi-stage PAs,” presented at the IEEE MTT-S Int. Microw. Symp. Workshop, Jun. 2000. [10] T. Sowlati and D. M. W. Leenaerts, “A 2.4-GHz 0.18-m CMOS selfbiased cascode power amplifier,” IEEE J. Solid-State Circuits, vol. 38, no. 8, pp. 1318–1324, Aug. 2003.

Changkun Park (S’03) received the B.S. and M.S. degrees in electrical engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2001 and 2003, respectively, and is currently working toward the Ph.D. degree at KAIST. His research interests include CMOS power amplifiers, polar transmitters, and RF electrostatic discharge (ESD) protection circuits.

Younsuk Kim (S’03) was born in Seoul, Korea, in 1969. He received the B.A., M.A., and Ph.D. degrees in electronic engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 1991, 1993, and 2006, respectively. His doctoral dissertation concerned CMOS RF power amplifiers with reconfigurable transformers. From 1995 to 2002, he developed voltage-controlled oscillator (VCO) modules for mobile phone in Samsung electromechanics. From 2002 to 2006, he was a graduate student involved with projects concerning MMIC circuits and RF power amplifiers. He is currently with the Samsung Electro-Mechanics Company Ltd., Suwon, Korea, where he is involved with the development of CMOS power amplifiers and polar transmitters.

Haksun Kim (M’93) received the B.S., M.S., and Ph.D. degree in electronics from Hankuk Aviation University, Goyang City, Korea, in 1986, 1990, and 1993, respectively. He has been with the Samsung Advanced Institute of Technology, the Ministry of Information and Communication, and Turbo Telecom. Since 1989, he has been a Professor with the Department of Radio-Wave Engineering, Hanbat National University, Daejeon, Korea. Since 2004, he has been Vice President of the Samsung Electro-Mechanics Company Ltd., Suwon, Korea. His research interests are circuits and systems, RF integrated-circuit design, which includes low/high date-rate wireless connectivity, wireless local area networks (WLANs), Wimax, millimeter waves, etc.

Songcheol Hong (S’87–M’88) received the B.S. and M.S. degrees in electronics from Seoul National University, Seoul, Korea, in 1982 and 1984, respectively, and the Ph.D. degree in electrical engineering from The University of Michigan at Ann Arbor, in 1989. In May 1989, he joined the faulty of the Department of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea. In 1997, he held short visiting professorships with Stanford University and Samsung Microwave Semiconductor. His research interests are microwave integrated circuits and systems including power amplifiers for mobile communications, miniaturized radar, millimeter-wave frequency synthesizers, as well as novel semiconductor devices.

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Terahertz Performance of Integrated Lens Antennas With a Hot-Electron Bolometer Alexei D. Semenov, Heiko Richter, Heinz-Wilhelm Hübers, Burghardt Günther, Andrey Smirnov, Konstantin S. Il’in, Michael Siegel, and Jugoslav P. Karamarkovic, Member, IEEE

Abstract—Radiation coupling efficiency and directive properties of integrated lens antennas with log-spiral, log-periodic, and double-slot planar feeds coupled to a hot-electron bolometer were experimentally studied at frequencies from 1 to 6 THz and compared with simulations based on the method of moments and physical-optics ray tracing. For all studied antennas, the modeled spectral dependence of the coupling efficiency fits to the experimental data obtained with both Fourier transform spectroscopy and noise temperature measurements only if the complex impedance of the bolometer is explicitly taken into account. At high frequencies, the radiation pattern of integrated antennas exhibits sidelobes, which are higher than those predicted by the antenna model. Index Terms—Integrated lens antenna, quasi-optical coupling, superconducting bolometers, terahertz receivers.

I. INTRODUCTION XPLORING THE terahertz frequency range is an important trend in planetary science, astronomy, and security research. For terahertz receivers, the planar integrated quasioptical technology is expected to be a preferable alternative to waveguide-based front ends. Although corrugated horns can be fabricated for frequencies of a few terahertz, they are getting more expensive to manufacture and the effect of misalignments becomes more severe when the frequency increases. Recent progress in nanostructuring and micromachining allows reliable production and alignment of planar antenna structures on dielectric lenses with an accuracy sufficient for the terahertz range. On the other hand, superconducting hot-electron bolometers (HEBs) [1] proved to be mixers of choice for frequencies above the superconducting energy gap. Thus, the combination of a planar integrated lens antenna with a HEB mixer should result in better performance of a terahertz heterodyne receiver. Log-periodic [2], [3], log-spiral [4]–[6], double-slot [7]–[9], and double-dipole [10] planar feed integrated with quartz or silicon lenses [11] have been shown to efficiently couple radiation in the sub-millimeter-wavelength range. Driven by a relative lack

E

Manuscript received April 20, 2006; revised September 8, 2006. The work of A. Smirnov was supported in part by the Parliament of Berlin, Germany, under a scholarship. A. D. Semenov, H. Richter, H.-W. Hübers, and B. Günther are with the German Aerospace Center, 12489 Berlin, Germany (e-mail: [email protected]). A. Smirnov is with the German Aerospace Center, 12489 Berlin, Germany, on leave from the Radiophysical Laboratory, Moscow State Pedagogical University, 119435 Moscow, Russia. K. S. Il’in and M. Siegel are with the Institute for Micro- and Nano-electronic Systems, University of Karlsruhe, 76187 Karlsruhe, Germany. J. P. Karamarcovic is with the Faculty of Civil Engineering and Architecture, University of Niˇs, 18000 Niˇs, Serbia. Digital Object Identifier 10.1109/TMTT.2006.889153

of experimental data, reliable modeling of the feed performance at terahertz frequencies becomes extremely important in designing quasi-optical front-ends. A semianalytical lumped-element technique [12] that has been implemented so far provides results inconsistent with the experimental data at frequencies above a few terahertz. The main reason, e.g., for a double-slot feed, is that this technique neglects frequency-dependent parasitic impedances appearing at the points where virtual lumped elements are connected to each other. As a result, an experimentally verified spectrum of the antenna response does not coincide with the computed frequency band. Although the discrepancy can be partly relaxed [13] taking into account the geometric inductance of the detector used for the experimental evaluation, the remaining inconsistency is rather large. This impedes the feed design and engineering of the entire integrated antenna. In this paper, we apply the method of moments (MoM) to the entire planar-feed structure including the feed itself and embedding and interconnecting elements. We compare the simulated and measured coupling efficiency and beam pattern of integrated lens antennas, which consist of an extended hemispherical dielectric lens and different planar feeds coupled to a HEB. II. PLANAR-FEED ANTENNAS The full-wave MoM reveals the current and charge density distribution in a planar-feed antenna via the numerical solution of Maxwell equations with boundary conditions along a virtual mesh superimposed on the model structure. The electromagnetic fields at an arbitrary observation point are then computed via scalar and vector potentials generated by the current and charge densities. We used the FEKO software package [14] that combines the accurate MoM with the physical optics approach and the uniform theory of diffraction to obtain fields radiated by our planar feed. The feeds were excited along an infinitely thin slit in the geometric center. The slit in the feed model substitutes the bolometer in the real structure. The model describes the feed with a finite thickness printed on the plane boundary of a dielectric half-space and includes resistive losses in the feed and losses in the dielectric. We applied this technique to self-complementary [15] logarithmic-spiral and log-periodic, as well as to double-slot planar-feed antennas. A. Log-Spiral Feed The full structure of the self-complementary log-spiral planar feed is shown in Fig. 1 (left panel). The feed includes the spiral arms and two bow arms connecting the spiral part of the feed with a coplanar transmission line. The symmetry axis of the coplanar line crosses the geometric center of the feed and is parallel to the excitation slit (Fig. 1, right panel). The parameter

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Fig. 1. (left) SP2 log-spiral feed. Dark color represents the substrate. (right) Distribution of the high-frequency current in the SP1 feed at 2.5 THz. The white background represents the substrate. The gray scale shows the surface current density in relative units. The white line in the center shows the position of the excitation slit. The slit is oriented at an angle of 45 with respect to x-axis and has a length of 1.2 m.

[16] was chosen to build the log-spiral arms of the with being the azimuth feed according to angle and being the distance from the geometric center of the spiral. The size of a spiral portion of the feed is defined by the is the diamouter and inner diameter. The outer diameter eter of the smallest circle that encompasses the spiral structure. is the diameter of the smallest The inner diameter circle at which the arms still obey the spiral equation. Terminals connecting the feed to the bolometer are formed inside this circle. We modeled performance of two gold planar feeds with a thickness of 70 nm printed on silicon. The feeds SP1 and SP2 m and m and were characterized by m and m, correspondingly. They were intuitively designed to cover the upper (from 1.6 to 5 THz) and lower (from 0.6 to 3 THz) terahertz frequency ranges. Feeds were excited with the voltage source whose strength was homogeneously distributed along the length of the slit. Fig. 1 (right panel) shows the MoM-modeled distribution of the high-frequency electric current in the central part of SP1 at 2.5 THz. Due to the skin effect, the high-frequency current flows mostly along the edges of the spiral arms. The impedance of the SP1 feed seen from the excitation slit is shown in the left panel of Fig. 2. In the whole frequency range, the impedance has a nonnegligible imaginary component. Generally, the field irradiated by log-spiral feeds is elliptically polarized. The axial ratio of the polarization ellipse and the azimuth position of the larger axis are shown in Fig. 2 (right panel) for the SP1 feed. At frequencies below 1 THz, the radiation properties of the feed are defined by the two bows. The radiated field is linearly polarized. Correspondingly, the axial ratio drops to zero. The difference between the and the symmetry axis of polarization direction might be caused by lateral offset of the the bows bow apexes. The real part of the impedance remains almost constant and substantially exceeds the imaginary component at frequencies up to 2.5 THz. At 6 THz, the magnitudes of the two components are almost equal. Therefore, in terms of the impedance, the feed cannot be qualified as frequency independent. Instead, we introduce a usable frequency interval from 1 to 6 THz. Within this frequency interval, the radiation of the feed is elliptically polarized and the imaginary part of

Fig. 2. (left) Real (squares) and imaginary (triangles) part of the impedance of a log-spiral feed SP1 on silicon. (right) Axial ratio and the azimuth angle () of the larger axis of the polarization ellipse.

Fig. 3. (left) Beam profile of the SP1 feed at the dielectric side for frequencies 2.5 THz (open symbols) and 4.3 THz (closed symbols). Scans are taken at the main azimuth angles  = 5 (triangles) and  = 95 (squares) for 2.5 THz and at  = 80 (triangles) and  = 10 (squares) for 4.3 THz. (right) Polar plot of the directional gain (elevation angle at the level 9 dB) for 2.5 THz (solid line) and 4.3 THz (dashed line).

0

0

the feed impedance is smaller than the real part. Thus, for our log-spiral feed on silicon, the rule-of-thumb to define the and lower and higher cutoff wavelengths becomes . The latter is close to the criteria [17] for the upper cutoff that occurs when the arm length equals the wavelength where is the relative permittivity of silicon. For all frequencies, the beam pattern of a log-spiral feed that (Fig. 3) exhibits a dip at an elevation angle at corresponds to the angle of total reflection the boundary between silicon and air [18]. The whole pattern rotates and becomes less symmetric with the frequency. At 2.5 THz, the azimuth angles corresponding to the largest and and . smallest width of the beam are close to The beamwidth ratio taken for these azimuth angles at the level of 9 dB increases from 1.2 at 1 THz to 1.7 at 5 THz, while an average width of the beam approximately equals 120 and does not practically change with the frequency. B. Log-Periodic Feed The structure of the self-complimentary log-periodic feed without bows is shown in Fig. 4. We studied a feed with , , and five teeth whose radii were defined according to and

(1)

where and the smallest radius m. The well-known spectral features of the log-periodic feed observed

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Fig. 4. (left) Layout of the log-periodic feed. (right) Central area of the feed imaged with an electron-beam microscope shows an accuracy of the feed manufacturing better than 0.1 m. Dark color represents the substrate.

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Fig. 6. (left) Layout of a double-slot feed antenna. Gray color denotes the gold layer. White color denotes the dielectric. (right) Distribution of the high-frequency current in the DS2 feed at 2.5 THz. The scale shows the surface current density in relative units.

TABLE I STUDIED DOUBLE-SLOT FEEDS



The double-slot feeds were originally designed according to [8], [12] L 0:33  , S = 0:17  , and W = 0:05  , where  is the central wavelength in free space. Expected central frequencies are 1.6, 2.5, and 1.9 THz for DS1, DS2, and DS3 feeds, respectively.

Fig. 5. (left) Real (solid line) and imaginary (dashed line) parts of the impedance of the log-periodic feed. Open and closed symbols show the axial polarization ratio and the azimuth angle of the main axis of the polarization ellipse, respectively. Stars with drop lines mark the frequencies calculated with (2). (right) Distribution of the high-frequency current in the log-periodic feed at 2.7 THz (top) and 2.5 THz (bottom).

at subterahertz frequencies [2], [19] are also present in the terahertz range. Fig. 5 shows the computed impedance, polarization ratio, and polarization angle (azimuth angle of the main axis of the polarization ellipse) of the 70-nm-thick gold log-periodic feed on silicon. The impedance oscillates and the polarization becomes linear at each local minimum of the real part of the impedance. The polarization angle oscillates between 160 –190 taking an edge position each time when the real part of the impedance reaches the minimum. This behavior is pretty similar to the oscillations of the polarization angle that was found [20] for log-periodic feeds at lower frequencies. The right panel in Fig. 5 shows the distribution of the high-frequency current in the feed at 2.7 THz (linear polarization) and 2.5 THz (elliptical polarization with the polarization ratio 0.1). The linear polarization occurs when the antenna acts as a pair of symmetrically disposed full-wave dipoles. Although the current distribution in Fig. 5 does not explicitly indicate the structural element acting as a dipole, it suggests the length of an equivalent dipole equal to the sum of the tooth length and the length of the metallic arc at the tooth’s position. Such dipoles would act as full-wave dipoles at frequencies

(2)

where is the speed of light in vacuum and the factor is the fitting parameter. In the frequency range from 1.5 to 5 THz, the best coincidence (Fig. 5) between occurrences of the linear polarization and frequencies from (2) was achieved for . Since all occurrences of the linear polarization are covered by (2), we do not anticipate full-wave dipoles having the double-tooth length [2]. An almost unnoticeable discrepancy at low frequencies may result from the change of the effective wavelength with the tooth’s width. The approach described here does not allocate the occurrence of linear polarization to a structural element of the feed, but rather provides an estimate of frequencies at which the linear polarization occurs. C. Double-Slot Feed The sizes and geometry of a double-slot feed are defined in Fig. 6. The black rectangle in the middle of the coplanar line depicts a bolometer connected to the feed. MoM simulations were made for feeds cut out in the center of a 70-nm-thick gold square on silicon with an overall size of 100 100 m . Parameters of the feeds used in this study are listed in Table I. The computed current distribution in the DS2 feed (Fig. 6, right panel) shows an excess current that appears around the joints between the coplanar waveguide and slots. Such distribution is inconsistent with the lumped-element semianalytical analysis, which relies on the sinusoidal current distribution along each slot with a minimum located in the center of the slot. The simulated impedance of the DS2 feed seen from the excitation slit is shown in Fig. 7. Although the coplanar waveguide noticeably modifies the impedance of the feed, the simulated frequency dependence has a certain similarity to the analytical frequency dependence of the impedance for two slots without a connecting coplanar waveguide [12], [21]. According to the

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Fig. 7. (left) Real (squares) and imaginary (triangles) components of the impedance computed for the DS2 feed. (right) Far-field radiation pattern of the DS6 feed at the dielectric side at 1.87 THz. The thick line shows results of the simulation, the thin line depicts the analytical pattern.

lumped-element model, the radiation coupling efficiency of the feed antenna reaches the maximum value when the imaginary component of the impedance becomes zero. For the DS2 feed, the corresponding resonance frequency appears approximately between 2.2 –2.5 THz. In the receiving mode, the frequency resulting in the best coupling will additionally depend on the impedance of the detector connected to the feed. The computed far-field beam profile of the DS6 feed almost coincides (Fig. 7) with the profile calculated analytically [7], [21] for the sinusoidal field distribution in the slots. The beam profile at 9 dB has a full width of approximately 80 and 120 in the - and -plane, respectively, indicating a Rayleigh distance less than 1 mm. Near the resonance frequency (1.8 THz) of the DS6 feed, the beam pattern in the -plane exhibits a dip that corresponds to the total internal reflection at the boundary between silicon and air. In the -plane, this effect is less pronounced and appears as a shoulder at the same elevation angle. These features are typical for all studied double-slot feeds. They do not appear in the analytically computed pattern since the analytical approach treats the feed suspended in a uniform media larger than one, but less with an effective permittivity than the relative permittivity of silicon. III. PHYSICAL-OPTICS (PO) RAY TRACING The ray tracing delivers the beam pattern of the entire integrated lens antenna. The field irradiated by the feed is used as an input for the ray-tracing procedure, which is based on the Kirchhoff integral presentation of the Huygens principle, and is known as PO ray tracing. Given the Rayleigh distance of less than 1 mm, the surface of a lens with a diameter of a few millimeters is located in the far-field of the feed. We, therefore, used the far-field result of the MoM simulations of the feed. The field of the feed at the lens surface inside the lens was decomposed into the - and -components (electric field perpendicular ( ) and parallel ( ) to the plane of incidence). Each component was then multiplied with an appropriate Fresnel transmission coefficient, and resulting components were combined again to obtain the electric -field and magnetic -field just outside the lens. The equivalent electric and magnetic sources outside the lens were defined [19] as and

(3)

Fig. 8. (left) Transformation of the antenna beam pattern caused by the change of the extension length for DS6 feed and 6-mm lens at 1.87 THz. Symbols show the best Gaussian fit to the computed pattern. (right) Transformation of the beam pattern caused by the shift of the feed from the geometric center of the lens with the extension length of 2.45 mm. Patterns from left to right correspond to 400 and 200 m and zero shift.

is the unit vector normal to the lens surface. The elecwhere tric field at the observation point at a distance from the running point on the surface, been the unit vector of this distance, was obtained by integrating over the lens surface

(4) with , , and been the free-space wave vector, the impedance of free space, and the angular frequency, respectively. The explicit description of this technique can be found in [15]. We computed the beam profile formed by an extended either 6 or 3 mm) with hemispherical silicon lens (radius different extension lengths . Results for the DS6 feed and a 6-mm lens at 1.87 THz are shown in Fig. 8. The width of the main lobe at 9 dB reaches the minimum value of 2.55 for mm, while the sidelobes drop to the lowest level at mm. An acceptable level ( 14 dB) of sidelobes and an almost Gaussian shape of the main lobe are both achieved for an intermediate extension length of 2.45 mm that brings of the feed to the most distant focal point a corresponding synthesized elliptical lens [7], [23]. For this optimal extension, the simulated beam profile of the integrated antenna is practically the same in both the - and -plane down to a 15-dB level. However, an accuracy of the feed positioning better than 100 m is required to keep the beam of the antenna rotationally symmetric. This is illustrated in the right panel of Fig. 8 that shows the influence of the shift of the feed on the beam pattern of the integrated antenna. A shift of the feed causes asymmetry of the beam and its deviation from the m, a computed deviation of optical axis. For the shift 4.5 was close to the value suggested by the geometric optics. A similar relation between the relative displacement of the feed and the deviation angle of the beam was computed earlier [22] for subterahertz frequencies in a broad range of extension length. For the log-spiral feed, we found the same optimal extension length as for the double-slot feed. The computed beam pattern of the antenna with the log-spiral feed remains asymmetric. The main azimuth angles of the antenna beam coincides with the

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main angles of the feed, while the beamwidth ratio decreases to 1.2 and does not practically depend on the frequency.

IV. EXPERIMENT We experimentally studied the performance of integrated lens antennas with NbN HEBs in the receiving mode. The log-spiral and double-slot devices were provided by the State Pedagogical University, Moscow, Russia; log-periodic devices were fabricated at the University of Karlsruhe, Karlsruhe, Germany. Shortly, the bolometers were made from B1 poly-crystalline NbN films with a nominal thickness of approximately 5 nm just above the superconand a sheet resistance of 600 ducting transition. They typically had a superconducting transition temperature of 9 K and a critical current density of cm at 4.2 K. Fabrication details are presented 5 10 elsewhere [23], [24]. HEBs were incorporated in different feed antennas. All feeds were printed on a 350- m-thick Si substrate from a 70-nm-thick gold film. The width of a typical bolometer amounted ten times its length. However, the typical was larger than the normal-state resistance sheet resistance and the accuracy of the manufacturing process indicated. An excess resistance appeared at the contacts between the bolometer and feed structure. The contact resistance to 0.5 . The substrate carrying the feed varied from 0.3 with the bolometer was glued with its backside onto the flat optically polished cut of an extended hemispherical silicon lens. The lens had a radius of either 6 or 3 mm. An extension of the lens together with the substrate thickness positioned the feed at the more distant elliptical focus. The lens and feed were mounted in a Dewar with optical access through a wedged TPX vacuum window and a cold (77 K) quartz filter. In order to prevent specular reflections from the filter, it was covered with a diamond powder. The HEB was operated as either heterodyne or direct detector. The lenses used for heterodyne measurements had an antireflection coating [25] optimized for the frequency of 2.5 or 1.87 THz. In the heterodyne regime, the IF signal was amplified and registered at 1.5 GHz. An optically pumped gas laser providing emission at frequencies 0.69, 1.63, 1.87, 2.53, 3.1, 4.3, and 5.2 THz was used as a local oscillator. The double-sideband noise temperature was measured by the -factor method. Hot and cold loads at 293 and 77 K alternatively covered the antenna beam. Thermal radiation and radiation of the local oscillator were superimposed by a 6- m-thick Mylar beam splitter. The beam pattern of the integrated antenna was measured with a high-pressure metal-halide lamp having a diameter of 5 mm. The lamp was moved in the far field of the integrated antenna, typically at a distance from 1.5 to 2 m from the feed. The corresponding angular resolution was better than 0.2 . The signal at the IF was recorded as a function of the lamp position. In addition, Fourier-transform spectra (FTS) with a resolution of 1/40 were acquired with the HEB operated in the middle of the superconducting transition as the direct detector. In this case, the Si lens had no antireflection coating. We used an interferometer with a 12- m-thick Mylar beam splitter and a mercury discharge lamp as radiation source.

Fig. 9. Far-field beam pattern of the integrated lens antennas (symbols) and the results of the MoM + PO simulation (lines). (left) Integrated antenna with the DS6 feed and 6-mm lens at 1.87 THz. E -plane: hollow symbols + dashed line; H -plane: solid symbols + solid line. (middle) SP1 feed and 6 mm at 2.5 THz for  = 45 (solid line + solid symbols) and  = 135 (dashed line + hollow symbols) (right) SP1 feed and 3-mm lens (without coating) at 4.3 THz; the azimuth angle  = 45 .

V. EXPERIMENTAL DATA VERSUS SIMULATIONS The measured beam pattern of the integrated lens antenna with the DS6 feed at 1.87 THz and with the SP1 feed at 2.5 and 4.3 THz are shown in Fig. 9 along with the computed beam profiles. At 1.87 THz, the width and shape of the main lobe, as well as the height of the first sidelobes, correspond fairy well with the results of the simulation. At 2.5 THz, the main lobe in the measured profile reasonably agrees with the simulations, whereas first sidelobes are noticeably higher that the level predicted by the simulations. At 4.3 THz, the simulated main lobe is almost twice as large as the measured main lobe and first sidelobes appear almost 10 dB higher than the simulation predicts. Due to the large distance between the lamp and antenna, the noise level was relatively high, not allowing us to clearly distinguish all sidelobes. However, the experimentally observable first sidelobes were always higher than the ray tracing suggested. The discrepancy increased with the frequency. The trend is well justified even with different feed types since all planar feeds have radiation patterns similar to that of the double-slot feed (Fig. 7, right panel). Possible reasons for the antenna beam to fracture could be internal reflections inside the lens and the deviation of the lens surface from the nominal shape. Internal reflections in the integrated lens antennas have been extensively studied during the last decade [26]–[28]. It has been shown that the reflections affect the beam pattern and the admittance of the antenna. The magnitude of the effect less than few decibels at the 15-dB level was found at frequencies below 1 THz. Even if one assumes that the strength of the effect increases with frequency, internal reflections cannot explain much stronger discrepancy that we found for our antennas. Additionally, we have found [29] that, up to 2.5 THz, the phase only slightly varies within the main lobe and does not show any ripples that would indicate a contribution of internal reflections. Studies of the influence of the surface roughness on the directive properties of optical elements date back to the 60th [29], [30]. It has been experimentally verified at frequencies around 10 GHz that statistically noncorrelated deviations from the nominal shape with raise the scattered background and an rms magnitude of the sidelobes to 14 dB. In order to keep sidelobes below this

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level for an integrated lens antenna working at 2.5 THz, a surface tolerance of better than 5 m is required. This value is already smaller than the rms roughness specified by the manufacturer of the lens. We, therefore, believe that the most plausible reason for enhanced sidelobes is nonidealities of the lens surface. Given such nonidealities, the relative rms accuracy will decrease with frequency that would explain a stronger beam distortion at higher frequencies. In order to compare the measured and simulated coupling efficiency of our integrated antennas, we assumed a frequency-independent conversion efficiency of the HEB and took into account the skin effect in the bolometer itself [23], [32]. The skin reeffect yields a complex impedance of the bolometer sulting in the following coupling efficiency of the antenna

Fig. 10. Simulated coupling efficiency versus frequency (solid lines), experimentally measured FTS spectra (dashed lines), and normalized reciprocal noise temperature of integrated antennas with (from left to right) DS1 (squares), DS6 (triangles), and DS2 (circles) feeds. The ovals bundle together data belonging to the same feed.

(5) is the impedance of the feed, is where the contact resistance and and are the skin depth and the normal-state resistivity of the bolometer, reis the absolute efficiency of the spectively. The quantity feed that is the full broadside (dielectric) irradiated power related to the power, which would be irradiated by a lossless feed connected to a load with a matched impedance. The reflection is defined by the impedance mismatch between efficiency the feed and the bolometer, whereas the conductive and dielecis provided by FEKO. The factor accounts tric efficiency for coupling losses, which have been estimated using the attenuation of the optical elements (beam splitter, filter, surface of the lens) [23]. Since the contact resistance for a particular specfor coupling imen is not well known, we used efficiency simulations. Given that the antenna beam was not truncated during the measurements, both the reciprocal system noise temperature and the FTS spectral weight are proportional to the simulated coupling efficiency with a fixed proportionality factor. For all studied feeds, we have found a reasonably good agreement between the measured FTS spectrum, system noise temperatures, and simulated coupling efficiency of the antenna. Typical data for studied feed types are presented in Figs. 10–12. For the integrated antennas with the double-slot feed (Fig. 10), the measured noise temperatures correlate better with the simulated efficiency than with the FTS spectra, which are systematically shifted to lower frequencies. We speculate that this shift may be due to the difference between the HEB impedance in the heterodyne and the direct detection mode. Another outcome of our study also noticed in [1] is that conventional design rules [8], [12] for the feed with the central , wavelength in free space , i.e., and , result in a central wavelength larger than . The difference becomes noticeable above 1 THz and gradually grows with the operation frequency.

Fig. 11. (left) Normalized coupling efficiency of the antenna with the SP1 feed concluded from the FTS (dashed line) and the noise temperature (symbols) measurements. The solid line shows the simulated coupling efficiency. (right) Noise temperature of the antenna with the SP2 feed and different bolometers having a width of 0.6 m (triangles), 1.4 m (circles), and 3.0 m (squares). The solid lines show the simulated noise temperature.

Fig. 12. Simulated coupling efficiency versus frequency (solid lines) and experimentally measured FTS spectra (dashed lines) for the integrated antenna with the log-periodic feed.

For antennas with log-spiral feeds, we found a good coincidence between simulated and measured coupling efficiency in a very broad frequency range (Fig. 11). The left panel compares the reciprocal system noise temperature measured in the heterodyne mode with the simulated efficiency of the antenna and the FTS spectrum. An improved efficiency of the antenna with the SP1 feed with respect to the FTS spectrum around 2.5 THz originates from the antireflection coating of the lens.

SEMENOV et al.: TERAHERTZ PERFORMANCE OF INTEGRATED LENS ANTENNAS WITH HEB

The right panel in Fig. 11 illustrates the influence of the bolometer size on the noise temperature of the integrated antenna with the HEB mixer. In the experiment, the normal was kept constant by prostate resistance of bolometers portionally scaling the width and the length. The geometry of the SP2 feed determines a useful frequency range from 0.3 to 2.5 THz. At 4 THz, we measured for the antenna with this feed a noise temperature that was an order of magnitude larger than at 2.5 THz. Larger internal terminals of the SP2 feed allow one to incorporate a larger bolometer that opens room for further improvement of the antenna performance. Although the physics of this effect is not yet understood, it has been reproducibly shown [23], [33] that the noise temperature of a HEB mixer itself decreases with the increase of the bolometer size. On the other hand, due to the skin effect, the imaginary part of the bolometer impedance [see (5)] grows slower with the frequency for a narrower bolometer. Correspondingly, the spectral response of the antenna with the bolometer flattens. However, the practical advantage of using a larger bolometer disappears already at 3 THz where the noise temperature of the antenna does not change any more with the bolometer size. Since conventional theories of the hot-electron effect [1] in superconductors do not suppose any frequency dependence of the intrinsic conversion efficiency of a HEB mixer, we used different frequency-independent intrinsic noise temperatures in order to simulate the coupling efficiency for bolometers with different sizes. The best fit shown in Fig. 11 was obtained for and an intrinsic noise temperature of 130, 340, and 800 K for the bolometers with the width 3.0, 1.4, and 0.6 m, correspondingly. The FTS spectrum and simulated coupling efficiency for the antenna with the log-periodic feed are shown in Fig. 12. Local maximums and minimums are clearly seen in the measured FTS spectrum. Their positions coincide with the positions of the maxima in the simulated coupling efficiency. However, the span of oscillations in the FTS spectrum and also the median spectral weight decrease with frequency faster than the simulations predict. Although less noticeable, a similar discrepancy was found for the log-spiral feeds (see Fig. 11). Thus, the effect is not specific for the feed. As we have shown earlier in this paper, the beam of the integrated antennas fractures at higher frequencies. This may contribute to the degradation of the efficiency since, at a higher frequency, a larger portion of the excitation energy will be directed to sidelobes. Although there is a discrepancy between measured and simulated coupling efficiency, our data do not show an anticipated in the range up to 5-THz [34] twofold degradation of the bolometer sensitivity due to quantum noise of the bolometer. Instead, overall good agreement between the simulated and measured coupling efficiency indicates that the decrease of the efficiency with frequency in the terahertz range is caused rather by the coupling ability of the antenna than by the bolometer itself. VI. CONCLUSION We have demonstrated that the full MoM applied to the entire feed structure and combined with the PO ray-tracing technique adequately models the performance of an integrated lens

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antenna with a HEB at frequencies up to 6 THz. The experimentally observed degradation of the coupling efficiency with frequency has been explained invoking the feed properties and the skin effect in the bolometer. Our experimental data do not indicate any noticeable contribution of the quantum noise to the system noise temperature. The experimentally observed deviation of the beam pattern from the model prediction increases with frequency. Although it has not been fully justified, we suggest that the effect is due to a nonideality of the lens surface. The model approach can be used as an engineering tool for the design of quasi-optical front-ends in state-of-the-art receivers. REFERENCES [1] A. D. Semenov, G. N. Gol’tsman, and R. Sobolevski, “Hot-electron effect in superconductors and its application for radiation sensors,” Superconduct. Sci. Technol., vol. 15, no. 4, pp. R1–R16, Apr. 2002. [2] M. M. Gitin, F. W. Wise, G. Arjavalingam, Y. Pastol, and R. C. Compton, “Broad-band characterization of millimeter-wave log-periodic antennas by photoconductive sampling,” IEEE Trans. Antennas Propag., vol. 42, no. 3, pp. 335–339, Mar. 1994. [3] E. Gerecht, C. F. Musante, Y. Zhuang, K. S. Yngvesson, G. N. Gol’tsman, B. M. Voronov, and E. M. Gershenzon, “NbN hot electron bolometric mixer—A new technology for low-noise THz receivers,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 12, pp. 2519–2527, Dec. 1999. [4] T. H. Büttgenbach, R. E. Miller, M. J. Wengler, D. M. Watson, and T. G. Phillips, “A broadband low-noise SIS receiver for submillimeter astronomy,” IEEE Trans. Microw. Theory Tech., vol. 36, no. 12, pp. 1720–1726, Dec. 1988. [5] A. D. Semenov, H.-W. Hübers, H. Richter, M. Birk, M. Krocka, U. Mair, K. Smirnov, G. N. Gol’tsman, and B. M. Voronov, “2.5 THz heterodyne receiver with NbN hot-electron-bolometer mixer,” Physica C, vol. 372–376, pt. 1, pp. 448–453, Aug. 2002. [6] A. Semenov, H. Richter, B. Günther, H.-W. Hübers, and J. Karamarkovic, “Integrated planar log-spiral antenna at terahertz waves,” in Proc. IEEE Int. Antenna Technol. Workshop, Singapore, Mar. 2005, pp. 197–200, IEEE Cat. 05EX980. [7] D. F. Filipovic, S. S. Gearhart, and G. M. Rebeiz, “Double-slot antennas on extended hemispherical and elliptical silicon dielectric lenses,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 10, pp. 1738–1749, Oct. 1993. [8] J. Zmuidzinas and H. G. Leduc, “Quasi-optical slot antenna SIS mixer,” IEEE Trans. Microw. Theory Tech., vol. 40, no. 9, pp. 1797–1804, Sep. 1992. [9] W. F. M. Ganzevles, L. K. Swart, J. R. Gao, P. A. J. de Korte, and T. M. Klapwijk, “Direct response of twin-slot antenna-coupled hot-electron bolometer mixer designed for 2.5 THz radiation detection,” Appl. Phys. Lett., vol. 76, no. 22, pp. 3304–3306, May 2000. [10] D. F. Filipovic, W. Z. Ali-Ahmad, and G. M. Rebeiz, “Millimeter-wave double-dipole antennas for high-gain integrated reflector illumination,” IEEE Trans. Microw. Theory Tech., vol. 40, no. 5, pp. 962–967, May 1992. [11] T. H. Büttgenbach, “An improved solution for integrated array optics in quasi-optical mm and submillimeter receivers: The hybrid antenna,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 10, pp. 1750–1761, Oct. 1993. [12] M. Kominami, D. M. Pozar, and D. H. Schaubert, “Dipole and slot elements and arrays on semi-infinite substrates,” IEEE Trans. Antennas Propag., vol. MTT-33, no. 6, pp. 600–607, Jun. 1985. [13] P. Focardi, A. Netto, and W. R. McGrath, “Coplanar-waveguide-based terahertz hot-electron-bolometer mixer—Improved embedding circuit description,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 10, pp. 2374–2383, Oct. 2002. [14] FEKO: A computer code for the analysis of electromagnetic problems. EM Software & Systems-S.A. (Pty) Ltd., Stellenbosch, South Africa, 2005. [15] Y. Mushiake, “Self-complementary antennas,” IEEE Antennas Propag. Mag., vol. 34, no. 6, pp. 23–29, Dec. 1992. [16] Y. P. Gousev, A. D. Semenov, E. V. Pechen, A. V. Varlashkin, R. S. Nebosis, and K. F. Renk, “Broadband coupling of terahertz radiation to an YBaCuO hot-electron bolometer mixer,” Superconduct. Sci. Technol., vol. 9, no. 9, pp. 779–787, Sep. 1996.

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[17] J. D. Dyson, “The equiangular spiral antenna,” IRE Trans. Antennas Propag., vol. AP-7, no. 4, pp. 181–187, Apr. 1959. [18] G. S. Smith, “Directive properties of antennas for transmission into a material half-space,” IEEE Trans. Antennas Propag., vol. AP-32, no. 3, pp. 232–246, Mar. 1984. [19] J. A. Balanis, Antenna Theory: Analysis and Design. New York: Wiley, 1982, pp. 448–450. [20] M. Nahum, “New design for antenna-coupled superconducting microbolometer,” Ph.D. dissertation, Phys. Dept., Univ. California at Berkeley, Berkeley, CA, 1992. [21] M. J. M. van der Vorst, “Integrated lens antennas for submillimeter wave applications,” Ph.D. dissertation, Dept. Elect. Eng., Eindhoven Univ. Technol., Eindhoven, The Netherlands, 1999, ISBN 90-386-1590-6. [22] D. F. Filipovic, G. P. Gauthier, S. Raman, and G. M. Rebeiz, “Off axis properties of silicon and quartz dielectric lens antennas,” IEEE Trans. Antennas Propag., vol. 45, no. 5, pp. 760–766, May 1997. [23] A. D. Semenov, H.-W. Hübers, J. Schubert, G. N. Gol’tsman, A. I. Elantiev, B. M. Voronov, and E. M. Gershenzon, “Design and performance of the lattice-cooled hot-electron terahertz mixer,” J. Appl. Phys., vol. 88, no. 11, pp. 6758–6767, Dec. 2000. [24] A. D. Semenov, K. Il’in, M. Siegel, A. Smirnov, S. Pavlov, H. Richter, and H.-W. Hübers, “Evidence of non-bolometric mixing in the bandwidth of a hot-electron bolometer,” Superconduct. Sci. Technol., vol. 19, pp. 1–6, 2006. [25] H.-W. Hübers, J. Schubert, A. Krabbe, M. Birk, G. Wagner, A. Semenov, G. Gol’tsman, B. Voronov, and E. Gershenzon, “Parylene antireflection coating of a quasi-optical hot-electron-bolometric mixer at terahertz frequencies,” Infrared Phys. Technol., vol. 42, pp. 41–47, 2001. [26] M. J. M. van der Vorst, P. J. I. de Maagt, and M. H. A. J. Herben, “Effect of internal reflections on the radiation properties and input admittance of integrated lens antennas,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 9, pp. 1696–1704, Sep. 1999. [27] M. J. M. van der Vorst, P. J. I. de Maagt, A. Netto, A. L. Reynolds, R. M. Heeres, W. Luinge, and M. H. A. J. Herben, “Effect of internal reflections on the radiation properties and input impedance of integrated lens antennas: comparison between theory and measurements,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 6, pp. 1118–1125, Sep. 2001. [28] A. P. Pavacic, D. L. del Rio, J. R. Mosig, and G. V. Eleftheriades, “Three-dimensional ray-tracing to model internal reflections in off axis lens antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 604–612, Feb. 2006. [29] H.-W. Hübers, A. D. Semenov, H. Richter, J. Schubert, S. Hadjiloucas, J. W. Bowen, G. N. Gol’tsman, B. M. Voronov, and E. M. Gershenzon, “Antenna pattern of the quasi-optical hot-electron bolometric mixer at THz frequencies,” in Proc. 12th Int. Space Terahertz Technol. Symp., Pasadena, CA, 2001, pp. 286–296. [30] J. Ruze, “Antenna tolerance theory—A review,” Proc. IEEE, vol. 54, no. 4, pp. 633–640, Apr. 1966. [31] R. J. Papa, “Conditions for the validity of physical optics in rough surface scattering,” IEEE Trans. Antennas Propag., vol. 36, no. 5, pp. 647–650, May 1988. [32] H. F. Merkel, P. Khosropanah, S. Cherednichenko, K. S. Yngvesson, A. Adam, and E. I. Kollberg, “Two-dimensional hot-spot mixer model for phonon-cooled hot-electron bolometers,” IEEE Trans. Appl. Superconduct., vol. 11, no. 1, pp. 179–182, Mar. 2001. [33] S. Cherednichenko, P. Khosropanah, E. Kollberg, M. Kroug, and H. Merkel, “Terahertz superconducting hot-electron bolometer mixers,” Physica C, vol. 372–376, pt. 1, pp. 407–415, Aug. 2002. [34] E. L. Kollberg and K. S. Yngvesson, “Quantum-noise theory for terahertz hot electron bolometer mixer,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 5, pp. 2077–2089, May 2006.

Alexei D. Semenov received the Ph.D. degree in solid-state physics from the Moscow Pedagogical University, Moscow, Russia, in 1984, and the Habilitation degree from the Moscow Institute of Radio and Electrical Engineering, Moscow, Russia, in 1994. From 1991 to 1996, he was a Researcher with the Institute of Applied Physics, University of Regensburg, Regensburg, Germany. From 1996 to 1997, he was a Researcher with the University of Las Vegas, Las Vegas, NV. In 1999, he became a Professor with

the Physical Department, Moscow Pedagogical University. In 2000, he joined the German Aerospace Center (DLR), Berlin, Germany. He has authored or coauthored over 100 papers in referred journals. His research interests include physics of nonequilibrium superconductors ant their application for radiation sensing. He is also interested in the development of terahertz receiving and imaging systems for atmospheric and security applications.

Heiko Richter was born in Schwäbisch Hall, Germany, on June 6, 1973. He received the Diploma degree in physics from the University of Karlsruhe, Karlsruhe, Germany, in 1999, and the Ph.D. degree in physics from the Technical University of Berlin, Berlin, Germany, in 2005. He is currently with the German Aerospace Center (DLR), Berlin, Germany, where he is involved in the field of terahertz and infrared sensors/optics.

Heinz-Wilhelm Hübers received the Diploma degree in physics and Ph.D. degree from the University of Bonn, Bonn, Germany, in 1991 and 1994, respectively. From 1991 to 1994, he was with the Max-Planck Institute for Radioastronomy, Bonn, Germany. He then joined the Institute of Planetary Research, German Aerospace Center (DLR), Berlin, Germany, where he is currently the Head of the Department of Terahertz and Infrared Sensors. His research interests are terahertz heterodyne receivers and their associated technology and physics. He is particularly interested in the development of detectors and lasers for the terahertz frequency range, which include superconducting hot electron bolometers and photon counters, as well as solid-state lasers such as quantum cascade and silicon lasers. He is currently involved in the development of heterodyne receivers for astronomy, atmospheric research, and security applications.

Burghardt Günther was born in Luckau, Germany, in 1962. He received the Diploma in electrical engineering from the Technical University of Dresden, Dresden, Germany, in 1988. In 1988, he joined the German Aerospace Center (DLR), Berlin, Germany, where he currently develops imaging systems for planetary science and security applications.

Andrey Smirnov was born in Moscow, Russia, in 1981. He received the Master of Science degree from the Moscow State Pedagogical University, Moscow, Russia, in 2005, and is currently working towards the Ph.D. degree at the German Aerospace Center (DLR), Berlin, Germany. From 2002 to 2005, he was with the Radio-physical Laboratory, State Pedagogical University, Moscow, Russia. His research interests include microwave electronics and receiver technology. Mr. Smirnov was the recipient of a 2005 scholarship presented by the Parliament of Berlin, Berlin, Germany.

SEMENOV et al.: TERAHERTZ PERFORMANCE OF INTEGRATED LENS ANTENNAS WITH HEB

Konstantin S. Il’in was born in Moscow, Russia, on March 24, 1968. He received the Ph.D. degree in solid-state physics from Moscow State Pedagogical University (MSPU), Moscow, Russia, in 1998. From 1997 to 1998, he was a Visiting Scientist with the Electrical and Computer Engineering Department, University of Massachusetts at Amherst, and with the Electrical Engineering Department, University of Rochester, Rochester, NY. From January 1998 to June 1999, he was an Assistant Professor with the Physics Department, MSPU. From 1999 to 2003, he was a Scientific Researcher with the Institute of Thin Films and Interfaces, Research Center Juelich, Juelich, Germany. In June 2003, he joined the Institute of Micro- and Nano-electronic Systems, University of Karlsruhe, Karlsruhe, Germany, where he currently develops technology of ultrathin films of conventional superconductors for receivers of electromagnetic radiation. His research interests include fabrication and study of normal state and superconducting properties of submicrometer- and nanometer-sized structures from ultrathin films of disordered superconductors.

Michael Siegel received the Diploma degree in physics and Ph.D. degree in solid-state physics from Moscow State University, Moscow, Russia, in 1978, and 1981, respectively. In 1981, he was with the University of Jena, where he was a Staff Member and later a Group Leader with the Superconductive Electronic Sensor Department. During this time, his research was focused on nonlinear superconductor–semiconductor devices for electronic applications. In 1987, he initiated research in thin-film high-temperature superconduc-

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tivity (HTS) for Josephson junction devices, mainly for the superconducting quantum interference device (SQUID), at the University of Jena. In 1991, he was with the Institute for Thin Film and Ion Technology, Research Center Juelich, where he was involved with the development and application of HTS Josephson junctions, SQUID, microwave arrays and mixers, and high-speed digital circuits based on rapid-single-flux-quantum logic. In 2002, he became a Full Professor with the University of Karlsruhe, Karlsruhe, Germany, where he is also Director of the Institute of Micro- and Nanoelectronic Systems. He studied transport phenomena in superconducting, quantum, and spin-dependent tunneling devices. He has authored or coauthored over 200 technical papers.

Jugoslav P. Karamarkovic (M’97) received the Diploma of Engineer and M.S. degree in engineering physics from the University of Belgrade, Belgrade, Serbia, in 1987 and 1991, respectively, and the Ph.D. degree in electronic engineering from the University of Niˇs, Niˇs, Serbia, in 1996, all in electrical engineering. From 1987 until 1991, he was a Research and Development Engineer with Ei-Semiconductors. He then joined the University of Niˇs, where he is currently a Professor of physics with the Faculty of Civil Engineering and Architecture. In 2001, he was a Guest Scientist with German Aerospace Center (DLR), Berlin, Germany. His research interests are analytical, numerical, and stochastic modeling of different physical processes.

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A New Mode-Multiplexing LINC Architecture to Boost the Efficiency of WiMAX Up-Link Transmitters Mohamed Helaoui, Student Member, IEEE, Slim Boumaiza, Member, IEEE, Fadhel M. Ghannouchi, Fellow, IEEE, Ammar B. Kouki, Senior Member, IEEE, and Adel Ghazel, Senior Member, IEEE

Abstract—This paper proposes a new amplifier architecture based on the outphasing technique intended for the efficiency enhancement of linear amplification with nonlinear components (LINC) transmitters. The proposed mode-multiplexing linear amplification with nonlinear components (MM-LINC) scheme operates according to the LINC concept for input signal magnitude drive levels below a certain threshold and as a balanced amplifier beyond this threshold. The setting of this threshold level influences the performance of the amplifier in terms of average power-added efficiency and linearity. A 2-W up-link transmitter prototype for worldwide interoperability for microwave access (WiMAX) applications was designed using this new architecture and optimized for a WiMAX signal with an 11.3-dB peak-to-average power ratio. The experimental results revealed a significant increase in power-added efficiency, from 6% for a LINC transmitter to 21% for the MM-LINC amplifier, while maintaining an error vector magnitude value under 8%, which is compliant with the standard requirement. Index Terms—Linear amplification with nonlinear components (LINC), outphasing, power amplifiers (PAs), worldwide interoperability for microwave access (WiMAX) transmitters.

I. INTRODUCTION ATA AND voice wireless communication networks are becoming more and more omnipresent in everyday life through various applications and services that take advantage of their inherent mobility. Many wireless standards were recently proposed for a wide range of frequency bands, in order to address the ever-increasing demand for spectrum efficiency. Therefore, various complex modulation schemes and multiple

D

Manuscript received May 1, 2006; revised September 30, 2006. This work was supported by the Informatics Circle of Research Excellence, by the National Sciences and Engineering Research Council of Canada, and by the Canada Research Chairs. M. Helaoui, S. Boumaiza, and F. M. Ghannouchi are with the Intelligent RF Radio Laboratory, Electrical and Computer Engineering Department, The University of Calgary, Calgary, AB, Canada T2N 1N4 (e-mail: [email protected]; [email protected]; [email protected]). A. B. Kouki is with the Communications and Microelectronics Laboratory, Department of Electrical Engineering, Ecole de Technologie Supérieure, Montreal, QC, Canada H3K 1K3 (e-mail: [email protected]). A. Ghazel is with the CIRTA’COM Research Laboratory, Physics, Electronics, and Propagation Department, École Supérieure des Communications de Tunis, Ariana 2088, Tunisia (e-mail: [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.2006.889318

access techniques, such as -ary quadrature amplitude modulation (M-QAM) ( up to 256) and orthogonal frequency division multiplex (OFDM), were adopted in these standards. Such modulations bring about highly varying envelope signals, which have a large peak-to-average power ratio (PAPR) [1], [2]. In order to meet the linearity requirements of the standards, the power amplifier (PA) of a transmitter’s RF front-end is forced to work at a very large backoff, where the power efficiency drops drastically. Power efficiency is of major concern when designing wireless transmitters for both base and mobile stations. In the particular case of the base station amplification stage, any improvement in the power efficiency contributes significantly to power-consumption reduction and, consequently, considerably reduces the energy bill. For mobile stations, mobility is a very important factor. Increasing the power efficiency for mobile station applications increases the battery life and, consequently, enhances their mobility parameters. Many techniques have been proposed to enhance the power efficiency of the wireless RF front-ends’ power stages. These techniques can be divided into two groups. The first group linearizes mildly nonlinear PAs (class AB) in order to reduce the power backoff and, consequently, improve the power efficiency. Among the various linearization techniques, digital predistortion [3]–[6] allows for a very interesting linearity versus efficiency tradeoff when applied to very high-power transmitters. However, its application to medium- and low-power transmitters is not advantageous because of the relatively high cost and the dc power consumption of the digital predistortion circuitry. The second group of power efficiency enhancement techniques uses advanced amplification schemes such as a Doherty amplifier, linear amplification with nonlinear components (LINC), and switching amplifiers. The Doherty amplifier [7]–[9] uses the load modulation technique to improve the power efficiency for large power backoffs from the saturation point. Although the Doherty technique shows very good efficiency performance, the amplified signal quality is deteriorated. Some researchers proposed to use a linearizer along with the Doherty amplifier to restitute the good quality of the signal, but this solution is not suitable for mobile stations. Furthermore, LINC using the outphasing technique is proposed in the literature [10]–[21] as an extraordinary means to achieve a linear amplification while using very nonlinear and power-efficient amplifiers. Its principle consists of dividing an amplitude and phase-modulated signal into two phase-modulated signals with a constant envelope. The combination of these

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lated combiner, this part of the energy is lost in the isolation resistance, which degrades the amplifier efficiency considerably [15]. In fact, to generate small power values, large signals are amplified by the PAs, which require high dc consumption; however, the most important part of the signal is dissipated as heat in the resistance. The system efficiency of a LINC system using an isolated combiner is equal to the product of the maximum amplifier efficiency and combiner efficiency. Given an ideal isolated combiner with a 3-dB loss, the instantaneous combiner efficiency is given by the following equation: Fig. 1. Principle of signal component decomposition of amplitude- and phasemodulated signal into two constant envelope signals in LINC transmitters.

signals after amplification results in a linearly amplified version of the input signal. However, it was shown that the combining process results in efficiency degradation when using an isolated combiner [13], [19], [20] and in linearity deterioration when using a lossless and nonisolated combiner [13], [15]. In this paper, a new architecture that is aimed at the enhancement of the LINC transmitter efficiency, when driven with highly varying envelope signals such as code division multiple access (CDMA) or orthogonal frequency-division multiplexing (OFDM), is proposed. This so-called mode-multiplexing linear amplification with nonlinear components (MM-LINC) architecture operates as a traditional LINC architecture when the magnitude of the signal is under a certain threshold, and it changes to a balanced architecture for higher magnitudes. In Section II, a review of the LINC theory is presented briefly. Section III details the proposed amplification architecture and studies its performance through simulations. In Section IV, continuous wave and modulated signal measurements are carried out to demonstrate the advantages of the proposed MM-LINC and compare its performance to that achieved using a traditional LINC system. II. TRADITIONAL LINC ARCHITECTURE The principle of the LINC amplification scheme consists of decomposing the highly varying envelope signal into two phasemodulated signals (Fig. 1). Given the input signal envelope, these two signals can be obtained according to the following two equations. (1) (2) is given by . and where represent the maximum and instantaneous magnitude values. and signals have constant envelopes, they can be As amplified using high-efficiency and nonlinear amplifiers. The sum of the two amplified signals results in a linearly amplified version of the original signal. should When combining the amplified signals, the signal be added destructively with its opposite in order to recover the amplified version of the input signal. In the case of an iso-

(3) Hence, the instantaneous combiner efficiency drops rapidly when the power of the original signal is reduced. Alternatively, the use of nonisolated combiners leads to a higher average combining efficiency. In fact, at low input signal levels, the impedance seen by each PA at the input of the nonisolated combiner is changed as a result of the load modulation effects. Consequently, the PAs deliver less power and consume less dc power; therefore, the total power efficiency is increased. While this combining technique demonstrates good efficiency performance, it introduces considerable distortions in the output signal due to the load modulation phenomenon [15], [16]. Some researchers proposed to use the phase predistortion technique to deal with the linearity degradation of such architecture [17]. However, this avenue has not been validated experimentally and seems risky since the addition of any predistortion processing to the signal would lead to a poor power-added efficiency, as in the case of the isolated combiner [16] III. MM-LINC ARCHITECTURE To alleviate the rapid efficiency deterioration, as a function of the input signal level, experienced in the LINC amplifier, a new approach is proposed herein. As shown in Fig. 2, the main idea consists of using both class B PAs in a balanced mode for high input signal levels and switching to the LINC mode at lower levels. Since the class B efficiency drops slower than the LINC efficiency, the use of the balanced mode postpones the fast efficiency drop of the LINC system to a higher backoff. A high-efficiency value is maintained up to and about the mean power region of the signal excitation. When the gain of the class B amplifier starts to drop for lower power values, the PAs switch to the LINC architecture mode. This allows the amplification system to maintain a high gain for the low power levels. Moreover, a linear gain is maintained during the whole power range corresponding to the LINC operating mode. This preserves good performance in terms of signal quality over the whole dynamic range of the signal. In this way, the MM-LINC architecture benefits from the good efficiency of class B amplifiers and the linearity of the LINC architecture, even when using highly nonlinear PAs. The equation of the signal decomposition is given as follows: (4)

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Fig. 3. Theoretical LINC and MM-LINC system efficiency variation as a function of input power.

Fig. 2. MM-LINC signal component separation.

where , is defined as previously. is the threshold for switching from one mode to another; . The value of it is a constant positive value lower than the threshold is chosen so that the efficiency versus linearity tradeoff is optimized. The setting of its value depends on the complex gain and power-efficiency characteristics of the PAs, the standard’s linearity requirements, and the signal characteristics, mainly PAPR. The proposed approach achieves the same instantaneous efficiency of a class B PA when the magnitude of the input signal is greater than the threshold. It admits a similar instantaneous efficiency to the LINC system for signal levels below the threshold value. The instantaneous power efficiency of an ideal class B PA can be estimated by the following equation: (5) Since the signal separation is performed for signal levels less than the threshold value , the signals and have a constant magnitude equal to when the system is operating in the LINC mode. Consequently, for this mode, the PA efficiency is equal to (6) and the combiner efficiency is given by (7) Using (5)–(7), the power efficiency of the proposed system can be written as

(8)

Fig. 3 plots and compares the efficiencies of MM-LINC and traditional LINC architectures versus the input power given by (9)

The efficiencies are calculated according to (5) and (8). In the traditional LINC system, the efficiency is equal to the combiner efficiency multiplied by the maximum efficiency of a class . The LINC system power efficiency drops rapidly B PA when the backoff increases. As an example, a 6-dB backoff leads to the LINC system’s efficiency dropping from 80% to 20%. This drastic fall in the instantaneous efficiency is the reason behind the limited average power efficiency of the LINC system when dealing with a complex modulated signal characterized with high PAPR (more than 6 dB). equal to 0.4 For illustrative purposes, a value for is chosen in the case of the MM-LINC architecture, and the corresponding efficiency trace is constructed according to (8). This curve is always above the efficiency curve of the traditional LINC. At 6-dB backoff, the efficiency of the MM-LINC system is equal to 40%, compared to 20% for the traditional LINC system. For a backoff equal to 8 dB, the MM-LINC power efficiency is almost 2.5 times the traditional LINC one (34% for the MM-LINC, compared to 13% for the traditional LINC). In the case of a modulated input signal, the efficiency gain in MM-LINC PAs depends mainly on the signal’s probability density function. A careful choice of the threshold parameter will ensure a high power-added efficiency where the input signal admits the higher probability. IV. VALIDATION To prove experimentally and validate the MM-LINC concept and to compare its performance to the traditional LINC architecture, two paired class B amplifiers were designed and built using transistors PTF 10107 from Ericsson Inc. Microelectronics, Morgan Hill, CA. Each amplifier provides 2 W of output power and 39 dB of linear gain at 1.96 GHz. As shown in Fig. 4, the signals and were generated using two synchronized arbitrary waveform generator sources. The two baseband signals were then up-converted to RF using the same local oscillator (LO). The two paths were adjusted in gain and phase in order to ensure a balanced structure. The amplified signals were then combined using a 3-dB isolated hybrid combiner. A. Continuous-Wave (CW) Measurements To validate the simulated results of Fig. 3, the same experiment as in Section III was carried out using two RF modules and two synchronized arbitrary waveform generators. A CW

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Fig. 4. Measurement setup of the MM-LINC architecture.

Fig. 6. Measured power response curves for LINC, class B, and MM-LINC architectures.

noting that the gain of the MM-LINC architecture is higher than the gain of the LINC architecture. This is caused by the gain compression of the class B near saturation. B. Measurement Using Modulated Signal

Fig. 5. Measured LINC, class B, and MM-LINC efficiency variation as function of input power backoff.

fed both paths. The output power of the system was varied by changing the input amplitudes of the CW in the balanced class B operating mode and by changing the phase between the carriers of both paths for the LINC operating mode. The power-added efficiency was measured at different input power levels. The system input power is defined as the RF power before signal and for decomposition in order to generate the signals a given outphasing angle. This system input power was measured by combining the RF signals and using a Wilkinson combiner. Fig. 5 shows the measured power-added efficiency of the LINC, class B, and MM-LINC architectures, as a function of the input power for a value of the normalized threshold . The LINC’s efficiency curve exhibited the same behavior as its corresponding simulated curve. The LINC efficiency dropped drastically from 52% at the maximum output to 15% at the 6-dB backoff. The MM-LINC’s efficiency curve followed the efficiency of a class B curve for the first decibels in the backoff region, where the efficiency decreased more slowly. At the 6-dB backoff, the efficiency dropped to 40%. Fig. 6 shows the system output power variation as a function of the system input power for LINC, class B, and MM-LINC transmitters. The MM-LINC power transfer curve indicated a gain compression for power levels close to the saturation point. Contrary to class B transfer function nonlinearity, this small nonlinearity did not greatly compromise the overall linearity since the occurrence of power levels in this region is not frequent when high envelope modulated signals are used. For low power levels, both architectures operated in the outphasing mode, which explains the good linearity. Finally, it is worth

To further demonstrate the power-efficiency improvement provided by the MM-LINC, a realistic modulated signal was considered. A typical signal for uplink WiMAX transmitters was generated using Advanced Design System computer-aided design (CAD) software from Agilent Technologies Inc., Palo Alto, CA. The generated signal was a 16-QAM OFDM modulated signal using 256 sub-carriers. The bandwidth of the signal was set to 1.75 MHz, and the code rate was fixed at 1/2, which fixed the data rate at 2.91 Mb/s. The resultant PAPR was equal to 11.3 dB. The modulated signal was separated using a digital component separator to generate the two constant envelope signals. Each signal was converted from digital to analog and then up-converted to feed the amplifiers. The amplified signals were combined using an isolated 3-dB hybrid combiner. The output signal was captured by a vector signal analyzer, and the time-domain waveforms were analyzed to study the performance of the MM-LINC. The error vector magnitude (EVM) was also computed to quantify the linearity performance of the new architecture. Fig. 7 shows the efficiency and EVM variation as a function of the normalized threshold. One can easily observe that the EVM decreased from 14% for a class B amplifier to 2% for a traditional LINC architecture, as the threshold level increased. The measured system power-added efficiency of 26% (class B ) increased up to 28% when was equal to 0.2 and and then decreased, reaching 6% with the traditional LINC system. The increase in the first part of the curve was due to the increase in gain and power of the system when increasing the value. In fact, the output power, which was equal to 32 dBm in the case of class B, increased up to 34 dBm for a value equal to 0.6 and, beyond it, starts to decrease to reach 31.5 dBm in the case of the LINC architecture. Fig. 8 shows the spectrum of the recombined signal at the combiner output of the MM-LINC architecture. The measurement was carried out for an output mean power of approximately 33 dBm. The out-of-band noise was canceled up to 45 dBc. This

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TABLE I PERFORMANCE COMPARISON OF CLASS B, LINC, AND MM-LINC ARCHITECTURES

V. CONCLUSION Fig. 7. Measured MM-LINC EVM and efficiency performances for different threshold values.

In this paper, a new architecture based on the outphasing technique has been proposed to enhance LINC efficiency performance, while maintaining acceptable linearity performance. The proposed architecture’s performances were validated using a 1.75-MHz WiMAX modulated signal. The 6% efficiency of the traditional LINC architecture was increased to 21% with the proposed method for a signal having an 11.3-dB PAPR. At these conditions, the linearity was kept within the standard constraints, the EVM was less than 8%, and the adjacent channel power ratio was better than 45 dBc. REFERENCES

Fig. 8. Measured MM-LINC output spectrum.

is an interesting result since no comparable results have been found in the open literature for measurements conducted with a LINC architecture using a highly varying envelope modulated signal. C. OPTIMIZING THE WiMAX UP-LINK TRANSMITTER The proposed MM-LINC architecture considers a tradeoff between linearity and efficiency by changing the value of the threshold from 0 to 1, which corresponds to class B and LINC mode of operations, respectively. The optimal value can be chosen according to the required linearity performance of the transmitter. Indeed, depending on the signal type, data rate, and transmission conditions, the transmitter can dynamically vary the threshold value to meet a certain tradeoff between signal quality (linearity) and power efficiency depending on the standard requirement and traffic conditions. In this experimental validation, the up-link WiMAX signal, with a PAPR equal to 11.3 dB and a bandwidth equal to 1.75 MHz, was considered. The value of was optimized so as to achieve the maximum power efficiency while meeting the standard’s linearity requirement. At equal to 0.4 and beyond, the EVM was less than 8%, hence, meeting the 8.4% EVM WiMAX requirement for the signal used. At this point, the power-added efficiency reached 21%, compared to the 6% efficiency obtained for the LINC architecture. Furthermore, the output power was equal to 33.5 dB, which is 2 dB higher than the output power of the traditional LINC architecture. Table I summarizes these measurement results.

[1] J. Tellado, “Peak to average power reduction for multicarrier modulation,” Ph.D. dissertation, Elect. Eng. Dept., Stanford Univ., Stanford, CA, 2000. [2] V. Tarokh and H. Jafrakhani, “On the computation and reduction of the peak-to-average power ratio in multicarrier communications,” IEEE Trans. Commun., vol. 48, no. 1, pp. 37–44, Jan. 2000. [3] J. K. Cavers, “Amplifier linearization using a digital predistorter with fast adaptation and low memory requirements,” IEEE Trans. Veh. Technol., vol. 39, no. 4, pp. 374–382, Nov. 1990. [4] S. P. Stapleton and F. C. Costescu, “An adaptive predistorter for power amplifier based on adjacent channel emissions,” IEEE Trans. Veh. Technol., vol. 41, no. 1, pp. 49–56, Feb. 1992. [5] M. Helaoui, S. Boumaiza, A. Ghazel, and F. M. Ghannouchi, “On the RF/DSP design-for-efficiency of OFDM transmitters,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 7, pp. 2355–2361, Jul. 2005. [6] ——, “Power and efficiency enhancement of 3G multicarrier amplifiers using digital signal processing with experimental validation,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 4, pp. 1396–1404, Apr. 2006. [7] W. H. Doherty, “A new high efficiency power amplifier for modulated waves,” Proc. IRE, vol. 24, no. 9, pp. 1163–1182, Sep. 1936. [8] F. H. Raab, “Efficiency of Doherty power-amplifier systems,” IEEE Trans. Broadcast., vol. BC-33, no. 3, pp. 77–83, Sep. 1987. [9] J. Sirois, S. Boumaiza, M. Helaoui, G. Brassard, and F. M. Ghannouchi, “A robust modeling and design approach for dynamically loaded and digitally linearized Doherty amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 9, pp. 2875–2883, Sep. 2005. [10] H. Chireix, “High power outphasing modulation,” Proc. IRE, vol. 23, no. 11, pp. 1370–1392, Nov. 1935. [11] D. C. Cox, “Linear amplification with nonlinear components,” IEEE Trans. Commun., vol. COM-23, no. 12, pp. 1942–1945, Dec. 1974. [12] S. C. Cripps, RF Power Amplifier Design. Boston, MA: Artech House, 2002, pp. 33–72. [13] F. Raab, “Efficiency of outphasing RF power-amplifier systems,” IEEE Trans. Commun., vol. COM-33, no. 10, pp. 1094–1099, Oct. 1985. [14] X. Zhang, L. E. Larson, and P. M. Asbeck, Design of Linear RF Outphasing Amplifiers. Boston, MA: Artech House, 2003. [15] A. Birafane and A. B. Kouki, “On the linearity and efficiency of outphasing microwave amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 7, pp. 1702–1708, Jul. 2004. [16] ——, “Phase-only predistortion for LINC amplifiers with Chireix-outphasing combiners,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 6, pp. 2240–2250, Jun. 2005. [17] F. J. Casadevall and A. Valdovinos, “Performance analysis of QAM modulations applied to the LINC transmitter,” IEEE Trans. Veh. Technol., vol. 42, no. 4, pp. 399–406, Nov. 1993.

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[18] M. Helaoui, S. Boumaiza, A. Ghazel, and F. M. Ghannouchi, “Digital compensation of branches imbalance effects in linc transmitters,” in 16th Int. Microelectron. Conf., Dec. 2004, pp. 688–691. [19] B. Stengel and W. R. Eisenstadt, “LINC power amplifier combiner method efficiency optimization,” IEEE Trans. Veh. Technol., vol. . 49, no. 1, pp. 229–234, Jan. 2000. [20] X. Zhang, L. E. Larson, P. M. Asbeck, and R. A. Langridge, “Analysis of power recycling techniques for RF and microwave outphasing power amplifiers,” IEEE Trans. Circuits Syst. II, Analog Digit, Signal Process., vol. 49, no. 5, pp. 312–320, May 2002. [21] G. Poitau and A. Kouki, “MILC: Modified implementation of the LINC concept,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2006, pp. 1883–1886.

Fadhel M. Ghannouchi (S’84–M’88–SM’93–F’07) received the Ph.D. degree in electrical engineering from the University of Montréal, Montréal, QC, Canada, in 1987. He is currently an iCORE Professor, a Canada Research Chair, and the Director of the iRadio Laboratory, Department of Electrical and Computer Engineering, The University of Calgary, Calgary, AB, Canada. He has held invited positions with several academic and research institutions in Europe, North America, and Japan. His has authored or coauthored over 300 publications. He holds seven patents. His research interests are in the areas of microwave instrumentation, modeling of microwave devices and communications systems, design and linearization of RF amplifiers, and SDR radio systems.

Mohamed Helaoui (S’06) received the B.Eng. and M.Sc.A. degrees in communications from the École Supérieure des Communications de Tunis, Ariana, Tunisia, in 2002 and 2003, and is currently working toward the Ph.D. degree at The University of Calgary, Calgary, AB, Canada. In 2002, he was a student member of the MEDIATRON Laboratory, École Supérieure des Communications de Tunis. From 2003 to 2004, he was with the Polygrames Research Center, École Polytechnique de Montreal. In 2005, he joined the iRadio Laboratory, The University of Calgary. His current research interests are digital signal processing, power-amplifier predistortion, power-efficiency enhancement for wireless transmitters, third-generation (3G)/fourth-generation (4G) transmitter optimization.

Ammar B. Kouki (S’88–M’92–SM’01) received the B.S. (with honors) and M.S. degrees in engineering science from Pennsylvania State University, Philadelphia, in 1985 and 1987, respectively, and the Ph.D. degree in electrical engineering from the University of Illinois at Urbana-Champaign, in 1991. From 1991 to 1998, he was with the Microwave Research Laboratory, École Polytechnique de Montréal, as a Post-Doctoral Fellow and Senior Microwave Engineer. In 1998, he joined École de Technologie Supérieure, Montréal, Montréal, QC, Canada, where he is currently a Full Electrical Engineering Professor and Director of the LACIME Laboratory. His research interests cover intelligent/efficient RF transceivers, linearization and efficiency enhancement techniques, computational electromagnetics, and multiple antenna systems.

Slim Boumaiza (S’00–M’04) received the B.Eng. degree in electrical engineering from the École Nationale d’Ingénieurs de Tunis, Tunis, Tunisia, in 1997, and the M.Sc. and Ph.D. degrees from the École Polytechnique de Montréal, Montréal, QC, Canada, in 1999 and 2004, respectively. In May 2005, he joined the Electrical and Computer Engineering Department, The University of Calgary, Calgary, AB, Canada, as an Assistant Professor and faculty member of the iRadio Laboratory. His research interests are in the areas of design of RF/microwave and millimeter-wave components and systems for wireless communications. His current interests include RF/digital signal processing (DSP) mixed design of intelligent transmitters, design, characterization, modeling and linearization of high-power amplifiers, reconfigurable and multiband RF transceivers, and adaptive DSP.

Adel Ghazel (SM’97) received the M.S. degree in systems analysis and digital processing and Ph.D. degree in electrical engineering from the École Nationale d’Ingénieurs de Tunis, Tunis, Tunisia, in 1990 and 1996, respectively. In 1993, he joined Ecole Superieure des Communications de Tunis (SUP’COM), Tunis, Tunisia, as a Telecommunications Professor and, in 1999, became the Head of the Physics, Electronics, and Propagation Department. He is currently the Dean of Planning and CIRTA’COM Research Laboratory Director. Since 2001, he has been the Program Manager of the Research and Development Center partner of the SST Division, Analog Devices, Boston, MA. He has authored or coauthored over 100 publications. His current research interests include very large scale integration (VLSI) and DSP circuits, algorithms, and architectures for communications systems.

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Effective Parameters for Metamorphic Materials and Metamaterials Through a Resonant Inverse Scattering Approach Nicolaos G. Alexopoulos, Fellow, IEEE, Chryssoula A. Kyriazidou, and Harry F. Contopanagos

Abstract—We present an inverse scattering approach for the bulk electromagnetic characterization of composite materials, based on a prior proof that artificial metallo-dielectric photonic crystals can be described by effective highly resonant response functions in a wide frequency range, including several passbands/bandgaps. The method becomes complete and unambiguous at high frequencies by employing the analytic continuation of the optical path length and a consistency criterion to ascertain the physical meaning of the extracted effective parameters. It may also be used as a fast simulator or as a measurement-based predictor of the performance of multilayered structures using the scattering matrix (simulated or measured) of a single monolayer of that material. The approach is applied for the characterization of metamorphic materials, which are recently introduced artificial structures that exhibit distinct macroscopic states of behavior as far as the reflected electromagnetic field is concerned. According to interconnect topologies of their scatterers, they appear, at a single frequency, as electric conductors, absorbers, amplifiers, and passive or active magnetic conductors. Detailed evaluations are given of the complex dispersive wave impedance, refractive index, and permittivity and permeability functions for each metamorphic state of a specific three-state metamorphic material. It is found that, as a rule, the electric and magnetic wall states are related to resonant permittivity and permeability values, respectively. Furthermore, the analysis reveals broad regions with negative values of permittivity or permeability. Both resonant and negative values of e , e occur within bandgaps or at band-edges. Finally, the approach is applied to the negative refractive index metamaterial composed of a cylinder and two split-ring resonators, which reveals the existence of a high-frequency band with negative group velocity. Index Terms—Effective permittivity, electromagnetic scattering inverse problems, permeability, reconfigurable architectures.

I. INTRODUCTION ETAMORPHIC materials [1], [2] are newly defined artificial structures composed of passive and active components that may be electronically reconfigured to make tran-

M

Manuscript received June 20, 2006; revised September 28, 2006. The work of N. G. Alexopoulos and C. A. Kyriazidou was supported by the Samueli Foundation and by the University of California at Irvine. N. G. Alexopoulos and C. A. Kyriazidou are with the Department of Electrical Engineering and Computer Science and the Integrated Nanosystems Research Facility, The Henry Samueli School of Engineering, University of California at Irvine, Irvine, CA 92697 USA (e-mail: [email protected]). H. F. Contopanagos is with the Institute for Microelectronics, National Center for Scientific Research “Demokritos,” 15310 Athens, Greece (e-mail: [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.2006.890074

sitions among a set of distinct electromagnetic states. Among those, we distinguish the states that target application-specific operations, reacting as an electric conductor, a magnetic conductor, a passband filter, or an absorber under the excitation of an incident electromagnetic field. These multifunctional structures have been introduced in the form of two superimposed lattices: a passive lattice of scattering elements and an active lattice of switches acting on and reconfiguring the former to produce a variety of basic unit cells or super-lattices in the crystal. The purpose of this study is to present the bulk material characterization of metamorphic structures in terms of (effective or ina pair of complex functions trinsic-wave impedance and refractive index) or in terms of (effective permittivity and permeability). a pair This is useful in order to reclassify the metamorphic structures in terms of transitions among fundamental values of their effective parameters, instead of just their backscattering response presented in [1] and [2]. Apart from simply being a restatement or an alternative description, the reduction of each metamorphic state to a pair of basic parameters is useful to reveal unusual dispersive properties not found in natural media such as permittivities, permeabilities, and refractive indices less than one or negative. In this sense, it promotes the physical intuition and application space and simplifies the design of metamorphic crystals. To this end, a general formalism based on inversion of -parameters, which has already been used in the past [3]–[10], is now being completed and perfected. Its function is twofold: first to determine the structures that accept a bulk description, and second, to extract the corresponding effective parameters. This method exceeds the specific metamorphic structures to treat general composite artificial crystals in a complete, unambiguous, and consistent manner. Traditional effective medium theories provide an effective bulk description, in terms of permittivity and permeability functions, for common liquids and solids, as well as for composite media in the quasi-static limit where constituents and lattice sizes are small compared to the propagating wavelength. Resonant structures with dispersive properties contain inclusions and unit cell sizes of the order of the wavelength and naturally fall out of the scope of such traditional approaches. In contrast to the general belief whereby the description of resonant composites cannot be reduced to a pair of effective parameters, a new approach has emerged some time ago, which expresses resonant physical properties in terms of dispersive permittivity and permeability functions [3]–[5]. This is essen-

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tially a resonant inverse scattering formalism that literally follows the experimental procedure used to derive and for natural nonresonant materials through the reflection and transmission of radiation incident on a finite planar slab of the material [7]. In order to successfully apply this approach as an inverse scattering method, an essential theorem was derived analytically in [3, Sections 5 and 6], which we repeat here for completeness. When an arbitrary number of identical rectangular arrays of scatterers (of thickness ) is stacked, the resulting scattering (analytically calculated in [3]) is identical to the scatmatrix of a homogeneous slab of equal thickness tering matrix even for resonant frequencies. This theorem proceeds by proving the equality (1) where is a set of geometrical parameters and electrical constants characterizing the artificial crystal host, scatterer, are effective permittivity and and unit cell, while permeability functions independent of that enter the standard for a homogeneous slab. functional form of Further, in [3]–[5], the scattering matrix was derived analytically for a finite-thickness crystal of disk metal implants immersed into a dielectric host, and these expressions were matched to those of an equivalent macroscopically homogeneous medium. Even though that procedure was proven in the form of the above theorem in those references, a simplified inversion and extraction of the effective parameters was possible because of the existence of an analytical solution. One could independently calculate the semi-infinite complex reflection coefficient by analytically taking the limit of the number of . This would allow decoupling and direct inverlayers . Further, the Bloch–Floquet propagation phase sion for was also analytically computed, which allowed a decoupled . inversion for Due to the complexity of the metallic constituents in metamorphic media, this paper bypasses the analytic derivations and applies the methodology derived in [3]–[5] for finite-thickness using numerical inputs directly. Even slab observables though the resonant inverse scattering extraction couples the -matrix elements, the methodology is conceptually very simple. Indeed, one simulation for any arbitrarily fixed suffices for the extraction since the theorem of (1) ensures the corresponding effective response functions to be independent of . This is true for systems containing bounded scatterers, similar to the ones in [3]–[5]. In more complicated systems with unbounded scatterers and mixtures, we postulate a general consistency criterion as a novel necessary and sufficient condition that discerns the structures with effective description, as well as the frequency regime for which this is valid. The consistency criterion specifies that the deduced effective parameters are physically meaningful only when they may be uniquely defined, irrespective of the thickness of the observable sample. Further, we impose the analytic continuation of the optical path length as specified by the inversion of a circular trigonometric function. The technique for phase continuation across Riemann sheets has already been applied for resonant structures within the analytical approach of [3], but is now

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given within the context of a full inversion formalism. This ensures that a continuous and single-valued refractive index may be deduced from the formalism for a slab of any thickness without restriction to electrically thin samples, as used by the authors in [6], who employed a similar scheme for characterizing negative refractive index metamaterials. Finally, we also discuss and resolve the issue of the phase sensitivities that appear in the treatment of electrically thick samples producing spurious narrow resonances for numerical and experimental -parameter inputs to the inversion algorithm. In Section II, we give a detailed account of the resonant inverse scattering approach and present the essential mathematical problems along with their resolution. We also introduce the consistency criterion, a general principle that specifies which structures accept an effective description. In Section III, we apply the resonant inverse scattering formalism and obtain the effective parameters of the three topological states of the metamorphic medium of [2], corresponding to three different interconnections of its metallic scatterers. We also validate the inverse scattering approach by comparing the effective parameters for the disk medium with the corresponding ones derived through the analytical method of [3] as they were later completed with higher order effects and validated in [11]. In Section IV, we assemble the separate topologies into a three-faceted entity, the three-state metamorphic material, and we examine the three-way transitions of the effective parameters as a function of frequency. As a particular illustration, in Section V, we show how this method is applied to a metamaterial that exhibits a region with a negative refractive index. Finally, Section VI contains concluding remarks. II. RESONANT INVERSE SCATTERING FORMALISM The resonant inverse scattering method allows us to distinguish the structures that have a bulk description as well as to obtain their unambiguous characterization regarding its electromagnetic parameters as follows. First, we assume that a structure is equivalent to a macroscopically homogeneous medium. Consequently it may be described in terms of effective response functions, i.e., either in (effecterms of the pair of the complex functions tive wave impedance and refractive index) or in terms of the pair (effective permittivity and permeability). In such systems, the scattering parameters , i.e., the reflection and transmission coefficients for a slab, assume the form of the corresponding formulas for a macroscopically homogeneous medium. and are the inputs or observables in our approach and may, in general, be obtained through analytical solutions, simulations, or measurements. All structures that we treat in this paper are electromagnetically symmetric, i.e., and, hence, the illumination side is immaterial. Secondly, we algebraically invert the system of equations for the complex quantities and in accordance with the experimental extraction of material parameters [7]. In this manner, we obtain a pair of effective parameters. In principle, one can stop here if one only used scatterers and interconnect topologies similar to the ones described in [3]–[5], as summarized above.

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The question may arise, however, whether the derived parameters provide an effective description of the medium for the specific metal scatter shapes/topologies used in simulations or measurements, which may be quite complicated. Certainly not every complex structure accepts an effective description and, moreover, those which do may not maintain it for the entire frequency regime. This leads us to formulate a third step, which should be viewed as a general criterion. Consistency criterion: A periodic or random structure does have a bulk description in a specific frequency regime only when it yields the same effective parameters for a slab of any thickness. A successful homogeneous description of a slab of any thickness is, in our view, the most fundamental and physically transparent phenomenological description of a composite material because it involves the most fundamental aspects of macroscopic scattering, i.e.: 1) transmission through the bulk; 2) diffraction by two terminating interfaces; and 3) a three-way power balance (reflection, transmission, loss). If the effective parameters are uniquely determined, independent of the slab thickness, then obviously they represent the correct effective parameters of the system. Of course, in principle, one would have to test infinite cuts of the structure, which, in practice, is impossible. What we do is to take two cuts of the various structures we treat. Given that the input -parameters for these two cuts are extremely different and still yield identical parameters, the chance that the bulk description does not hold for some other cuts is really minimal. Hence, our approach may be incompletely inductive (mathematically speaking), but completely inductive (probabilistically speaking). Equivalently, we may produce the effective parameters for one specific slab, use them to predict the scattering matrix for a slab of different thickness, and finally, compare these to measured or simulated results. This criterion is, therefore, a consistency test, functioning as a necessary and sufficient condition, which will obviously reject the structures that do not accept a bulk description. For periodic structures, the slab thickness should be an integer multiple of . the monolayer period , i.e., Our starting point is the slab reflection and transmission complex coefficients under normal incidence, which assume the form of the corresponding formulas for a macroscopically homogeneous medium

(2)

This is an algebraic system of two equations with two unknowns: the wave impedance and the refractive index , which is obtained by inverting the auxiliary unknown used in (2). The known inputs are the complex quantities and , obtained through analytical solutions, simulations, or measurements.

Our method consists of algebraically inverting this system to obtain the effective parameters. The exact solution is

(3)

(4) where

(5) From (3), we conclude that the reflection coefficient for the planar semi-infinite medium is given by (6) The inversion formulas of (3)–(5), although exact, will only be as accurate as the fundamental inputs, which will carry measurement or computational uncertainties. In general, they present the following three types of problems. 1) The sign ambiguity of (3) is specified according to the requirement that, for all passive structures, we must have or, equivalently, . 2) Phase sensitivities develop when the function obtains the indeterminate form 0/0. This happens when the material reaches Fabry–Perot resonances, i.e., for freand . As we will quencies when see below, the analytically computed -parameters exhibit accurately matched phases that reduce the indefinite form to its well-defined limiting value. However, in numerically and experimentally derived -parameters, even slight phase inconsistencies produce spurious narrow resonances that, once recognized, may be subtracted. 3) Multivaluedness of the inverse trigonometric function and analytic continuation of phase: In addition to the mathematical problems mentioned above, a very important problem arising when we obtain through (4) is that the inverse trigonometric function provides a phase in a restricted range within the first Riemann sheet only. However, it is important to obtain unrestricted ranges through this inversion as the phase directly determines the refractive index of the medium, which evolves continuously as a function of frequency. We, therefore, need to restore the analyticity of phase upon this inversion, thereby restoring the analyticity of the complex dispersive refractive index. This is important in order to obtain a physical refractive index respecting analyticity, the Kramers–Kronig relations, and causality

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Fig. 2. Normal plane-wave incidence on a three-layer disk medium. Analytically (thin black line) and numerically (thick black line) calculated reflected power. Analytically (thin grey line) and numerically (thick grey line) calculated transmitted power.

Fig. 1. (a) First metamorphic state: disk medium. (b) Second metamorphic state: shorted-disk medium.

of signals transmitted through the medium. We emphasize that when analyticity is restored this way, no restrictions apply to the thickness of the slab of material (or, equivalently, the frequency range of the effective parameters). This bypasses the restrictions of thin slabs mentioned in [6], and makes the method completely general. We will refer to this procedure as the analytic continuation or rectification of phase in the remainder of this paper. III. EFFECTIVE PARAMETERS FOR THE METAMORPHIC MEDIUM We apply the resonant inverse scattering method to the metamorphic medium of [1] and [2]. The passive lattice consists of arrays of metal disks immersed into a dielectric host and interconnected by very thin metal strips, which are supplied with electronic switches. The open (closed) position of all the switches leads to the first (second) state of the metamorphic medium, depicted in Fig. 1(a) and (b). In the following, we use , a three-layer metamorphic medium with mm, , and thickness mm. All numerical results in this paper were obtained by High Frequency Structure Simulator (HFSS) full-wave simulations. A. First State of the Metamorphic Crystal: The Disk Medium We extract the effective parameters through resonant inverse scattering for the disk medium [see Fig. 1(a)], which constitutes the first state of our metamorphic system. These have already been derived in terms of a forward analytical method [3]–[5].

Comparison with the ones derived here through inverse scattering illustrates the nature of the aforementioned problems and validates the approach. In addition, the disk medium serves as testing ground at still another level since we are able to compare the effective parameters deduced through resonant inverse scattering with either analytical or numerical -parameter inputs. The analytically and numerically computed reflected (transmitted) power, presented in Fig. 2, are in excellent agreement. The corresponding complex -parameters are given in Fig. 3(a) and (b). The analytical results contain higher order corrections as they have been computed in [11], where they have also been validated experimentally. An independent numerical validation has also been produced in [16]. Fig. 4 presents the wave impedance obtained by inverting the numerically and analytically computed -parameters. Notice that the inverse scattering formulas take the wildly varying functions of Fig. 3 and produce an almost constant value from dc to 17 GHz where the first bandgap starts. In this regime, the structure behaves as a homogeneous dielectric. In fact, constant values for and are also obtained if we apply the formalism in the case of a common homogeneous dielectric slab. When the inverse scattering formalism is being fed by analytical -parameters, no phase sensitivities appear for the whole regime. For the analytically computed complex -parameters of and are perfectly tuned Fig. 3, the phases of at the location of the Fabry–Perot resonances at approximately obtains 7, 13.3, 28.8, and 31.9 GHz. It follows that its extremum, while the other three curves go through zero at the exact same resonant frequency point. In contrast, for the numerically computed -parameters, these curves are detuned and their phases are slightly inaccurate. It follows that the effective parameters carry spurious narrowband resonances, which may be identified and subtracted. To illustrate this point, we have left those resonances in Fig. 4. The real part of the refractive index is defined as the ratio between the phase change for the wave traveling within a fi, and nite piece of the crystal of thickness ,

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1

Fig. 5. Nonrectified phase  (thick black line),  and rectified phase (thick grey line).

Fig. 3. Comparison of the analytically and the numerically calculated: (a) complex S and (b) complex S . Real/imaginary parts of analytical S -parameters are in thin black/grey line and real/imaginary parts of the numerical S -parameters in thick black/grey line.

Fig. 4. Real (black line) and imaginary (grey line) parts of the wave impedance from numerically calculated S -parameters of the disk medium. Spurious resonances due to phase detuning are left for illustrative purposes.

the phase change of the wave traveling the same distance within . The optical path within the crystal is free space derived by inverting the cosine function [see (4)], a process that involves certain fine points.

0 1

(thin black line),

First, there exists an overall sign ambiguity for , which we choose to be positive in the limit of vanishing scatterer size, corresponding to a traveling wave in the positive direction and yielding the expected positive refractive index of the host dielectric at dc. This is true even for metal scatterers that are unbounded and, hence, their size does not vanish at dc because, in is similar to that of a homogeneous metal, and that case, is still positive. Secondly and most importantly, we impose analyticity on function to unambiguously specify the branch of the the multivalued inverse circular function. Branch cuts are constructed to restrict the phase values within the first Riemann sheet, which then become discontinuous across the is a physical quantity that evolves cut line. However, continuously with frequency (or ). Accordingly, the phase from which originates needs to be rectified across the branch-cuts where sharp discontinuities appear. We demonstrate in detail the procedure of analytic continuation in Fig. 5 where we present the nonrectified phase for the function rises from the three-layer disk medium. The . zero value at dc and exhibits sharp discontinuities at The discontinuities appear clearly amplified in the function , also plotted in Fig. 5. The rectification algorithm is based on the principle of analyticity and the Kramers–Kronig relations across and produces a continuous analytical function the phase discontinuities, plotted via thick gray line in Fig. 5. Notice that for metamaterials with negative group velocity, we will observe smooth transitions of the nonrectified phase from increasing to decreasing values, without discontinuities, which will remain unaffected by the rectification algorithm. The refractive index obtained according to (4) is given in increases smoothly starting from Fig. 6. We observe that its dc value until it reaches the first bandgap at 17 GHz. Within the bandgaps, i.e., within the ranges (17 and 27 GHz) and (33 and 42 GHz), we observe a dispersion that is anomalous, in the sense that the index of refraction is decreasing. Within the passagain becomes band regime between the two bandgaps, is approxa smoothly increasing function. The swing of imately 45% and increases with larger disk size.

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Fig. 6. Refractive index for the disk medium. Analytically (thin black line) and numerically (thick black line) calculated real parts. Analytically (thin grey line) and numerically (thick grey line) calculated imaginary parts.

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Fig. 8. Real (black line) and imaginary (grey line) parts for the permeability derived from analytically calculated S -parameters for the disk medium.

of layers. Here, we will show that this is true using resonant inverse scattering. In other metamorphic states of this material, where no forward analytic derivation exists, this consistency test operates as proof of the validity of the effective description and will be supplied as well. The -parameters for a single-layer disk medium are significantly different when compared to the three-layer structure presented in Fig. 3. Yet, as we show in Fig. 9, using analyt(and, hence, also ical inputs, the effective functions ) are identical. The same results follow if we use numerical inputs. In other words, the inversion formalism applied to the scattering parameters off a slab consisting of a single layer of the disk medium yields the exact same parameters as for the three-layered medium shown before. Thus, the consistency criterion is validated. Fig. 7. Real (black line) and imaginary (grey line) parts for the permittivity derived from analytically calculated S -parameters for the disk medium.

B. Second State of the Metamorphic Crystal: The Shorted-Disk Medium

The effective permittivity and permeability functions are readily given as and are plotted in Figs. 7 and 8. We present the results obtained by inverting the analytical -parameters since the corresponding ones with numerical inputs are essentially identical. Notice that we recover the main characteristics observed in [3]. In particular, the electric and magnetic conductor behavior of the crystal at the band-edges of each bandgap is now expressed in terms of resonant values of the permittivity and permeability, respectively. Of particular interest are the frequency regimes and , which appear with negative values for within the bandgaps. These are absent for the smaller aspect ratios and disk filling fractions of the crystal used in [3]. Consistency criterion for the effective description of the disk medium. In [3], we had shown analytically, by solving the forwardscattering problem for the disk medium, that the effective response function derivation holds independently of the number

In the second state of the metamorphic material, depicted in Fig. 1(b), the disks are interconnected by thin metal strips. Fig. 10 presents the numerically calculated reflected and transmitted powers for the three-layer shorted-disk medium. We repeat the application of the resonant inverse scattering formalism to the numerically computed -parameters for the one- and three-layer structures. The outcome of (3)–(5) is processed as explained earlier. We smooth out the data by rejecting the points that reflect the narrowband phase sensitivities and we impose the analyticity of the phase. The medium’s complex wave impedance and refractive index, as extracted from the one- and three-layer inputs, are presented in Figs. 11 and 12, respectively. An equivalent description of the medium can be given in terms of effective permittivity and permeability functions, shown in Figs. 13 and 14. We observe that the effective parameters are identical, independent of the number of layers. Due to this property, the consistency criterion is satisfied and the effective description of the shorted-disk medium is established.

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Fig. 10. Reflected (black line) and transmitted (grey line) HFSS-calculated power (in decibels) for a three-layer shorted-disk medium.

Fig. 9. Uniqueness (consistency) of the: (a) wave impedance and (b) refractive index extracted through inverse scattering from one-layer (thin black/grey line) and three-layer (thick black/grey line) disk medium.

We notice that the second metamorphic state acts as a metal, from dc up to the “plasma” frequency (at around 17 GHz) of the effective apertures formed by the “negative printing” of the shorted disks. In particular, Fig. 13 provides a large negative permittivity, while Fig. 14 gives a constant permeability ( 1) in that frequency range. From 18 to 27 GHz, the metamorphic state is matched to air, consistently with being an excellent passband filter, as shown in Fig. 10. The refractive index extracted in Fig. 12 shows a phase-advanced dielectric in this frequency regime. The second bandgap of this state starts with a strong resonant permeability at the left band-edge at around 27 GHz and then the value of the permeability remains negative throughout the bandgap, which extends to 31 GHz. The negative values of can become larger (in absolute value) with larger disks or other scatterers of a larger cross-sectional area. According to the information extracted from the effective parameters (Figs. 11–14), at the frequency region above 31 GHz, the medium is again a low-loss dielectric. This seems to contradict Fig. 10 where the transmission through three layers shows a minimum at 31 GHz, indicating a position in the middle of a bandgap. This is due to the fact that the frequency regimes

Fig. 11. Wave impedance for the shorted-disk medium by inverting the S -parameters for two slabs. (i) Real (thick black line) and imaginary (thick grey line) parts for a three-layer medium. (ii) Real (thin black line) and imaginary (thin grey line) parts for a one-layer medium.

where propagation is prohibited shift according to the number of layers. The precise bandgap edges are an intrinsic property of the infinite medium and would appear in the transmission through a slab of many layers, which is not the case for the system of the three monolayers shown in Fig. 10. Yet, through the resonant inverse scattering extraction of the effective parameters, the band-edges can be precisely located in frequency because these functions are independent of the number of layers. This indicates that simulations or measurements of a thin sample of a photonic crystal, even a single monolayer, can provide the precise bandgap location. C. Hole Medium Before we embark on the third metamorphic state, we will treat the hole medium shown in Fig. 15. This is the Babinet complementary system of the disk medium. Similar to the shorteddisk medium, it contains unbounded scatterers and falls in the same category of the “phase-advanced dielectrics.” What makes

ALEXOPOULOS et al.: EFFECTIVE PARAMETERS FOR METAMORPHIC MATERIALS AND METAMATERIALS

Fig. 12. Refractive index for the shorted-disk medium by inversion of the S -parameters for two slabs. (i) Real (thick black line) and imaginary (thick grey line) parts for a three-layer medium. (ii) Real (thin black line) and imaginary (thin grey line) parts for a one-layer medium.

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Fig. 14. Permeability for the shorted-disk medium computed for two slabs of different thickness. (i) Real/imaginary (thick black/grey line) parts for a threelayer medium. (ii) Real/imaginary (thin black/grey line) parts for a one-layer medium.

Fig. 13. Permittivity for the shorted-disk medium computed for two slabs of different thickness. (i) Real/imaginary (thick black/grey line) parts for a threelayer medium. (ii) Real/imaginary (thin black/grey line) parts for a one-layer medium.

it particularly interesting is the fact that the effective response of this system is almost identical to the shorted-disk medium. Indeed the reflected and transmitted power response, shown in Fig. 15(b), has a striking similarity with the one in Fig. 10 up to 30 GHz. Through resonant inverse scattering, we extract the effective response functions, depicted in Figs. 16 and 17. We observe that they are very similar to the ones for the shorted-disk state, even though the geometrical details of the scatterers are very dissimilar. Apart from small details, we see that topology fixes the effective parameters rather than shape of the metal scatterers. D. Third State of the Metamorphic Crystal: The Disk-Shorted Disk-Disk Medium The third metamorphic state, complementing the two-way metamorphism of Fig. 1, has been constructed by shorting the

Fig. 15. (a) Hole medium. (b) Reflected (black line) and transmitted (grey line) HFSS-calculated power (in decibels) for a three-layer hole medium.

middle disk array [1], [2], as shown in Fig. 18(a). The filtering response of the three-layer disk–shorted disk–disk (D–S–D) structure is shown in Fig. 18(b). The material characterization of this third metamorphic state is highly nontrivial since we are now faced with a super-lattice,

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Fig. 16. (a) Real (black line) and imaginary (grey line) parts of the: (a) wave impedance and (b) refractive index for a three-layer hole medium.

i.e., a periodic lattice whose unit cell expands in more than one physical layers. The system D–S–D contains mixed elements of the disk and shorted-disk media. It is a finite piece cut from a broader crystal that needs to be identified since there are various options. First we examine whether it forms part of the system D–S–D–S–D, i.e., of the simply alternating crystal with nonsymmetric unit cell consisting of a disk array followed by a shorted-disk array. Thus, we consider the original D–S–D system as a piece of the alternating crystal, cut along a specific layer, that of the “D” interface. We do not find consistent, i.e., unique, parameters for slabs of different thickness, which indicates that the D–S–D structure cannot be considered as a piece of this alternating crystal. This is not surprising since the D–S–D–S–D system is a symmetric structure, but does not contain an integer number of periods. Next we examine whether the system D–S–D is the monolayer of a larger material formed by the succession of two or more such monolayers. We show that this is indeed the case and we call the resulting medium of repeated such monolayers the D–S–D medium. For this, we first extract the effective wave

Fig. 17. Real (black line) and imaginary (grey line) parts of the: (a) permittivity and (b) permeability for a three-layer hole medium.

impedance and refractive index through inverse scattering with input the -parameters of the D–S–D structure, i.e., for one monolayer of this super-lattice. We then repeat the procedure for the D–S–D–D–S–D structure formed out of two monolayers of the super-lattice. The inverse scattering outputs are depicted in Figs. 19 and 20. We observe an excellent match of the refractive index outputs and a very good agreement in the wave impedance ones. It is obvious that the medium respects the consistency criterion and, thus, accepts an effective medium characterization at least up to 40 GHz, a range that includes seven bandgaps. Figs. 21 and 22 present the complex permittivity and permeability functions of the D–S–D Medium. We notice that the third metamorphic state acts as a metal, from dc up to the “plasma” frequency (at around 6 GHz). In that frequency range, Fig. 21 exhibits a large negative permittivity, while Fig. 22 gives a constant permeability ( 1). The medium then presents a succession of nonharmonically placed narrow passbands, where the permittivity and permeability functions obtain real values. The series of narrow passbands within an otherwise reflecting medium is best recognized in the plot of the refractive index (Fig. 20). While the first

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Fig. 18. (a) Third state of the metamorphic material with a shorted middle disk array. (b) Reflected (black line) and transmitted (grey line) HFSS-calculated power for the three-layer D–S–D structure.

of them appears clearly at 6 GHz, the second narrow passband is not formed in the monolayer power response that we see in Fig. 18. Nonetheless, it is apparent in the effective parameters since they reflect the response of the infinite medium. It will also appear in the response of crystals with a larger number of unit monolayers. Similar to the previously analyzed systems, each bandgap is framed by a resonant value of the permittivity at one band-edge and a resonant value of the permeability at the other band-edge. Thus, the first bandgap ends at a magnetic wall characterized by a strong resonant permeability. The second bandgap starts right after the first narrow passband with a strong negative permeability at the left band-edge. It ends at approximately 12 GHz with a resonant permittivity, indicating the existence of an electric wall. Similarly, the third bandgap starts with a strong negative permittivity, ending at a resonant permeability at 22.5 GHz and, thus, it continues. It is interesting to notice the region of negative permeability within the second bandgap, as well as the region of negative permittivity within the third bandgap. IV. METAMORPHISM OF EFFECTIVE PARAMETERS The formal classification of metamorphic states has been given previously in [1] and [2] in terms of the reflection coeffi, or equivcient at the interface of a semi-infinite medium alently, in terms of the medium’s effective wave impedance

Fig. 19. (a) Real part of the wave impedance through inverse scattering for a three-layer D–S–D structure (black line) and a six-layer D–S–D–D–S–D structure (grey line). (b) Imaginary part of the wave impedance through inverse scattering for a three-layer D–S–D structure (black line) and a six-layer D-S–D–D–S–D structure (grey line).

. Within the set of fundamental metamorphic states, we have distinguished the electric wall state corresponding to , the magnetic wall state corresponding , i.e., to high-impedance values, and to , finally, the bandpass state, defined as corresponding to an effective wave impedance matched to free space. While these distinct states have a special maximal weight due to their applicability, they form part of a continuum of states, each with a separate electromagnetic functionality, . Highly dispercorresponding to other physical values of sive media, like photonic crystals, evolve through this set of states as a function of frequency. A metamorphic medium has by definition the capability of making transitions among these fundamental states at a single frequency point. By appropriately switching on and off the connectivity among metallic implants on the various layers, we have constructed three topological states: the disk medium (first state), the shorted-disk medium (second state), and the D–S–D medium (third state). Each topology possesses specific electromagnetic properties and evolves differently with frequency, passing through a set of metamorphic states. Thus, each

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Fig. 21. Real (black line) and imaginary (grey line) parts of the permittivity for the D–S–D medium.

Fig. 20. (a) Real part of the refractive index through inverse scattering for a three-layer D–S–D structure (black line) and a six-layer D–S–D–D–S–D structure (grey line). (b) Imaginary part of the refractive index through inverse scattering for a three-layer D–S–D structure (black line) and a six-layer D–S–D–D–S–D structure (grey line).

topology at different frequency positions becomes a magnetic conductor, an electric conductor, etc. By bringing the three topologies together in a three-faceted entity, we form a three-state metamorphic material, which, at each frequency, makes transitions among three distinct metamorphic states. In particular, at 31 GHz it transitions among the maximal metamorphic states, that of a magnetic conductor, an electric conductor, and a passband filter. We have thus far considered the dispersive properties for each topology separately. Presently, it is of interest to gather each effective parameter for all three topologies in a unique plot. Thus, we construct the dispersive graphs for the metamorphic medium as a whole. They will allow us to evaluate the magnitude of the parameter transitions at each frequency point. The wave impedance across frequencies for the three-state metamorphic medium is given in Fig. 23(a), combining results previously presented in Figs. 4, 11, and 19. Of particular interest is the region around 31 GHz where the wave impedance transitions between the values 1 (passband of disk state), 0 (electric conductor of the shorted-disk state), and 8 (high-impedance magnetic conductor of D–S–D state). The refractive index

Fig. 22. Real (black line) and imaginary (grey line) parts of the permeability for a three-layer D–S–D medium.

of the three-state metamorphic medium is given in Fig. 23(b). The most significant transition is appears at 18 GHz where the of state 1 becomes in state 2. At 31 GHz, we have a large, but less significant transition. The various metamorphic states may alternatively be ex, presented pressed in terms of values of the parameters of the various systems disin Fig. 24. The derived cussed above exhibits some common characteristics, which may be summarized as follows. and 1) An electric wall corresponds to resonant values of . near zero values of and 2) A magnetic wall corresponds to near zero values of resonant values of . 3) A passband state corresponds to real and equal finite values . of In the systems that we have examined, the frequency regimes with negative values of the permittivity or permeability are within the bandgap where transmission is prohibited. Similarly, appear at the bandgap edges the resonant values of where transmission is generally reduced.

ALEXOPOULOS et al.: EFFECTIVE PARAMETERS FOR METAMORPHIC MATERIALS AND METAMATERIALS

Fig. 23. Real part of the: (a) wave impedance and (b) refractive index for the three-state metamorphic medium. First metamorphic state (disk state) (thick black line), second state (shorted-disk state) (thin black line), third state (D–S–D state) (thick grey line).

V. ANALYSIS OF A NEGATIVE GROUP VELOCITY MATERIAL For completeness, here we apply our resonant inverse-scattering formalism to the negative refractive index medium, shown in Fig. 25(a). This is the original medium, introduced in [12]–[15], consisting of a split-ring resonator in parallel with a conducting cylinder, but we have immersed it into a standard . We show Rogers host dielectric with that our formalism works well for such metamaterials that present their own unique properties. It is for this metamaterial that Smith et al. [6] applied a similar characterization approach. However, in this paper, we are able to treat electrically thick material samples and, thus, extend the characterization to higher frequencies, as a result of the analyticity of our resonant inverse scattering formalism. The unit cell dimensions are in accordance with the scale in mm. the remainder of this paper: The system is excited by a plane wave with the -field parallel to the cylinder axis and the -field perpendicular to the plane of the split-ring resonator. We use as inputs to the resonant inverse

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Fig. 24. Real part of the: (a) permittivity and (b) permeability for the threestate metamorphic medium. First metamorphic state (disk state) (thick black line), second state (shorted-disk state) (thin black line), third state (D–S–D state) (thick grey line).

scattering method the -parameters for the three-layer structure, presented in Fig. 25(a). Fig. 25(b) presents the effective refractive index of the medium in a broad frequency range that includes the regime of negative refraction and extends the characterization to higher frequencies. The thickness of the sample at the highest frequency of the plot is 1.25 wavelengths. We observe the expected region of the negative refractive index, which is in agreement with the results of [12]–[15]. This is located within the bandgap exhibited by the cylinder medium that extends from dc up to approximately 22 GHz. The regime of negative refraction is followed by a second bandgap that extends up to 10.5 GHz. Of particular interest is a second band corresponding to negative group velocity, which appears at 19.4 GHz and is located within the third bandgap of the structure. VI. DISCUSSION OF THE RESULTS We have calculated the effective parameters of a previously introduced three-way metamorphic structure [1], [2], whence its characterization as a three-state metamorphic material. The

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Fig. 25. (a) Three-layer split-ring resonator metamaterial. (b) Real (black line) and imaginary (grey line) parts of the refractive index.

structure is metamorphic in the sense that it reconfigures its topology and transitions among the three states by means of a lattice of active switches interlaced within the lattice of passive metallic elements. The switches redefine the interconnections among the periodic metallic implants that are embedded within a dielectric host. To obtain the material characterization, we have advanced a prior method [3]–[10] of extracting the effective description of artificial crystals with unit-cell size comparable to the wavelength. The method is essentially an inverse scattering approach that directly processes the system observables, i.e., the experimentally, numerically, or analytically derived -parameters for a planar slab of the material. We have introduced two main novelties. We have postulated the consistency criterion stating that the effective description of structures is physically meaningful only when it is unique, i.e., independent of the slab thickness whose -parameters are inverted (provided that we cut the crystal with an integer number of unit cells). We have imposed the analyticity of the optical path length, which extends the applicability of the method, beyond electrically thin samples, to slabs of any thickness. In this manner. a complete inverse scattering framework is formed at high frequencies, capable of unifying the treatment of artificial dielectrics beyond the first passband/bandgap. The approach attributes an effective parameter description in terms

or to those systems that of the pair satisfy the aforementioned consistency criterion. We have applied the approach to the three states of our metamorphic material: the disk medium, the shorted-disk medium. and the alternating disk–shorted disk–disk medium. We have examined structures of unit cell size comparable to a half-wavelength, i.e., for broadband frequency ranges, including several bandgaps. Previously [3]–[5], we have shown the possibility of assigning effective parameters to composites of bounded scatterers. Here, we have been able to verify these results and show that this is possible for unbounded scatterers and super-lattices. The main purpose of this paper is to deduce the material parameters of the metamorphic system. In this sense, we have been able to identify the maximal metamorphic states with specific . In particular, the electric conresonant values of ductor state is characterized by resonant permittivity values and near zero permeability, while the magnetic conductor state exhibits mostly high permeability value and near zero permittivity. or Of particular interest are the regions of negative , which are located within the bandgaps. Similarly in, particularly teresting is the occurrence of resonant values of in the case of the permeability, which appear at the band-edges. For completeness, we have also treated the negative refractive index metamaterial of [12]–[15] and have found the same characterization in the negative refraction band with these references. In addition, we have found a similar band, at much higher frequencies, characterized by a positive refractive index, but negative group velocity. What adds even more practical value to the current resonant inverse scattering method is that once the effective response functions have been extracted using -parameter inputs of a thin crystal (e.g., a monolayer), we can use these functions in the analytic formulas of (3)–(5) to deduce the -parameters for a crystal of arbitrary thickness (as an integer multiple of monolayers). Hence, our method provides a significant shortcut in computational time or fabrication cost for realistic designs requiring many layers to achieve a certain scattering response. The method would be particularly useful in the terahertz, infrared, or optical regime, where many monolayers can be (and are) used for specific filtering functions. REFERENCES [1] C. A. Kyriazidou, H. F. Contopanagos, and N. G. Alexopoulos, “Metamorphic electromagnetic media,” in Proc. 9th Int. Electromagn. in Adv. Applicat. Conf., Turin, Italy, Sep. 12–16, 2005, pp. 965–968. [2] ——, “Metamorphic materials: bulk electromagnetic transitions in electronically reconfigurable composite media,” J. Opt. Soc. Amer. A, Opt. Image Sci., vol. 23, pp. 2961–2968, 2006. [3] H. F. Contopanagos, C. A. Kyriazidou, W. M. Merrill, and N. G. Alexopoulos, “Effective response functions for photonic bandgap materials,” J. Opt. Soc. Amer. A, Opt. Image Sci., vol. 16, pp. 1682–1699, 1999. [4] C. A. Kyriazidou, H. F. Contopanagos, W. M. Merrill, and N. G. Alexopoulos, “Effective permittivity and permeability functions of photonic crystals,” in Proc. IEEE AP-S Int. Symp., 1999, pp. 1912–1916. [5] ——, “Artificial versus natural crystals: Effective wave impedance for printed photonic bandgap materials,” IEEE Trans. Antennas Propag., vol. 48, no. 1, pp. 95–106, Jan. 2000. [6] D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous materials,” Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top., vol. 71, pp. 036617–036628, 2005.

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[7] W. B. Weir, “Automatic measurement of complex dielectric constant and permeability at microwave frequencies,” Proc. IEEE, vol. 62, no. 1, pp. 33–36, Jan. 1974. [8] D. R. Smith, S. Schultz, P. Marcos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B, Condens. Matter, vol. 65, pp. 195104–195109, 2002. [9] S. O’Brien and J. B. Pendry, “Magnetic activity at infrared frequencies in structured metallic photonic crystals,” J. Phys., Condens. Matter, vol. 14, pp. 6383–6394, 2002. [10] R. W. Ziolkowski, “Design, fabrication and testing of double negative metamaterials,” IEEE Trans. Antennas Propag., vol. 51, no. 7, pp. 1516–1529, Jul. 2003. [11] C. A. Kyriazidou, H. F. Contopanagos, and N. G. Alexopoulos, “Monolithic waveguide filters using printed photonic bandgap materials,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 2, pp. 297–307, Feb. 2001. [12] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2075–2084, Nov. 1999. [13] D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett., vol. 84, pp. 4184–4187, 2000. [14] D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett., vol. 85, pp. 2933–2936, 2000. [15] D. R. Smith, D. C. Vier, N. Kroll, and S. Schultz, “Direct calculation of permeability and permittivity for a left-handed metamaterial,” Appl. Phys. Lett., vol. 77, no. 14, 2000. [16] F. Hirtenfelder, T. Lopetegi, M. Sorolla, and L. Sassi, “Designing components containing photonic bandgap structures using time domain field solvers,” Microw. Eng., pp. 23–29, Mar. 2002. Nicolaos G. Alexopoulos (M’68–SM’82–F’85) was born in Athens, Greece, on March 30, 1942. He graduated from the Eighth Gymnasium of Athens, Athens, Greece, in 1959. He received the B.S.E.E., M.S.E.E., and Ph.D. degrees from The University of Michigan at Ann Arbor, in 1965, 1967, and 1968, respectively. He joined the School of Engineering and Applied Science, University of California at Los Angeles (UCLA), where he was a member of the faculty of the Electrical Engineering Department from 1969 to 1996. While with UCLA, he was Associate Dean for Faculty Affairs from 1986 to 1987 and Chair of the Electrical Engineering Department from 1987 to 1992. Since January 1997, he has been a Professor with the Department of Electrical Engineering and Computer Science and Dean of The Henry Samueli School of Engineering, UCLA. He has served over the years as a consultant to a variety of foreign corporations and the U.S. Government. In addition, he has been on the Editorial Board of various professional journals and, more recently, served as Editor-in-Chief of Electromagnetics. He has authored over 250 refereed journal and conference proceedings papers. His research interests are electromagnetic theory and its applications in 3-D modeling of integrated microwave and millimeter-wave circuits. His recent research activities have focused on the theory and design of printed circuit antennas, microstrip arrays, scattering (radar cross section) from antennas in magneto-dielectric thin films, nonreciprocal materials, the interaction of electromagnetic waves with metamorphic media, and computational electromagnetics.

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Dr. Alexopoulos was co-recipient of the IEEE S. E. Schelkunoff Prize Best Paper Award in 1985 and in 1998. In 2000, he was named Engineer of the Year by the Orange County Section of the IEEE. In 2005, the National Technical University of Athens presented him with an honorary doctorate for his numerous contributions to the global engineering community.

Chryssoula A. Kyriazidou received the B.S. degree in physics from the University of Athens, Athens, Greece, in 1984, the M.S. and Ph.D. degrees in physics from The University of Michigan at Ann Arbor, in 1989 and 1992, respectively, and the Ph.D. degree in electrical engineering from the University of California at Los Angeles (UCLA), in 1999. She was a Research Associate with the High Energy Physics Group, Brookhaven National Laboratory, and with the Argonne National Laboratory. In 1999, she became a Research Engineer with the Electrical Engineering Department, UCLA. In 2000, she joined the Kimalink Corporation, and subsequently then Broadcom Corporation. In 2003, she returned to Greece and has been a private consultant in RF integrated circuit (RFIC) and microwave and electromagnetics engineering. Since 2004, she has been with the University of California at Irvine (UCI). She has authored 24 publications in the area of electrical engineering and physics. She holds or coholds 11 U.S. patents with five U.S. patents pending. Her interests include modeling and design of microwave integrated circuits, wireless communication antennas and systems, composite electromagnetic materials, and accelerated full-wave codes for electromagnetic simulations. Her research has also covered the application of full-wave electromagnetic methods to the design and measurement of onand off-chip passive and active components and the study of on-chip parasitics and electromagnetic interference in submicrometer CMOS technology for radio transceivers in single-chip wireless applications.

Harry F. Contopanagos received the B.Sc. degree in physics from the University of Athens, Athens, Greece, in 1984, and the M.S. and Ph.D. degrees in physics (with top honors, specializing in theoretical high-energy physics) from The University of Michigan at Ann Arbor, in 1989 and 1991, respectively. Until 1996, he was a Researcher, initially with the Institute for Theoretical Physics, State University of New York at Stony Brook, and then with the Argonne National Laboratory. He has been a Senior Researcher with The Henry Samueli School of Engineering and Applied Science, University of California (UCLA), Senior Scientist with Hughes Research Laboratories, Malibu, CA, Principal Scientist with the Broadcom Corporation, Irvine, CA, and Director of Advanced Research and Development with the Ethertronics Corporation, San Diego, CA. Since September 2004, he has been a Senior Scientist with the Institute for Microelectronics (IMEL), National Center for Scientific Research “Demokritos,” Athens, Greece. He has authored or coauthored numerous publications in these areas. He holds or coholds 20 U.S. and international patents. Since 1996, he has been involved with electrical engineering with his research focused on areas spanning electromagnetics and microwave engineering, artificial materials and photonic crystals, wireless front ends, antennas, and high-frequency analog integrated circuits.

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Application of Total Least Squares to the Derivation of Closed-Form Green’s Functions for Planar Layered Media Rafael R. Boix, Member, IEEE, Francisco Mesa, Member, IEEE, and Francisco Medina, Senior Member, IEEE

Abstract—A new technique is presented for the numerical derivation of closed-form expressions of spatial-domain Green’s functions for multilayered media. In the new technique, the spectral-domain Green’s functions are approximated by an asymptotic term plus a ratio of two polynomials, the coefficients of these two polynomials being determined via the method of total least squares. The approximation makes it possible to obtain closed-form expressions of the spatial-domain Green’s functions consisting of a term containing the near-field singularities plus a finite sum of Hankel functions. A judicious choice of the coefficients of the spectral-domain polynomials prevents the Hankel functions from introducing nonphysical singularities as the horizontal separation between source and field points goes to zero. The new numerical technique requires very few computational resources, and it has the merit of providing single closed-form approximations for the Green’s functions that are accurate both in the near and far fields. A very good agreement has been found when comparing the results obtained with the new technique with those obtained via a numerically intensive computation of Sommerfeld integrals. Index Terms—Green’s functions, layered media, Sommerfeld integrals.

I. INTRODUCTION

T

HE application of the method of moments to the solution of mixed-potential integral equations has proven to be a powerful numerical tool in the analysis of planar circuits and antennas [1], [2], as well as in the study of the scattering properties of objects that are partially/completely buried in earth [3], [4]. In fact, current commercial software products that are extensively used in the design of planar circuits and antennas (such as Ansoft’s Ensemble, Zeland’s IE3D, and Agilent’s Momentum) are based on a mixed-potential integral-equation approach. One crucial step in the application of the method of moments to mixed-potential integral equations is the numerical computation of spatial-domain Green’s functions for the scalar and vector potentials in multilayered media [5], [6]. These Green’s functions can be expressed as infinite integrals of spectral-domain Manuscript received May 15, 2006; revised September 27, 2006. This work was supported by the Spanish Ministerio de Educación y Ciencia and European Union FEDER funds under Project TEC2004-03214 and by the Junta de Andalucía under Project TIC-253. R. R. Boix and F. Medina are with the Microwaves Group, Department of Electronics and Electromagnetism, College of Physics, University of Seville, 41012 Seville, Spain (e-mail: [email protected]). F. Mesa is with the Microwaves Group, Department of Applied Physics 1, Escuela Técnica Superior de Ingeniería Informática, University of Seville, 41012 Seville, Spain (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.889336

Green’s functions that are commonly known as Sommerfeld integrals. Owing to the slowly decaying and highly oscillating behavior of the functions to be integrated, the brute-force numerical computation of Sommerfeld integrals is a time-consuming process. Therefore, in order to save CPU time in the solution of mixed-potential integral equations, an extensive research has been carried out over the last two decades to accelerate the computation of Sommerfeld integrals. Some of the methods proposed for the fast computation of Sommerfeld integrals are based on tailor-made numerical integration techniques. These techniques include the weighted-average algorithm [5], [7] and related extrapolation algorithms [8], the integration along the imaginary axis of the spectral complex plane [7], [9], the integration along the steepest descent path [10], and the integration with a window function as a convolution kernel [11]. Although these techniques are all powerful numerical tools for the computation of Sommerfeld integrals, they have to be repeatedly used in the application of the method of moments as the distance between source and field points changes, which limits their numerical efficiency. One alternative, which partially alleviates this problem, is to combine numerical integration with the use of the fast Hankel transform [12]. A different approach to the computation of Sommerfeld integrals is based on the application of asymptotic methods such as the steepest descent method and the stationary phase method. These methods lead to asymptotic closed-form expressions of Green’s functions that are valid in a wide range of distances between source and field points (typically above one wavelength) [10], [13], [14]. However, these expressions have been obtained only for simple layered media containing two or three different materials, and extension of the methods to layered media with an arbitrary number of materials seems to be a difficult task. One method that has reached a prominent position in the computation of Green’s functions for multilayered media is the discrete complex image method [15]–[31]. In this method, the spectral-domain Green’s functions are approximated in terms of certain functions for which Sommerfeld integrals can be obtained in closed form. The method avoids numerical integration and provides results that are valid in a wide range of distances between source and field points. However, there is no common agreement in the functions that should be used in the approximation of the spectral-domain Green’s functions. For some researchers, the spectral-domain Green’s functions should be exclusively approximated in terms of complex exponentials that are fitted by means of the generalized pencil of functions or matrix pencil method [18], [19], [25], [26], [28], [30], [31]. One

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BOIX et al.: APPLICATION OF TOTAL LEAST SQUARES TO DERIVATION OF CLOSED-FORM GREEN’S FUNCTIONS FOR PLANAR LAYERED MEDIA

drawback of this approach is that unpredictable large errors arise in the far-field computation of the spatial-domain Green’s functions [27], [30]–[32]. Recently, Yuan et al. [31] have found that the value of (horizontal separation between source and field points) marking the onset of far-field numerical errors can be made much larger by increasing the number of complex exponentials used in the approximations, but it may lead to a considerable CPU time consumption (see [31, Table I]). In accordance with the results shown in [27] and [29], the discrete complex image method improves its stability when the complex exponentials used in the approximations of the spectral Green’s functions are accompanied by both a quasi-static term contributing the near field and a surface-waves term accounting for the far field. This is the approach originally proposed in [16] and followed in [17], [20]–[24], [27], and [29]. The drawback of this approach is that the surface-waves term contains poles of the spectral-domain Green’s functions, as well as residues of the Green’s functions at these poles, and the accurate determination of both the poles and the residues requires time-consuming algorithms [24], [29]. It has also been stated that the Hankel functions of the surface-waves term introduce near-field singularities in the spatial-domain Green’s functions that do not exist when source and field points are placed in different horizontal ) [20], [23], planes of a multilayered media (i.e., when [24]. Fortunately, this latter problem can be solved by using the error analysis technique proposed in [29]. A few years ago, Okhmatovski and Cangellaris introduced a new numerical method for the derivation of closed-form expressions for multilayered media Green’s functions [32]. In their proposal, the spectral-domain Green’s functions are represented in pole-residue form, and the poles and residues are numerically computed by means of a finite-difference approximation of the boundary value equations in the spectral domain. As a result of this, the spatial-domain Green’s functions are expressed as finite series of Hankel functions representing cylindrical waves. The main drawback of this method is that the computational cost becomes very high in the near-field region since the number of terms required for the accurate approximation of the Green’s . More recently, Canfunctions grows very quickly as gellaris and his collaborators proposed an alternative formulation of the method where the poles and residues of the spectral-domain Green’s functions are computed by means of an iterative algorithm called the vector-fitting algorithm [33], [34]. Although this new approach makes it possible to reduce the number of Hankel functions used in the approximation of the spatial-domain Green’s functions, the vector-fitting algorithm is computationally demanding and does not suffice to eliminate the near-field numerical problems of the original method, as recognized in [34]. In this paper, a new numerical technique is presented for the derivation of closed-form Green’s functions both in the spectral and spatial domains. The technique borrows several ideas from [32]–[34], but it does not share the drawbacks of the approaches followed in these papers. In the new technique, every spectral-domain Green’s function is approximated in terms of an asymptotic term plus a fraction of two polynomials. The asymptotic terms are chosen in such a way that their spatial-domain counterparts account for the singular or quasi-singular be-

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[34], havior of the spatial-domain Green’s functions as [35]. Concerning the coefficients of the polynomials of the approximation, they are computed via the method of total least squares following an approach similar to that reported in [36]. Once an explicit expression for the fraction of polynomials is available, this expression is converted into a partial fraction expansion (pole-residue form), which makes it possible to write its spatial counterpart as a short series of Hankel functions. These , Hankel functions introduce nonexisting singularities as but these singularities are eliminated by suppressing some of the polynomials coefficients. As a result, our technique leads to single closed-form expressions of the spatial Green’s functions that are accurate both in the near and far fields (i.e., in the whole range of values of ), as it happens with the expressions of [29]. However, the CPU time required by the new technique is much smaller than that required in [29] for the following two main reasons. • Although a singular-value decomposition is carried out by both the method of total least squares in the new technique and the matrix pencil method in the discrete complex image technique, the size of the matrix required by total least squares in the new technique is much smaller than that handled by the matrix pencil method in the discrete complex image technique, which considerably reduces the number of operations. • The new technique only requires to compute the poles and residues of a fraction of polynomials, which is much simpler than computing the poles and residues of the spectral-domain Green’s functions of an arbitrary multilayered media [29]. This paper is organized as follows. The derivation of closed-form expressions of multilayered media Green’s functions is presented in Section II, which also describes in detail the strategies followed for eliminating nonexisting singularities both in the spectral- and spatial-domain approximations. Section III provides numerical results for the spatial-domain Green’s functions of scalar and vector potentials both in the near and far fields. These results are compared with results obtained via numerical computation of Sommerfeld integrals, and good agreement is found in all cases. Conclusions are summarized in Section IV. II. THEORY Let be the coordinates of an arbitrary field point in be the coordinates a multilayered substrate, and let of an infinitesimal electric dipole source embedded in the multilayered substrate (since the treatment of magnetic sources is analogous to that of electric sources by virtue of the duality principle [6], only electric sources will be considered in this study). With respect to a reference frame centered at the source point, the radial and angular cylindrical coordinates of the field point will be (see [6, eq. (36)]) (1) (2)

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Among the different mixed-potential integral-equation formulations [6], it will be the formulation C of Michalski and Zheng [37] what will be used in this study. For the aforemenbe tioned electric source in a multilayered substrate, let a generic function representing either the corrected scalar poof [37] or any of the diagonal eletential Green’s function ments of the corrected dyadic vector-potential Green’s function of [37] ( is defined in terms of the traditional vector-poand the so-called correction factor tential Green’s function ). In the frame of formulation C, the functions can be written as Sommerfeld integrals of the type [37]

(3) where

is the spectral-domain counterpart of , is the Bessel function of first kind and order zero, and is an integration path in the first quadrant of the complex -plane that detours around the poles and branch points of (see [8, Fig. 1]). Let be a generic function representing any of the . In formulation C, the functions off-diagonal elements of can be all expressed in terms of Sommerfeld integrals of the type [37]

(4) is the spectral-domain counterpart of , are multiplying factors ( appears in and , and appears in and ), and is the Bessel function of first kind and order 1. In [32], Okhmatovski and Cangellaris used a finite-difference and scheme to prove that the spectral-domain functions can be approximately written as pole-residue finite series of the form where

or

(5) When (5) is introduced in (3) and (4), closed-form expresand are obtained in terms of finite sums sions of of Hankel functions [32]. The problem with these finite sums for obtaining accuis that they require a very large value of and as [32]. In an attempt rate values of to overcome this problem, Kourkoulos and Cangellaris have recently suggested that the Hankel functions series should be combined with a closed-form quasi-static term accounting for the near-field contributions to the Green’s functions [34]. Unfortunately, the introduction of this quasi-static term does not always suffice to solve the near-field convergence problems reported in [32]. The origin of these numerical problems is that the Hankel

functions used in [32] and [34] introduce nonphysical singular, which is a well-known phenomenon [20], [23], ities as [24], [29], [34]. If a quasi-static term is added to the Hankel functions, as in [34], the near-field problems are only eliminated in the particular cases where the Green’s functions are singular (e.g., shows this behavior when the source and as ) field points are in the same horizontal plane, i.e., when since, in those cases, the quasi-static term is also singular, and this singularity dominates over the Hankel functions singularities [29]. However, when the Green’s functions are not singular (e.g., when the source and field points are in different as horizontal planes, i.e., when ), the quasi-static term is not singular either and, therefore, it cannot mask the nonphysical singularities introduced by the Hankel functions. In the following, a new approach is proposed for calculating closed-form spatial-domain Green’s functions. Our proposal is similar to that of [34], but does not suffer from inaccuracies related to Hankel functions singularities. In the new approach, and are the spectral-domain Green’s functions written as (6) denotes the asymptotic behaviors where of for large values of . Note that, in (6), every spectral-domain Green’s function is approximated by means of the pole-residue representation of (5) plus one asymptotic term. In accordance with the explanations of [38], the and determine the behavior asymptotic terms of and in the vicinity of , respectively. If (6) is to provide an accurate closed-form representation of the spatial-domain Green’s functions, the asymptotic terms have to satisfy the following conditions. • They must have closed-form inverse Hankel transforms, i.e., the Sommerfeld integrals arising from the introduction in (3) and (4) must have closed-form of expressions. • Their spatial-domain counterpart should reproduce the sinand as gular or quasi-singular behavior of . • They should not have spectral-domain singularities difand . ferent from those of The asymptotic expressions defined in [34] and [35] for the spectral-domain Green’s functions of a source in a multilayered since they substrate are good candidates for fulfill the first two conditions (they have a closed-form inverse Hankel transform and they account for spatial-domain near-field singularities). However, they do not satisfy the third condition. In fact, the spectral asymptotic expressions proposed in [34] and [35] have “additional” singularities (either at or , where is the free-space wavenumber and at and are the relative permittivity and permeability of the source layer) that are not present in and . Thus, the direct use of the spectral asymptotic expressions of [34] and [35] in (6) would raise numerical inaccuracies. Demuynck et al. addressed a possible solution to the problem of singularities in the spectral asymptotic expressions. Their solution consists of

BOIX et al.: APPLICATION OF TOTAL LEAST SQUARES TO DERIVATION OF CLOSED-FORM GREEN’S FUNCTIONS FOR PLANAR LAYERED MEDIA

multiplying these expressions by a factor that cancels out the “additional” singularities and, at the same time, keeps the same [38]. In this paper, we have asymptotic behavior for large applied the cancellation technique of [38] to the asymptotic expressions of [35], and the new resulting asymptotic expressions have been adopted for playing the role of in (6). These new asymptotic expressions are no longer singular and satisfy the aforementioned three conditions. In Appendix I, are presented in the the expressions of particular case of the spectral-domain Green’s functions for a simple one-layer microstrip structure. of (6) have Once the asymptotic terms and ( ; been chosen, the coefficients ) have to be obtained. In [32], these coefficients are computed by means of a quasi-analytical method resulting from a finite-difference scheme. However, Kourkoulos and Cangellaris [34] employ an iterative numerical algorithm (the vectorfitting algorithm) based on pole relocation of complex starting poles [39]. In this paper, the approach used for the computaand is also numerical, and it is based on the tion of method of total least squares. In order to apply this method, the pole-residue term of (6) must be written as a fraction of two polynomials [36] as follows: (7) and where and able of degrees nomials can be written as

are polynomials in the vari, respectively. These two poly-

(8) (9) where and are unknown coefficients that are related and of (6). When (8) and (9) are to the coefficients introduced in (7), and the resulting expression is multiplied by , it is obtained after some rearrangements that

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Fig. 1. Path chosen in the complex k =k -plane when applying the method of total least squares to (10).

equations can be solved via a singular-value decomposition for obtaining the minimum total least squares solution. Since the spectral-domain Green’s functions of lossless multilayered substrates have surface-wave singularities and branch-point singularities along the real axis of the complex -plane [5], in the application of the method of total least squares, it is convenient to sample (10) along a path in the complex -plane that detours around the singularities [16]. In this study, we have chosen a path, i.e., , that satisfies this last condition and that asymptotically approaches the real axis of the -plane; specifically (11) where determines the maximum imaginary value taken by the path. The path defined by (11) is shown in Fig. 1. In our expeand rience, a good choice is to take in (11), where is the maximum wavenumber among the layers of the multilayered substrate. Anyway, numerical simulations have proven that the results obtained via the method of total least squares (see Section III) are not appreciably sensitive to slight variations in the aforementioned values of and . Once a total least squares solution has been obtained for the and of (10), the coefficients of (6) (i.e., coefficients the squared poles of the pole-residue term) can be determined as the complex roots of the polynomial . These roots can be readily extracted from the polynomial coefficients by computing the eigenvalues of the companion matrix, as explained in can also be obtained in terms of [41]. The coefficients by using the standard residue definition

(12)

(10) Each of the equations of (10) poses one problem of linear paand , and rameter estimation with unknown parameters for each of these problems, there is a best solution in the total least squares sense. In order to obtain that solution, the approximate expression of (10) is enforced to be exactly satisfied at different values of the complex variable , , which yields an overdetermined system of linear equations. As explained in [40], this linear system of

Regarding the computation of the coefficients and of (6), the application of the method of total least squares has two advantages over the vector-fitting algorithm [34]. First, whereas the method of total least squares generates the poles and residues in one step, the vector-fitting algorithm carries out an iterative search, the computation time required by each iteration being comparable to that required by the method of total least squares as a whole. The second advantage is that the method of total (the number of terms releast squares gives an estimate for tained in the pole-residue series) from the number of nonzero

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singular values obtained in the singular-value decomposition [36]. In Section III, it will be shown that values of lower than 13 have always been found sufficient to obtain accurate approxi. These values of are certainly mations of smaller than those handled in [33] and [34], and considerably smaller than those reported in [32]. When the approximation proposed in (6) is introduced in (3) and (4), and the Sommerfeld integrals are calculated [42], the following expressions are obtained:

where is Euler’s constant [43]. If (15) is introduced in (13), as is obtained: the following expansion of

(13)

(16)

(14)

Equation (16) shows that the Hankel functions of (13) introduce . However, note that this loga logarithmic singularity as is enarithmic singularity can be suppressed if can be reforced. Looking at (6) and (7), the sum lated to the coefficients of the polynomial in the following way:

are Hankel functions of the second where and can be written in kind and order , and closed form in terms of functions of the type shown in (45) and (47). As mentioned above, the complex roots of the polynomial provide the numerical values of ( ; ). Since the complex numbers

and

( ; ) are all poles of [see (6)], the used in (13) and (14) has to be chosen in sign of the poles , which ensures that the such a way that and fulfills the causality solution obtained for and radiation conditions [34]. and of As commented above, the functions and as (13) and (14) reproduce the behavior of . In fact, and may have singularities as , and these singularities coincide with those of and (as stated above, the singularities may be present when the source and field points are in the same horizontal plane , but they never appear when the source and field points are in different horizontal planes ). The Hankel functions and of (13) and (14) also have singu, but these singularities are not shared by larities as and , and this may have a detrimental effect on the accu[20], [23], [24]. In the remainder racy of (13) and (14) as of this section, it will be shown that the coefficients can be chosen in such a way that the sums of Hankel functions of (13) and (14) have a nonsingular smooth behavior in the vicinity . In that case, the series of Hankel functions will not of mask the correct behavior of and as , which and . is provided by In order to show the problem of Hankel functions singularities in the case of (13), the following expansion of as [43] is used:

(15)

(17)

is enforced, the coefficient Therefore, if of must be set to zero. According to this result, in order to avoid the effect of Hankel functions singularities as in (13), a polynomial in of degree should be . Thus, used in the numerator of the approximation (7) of it is convenient to approximate as

(18) where (19) The procedure designed for avoiding the effect of Hankel functions singularities in (13) can also be applied to (14). In this latter case, the analysis of the singularities suggests to use the as following expansion of the functions [43]:

(20)

BOIX et al.: APPLICATION OF TOTAL LEAST SQUARES TO DERIVATION OF CLOSED-FORM GREEN’S FUNCTIONS FOR PLANAR LAYERED MEDIA

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Equation (20) must then be introduced in (14) in order to obtain as the expansion of

Fig. 2. One-layer substrate microstrip structure studied in this paper. The source point is in the air.

degree written as

. Thus, the approximation of

should be

(24) where (21) (25) Equation (21) shows that the Hankel functions of (14) introduce as . This singularity can a singularity of the type be eliminated provided that is enforced. However, this condition does not always suffice to ensure that the Hankel functions of (14) do not mask the behavior of as . In fact, since as and (see [21, eq. (24)]) and as and , may tend to zero faster than as . If has to go to zero faster than (or at least with the same decay rate), it is then necessary that in (21), but also . not only and can be The two finite sums related to the coefficients of the polynomial in the following way:

It should be noted that the pole-residue terms of (6) and (7) were for large . Howoriginally proposed to decay at a rate ever, in order to avoid problems in the approximation of the spa, it has now been proven tial-domain Green’s functions as that the pole-residue term of should decay at a rate for large , and the pole-residue term of at a rate . This conclusion can be connected with the error analysis carried out in [29] where it was shown that singularity problems in the application of the discrete comencountered for plex image method can be solved when the surface-wave term of decays at a rate for large , and the surface-wave decays at a rate . Whereas the former result term of is coherent with that obtained in this paper, there is a discrepancy in the latter result. In our opinion, the decay rate imposed may be stricter than in [29] to the surface-wave term of necessary. In fact, if that surface-wave term is enforced to decay , this may probably suffice to eliminate the singuat a rate larity problems discussed in [29]. III. NUMERICAL RESULTS

(22)

(23) and are both Therefore, if enforced, the coefficients and of must be set to zero. In conclusion, in order to avoid the effect of Hankel functions singularities in the approximation (14) of as , the polynomial of the numerator in the approximation (7) of has to be a polynomial in of

For demonstrative purposes, the method of Section II has been applied to the computation of the Green’s functions of the simple microstrip structure of Fig. 2 when the value of the relaand the value of the substrate thicktive permittivity is ness is mm. The results are presented in Figs. 3–11. Fig. 3 shows results for the spectral-domain scalar potential Green’s function along the path of Fig. 1. An excellent agreement is found when the exact values of the spectral-domain Green’s function are compared with the values arising from the application of the method of total least squares (18) (in all the figures in this section, the results obtained via the method of total least squares will be denoted as TLS). The differences between the two sets of results are always found to be below 0.03%. Since the approximation provided by the spectral-domain expression (18) is very accurate, the approximation provided by its spatial-domain counterpart (13) should also be very accurate. This is verified in Fig. 4 where the results obtained with the closed-form expression (13) are compared with results obtained via numerical integration of Sommerfeld integrals. Again, the

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Fig. 3. Real and imaginary parts of spectral-domain Green’s function K along path C of Fig. 1. The exact results (solid and dashed line) are compared with those obtained via the method of TLS following (18) ( , ). f = 4:075 GHz, z = z = 0, A = 0:1, T = 2:2, M = 12, N = 27.

2 3

Fig. 4. Magnitude of the spatial-domain Green’s function K . Numerical inte= gration results ( ) are compared with those obtained via (13) when c 0 ( ). f = 4:075 GHz, z = z = 0, A = 0:1, T = 2:2, M = 12, N = 27.



2

Fig. 5. Magnitude of the maximum relative error detected along path C of (11) in the total least squares approximation of K in Fig. 3. The maximum relative error is plotted versus the number of pole-residue terms retained in (6) (A = 0:1, T = 2:2, N = 2M + 3).

agreement between the two sets of results is excellent for both small and large values of . The structure studied in Figs. 3 and 4 has already been analyzed in [9, Fig. 6(a)]. This structure was claimed to be troublesome because the spectral-domain Green’s function has a pole in the real axis of the complex -plane . According to that is very close to the branch point at [9], this has an influence on the terms contributing the far-field behavior of the spatial-domain Green’s function. Fortunately, Figs. 3 and 4 clearly show that the aforementioned problem does

Fig. 6. Real and imaginary parts of spectral-domain Green’s function K along path C of Fig. 1. The exact results (solid and dashed lines) are compared with those obtained via (18) ( , ). f = 25 GHz, z = 0:5 mm, z = 0:5 mm, A = 0:1, T = 2:5, M = 12, N = 27.

0

23

Fig. 7. Magnitude of the spatial-domain Green’s function K . Numerical integration results ( ) are compared with those obtained via (13) when = 0 ( ). f = 25 GHz, z = 0:5 mm, z = 0:5 mm, A = 0:1, c T = 2:5, M = 12, N = 27.

2



0

Fig. 8. Magnitude of the spatial-domain Green’s function K . The results = 0 ( ) are compared with those obtained via obtained via (13) when c 0:5 mm, = 0 ( ). f = 25 GHz, z = 0:5 mm, z = (13) when c A = 0:1, T = 2:5, M = 12, N = 27.

6

2



0

not affect the approximations provided by (13) and (18). Fig. 4 shows that only 12 Hankel functions are needed in (13) for obtaining accurate values of the spatial-domain Green’s function , and this number of Hankel along six decades of the variable functions is roughly half of that handled in [34] for a similar accuracy, and one order of magnitude smaller than that handled in [32]. The computation of the total least squares results plotted in Fig. 4 also requires the singular-value decomposition of a

BOIX et al.: APPLICATION OF TOTAL LEAST SQUARES TO DERIVATION OF CLOSED-FORM GREEN’S FUNCTIONS FOR PLANAR LAYERED MEDIA

Fig. 9. Real and imaginary parts of spectral-domain Green’s function K along path C of Fig. 1. The exact results (solid and dashed line) are compared with those obtained via (24) ( , ). f = 11 GHz, z = 1 mm, z = 0 mm, A = 0:1, T = 2:3, M = 13, N = 29.

23

Fig. 10. Magnitude of the spatial-domain Green’s function K . Numerical integration results ( ) are compared with those obtained via (14) when c = c = 0 ( ), and with those obtained via (14) when = 0 and c = 0 ( ). f = 11 GHz, z = 1 mm, z = 0 mm, c A = 0:1, T = 2:3, M = 13, N = 29.

2

6



6

5

Fig. 11. Magnitude of the spatial-domain Green’s function K . Numerical integration results ( ) are compared with those obtained via (14) when c = c = 0 ( ). f = 11 GHz, z = z = 0 mm, A = 0:1, T = 2:3, M = 13, N = 29.

2



24 27 matrix. According to [32, Sec. V], this singular-value decomposition needs a computation time that is considerably smaller than that needed by the singular-value decomposition of the matrices usually involved in the application of both the discrete complex image method and the method of [32].

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In Fig. 5, the maximum relative error made in the TLS apof Fig. 3 is proximation of the spectral Green’s function [ denotes the number of poleplotted as a function of residue terms used in (6)]. This maximum relative error is obtained by computing the relative error between the exact and along 200 points of the path of approximate values of Fig. 1. The results of Fig. 5 show that the accuracy of the apincreases, and proximation obtained via (18) increases as . the maximum relative error remains below 0.1% for are not shown because there appear zero Values of singular values in the corresponding singular-value decompositions. It has then been checked that the method of total least squares has a robust convergence pattern concerning the number of pole-residue terms included in the approximation of the spectral-domain Green’s functions. component of the spectral-domain Fig. 6 shows the of Fig. 1. vector-potential Green’s function along the path are compared with the values provided The exact values of by (18), and excellent agreement is found (relative errors less GHz, the specthan 0.1%). At the operating frequency presents three TE poles on tral-domain Green’s function the real axis of the complex -plane [7]. The three poles can be identified as peaks of the real and imaginary parts of in Fig. 6. Fig. 7 shows the spatial-domain counterpart of the approximated via (18). spectral-domain Green’s function This spatial-domain counterpart is compared with the values of obtained via numerical integration of Sommerfeld integrals, and good agreement is found along eight decades of the . Note that the spatial-domain Green’s function variable plotted in Fig. 4 has a singularity of the type as (in that case, the source and field points are in the same horizontal of Fig. 7 plane), but the spatial-domain Green’s function (the source and field points are now in is not singular as different horizontal planes). As commented upon in Section II, via (13) for in order to obtain an accurate approximation of the case treated in Fig. 7, it is crucial to remove the logarithmic . Fig. 8 shows the singularity of the Hankel functions as via the method of Section II when the results obtained for Hankel functions singularity is removed , and . As shown in Fig. 7, when it is not removed match those obtained the results obtained when from numerical integration and, therefore, they can be assumed to be virtually correct. However, the results obtained when in Fig. 8 are not correct because they deviate from for . In fact, those obtained when , the larger the deviation is. Fig. 8 the smaller the value of shows that the curve obtained for the case becomes . Since the a straight line with negative slope for vertical scale in Fig. 8 is linear and the horizontal scale is logarithmic, this fact points out that the approximation obtained in the case has a logarithmic singularity for [which has not been removed from the Hankel functions of , and (13)]. This logarithmic singularity is not present in in the case justifies why the approximation obtained for is not correct. Whereas Fig. 8 illustrates how the technique of Section II provides accurate closed-form expressions of the spatial-domain Green’s functions for small , Table I(A) and (B) is introduced

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TABLE I NORMALIZED POLES AND RESIDUES OF THE SPECTRAL-DOMAIN GREEN’S FUNCTION ( ) STUDIED IN FIG. 6. (A) RESULTS OBTAINED BY USING THE APPROXIMATED EXPRESSION OF (18). (B) RESULTS OBTAINED BY USING THE EXACT EXPRESSION VIA A NUMERICAL ALGORITHM

K k

to explain why these closed-form expressions are accurate for large . Table I(A) and (B) is similar to [34, Table I]. Table I(A) shows the normalized poles and the normalized residues of the obtained via (18) for the strucapproximate expression of ture of Figs. 6–8 (in accordance with (6), the residue of this apat the pole will be given by proximate expression of ). The sign of these poles and residues has been chosen so that they can be used in the finite series of Hankel functions of (13). Since for large , the Hankel functions of (13) involving poles with large negative imaginary part will be negligible for large , and with very small those Hankel functions involving poles imaginary part will not be negligible. In Table I(A), the imaginary part of poles 5, 7, and 8 are at least seven orders of magnitude smaller than the remaining ones. Therefore, the Hankel functions involving these three poles will prevail over the rest of Hankel functions of (13) for large . The three Hankel functions will also prevail over the function of (13) for large [the , whereas behavior of these Hankel functions is of the type is of the type , as can be demonstrated from that of (45)]. Thus, the approximate expression given by (13) for the of Fig. 7 in the far-field region will be Green’s function dominated by a linear combination of the three Hankel functions involving poles 5, 7, and 8 of Table I(A). As is well known, an asymptotic analysis of the Sommerfeld integral (3) for large shows that the far-field expression of can be expressed as a linear combination of Hankel functions involving the poles of on the real axis the exact spectral-domain Green’s function of the proper sheet of the complex -plane, and each of these Hankel functions is weighted by the residue of at the corresponding pole (this linear combination of Hankel functions is, of [9, eqs. (3) and (4)]). The values of for instance, the term these poles and residues have been computed for the particular spectral-domain Green’s function of Fig. 6 by means of an appropriate numerical algorithm [44], and the results are shown in Table I(B). The comparison of Table I(A) and (B) points out

that the real part of poles 5, 7, and 8 of Table I(A) coincide with those of the poles of Table I(B) within six significant figures. The real part of the residues at the aforementioned poles also coincide within four significant figures. Since the imaginary parts of all these poles and residues are negligible, the expression of inferred from (13) for large matches with high accuracy the equivalent asymptotic closed-form expression arising from the Sommerfeld integral of (3). This coincidence in the approximate and exact asymptotic expressions justifies that the results obtained for the spatial-domain Green’s functions in Figs. 4 and 7 are not only accurate for small , but also for large . Coming back to the discussion on the poles and residues 5, 7, and 8 of Table I(A) and those of Table I(B), it should be pointed out that the imaginary part of all these poles and residues should be zero. However, it is not zero owing to the finite precision of the numerical algorithms employed [34]. For practical purposes, this imaginary part is negligible and has been set to zero in the deof Fig. 7 by means of (13). termination of the values of Please note that the imaginary part of pole 5 of Table I(A) is positive, which seems to contradict the criterion proposed for the choice of the poles used in (13) and (14). Again, this situation is a consequence of the finite precision of the method of total least squares, and it may occur when we apply the method to a lossless medium (such as that shown in Fig. 2) and one comof the polynomial has a very small plex root imaginary part (i.e., when ). In such a case, the sign of the pole to be used in (13) and (14) since this pole tries to must be chosen so that match a real pole of the original spectral-domain Green’s function, and the corresponding Hankel function represents a traveling cylindrical wave with positive phase velocity (a consemay be positive as it hapquence of this choice is that pens with pole 5 of Table I). In this paper, we have only used when this modified criterion for the choice of the sign of and the medium is lossless. Numerical simulations have shown that, in the case of practical

BOIX et al.: APPLICATION OF TOTAL LEAST SQUARES TO DERIVATION OF CLOSED-FORM GREEN’S FUNCTIONS FOR PLANAR LAYERED MEDIA

Fig. 12. Magnitude of spatial-domain Green’s functions for the four-layer substrate microstrip structure shown in [24, Fig. 1]. Numerical integration results ( ) are compared with those obtained via (13) when c = 0 and via (14) = c = 0 (solid lines). f = 11 GHz, z = 0:4 mm, when c z = 1:4 mm, A = 0:1, T = 3:7, M = 13, N = 29.

2

lossy media, the standard criterion for the choice of the sign of applies, and all the poles used in (13) and (14) turn out to have a negative imaginary part. In the lossy case, the poles that control the far-field behavior of the spatial-domain Green’s functions [i.e., the poles that play the role of 5, 7, and 8 in Table I(A)] also match with great accuracy the corresponding poles of the exact spectral-domain Green’s functions [i.e., the poles that play the role of the poles in Table I(B)] both in the real and imaginary parts. component of the In Fig. 9, results are presented for the spectral-domain vector-potential Green’s function along the are compared with path of Fig. 1. The exact values of the values provided by (24), and excellent agreement is found once again. In this case, the spectral-domain Green’s function has two TM poles and one TE pole on the real axis of the complex -plane [7]. Fig. 10 shows the spatial counterpart of the spectral-domain Green’s function plotted in Fig. 9. obtained from Specifically, Fig. 10 shows the results of (14) when the Hankel functions singularities are removed, and also when these singularities are not removed. The two sets of results are compared with results obtained via numerical integration of Sommerfeld integrals. As happens with the Green’s functions plotted in Figs. 7 and 8, the Green’s function plotted in Fig. 10 is not singular as . Owing to this, (i.e., the results shown in Fig. 10 when when the singularities are removed) are very accurate along , whereas the results obtained eight decades of the variable and (the singularities are not when removed) are incorrect for . The comparison of Figs. 8 and 10 shows that the error found in the approximation when the singularities are not removed is much larger of . This fact can be than that found in the approximation of attributed to the -type singularity of the Hankel functions , which is much stronger than the logarithmic . In Fig. 11, singularity of the Hankel functions we show another plot of in the particular case where this (which occurs Green’s function component is singular as when the source and field points are placed at the air-dielectric interface of Fig. 2 as pointed out in [21] and [22]). Note that

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obtained via (14) match those obtained by the values of means of numerical integration both for low and for large . as [21] is very In fact, the singularity of the type well reproduced by the function of (14). Finally, Fig. 12 shows results for the spatial-domain Green’s functions of the four-layer microstrip structure analyzed in [24, Fig. 1]. The source point is in the second layer and the field point is in the fourth layer. The Green’s functions of Fig. 12 are also plotted in [24, Fig. 3]. It can be checked that excellent agreement exists between the results of [24], the results obtained by means of the closed-form expressions of (13) and (14), and the results obtained via numerical integration of Sommerfeld integrals. This clearly demonstrates that the technique presented here is also valid and very accurate for the computation of the Green’s functions of arbitrary multilayered media. IV. CONCLUSIONS A new systematic technique has been introduced for the determination of closed-form expressions of the spatial-domain Green’s functions for the scalar and vector potentials associated with the application of the mixed-potential integral equation in multilayered media. In the new technique, the Green’s functions are fitted by means of a near-field term plus a finite sum of Hankel functions. The fitting coefficients are obtained by applying the method of total least squares. In the fitting process, special care is taken to remove the singularities of the Hankel functions as the horizontal separation between source and field points goes to zero. The results obtained with the new technique have been compared with numerical integration results. Excellent agreement has been found between the two sets of results both in the near- and the far-field regions. The removal of Hankel functions singularities has proven to be crucial for obtaining accurate results for the near field. Concerning the far field, the leading terms of the closed-form expressions have been found to match those obtained via analytic asymptotic techniques. A key point of the current technique is that it provides accurate results for a very wide range of distances with a computational expense that may be considerably smaller than that required by related previous techniques. APPENDIX The expressions of the spectral Green’s functions , as well for as the elements of the spectral dyadic Green’s function the one-layer substrate microstrip structure of Fig. 2 are presented here. Also included are the asymptotic expressions of the spectral Green’s functions that have been employed in (18) and (24) for obtaining the numerical results of Section III. is given by The spectral scalar potential Green’s function the following:

(26)

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(35) (27) where expressions of

and

and the asymptotic expressions of

are

. The asymptotic

can then be written as

(36) (28) (37) (29) where (in an arbitrary multilayered medium should be taken as , with being the maximum wavenumber among the layers of the medium [38]). Note has been deliberately introduced in that the factor (28) and (29) in order to avoid the singularity of the function when [38]. Since the function is not singular when , the function should not be singular either [remember that is involved in the approximation of , as shown in (6)]. are and [37]. The The diagonal elements of expressions of can be written as

(30)

(31) and the asymptotic expressions of

are given by

The off-diagonal elements of [37]. The expressions of

are

, and

,

, and are

(38)

(39) and are the Cartesian spectral variables associwhere via ated with and , respectively (and related to ). The asymptotic expressions of and are

(32)

(33) (40) The expressions of

are

(34)

(41)

BOIX et al.: APPLICATION OF TOTAL LEAST SQUARES TO DERIVATION OF CLOSED-FORM GREEN’S FUNCTIONS FOR PLANAR LAYERED MEDIA

Once again, the factor has been introduced in (40) and are singular when and (41) to avoid that [38]. Finally, and are related and by means of to

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functions of the type [42]

(47) (42)

(43) and and, as a consequence of these relations, . Looking at (28), (29), (32), (33), (36), and (37), it can be ob(i.e., served that the functions denoted in this paper by , and ) are all linear combinations of functions of the type

(44) where . It means that the functions of (13) will all be linear combinations of functions of the type [42]

(45) If , the functions show a singularity of the type as . In accordance with (28), (29), (32), (33), (36), and , the functions (37), this means that, in the case will show the same singularity as . According to (3) and (13), the singularity will also be present in , which is in described in agreement with the near-field behavior of [20]. According to (40) and (41), the functions denoted (i.e., and in this paper by ) are functions of the type (46) where

. Therefore, the functions

of (14) will be

as and , but It can be shown that as and . Looking at (40) and (41), (i.e., the source and this means that in the case field and points are placed at the interface between the air and will show the dielectric layer in Fig. 2), the functions a singularity of the type as . According to (4) and , which is in (14), this singularity will be reproduced in predicted in [21]. agreement with the behavior of REFERENCES [1] J. R. Mosig, “Arbitrarily shaped microstrip structures and their analysis with a mixed potential integral equation,” IEEE Trans. Microw. Theory Tech., vol. 36, no. 2, pp. 314–323, Feb. 1988. [2] R. C. Hall and J. R. Mosig, “The analysis of arbitrarily shaped aperturecoupled patch antennas via a mixed-potential integral equation,” IEEE Trans. Antennas Propag., vol. 44, no. 5, pp. 608–614, May 1996. [3] K. A. Michalski and D. Zheng, “Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media—Part II: Implementation and results for contiguous half-spaces,” IEEE Trans. Antennas Propag., vol. 38, no. 3, pp. 345–352, Mar. 1990. [4] S. Vitebskiy, K. Sturgess, and L. Carin, “Short-pulse plane-wave scattering from buried perfectly conducting bodies of revolution,” IEEE Trans. Antennas Propag., vol. 44, no. 2, pp. 143–151, Feb. 1996. [5] J. R. Mosig, “Integral equation technique,” in Numerical Techniques for Microwave and Millimeter-Wave Passive Structures, T. Itoh, Ed. New York: Wiley, 1989, pp. 133–213. [6] K. A. Michalski and J. R. Mosig, “Multilayered media Green’s functions in integral equation formulations,” IEEE Trans. Antennas Propag., vol. 45, no. 3, pp. 508–519, Mar. 1997. [7] J. R. Mosig and F. E. Gardiol, “Analytical and numerical techniques in the Green’s function treatment of microstrip antennas and scatterers,” Proc. Inst. Elect. Eng., vol. 130, no. 2, pt. H, pp. 175–182, Mar. 1983. [8] K. A. Michalski, “Extrapolation methods for Sommerfeld integral tails,” IEEE Trans. Antennas Propag., vol. 46, no. 10, pp. 1405–1418, Oct. 1998. [9] J. R. Mosig and Álvarez-Melcón, “Green’s functions in lossy layered media: Integration along the imaginary axis and asymptotic behavior,” IEEE Trans. Antennas Propag., vol. 51, no. 12, pp. 3200–3208, Dec. 2003. [10] T. J. Cui and W. C. Chew, “Fast evaluation of Sommerfeld integrals for EM scattering and radiation by three-dimensional buried objects,” IEEE Trans. Geosci. Remote Sens., vol. 37, no. 3, pp. 887–900, Mar. 1999. [11] W. Cai and T. Yu, “Fast calculations of dyadic Green’s functions for electromagnetic scattering in a multilayer medium,” J. Comput. Phys., vol. 165, pp. 1–21, 2000. [12] L. Tsang, C. J. Ong, C. C. Huang, and V. Jandhyala, “Evaluation of the Green’s function for the mixed potential integral equation (MPIE) method in the time domain for layered media,” IEEE Trans. Antennas Propag., vol. 51, no. 7, pp. 1559–1571, Jul. 2003. [13] S. Barkeshli, P. H. Pathak, and M. Marin, “An asymptotic closed-form microstrip surface Green’s function for the efficient moment method analysis of mutual coupling in microstrip antennas,” IEEE Trans. Antennas Propag., vol. 38, no. 9, pp. 1374–1383, Sep. 1990. [14] Y. Brand, A. Álvarez-Melcón, J. R. Mosig, and R. C. Hall, “Large distance behavior of stratified media spatial Green’s functions,” in IEEE AP-S Int. Symp. Dig., Montreal, QC, Canada, Jul. 1997, pp. 2334–2337. [15] D. G. Fang, J. J. Yang, and G. Y. Delisle, “Discrete image theory for horizontal electric dipoles in a multilayered medium,” Proc. Inst. Elect. Eng., vol. 135, no. 5, pt. H, pp. 297–303, Oct. 1988. [16] Y. L. Chow, J. J. Yang, D. G. Fang, and G. E. Howard, “A closed-form spatial Green’s function for the thick microstrip substrate,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 3, pp. 588–592, Mar. 1991.

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[17] R. A. Kipp and C. H. Chan, “Complex image method for sources in bounded regions of multilayer structures,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 5, pp. 860–865, May 1994. [18] M. I. Aksun, “A robust approach for the derivation of closed-form Green’s functions,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 5, pp. 651–658, May 1996. [19] N. Kinayman and M. I. Aksun, “Efficient use of closed-form Green’s functions for the analysis of planar geometries with vertical connections,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 5, pp. 593–603, May 1997. [20] C. H. Chan and R. A. Kipp, “Application of the complex image method to multilevel, multiconductor microstrip lines,” Int. J. Microw. Millimeter-Wave Comput.-Aided Eng., vol. 7, no. 5, pp. 359–367, 1997. [21] ——, “Application of the complex image method to characterization of microstrip vias,” Int. J. Microw. Millimeter-Wave Comput.-Aided Eng., vol. 7, no. 5, pp. 368–379, 1997. [22] N. Hojjat, S. Safavi-Naeini, R. Faraji-Dana, and Y. L. Chow, “Fast computation of the nonsymmetrical components of the Green’s function for multilayer media using complex images,” Proc. Inst. Elect. Eng.—Microw. Antennas Propag., vol. 145, no. 4, pp. 285–288, Aug. 1998. [23] N. Hojjat, S. Safavi-Naeini, and Y. L. Chow, “Numerical computation of complex image Green’s functions for multilayer dielectric media: Near-field zone and the interface region,” Proc. Inst. Elect. Eng.–Microw. Antennas Propag., vol. 145, no. 6, pp. 449–454, Dec. 1998. [24] F. Ling and J. M. Jin, “Discrete complex image method for Green’s functions of general multilayer media,” IEEE Microw. Guided Wave Lett., vol. 10, no. 10, pp. 400–402, Oct. 2000. [25] Y. Liu, L. W. Li, T. S. Yeo, and M. S. Leong, “Application of DCIM to MPIE–MoM analysis of 3-D PEC objects in multilayered media,” IEEE Trans. Antennas Propag., vol. 50, no. 2, pp. 157–162, Feb. 2002. [26] Y. Ge and K. P. Esselle, “New closed-form Green’s functions for microstrip structures-theory and results,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 6, pp. 1556–1560, Jun. 2002. [27] N. V. Shuley, R. R. Boix, F. Medina, and M. Horno, “On the fast approximation of Green’s functions in MPIE formulations for planar-layered media,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 9, pp. 2185–2192, Sep. 2002. [28] P. Yla-Oijala and M. Taskinen, “Efficient formulation of closed-form Green’s functions for general electric and magnetic sources in multilayered media,” IEEE Trans. Antennas Propag., vol. 51, no. 8, pp. 2106–2115, Aug. 2003. [29] S. A. Teo, S. T. Chew, and M. S. Leong, “Error analysis of the discrete complex image method and pole extraction,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 2, pp. 406–413, Feb. 2003. [30] M. I. Aksun and G. Dural, “Clarification of issues on the closed-form Green’s functions in stratified media,” IEEE Trans. Antennas Propag., vol. 53, no. 11, pp. 3644–3653, Nov. 2005. [31] M. Yuan, T. K. Sarkar, and M. Salazar-Palma, “A direct discrete complex image method from the closed-form Green’s functions in multilayered media,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 3, pp. 1025–1032, Mar. 2006. [32] V. I. Okhmatovski and A. C. Cangellaris, “A new technique for the derivation of closed-form electromagnetic Green’s functions for unbounded planar layered media,” IEEE Trans. Antennas Propag., vol. 50, no. 7, pp. 1005–1016, Jul. 2002. [33] ——, “Evaluation of layered media Green’s functions via rational function fitting,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 1, pp. 22–24, Jan. 2004. [34] V. N. Kourkoulos and A. C. Cangellaris, “Accurate approximation of Green’s functions in planar stratified media in terms of a finite sum of spherical and cylindrical waves,” IEEE Trans. Antennas Propag., vol. 54, no. 5, pp. 1568–1576, May 2006. [35] E. Simsek, Q. H. Liu, and B. Wei, “Singularity subtraction for evaluation of Green’s functions for multilayer media,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 216–225, Jan. 2006. [36] R. S. Adve, T. K. Sarkar, S. M. Rao, E. K. Miller, and D. R. Pflug, “Application of the Cauchy method for extrapolating/interpolating narrowband system responses,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 5, pp. 837–845, May 1997. [37] K. A. Michalski and D. Zheng, “Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media—Part I: Theory,” IEEE Trans. Antennas Propag., vol. 38, no. 3, pp. 335–344, Mar. 1990. [38] F. J. Demuynck, G. A. E. Vandenbosch, and A. R. Van de Capelle, “The expansion wave concept—Part I: Efficient calculation of spatial Green’s functions in a stratified dielectric medium,” IEEE Trans. Antennas Propag., vol. 46, no. 3, pp. 397–406, Mar. 1998. [39] B. Gustavsen and A. Semylen, “Rational approximation of frequency domain responses by vector fitting,” IEEE Trans. Power Del., vol. 14, no. 3, pp. 1052–1061, Jul. 1999.

[40] J. Rahman and T. K. Sarkar, “Deconvolution and total least squares in finding the impulse response of an electromagnetic system from measured data,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 4, pp. 416–421, Apr. 1995. [41] A. Edelman and H. Murakami, “Polynomial roots from companion matrix eigenvalues,” Math. Comput., vol. 64, no. 210, pp. 763–776, 1995. [42] L. S. Gradshteyn and L. M. Ryzhik, Table of Integrals, Series and Products, 6th ed. San Diego, CA: Academic, 2000. [43] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, 9th ed. New York: Dover, 1970. [44] R. Rodríguez-Berral, F. Mesa, and F. Medina, “Systematic and efficient root finder for computing the modal spectrum of planar layered waveguides,” Int. J. Microw. Millimeter-Wave Comput.-Aided Eng., vol. 14, pp. 73–83, Jan. 2004.

Rafael R. Boix (M’96) received the Licenciado and Doctor degrees in physics from the University of Seville, Seville, Spain, in 1985 and 1990 respectively. Since 1986, he has been with the Electronics and Electromagnetism Department, University of Seville, where, in 1994, he became an Associate Professor. His current research interests are focused on the numerical analysis of periodic electromagnetic structures with applications to the design of frequency-selective surfaces and electromagnetic bandgap passive circuits.

Francisco Mesa (M’93) was born in Cádiz, Spain, on April 1965. He received the Licenciado and Doctor degrees in physics from the University of Seville, Seville, Spain, in 1989 and 1991, respectively. He is currently an Associate Professor with the Department of Applied Physics 1, University of Seville. His research interest is focused on electromagnetic propagation/radiation in planar lines with general anisotropic materials.

Francisco Medina (M’90–SM’01)was born in Puerto Real, Cádiz, Spain, in November 1960. He received the Licenciado and Doctor degrees in physics from the University of Seville, Seville, Spain, in 1983 and 1987 respectively. From 1986 to 1987, he spent the academic year with the Laboratoire de Microondes de l’ENSEEIHT, Toulouse, France. From 1985 to 1989, he was an Assistant Professor with the Department of Electronics and Electromagnetism, University of Seville, where, since 1990, he has been an Associate Professor of electromagnetism. He is also currently Head of the Microwaves Group, Department of Electronics and Electromagnetism, University of Seville. He is on the Editorial Board of the International Journal of RF and Microwave ComputerAided Engineering. He is also a reviewer for the Institution of Electrical Engineers (IEE), U.K., and American Physics Association journals. His research interest includes analytical and numerical methods for guiding, resonant, and radiating structures, passive planar circuits, periodic structures, and the influence of anisotropic materials (including microwave ferrites) on such systems. He is also interested in artificial media modeling and design. Dr. Medina was a member of the Technical Programme Committees (TPC) of the 23rd European Microwave Conference, Madrid, Spain (1993), ISRAMT’99, Málaga, Spain (1999), and Microwaves Symposium’00, Tetouan, Morocco (2000). He was co-organizer of the “New Trends on Computational Electromagnetics for Open and Boxed Microwave Structures” Workshop, Madrid, Spain (1993). He is a member of the Massachusetts Institute of Technology (MIT) Electromagnetics Academy. He is on the review board of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. He also acts as a reviewer for other IEEE publications. He was the recipient of a 1983 research scholarship presented by the Spanish Ministerio de Educación y Ciencia (MEC). He was also the recipient of a scholarship presented by the French Ministère de la Recherche et la Technologie.

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281

An in situ Tunable Diode Mounting Topology for High-Power -Band Waveguide Switches

X

Thomas Sickel, Member, IEEE, Petrie Meyer, Member, IEEE, and Pieter W. van der Walt, Senior Member, IEEE

Abstract—An in situ tunable diode mounting topology for waveguide switches is presented and utilized to design and fabricate two evanescent-mode -band switching modules of approximately 15% and 25% fractional bandwidth. The ability of the mounting topology to operate in a high-power environment is verified in an evanescent-mode -band switch using six packaged p-i-n diodes, successfully reflecting 4 kW of pulsed power. Index Terms—Diode mount, evanescent mode, high power, p-i-n diodes, waveguide switch, -band.

I. INTRODUCTION

F

OR THE switching of high-power signals at -band frequencies, p-i-n diodes acting as switching elements in waveguide are widely used [1]–[4]. The introduction of diodes into waveguide structures requires two integrated parts, namely, a mounting mechanism and a dc-feed. The mounting mechanism acts as the physical connection between the diodes and waveguide, while the dc-feed supplies both positive and negative bias. The dc-feed has to allow dc power to reach the diodes without allowing RF energy to escape the system and without negatively impacting the signal flow of the switch in the off state. The mount/feed structure should provide good thermal and RF grounding between the diodes and waveguide. For large diodes, the mounting structure usually introduces parasitic inductance and capacitance, which negatively affects both isolation and insertion loss of the switch if not explicitly included in the design. To meet these requirements, the mount/feed structures in the waveguide are usually fairly complex and limit performance to bandwidths (BWs) of up to only 10%. Tuning of resonances are also extremely difficult due to the fixed dc-feed structures, and normally involves partial disassembly. Various mechanisms have been proposed for mounting structures. In [1], the diode is embedded in a coaxial stub, which shunts the signal to ground with the diodes in the on state, and offers a high shunt impedance to the signal when in the off state. Bias is introduced through a sliding contact coaxial post decou-

Manuscript received June 22, 2006; revised October 10, 2006. This work was supported by the National Research Foundation of South Africa, by the Wilhelm Frank Trust, and by Reutech Radar Systems. T. Sickel and P. Meyer are with the Department of Electrical and Electronic Engineering, University of Stellenbosch, Stellenbosch 7600, South Africa (e-mail: [email protected]; [email protected]). P. W. van der Walt was with the Department of Electrical and Electronic Engineering, University of Stellenbosch, Stellenbosch 7600, South Africa. He is now with Reutech Radar Systems, Stellenbosch 7600, South Africa. Digital Object Identifier 10.1109/TMTT.2006.889146

pled by a dielectric sleeved dc block. Isolation of 30 dB, insertion loss of 0.8 dB, and 1.6 : 1 voltage standing-wave ratio (VSWR) over a 8% BW at -band using two post-mounted diodes was achieved. The design of Dawson and DeLoach [2] utilizes a piece of coaxial line with attributed inductance, in series with the p-i-n diode in a reduced height waveguide. In the forward-bias state, the parallel resonance of p-i-n diode inductance and capacitance presents an open circuit required for filtering, while in the reverse-bias state, the inductance of the stub and capacitance of the p-i-n are series resonant, yielding a high transmission loss. Short bias pulses from an external source are introduced through a trapping arrangement at one of the diode terminals. Isolation of 12 dB over a 7% band and 0.4-dB insertion loss using a single p-i-n diode was achieved with the design. The single-pole single-throw (SPST) microwave switch presented by Bakeman and Armstrong [3] switches from a dielectric shunt across the waveguide in the low attenuation state to a resistive shunt across the waveguide when operating in the isolation state by using a grid of diodes mounted in the cross section of the waveguide. Bias lines running through the narrow wall of the waveguide, are perpendicular to the electric field and are, therefore, decoupled from the microwave signal. Isolation of 12 dB and insertion loss less than 1 dB was obtained over an octave BW using a single silicon slice with 600 p-i-n diodes. The p-i-n diode anti-transmit–receive (ATR) tube described by Sarkar [4] utilizes a stepped post inductance, gap capacitance, and p-i-n diode capacitance to produce series and parallel resonance conditions when reverse and forward biased, respectively. Isolation of 27 dB and insertion loss less than 1 dB was achieved over a 5% BW using a single p-i-n diode. Tuning out the diode capacitance to present pure open or short circuits to the waveguide causes significant reductions in BW, especially for high-power diodes with large capacitance. Instead, the diodes can be incorporated into filter structures as capacitive elements in the reverse-bias state and low resistances in the forward-bias state. This approach is particularly well suited to fin-line implementations [5] and evanescent-mode waveguides [6]. This paper presents a new diode mount/feed topology for waveguide structures. The topology allows for in situ tuning of the switching element capacitance without diminishing any of the advantages of classical post mounted diodes. This is a huge advantage in terms of manufacturing time and cost. Two single module switches using two packaged p-i-n diodes each are presented, as well as one three-module switch using a total of six packaged p-i-n diodes. The single-module switches produced 22-dB return loss and 0.8-dB insertion loss when oper-

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Fig. 2. Basic model of evanescent-mode waveguide switch and p-i-n diode branch setup.

the top and bottom of the central-ground plate, respectively. Feed lines, constructed from 0.1-mm copper shimstock with added circular pads at the ends, feed dc bias to the top and bottom diodes. The circular pads serve as one side of tunable air-gap capacitances formed between the pads and tuning rotors entering from the top and bottom of the waveguide. The lumped-element model of the mount/feed structure con, as well as a parallel combisists of two air-gap capacitances and to model the horizontal central-ground nation of plane. The influence of both the ground and feed lines in the mount are reduced by having them enter through the narrow wall of the waveguide, which makes the larger central-ground plate exactly perpendicular, and the smaller dc-feed lines roughly perpendicular, to the electric field orientation. III. PRINCIPLE OF OPERATION

Fig. 1. Horizontal-ground horizontal-feed air-gap design. (a) Physical setup. (b) Lumped-element model. (c) Prototype.

ating in the low-loss state, and 18-dB isolation and 1.1-dB return loss when operating in the isolation state, over a fractional BW of 15%; and 15-dB return loss and 1.53-dB insertion loss when operating in the low-loss state, and 24.4-dB isolation and 0.913-dB return loss when operating in the isolation state, over a fractional BW of 25%. The three-module switch produced 15.73-dB return loss and 1.23-dB insertion loss when operating in the low-loss state, and 62-dB isolation and 1.29-dB return loss when operating in the isolation state, over a fractional BW of 21%. II. PROPOSED DIODE MOUNTING TOPOLOGY The topology is based on a standard construction of two back-to-back diodes mounted on a vertical post in a waveguide with a central horizontal ground extending to the waveguide wall, and separated by a fixed capacitive gap. The proposed diode mounting topology is shown in Fig. 1. The central-ground protruding horizontally from the narrow wall of the waveguide is manufactured from 0.5-mm copper shimstock, and is crucial in the mechanical stability and thermal aspects of the mount. Rigidity of the structure and good heat flow from the diodes to the waveguide determines the minimum thickness and width requirement of the plate. Two p-i-n diodes are soldered onto

The mounting structure is illustrated through its application to an evanescent-mode -band waveguide switch, as the ability to tune the capacitance is most useful in a device where the diodes are embedded in a shunt-coupled filter topology. Evanescent-mode devices were popularized by Craven and Mok in the 1970s [6], and commercially available switches operating below-cutoff have excellent wideband performance. The basic structure of an evanescent-mode switch is shown in Fig. 2. Two reduced-height -band guides are separated by a section of the evanescent-mode guide with two diodes mounted vertically in the center of the evanescent-mode section. By varying the length of the evanescent-mode section, the BW of the structure may be adjusted. The equivalent lumped-element circuit model of the evanescent-mode waveguide and almost linear nature of the p-i-n diode in forward and reverse bias allows conventional circuit analysis and synthesis to be employed in the design. A simple equivalent circuit consisting of a -network of inductors can be used to represent a section of a below-cutoff guide, which may be rearranged to represent shunt inductances, , coupled by a inverter of reactance value , [6]. By using the reverse-bias state of the diodes to introduce suitable capacitive obstacles at correct intervals along the evanescent-mode waveguide, a microwave analog of a lumped inductance bandpass filter with series inductance coupling results. The equivalent-circuit model of the complete evanescent-mode waveguide switch is shown in Fig. 3, where the central capacitance is formed by the p-i-n diode mount with the p-i-n diodes in the reverse-bias state. The junctions, known as -plane steps, are modeled by a transformer of turns ratio and susceptance

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-BAND WAVEGUIDE SWITCHES

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Fig. 3. Equivalent-circuit model of evanescent-mode p-i-n diode switch.

Fig. 4. Forward- and reverse-bias conditions of p-i-n diode branch.

[7]. Brass rotors at the junctions implement the end capacitances, forming broadband resonators with the junction susceptances. When reverse biased, the capacitances of the p-i-n diodes resonate in parallel with effective guide inductances to form the central resonator of the filter, as shown in Fig. 4. Resonances are indicated by arrows. The diode mount creates additional capacitance to resonate in series with the combined p-i-n diode inductances . In the forward-bias state, the seand comries resonance between the mount capacitance bined p-i-n diode inductances remains as in the reversebias state. The resistive nature of the -region in the forwardbias state results in a thoroughly detuned parallel resonator section, also shown in Fig. 4. Capacitive obstacles are introduced at the junctions between the propagating and evanescent-mode guide to tune out junction susceptances, resulting in a third-order filter response, when using a single central diode mount, as shown in Fig. 2. IV. SWITCH DESIGN AND RESULTS Three switches are designed. Two of the modules are designed to operate over approximately 15% and 25% fractional BW at -band, respectively, using only a single diode mount with two diodes per mount. The third module is designed using three diode mounts employing a total of six diodes. The mount/ feed structure is presented with measured and simulated data of positive and negative p-i-n diode bias conditions. The designed switch was fabricated in an aluminium waveguide using high-power p-i-n switching diodes. As the number of untuned p-i-n diode branches increase in a higher order system, the isolation increases due to a greater distance between the junction

Fig. 5. 15% fractional BW structure—measurement versus theory. (a) Switch transmission. (b) Switch isolation.

resonators and high attenuation characteristics of the evanescent mode. Switch transmission and isolation conditions were

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Fig. 7. Fifth-order switch.

Fig. 6. 25% fractional BW structure—measurement versus theory. (a) Switch transmission. (b) Switch isolation.

measured on an HP8510c network analyzer at 10-V bias and 20-mA bias per p-i-n diode, respectively. High-power isolation conditions were measured with an increased diode bias of

150 mA per diode. All simulations were performed using CST Microwave Studio and AWR Microwave Office. The evanescent guide length on either side of the diode may be changed to control the BW of the structure. The first singlemount switch achieved a fractional BW of 15% in the 15% BW structure with 22-dB return loss and 0.8-dB insertion loss when operating in the low-loss state, and 18-dB isolation and 1.1-dB return loss when operating in the isolation state, as shown in Fig. 5; and the second single mount switch achieved a fractional BW of 25.5% in the 25% BW structure with 15-dB return loss and 1.53-dB insertion loss when operating in the low-loss state, and 24.4-dB isolation and 0.913-dB return loss when operating in the isolation state, as shown in Fig. 6. Good insertion- and return-loss characteristics when operating in the low-loss and isolation states, respectively, as well as the simple in situ tuning ability, ensure the design as a favorable choice in the implementation of a higher order switch structure. To obtain an increased level of attenuation for the purpose of higher power handling, a higher order switch is designed, incorporating a total of six diodes using three diode mounts, as shown in Fig. 7. Measured data of first-order sections is used to build up accurate lumped-element models of diode mounts, evanescent-mode guide sections, and junctions between evanescent-mode and propagating guide sections. Sections are cascaded with dimensions determined by an optimized approach, where evanescent guide lengths, mount capacitances, and junction capacitances are tuned for optimal switch operation in low-loss and isolation states. The addition of junction tuning rotors on the input and output of the switch results in a fifth-order filter structure. The switch achieves a switch fractional BW of 21% with 15.73-dB return loss and 1.23-dB insertion loss with a 0.155-dB amplitude flatness when operating in the low-loss state, and 62-dB isolation and 1.29-dB return loss when operating in the isolation state, as shown in Fig. 8. The higher order switch successfully reflected an input pulse level of 4 kW at 9.5 GHz for a 24- s pulsewidth at 4.8% duty cycle. A good agreement between measurement and theory is obtained for low-loss and isolation states. At extremely low power levels, as seen in the isolation state of the fifth-order switch in

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V. CONCLUSION The proposed diode mounting topology allows simple tuning of the resonant frequency of the structure, employing ground and feed lines entering through the narrow walls of the waveguide, perpendicular to the electric field orientation. The three switches show good transmission and isolation characteristics over a relatively wide BW at -band. The final six-diode switch achieves an isolation of greater than 60 dB across the entire measured band (6.7 13 GHz) when operating in the isolation state, and 15.73-dB return loss and 1.23-dB insertion loss over a fractional BW of 21% when operating in the low-loss state. The mount displays good thermal and RF grounding properties between the diodes and waveguide with a measured successful reflection of an input pulse level of 4 kW at 9.5 GHz for a 24- s pulsewidth at 4.8% duty cycle. The diode mounting topology has only been implemented in evanescent-mode structures, however, it may possibly be implemented in propagating structures with minor adjustments. ACKNOWLEDGMENT The authors wish to thank Computer Simulation Technology (CST), Darmstadt, Germany, and Applied Wave Research (AWR), El Segundo, CA, for the use of their software, as well as W. Croukamp and L. Saunders, both with Sentrale Elektroniese Dienste (SED), University of Stellenbosch, Stellenbosch, South Africa, for manufacturing the prototypes. REFERENCES [1] J. White, Microwave Semiconductor Engineering. New York: J. F. White, 1995. [2] R. Dawson and B. DeLoach, “A low-loss 1-nanosecond 1-watt -band switch,” in IEEE MTT-S Int. Microw. Symp. Dig., 1966, pp. 146–150, 66.1. [3] P. Bakeman, Jr. and A. Armstrong, “Fast, high power, octave bandwidth, -band waveguide microwave switch,” in IEEE MTT-S Int. Microw. Symp. Dig., 1976, pp. 154–156, 76.1. [4] B. Sarkar, “Biased PIN FDR for 45 kW, -band duplexing,” in IEEE MTT-S Int. Microw. Symp. Dig., 1979, pp. 241–242, 79.1. [5] H. Meinel and B. Rembold, “New millimeter-wave fin-line attenuators and switches,” in IEEE MTT-S Int. Microw. Symp. Dig., 1979, pp. 249–252, 79.1. [6] G. Craven and C. Mok, “The design of evanescent mode waveguide bandpass filters for a prescribed insertion loss characteristic,” IEEE Trans. Microw. Theory Tech., vol. MTT-19, no. 3, pp. 295–308, Mar. 1971. [7] L. Lewin, Advanced Theory of Waveguides. New York: Iliffe and Sons, 1951.

X

X

Fig. 8. Fifth-order switch—measurement versus theory. (a) Switch transmission. (b) Switch isolation.

Fig. 8(b), the relatively simple equivalent-circuit model is unable to accurately model the measured response.

X

Thomas Sickel (S’02–M’02) was born in Cape Town, South Africa, on June 21, 1980. He received the B.Eng. and Ph.D. degrees in electronic engineering from the University of Stellenbosch, Stellenbosch, South Africa, in 2002 and 2005, respectively. In 2006, he joined the Faculty of Engineering, University of Stellenbosch, where he is currently a PostDoctoral Researcher. His current research interests include thermal analysis and transient modeling of p-i-n diodes.

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Petrie Meyer (S’87–M’88) was born in Bellville, South Africa, in 1965. He received the B.Eng., M.Eng., and Ph.D. degrees from the University of Stellenbosch, Stellenbosch, South Africa, in 1986, 1988, and 1995, respectively. In 1988, he joined the Faculty of Engineering, University of Stellenbosch, where he is currently a Professor of microwave engineering. His current research interests include electromagnetic (EM) analysis, passive devices, and mathematical modeling techniques.

Pieter W. van der Walt (M’80–SM’91) was born in Germiston, South Africa, on April 12, 1947. He received the B.Sc., B. Eng., M. Eng., and Ph.D. degrees in electronic engineering from the University of Stellenbosch, Stellenbosch, South Africa, in 1970, 1973, and 1982, respectively. In 1971, he joined the Department of Electrical and Electronic Engineering, University of Stellenbosch, and served as Dean from 1993 to 2002. After his retirement from the University of Stellenbosch, he joined Reutech Radar Systems, Stellenbosch, South Africa, as Technology Executive responsible for RF Technology. His main interests include network synthesis and linear and nonlinear circuit design. Prof. van der Walt is a senior member of the South African Institute of Electrical Engineers (SAIEE), and is currently chairman of the IEEE Antennas and Propagation (AP)/Microwave Theory and Techniques (MTT) Society Chapter of the South Africa section.

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Novel Balanced Coupled-Line Bandpass Filters With Common-Mode Noise Suppression Chung-Hwa Wu, Student Member, IEEE, Chi-Hsueh Wang, Member, IEEE, and Chun Hsiung Chen, Fellow, IEEE

Abstract—Novel balanced coupled-line bandpass filters, using suitable balanced coupled-line sections and quarter-wavelength resonators, are proposed. For design purposes, the differentialand common-mode equivalent half-circuits are established. Based on these circuits, a better balanced filter structure is implemented so that the desired differential-mode response may be realized and the level of common-mode noise may be minimized simultaneously. Besides, a suitable capacitive or inductive cross-coupled effect is introduced so as to create two transmission zeros for improving the filter selectivity; however, it also enhances the signal imbalance and degrades the common-mode rejection. In this study, various second- and fourth-order balanced filters are implemented to discuss the associated differential-mode responses and the signal-imbalance phenomena resulted from the cross-coupled effect. Specifically, the fourth-order filter with a common-mode rejection ratio of 40 dB within the passband is demonstrated and examined. Index Terms—Balanced filter, common-mode noise suppression, coupled-line bandpass filter, quarter-wavelength resonator.

I. INTRODUCTION ALANCED circuits are important in building a modern communication system. Analog signals processed by a communication system are degraded by two different types of noises, namely, the environmental noise and device electronic noise. The former refers to the random disturbances that a circuit experiences through the dc power supply, ground lines, or substrate coupling. The latter includes the thermal noise, shot noise, and flicker noise, which come from the internal active device [1]. A noise limits the minimum signal level that a circuit can process with acceptable quality. The most important advantage of the balanced circuits with differential operation is the higher immunity to the environmental noise when compared with the unbalanced circuits with single-ended signaling. Over the past few years, several balanced circuit topologies have been developed to connect with the monopole antenna for which a balun is needed to convert a balanced signal into an unbalanced one. Typical balun configurations reported include those based on distributed [2], [3], lumped [4], or lumped-distributed elements [5]–[7]. Some balun topologies offering bandpass-type transmission characteristics were also discussed in [8] and [9]. After that, several single-to-differential bandpass filters

B

Manuscript received October 12, 2006; revised October 26, 2006. This work was supported by the National Science Council of Taiwan, R.O.C., under Grant NSC 95-2752-E-002-001-PAE, Grant NSC 95-2219-E-002-008, and Grant NSC 95-2221-E-002-196. The authors are with the Department of Electrical Engineering and Graduate Institute of Communication Engineering, National Taiwan University, Taipei 106, Taiwan, R.O.C. (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.889147

were proposed to replace the conventional balun together with a bandpass filter. In [10], a second-order combline filter in series with the conventional Marchand balun was realized in the low-temperature co-fired ceramic (LTCC) substrate. Based on the same concept, a modified version was presented in [11]. In [12], by introducing a capacitive loading, an additional transmission zero was created to optimize the filter response. Recently, an LTCC circuit architecture that combines the functionalities of balun and bandpass filter was proposed in [13]. Owing to the adoption of lumped elements, the realization of the balanced-to-unbalanced filter in [13] is not easy in a higher microwave region. Traditional off-chip radiating elements suffer from additional interconnect losses and cannot be duplicated like the on-chip antennas. The small wavelength at millimeter-wave frequency and the mature CMOS fabrication technology make it possible to use on-chip metal layers to fabricate small integrated antennas [14], [15]. Although still faced with a problem of improving the antenna efficiency, there will be more and more integrated circuits implemented in the fully differential form under the trend of system-on-chip (SOC). The on-chip integrated antenna usually takes the form of a dipole, which needs a balanced filter for differential operation. A well-designed differential-to-differential balanced bandpass filter should exhibit the desired bandpass frequency response in differential operation and should be able to reduce the level of the common-mode signal so as to increase the signal-to-noise-ratio in the receiver and to improve the efficiency of the dipole antenna in the transmitter. Pervious studies on differential-to-differential balanced filter designs are rather limited. In [16], a 40-GHz balanced filter was patch resonator and proposed based on a half-wavelength transmission lines. That filter four quarter-wavelength structure presents a good common-mode rejection ability, but it has the drawback of bulky size. Recently, a 2-GHz balanced filter [17] was realized based on conventional parallel-coupledline structures [18]–[22]. Compared with [16], the coupled-line balanced filter in [17] exhibits the advantages of a compact size and simple synthesis procedure; however, it shows a poor common-mode rejection ability, which is crucial in the balanced filter design. In this study, novel balanced coupled-line structures are proposed to implement the balanced filters so as to give an excellent common-mode rejection in addition to providing the desired differential-mode bandpass response. The desired differentialand common-mode responses are realized by a suitable choice of balanced coupled-line sections. Specifically, by properly designing the circuits associated with the common-mode signal, a very low common-mode signal level around (the center frequency of the differential-mode response) is achieved without any degradation of insertion loss in differential-mode operation.

0018-9480/$25.00 © 2007 IEEE

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Fig. 1. Type-I balanced coupled-line section. (a) Circuit structure. (b) Differential-mode equivalent circuit. (c) Common-mode equivalent circuit.

coupled-line Being composed of a quarter-wavelength section with its two ends shorted, the odd-mode equivalent circuit shown in Fig. 1(b) or 2(b) exhibits a bandpass frequency response with the center frequency at . On the other hand, the even-mode equivalent circuit shown in Fig. 1(c) or 2(c) would present the all-stop frequency response at because it consists of a coupled-line section with one end shorted and the other end opened. Theoretically, Figs. 1(c) and (2c) with ideal ground both present the all-stop characteristics under common-mode operation. However, in the practical case, the shorted-ends of coupled lines will be realized by the nonideal via-holes, which may result in different behaviors in the common-mode operation. To minimize the common-mode noise, the coupled-line sections should properly be chosen. III. SECOND-ORDER FILTERS

Fig. 2. Type-II balanced coupled-line section. (a) Circuit structure. (b) Differential-mode equivalent circuit. (c) Common-mode equivalent circuit.

For design purposes, the equivalent half-circuits are also established so that both differential- and common-mode frequency responses may be discussed separately. In this study, two types of second-order balanced filters are realized using the microstrip structure. To further improve the filter response, two types of fourth-order balanced filters are realized based on the novel dual-metal-plane structure consisting of both a microstrip and a coplanar waveguide. The proposed balanced filter circuits are implemented, measured, and carefully examined. II. BALANCED COUPLED-LINE SECTIONS A realization of the structure, which gives both desired differential-mode bandpass response and minimum common-mode noise level, is essential in developing a balanced filter. Figs. 1(a) and 2(a) show two types (types I and II) of balanced coupledline sections, which are capable of realizing such desired responses. Among these two structures, the most noticeable difference is in the positions of input/output ports and shorted ends. In Fig. 1(a), the differential input ports are through the elements and of the balanced coupled-line section, and the differential output ports are through the elements and . Alternatively, in Fig. 2(a), the differential input ports are through the elements and , and the differential output ports are through the elements and . As to the locations of short ends, they are on the left-hand side of Fig. 1(a), whereas they are on the right-hand side of Fig. 2(a). For balanced operation, the balanced coupled-line sections may be decomposed into two equivalent circuits under odd- and even-mode excitations. Under odd-mode excitation, although the structures in Figs. 1(a) and 2(a) are quite different in circuit layouts, they can both be represented by the same equivalent circuits, as shown in Figs. 1(b) and 2(b). However, under evenmode excitation, the structures in Figs. 1(a) and 2(a) should be represented by the two different equivalent circuits shown in Figs. 1(c) and 2(c), respectively.

A well-designed balanced bandpass filter should possess the capability of reducing the level of common-mode noise in addition to providing the desired bandpass frequency response in differential-mode operation. In [17], only the differential-mode operation was discussed without considering the common-mode response. Actually, both differential- and common-mode responses should be taken into consideration so as to give a good common-mode rejection. Based on the balanced coupled-line structures shown in Figs. 1(a) and 2(a), two types of second-order balanced filters may be developed to give a good common-mode rejection ability. Fig. 3 shows the layouts of proposed second-order balanced filters (types I and II), and their corresponding differential- and common-mode equivalent half-circuits are shown in Figs. 4 and 5. Specifically, by decomposing the balanced filter structures into two different equivalent half-circuits under odd- and even-mode excitations, one may discuss the differential- and common-mode responses accordingly. The proposed balanced filters (Fig. 3) are simulated by the commercial software Sonnet and AWR Microwave Office (MWO). All the circuits are fabricated on the FR4 substrate , , and the thickness mm). The bal( anced filter, as a four-port device, is first measured by Agilent’s E5071B network analyzer to give the standard four-port -parameters . The two-port differential- and common-mode -parameters and may then be extracted from the four-port -parameters , as given by [23] (1) (2)

Basically, the mixed-mode composed into

-parameters (

) may be de(3)

in which (4)

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Fig. 5. Common-mode equivalent half-circuits of the proposed second-order balanced filters in Fig. 3. (a) Type I. (b) Type II.

A. Differential-Mode Response Fig. 3. Physical layouts of the proposed second-order balanced filters. (a) Type I (S = 20:1 mm, S = 17:5 mm, L = 4 mm, W = 1:9 mm, W = 1:9 mm, W = 1:7 mm, G = 0:5 mm, G = 0:3 mm, G = 0:3 mm, D = 1 mm). (b) Type II (S = 17:8 mm, S = 16:8 mm, S = 2:7 mm, W = 1:9 mm, W = 1:9 mm, W = 1:7 mm, G = 0:3 mm, G = 0:5 mm, G = 0:3 mm, D = 1 mm).

Fig. 4. Differential-mode equivalent half-circuit of the proposed second-order 6= 0. (b) M = 0. balanced filters in Fig. 3. (a) M

For the balanced filters shown in Fig. 3, the corresponding differential-mode equivalent half-circuit [shown in Fig. 4(a)] is composed of two bandpass-response coupled-line sections [shown in Figs. 1(b) or (2b)], which are connected by a series . This capacitor is realized by the open-ended capacitor edge coupled structure associated with the gap in Fig. 3(a) or the gap in Fig. 3(b). In addition, a mutual inductance is introduced to provide a cross-coupled path for creating two transmission zeros. As shown in Fig. 3, the cross-coupled inductance for each type is achieved with different physical structures. In the type-I filter, due to the virtual short in the center line, the inductive cross-coupling is realized along the coupled-line sections associated with the gap in Fig. 3(a). On the other hand, the inductive cross-coupling is formed between two via-holes associated with the gap in Fig. 3(b). Physically, the location of the transmission zeros may simply be adjusted by controlling the mutual inductance through varying the gapwidth [of Fig. 3(a)] or [of Fig. 3(b)]. With the cross-coupled inductance neglected , the differential-mode equivalent half-circuit Fig. 4(a) for the second-order filters composed of resonators may further be represented by the one in Fig. 4(b). The design procedures for the filter in Fig. 4(b) are well documented in [21] and [22]. For the differential-mode operation, the second-order filter is designed with a center frequency GHz, a 3-dB bandwidth of 10%, and a characteristic impedance based on the maximally flat response. The corresponding circuit parameters are obtained as follows:

(5) represent the two-port differential- and common-mode -parameters, respectively. Similar expressions may be used for the and . For an ideal two-port cross-mode -parameters balanced structure, the cross-mode -parameters and would be a two-by-two matrix of zero elements.

at

GHz

pF Based on these parameters, the type-I and type-II filters are fabricated on the FR4 substrate with the layouts shown in

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Fig. 6. Measured and simulated differential-mode responses of the proposed second-order filters in Fig. 3. (a) Type I. (b) Type II.

Fig. 3(a) and (b). These filters are simulated using the full-wave simulator Sonnet, and are measured using the four- to two-port conversion technique, as given by (1)–(5). The narrowband measured and simulated differential-mode responses are shown in Fig. 6. Good agreement between measured and simulated results is observed with only a slight deviation in bandwidth. For the type-I filter, the measured center frequency is at 2 GHz, with a minimum differential-mode insertion loss of 2.9 dB and a 3-dB bandwidth of 10.5%. For the type-II filter, the measured center frequency is at 2 GHz with a minimum differential-mode insertion loss of 2.3 dB and a 3-dB bandwidth of 12%. B. Common-Mode Response For the second-order balanced filter structures shown in Fig. 3(a) and (b), the corresponding common-mode equivalent half-circuits are illustrated in Fig. 5(a) and (b), respectively. By neglecting the loaded inductances associated with the via-holes and the cross-coupled effect due to or , the two equivalent half-circuits would exhibit the all-stop frequency responses in common-mode operation since these circuits are mainly composed of the all-stop coupled-line sections, as shown in Figs. 1(c) and 2(c). The existence of via-hole inductances and cross-coupled effect destroys the all-stop characteristic and causes the unwanted signal-imbalance ( and ) around (the passband center frequency of differential-mode operation). To demonstrate the effects of the loaded inductors and the cross-coupled path (path 2), the common-mode insertion

Fig. 7. Simulated common-mode responses of the signals along paths 1 and 2 of the circuit models shown in Fig. 5 with L = 0:1 nH. (a) Type I. (b) Type II.

losses along two signal paths are simulated by the MWO circuit simulator and are shown in Fig. 7(a) and (b). Here, path 1 represents the main signal path primarily composed of two all-stop coupled-line sections. Also included in path 1 is the sealone, as in Fig. 5(b), or the capacitor ries capacitor plus the loaded inductors , as in Fig. 5(a). Path 2 denotes the cross-coupled path through the cross-coupled capacitance [see Fig. 5(a)] or the cross-coupled mutual inductance [see Fig. 5(b)]. Fig. 7 indicates that the common-mode response is mainly determined by the signal through path 2; GHz is largely influenced however, its response around by the signal through path 1. For the type-I filter [see Fig. 3(a)], the resonator under the transmission common-mode operation is composed of a line section with one end shorted and another end opened; here, is the guided wavelength corresponding to the frequency GHz. Therefore, the center frequency of the common-mode response of the type-I filter would be the same as that of the . Besides, the signal along differential-mode response path 1 presents two peaks around , which produce two spurs in the resultant common-mode response around , thereby degrading the common-mode rejection ability. For the type-II filter, the resonator under common-mode operation behaves as a open-end resonator, where is the guided wavelength corresponding to the frequency GHz. Therefore, the central frequency of the common-mode response would be twice the differential-mode response . Although the signal along path 2 also shows a peak around , its contribution to the resultant common-mode response is negligible, as depicted in Fig. 7(b).

WU et al.: NOVEL BALANCED COUPLED-LINE BANDPASS FILTERS WITH COMMON-MODE NOISE SUPPRESSION

Fig. 8. Measured differential- and common-mode frequency responses of the proposed second-order filters in Fig. 3. (a) Type I. (b) Type II.

The existence of loaded inductors is unavoidable in practice realization. However, the type-II design may push the center frequency of the common-mode response to the higher frequency band , thereby improving the common-mode rejection ability. The wideband differential- and common-mode measured responses are shown in Fig. 8(a) and (b) for comparison. In the type-I design, the minimum common-mode insertion loss is 16.8 dB around ; while in type-II design, it is 28 dB around , and all below 25 dB from 0.5 to 8.5 GHz. As a result, an improvement of 11.2 dB in common-mode noise rejection around could be achieved by using the type-II structure. IV. FOURTH-ORDER FILTERS Intuitively, the rejection level of the common-mode signal can further be suppressed by increasing the filter order. A fourthorder balanced filter may be achieved by introducing two additional series capacitors at the input/output ends of the proposed filter structures in Fig. 3. Figs. 9 and 10 show the layouts of the proposed fourth-order filters. The corresponding differential- and common-mode equivalent half-circuits are also shown in Figs. 11 and 12. Specially, the series input/output capacitors , to realize the -inverter, are implemented by the metal–insulator–metal (MIM) structure so as to achieve a higher capacitance value. As shown in Figs. 9 and 10, the proposed fourth-order balanced filters are constructed using the dual-metal-plane structures. Here, the microstrip coupled-line sections are lo-

291

Fig. 9. (a) 3-D physical layout and (b) top-/bottom-plane layout of the proposed fourth-order type-I balanced filter (S = 15:1 mm, S = 17:7 mm, S = 3 mm, W = 1:1 mm, W = 1:1 mm, W = 2:1 mm, W = 3:2 mm, G = 0:5 mm, G = 0:3 mm, G = 1:8 mm, D = 1 mm).

Fig. 10. Top-/bottom-plane layout of the proposed fourth-order type-II balanced filter (S = 15:7 mm, S = 16:6 mm, S = 3:5 mm, W = 1:1 mm, W = 1:1 mm, W = 2:1 mm, W = 3:2 mm, G = 0:3 mm, G = 0:3 mm, G = 1:7 mm, D = 1 mm).

cated on the top metal plane, and the coplanar waveguide (CPW) input/output transmission lines together with the ground plane of the microstrip are on the bottom metal plane. The MIM capacitor is achieved between the CPW input/output transmission line on the bottom metal plane and the microstrip coupled-line section on the top metal plane. A. Differential-Mode Response Fig. 11 demonstrates the odd-mode equivalent half-circuit of the fourth-order balanced filters. The design of the filter under odd-mode operation may follow the similar procedures, as mentioned in Section III. These filters are designed with

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Fig. 11. Differential-mode equivalent half-circuit of fourth-order balanced filters in Figs. 9 and 10.

Fig. 13. Measured differential- and common-mode frequency responses of the proposed fourth-order filters. (a) Type I. (b) Type II.

B. Common-Mode Response

Fig. 12. Common-mode equivalent half-circuits of fourth-order balanced filters in Figs. 9 and 10. (a) Type-I. (b) Type-II.

GHz, dB bandwidth , and . The corresponding circuit parameters are obtained based on the fourthorder maximally flat response as follows:

at GHz at GHz pF pF For the fourth-order filters shown in Figs. 9 and 10, the cross-coupled effect is also introduced between input/output resonators so as to enhance the filter selectivity. The wideband differential-mode measured responses are shown in Fig. 13. is at 1.99 GHz with a In the type-I design, the measured minimum differential-mode insertion loss of 3.73 dB and a 3-dB bandwidth of 11%. For the type-II filter, the measured is at 2.05 GHz with a minimum differential-mode insertion loss of 3.98- and a 3-dB bandwidth of 9.7%.

Shown in Fig. 12 are the corresponding even-mode equivalent half-circuits of the fourth-order balanced filters. Based on the same concept as mentioned in Section III, two signal paths are introduced to discuss the common-mode responses, which are shown in Fig. 14. Paying attention to the response for path 1 in Fig. 14, it is found that the frequency response of each type along path 1 is similar to that of the second-order one. However, having benefited from the higher order structure, the level of the common-mode signal along path 1 has been further suppressed below 50 dB so that its contribution to the resultant commonmode response is of minor significance. The common-mode responses of fourth-order filters (Figs. 9 and 10) are mainly dominated by the signal along the cross-coupled path (path 2). The even-mode equivalent half-circuits in Fig. 12 are useful in discussing this signal along path 2. Note that the type-II filter is primarily composed of resonators with one end shorted and another end opened [see Fig. 12(b)], making its resonant frequency equal to ( 2 GHz). Thus, both differential- and common-mode responses would have the same passband around . This explains why the type-II filter has a poor common-mode rejection around . Alternatively, the type-I filter is based on the open-end transmission line resonator, which would push the passband center frequency up to ( 4 GHz), as shown in Fig. 14(a). Therefore, the type-I design (Fig. 9) would achieve better common-mode rejection around .

WU et al.: NOVEL BALANCED COUPLED-LINE BANDPASS FILTERS WITH COMMON-MODE NOISE SUPPRESSION

Fig. 14. Frequency responses of the two signal paths with L fourth-order design. (a) Type I. (b) Type II.

= 0:1 nH for

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Fig. 16. Measured CMRR for: (a) second- and (b) fourth-order balanced filters.

common-mode insertion losses pressed as

and

may be ex-

(7) (8) Fig. 15. Practical balanced circuit.

where The wideband common-mode measured responses are also shown in Fig. 13. The minimum common-mode insertion loss is 42 dB at 1.97 GHz in the type-I design, and minimum commonmode insertion loss is 15 dB at 1.95 GHz in the type-II design.

(9) If the balanced circuit were ideal so that

V. COMMON-MODE REJECTION RATIO (CMRR) To provide a simple figure-of-merit for characterizing the implement balanced filters, the CMRR defined by dB

(6)

is adopted in this study. Basically, the value of the CMRR may be used to quantitatively discuss the degree of resemblance between the implemented balanced filter and the ideal balanced one. For the well-designed balanced circuit (Fig. 15) to realize an ideal balanced one, the corresponding differential- and

and

(10)

its CMRR would approach infinity, implying that the commonmode signal would completely be suppressed. Thus, in implementing a balanced filter, it is better to nearly meet the balanced conditions (10) so that the undesired signal imbalance ( and ) may keep to a minimum. The CMRR provides an important figure-of-merit for a meaningful characterization of balanced circuits. Fig. 16 shows the measured responses of the CMRR for the implemented secondand fourth-order filters. Specifically, the second-order type-II design has a maximum CMRR of 25.7 dB at 1.97 GHz with all

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TABLE I PERFORMANCE COMPARISON WITH PREVIOUS STUDIES

CMRR values above 22.5 dB from 1.88 to 2.12 GHz. As to the fourth-order type-I filter, it has a maximum CMRR of 52.2 dB at 2.03 GHz and all CMRR values are above 40 dB from 1.93 to 2.12 GHz, which is almost twice the second-order design. Four filter circuits realized in this study are summarized and compared with the previous works in Table I. The corresponding and around are also values for illustrated in Table I to demonstrate the signal-imbalance phenomena. The CMRR of type-I and type-II second-order designs are roughly 10 and 17.5 dB higher than that in [17], respectively. This demonstrates that balanced filters with excellent CMRR may be achieved through proper arrangement of the circuit configuration. In addition, the same technique is extended to realize the fourth-order filters for better CMRR. The well-designed fourth-order type-I filter shows an improvement of 35 dB in CMRR when compared with the one in [17].

VI. CONCLUSIONS Four novel balanced filters based on balanced coupled-line sections have been proposed and carefully examined. The adoption of the symmetrical structure and balanced coupled-line sections makes it possible to realize a balanced filter, which gives the desired differential-mode bandpass response, and also minimize the common-mode signal level on the other hand. The cross-coupled effect is also introduced to improve the filter selectivity; however, it enhances the signal imbalance and degrades the common-mode rejection. Specifically, by properly choosing the filter configuration, one may push the commonmode spurs to the higher frequency so that a filter with good selectivity and excellent CMRR may be achieved simultaneously. The CMRR of the well-designed fourth-order type-I balanced filter is above 40 dB around the differential-mode passband, which is very attractive for balanced topology applications.

REFERENCES [1] B. Razavi, Design of Analog CMOS Integrated Circuit. Boston, MA: McGraw-Hill, 2001. [2] D. Raicu, “Design of planar, single-layer microwave baluns,” in IEEE MTT-S Int. Microw. Symp. Dig., 1998, pp. 801–804. [3] A. M. Pavio and A. Kikel, “A monolithic or hybrid broadband compensated balun,” in IEEE MTT-S Int. Microw. Symp. Dig., 1990, pp. 483–486. [4] K. Nishikawa, I. Toyoda, and T. Tokumitsu, “Compact and broadband three-dimensional MMIC balun,” IEEE Trans. Microw. Theory Tech, vol. 47, no. 1, pp. 96–98, Jan. 1999. [5] S. P. Ojha, G. R. Branner, and B. P. Kumar, “A miniaturized lumped-distributed balun for modern wireless communication systems,” in Proc. IEEE Midwest Circuits Syst. Symp., 1996, pp. 1347–1350. [6] B. P. Kumar, G. R. Branner, and B. Huang, “Parametric analysis of improved planar balun circuits for wireless microwave and RF applications,” in Proc. IEEE Midwest Circuits Syst. Symp., 1998, pp. 474–475. [7] C. W. Tang and C. Y. Chang, “A semi-lumped balun fabricated by low temperature co-fired ceramic,” in IEEE MTT-S Int. Microw. Symp. Dig., 2002, pp. 2201–2204. [8] Y. C. Leong, K. S. Ang, and C. H. Lee, “A derivation of a class of 3-port baluns from symmetrical 4-port networks,” in IEEE MTT-S Int. Microw. Symp. Dig., 2002, pp. 1165–1168. [9] K. S. Ang, Y. C. Leong, and C. H. Lee, “Analysis and design of miniaturized lumped-distributed impedance-transforming balun,” IEEE Trans. Microw. Theory Tech, vol. 51, no. 3, pp. 1009–1017, Mar. 2003. [10] D. W. Yoo, E. S. Kim, and S. W. Kim, “A balance filter with DC supply for Bluetooth module,” in Proc. Eur. Microw. Conf., 2005, pp. 1239–1242. [11] M. C. Park, B. H. Lee, and D. S. Park, “A laminated balance filter using LTCC technology,” in Proc. Asia–Pacific Microw. Conf., 2005, pp. 4–7. [12] R. Kravchenko, K. Markov, D. Orlenko, G. Sevskiy, and P. Heide, “Implementation of a miniaturized lumped-distributed balun in balanced filtering for wireless applications,” in Proc. Eur. Microw. Conf., 2005, pp. 1303–1306. [13] L. K. Yeung and K. L. Wu, “An LTCC balanced-to-unbalanced extracted-pole bandpass filter with complex load,” IEEE Trans. Microw. Theory Tech, vol. 54, no. 4, pp. 1512–1518, Apr. 2006. [14] A. Babakhani, X. Guan, A. Komijani, A. Natarajan, and A. Hajimiri, “A 77 GHz 4-element phased array receiver with on-chip dipole antenna in silicon,” in Int. Solid-State Circuits Conf. Tech. Dig., San Francisco, CA, Feb. 2006, pp. 180–181. [15] C.-H. Wang, Y. H. Cho, C. S. Lin, H. Wang, C. H. Chen, D. C. Niu, J. Yeha, C. Y. Lee, and J. Chern, “A 60 GHz transmitter with integrated antenna in 0.18 m SiGe BiCMOS technology,” in Int. Solid-State Circuits Tech. Dig., San Francisco, CA, Feb. 2006, pp. 186–187. [16] A. Ziroff, M. Nalezinski, and W. Menzel, “A 40 GHz LTCC receiver module using a novel submerged balancing filter structure,” in Proc. Radio Wireless Conf., 2003, pp. 151–154. [17] Y.-S. Lin and C. H. Chen, “Novel balanced microstrip coupled-line bandpass filters,” in URSI Int. Electromagn. Theory Symp., 2004, pp. 567–569. [18] S. B. Cohn, “Parallel-coupled transmission-line-resonator filters,” IRE Trans. Microw. Theory Tech., vol. MTT-6, no. 7, pp. 223–231, Apr. 1958. [19] C.-Y. Chang and T. Itoh, “A modified parallel-coupled filter structure that improves the upper stopband rejection and response symmetry,” IEEE Trans. Microw. Theory Tech, vol. 39, no. 2, pp. 310–314, Feb. 1991. [20] C.-H. Wang, Y.-S. Lin, and C. H. Chen, “Novel inductance-incorporated microstrip coupled-line bandpass filters with two attenuation poles,” in IEEE MTT-S Int. Microw. Symp. Dig., 2002, pp. 1979–1982. [21] Y.-S. Lin, C.-H. Wang, C. H. Wu, and C. H. Chen, “Novel compact parallel-coupled microstrip bandpass filters with lumped-element K -inverters,” IEEE Trans. Microw. Theory Tech, vol. 53, no. 7, pp. 2324–2328, Jul. 2005. [22] C.-C. Chen, Y.-R. Chen, and C.-Y. Chang, “Miniaturized microstrip coupled-line bandpass filters using quarter-wave or quasi-quarter-wave resonators,” IEEE Trans. Microw. Theory Tech, vol. 51, no. 1, pp. 120–131, Jan. 2003. [23] D. E. Bockelman and W. R. Eisenstant, “Combined differential and common-mode scattering parameters: Theory and simulation,” IEEE Trans. Microw. Theory Tech, vol. 43, no. 7, pp. 1530–1539, Jul. 1995.

WU et al.: NOVEL BALANCED COUPLED-LINE BANDPASS FILTERS WITH COMMON-MODE NOISE SUPPRESSION

Chung-Hwa Wu (S’06) was born in Tainan, Taiwan, R.O.C., in 1982. He received the B.S. degree in electrical engineering from National Chung Hsing University, Taichung, Taiwan, R.O.C., in 2004, and is currently working toward the Ph.D. degree at the Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan, R.O.C. His research interests include the design and analysis of microwave filters and passive circuits.

Chi-Hsueh Wang (S’02–M’05) was born in Kaohsiung, Taiwan, R.O.C., in 1976. He received the B.S. degrees in electrical engineering from National Cheng Kung University, Tainan, Taiwan, R.O.C., in 1997, and the Ph.D. degree from National Taiwan University, Taipei, Taiwan, R.O.C., in 2003. He is currently a Post-Doctoral Research Fellow with the Graduate Institute of Communication Engineering, National Taiwan University. His research interests include the design and analysis of microwave and millimeter-wave circuits and computational electromagnetics.

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Chun Hsiung Chen (SM’88–F’96) was born in Taipei, Taiwan, R.O.C., on March 7, 1937. He received the B.S.E.E. and Ph.D. degrees in electrical engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 1960 and 1972, respectively, and the M.S.E.E. degree from National Chiao Tung University, Hsinchu, Taiwan, R.O.C., in 1962. In 1963, he joined the Faculty of the Department of Electrical Engineering, National Taiwan University, where he is currently a Professor. From August 1982 to July 1985, he was Chairman of the Department of Electrical Engineering, National Taiwan University. From August 1992 to July 1996, he was the Director of the University Computer Center, National Taiwan University. In 1974, he was a Visiting Scholar with the Department of Electrical Engineering and Computer Sciences, University of California at Berkeley. From August 1986 to July 1987, he was a Visiting Professor with the Department of Electrical Engineering, University of Houston, Houston, TX. In 1989, 1990, and 1994, he visited the Microwave Department, Technical University of Munich, Munich, Germany, the Laboratoire d’Optique Electromagnetique, Faculte des Sciences et Techniques de Saint-Jerome, Universite d’Aix-Marseille III, Marseille, France, and the Department of Electrical Engineering, Michigan State University, East Lansing, respectively. His areas of interest include microwave circuits and computational electromagnetics.

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Planar Realization of a Triple-Mode Bandpass Filter Using a Multilayer Configuration Cesar Lugo, Member, IEEE, and John Papapolymerou, Senior Member, IEEE

Abstract—A triple-mode bandpass filter has been designed using a multilayer approach. The filter topology consists of a square loop resonator located inside a resonant cavity. The walls of the cavity were designed using periodic metal vias that connect the top and bottom ground planes for an overall planar configuration. A filter prototype was fabricated and measured, producing a fully canonical filtering function with three resonant poles and three transmission zeros. The filter yielded an insertion loss of 1.1 dB at the passband centered at 5.8 GHz. Index Terms—Degenerate modes, dual-mode filter, dual-mode resonator, triangular loop resonator, triple-mode resonator.

I. INTRODUCTION OR YEARS, multimode resonators have been used in filter synthesis for their ability to reduce the number of resonating elements. Waveguide dual-mode filters have been extensively designed for satellite applications [1] and cellular base stations [2]. Multimode attractive characteristics include high performance in response selectivity, inherent size reduction, and advanced elliptic or quasi-elliptic responses with asymmetrical characteristics. Triple-mode filters have been reported using dielectric filled waveguide resonators of rectangular and cylindrical shape. For the most part, triple-mode operation is achieved by a dielectric filled structure inside a resonant metal cavity. This cavity is then perturbed using tuning and coupling elements such as screws or metal rods [3]. Mode degeneracy occurs when different cavity modes resonate at the same frequency. In this situation, it is necessary for each mode to produce a distinct field pattern. This allows the coupling elements to interfere with the field paths differently from mode to mode and produce a split in the resonant frequencies. , , , , and the hybrid The modes have been used extensively in the past [2]–[5]. The electromagnetic problem of dual-mode resonance has been successfully translated from waveguide realizations to planar technology. This has produced a range of microstrip patches and loops of circular [6], square [7] and triangular shapes [8]. However, one disadvantage in planar dual-mode resonators is their loss. This is attributed to conductor loss, radiation loss, and excitation feed coupling. Despite efforts to improve dual-mode performance [9], single-mode resonators usually result in lower

F

Manuscript received September 16, 2006; revised October 16, 2006. This work was supported by the Georgia Electronic Design Center. The authors are with the School of Electrical and Computer Engineering Georgia Institute of Technology, Atlanta, GA 30332 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2006.889148

Fig. 1. Proposed topology.

loss (in some cases, approximately 2 dB) [10]. This leaves selectivity and size reduction as the sole advantage in planar dualmode filters. A triple-mode resonator is difficult to realize in planar form. This is due to all of the inherent 3-D requirements. For example, the tuning and coupling elements used for modal coupling are, for the most part, required to exist at angles that perturb the modes in all three dimensions. For this reason, triplemode filters have only been accomplished in waveguide form. In this study, a multilayer approach is proposed for the first time to produce a triple-mode resonator. To the best of our knowledge, this filter represents the first triple-mode filter in planar configuration. The resulting prototype approaches the performance of waveguide filters in terms of loss, while keeping a compact size profile comparable to standard planar filters. II. TRIPLE-MODE RESONATOR PRINCIPLE A. Proposed Topology The proposed topology is shown in Fig. 1. The main resonant element consists of a square loop resonator placed inside a resonant metal cavity. The resonant loop closely follows the conventional design of a microstrip dual-mode square loop resonator , where is the guided wave[7] with side dimensions of length at the resonant frequency . The input/output ports are placed at a 90 excitation angle with linewidth corresponding to a strip transmission line with characteristic impedance . Fig. 2 shows the side view and the different layers of the filter. The filter was designed using four dielectric layers of various thickness constituting the different horand mm. The metal izontal levels

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Fig. 2. Filter cross section.

layer mm corresponds to the bottom ground plane. The mm and the ports are placed resonant loop is located at at mm. A set of perturbations are introduced in all three dimensions by placing a square patch on a higher layer mm with dimensions extending in the – -direction. This patch is then connected to the resonant loop using a mm and metal via extending vertically between layers mm. As will be discussed in detail, this set of perturbations are responsible for the split in resonant frequencies of three degenerate modes. The initial cavity dimensions are calculated using the eigenmode solver in the High Frequency Structure Simulator (HFSS) software package. The eigenmode frequencies of the dielectric filled cavity alone (without resonant loop or ports present) are mm (total layer thickinvestigated with dimensions . The wavenumber and resonant frequency ness) and for a lossless rectangular waveguide cavity with resonant and are given by modes (1) (2) where is the speed of light. In this first analysis, we seek the dimensions and that produce the lowest resonant mode at a frequency of GHz. Planar resonant square loops are treated as waveguide cavities with dual-mode operation ex, where is perpendicular panded by the degenerate modes to the ground plane. A natural choice for the cavity dimensions is to find the dom. Setting in (2) with inant resonant mode , the dimensions of the cavity are found mm. This initial calculation has no deto be pendency on or total height of the cavity. The dimension and an appropriate sois deliberately chosen to be lution is found for the triple-mode operation. As will be shown later in this discussion, these initial dimensions require little to no adjustment to obtain the final response. Fig. 3 shows the top view of the resonator with the cavity walls replaced by metal vias. The corner patches are introduced to adjust the resonant frequencies produced by the square loop. B. Electric Field Distribution In planar dual-mode resonators with square loop topologies, the modes are split when a perturbation patch or cut

Fig. 3. Top view of proposed resonator including cavity with periodic viawalls.

is introduced at any of the corners located along the symmetrical plane [7]. The electric field distributions seen in the 2-D case undergoes a natural rotating behavior. For example, in the absence of perturbation elements (corner patches or cuts), the with orthogonal modes are related to the cavity modes with source excitation at source excitation at port 1, and port 2 [11]. These modes with identical resonant frequencies are orthogonal and produce electric field patterns with two maxima or along located along the top and bottom loop arms . When a perturbation is introduced, the the side arms electric field maxima migrate to the corners with highest capacitance. In the case of square loop resonators, this corresponds to the perturbation patch along the plane of symmetry and the opposite corner. In the current design, a 3-D characteristic is added with the introduction of the resonant cavity. The perturbation placed along the symmetrical plane also occurs in a 3-D fashion by the existing via connection between the perturbation patch (level mm) and the resonant loop (level mm). The resonator is now expanded by the resonant loop modes and and the cavity mode . A third-order degeneracy is forced when both the square loop and resonant cavity are designed with equal resonant frequencies. This condition is met at at , the design stage given that the loop arm length and cavity dimensions are found with and the in (2). The introduction of the patch elements and and the perturbation via causes a resonant frequency split between de, , and . generate modes The electric field distribution inside the metal cavity undergoes a set of rotations similar to the 2-D case. One noticeable difference, however, is that the 3-D resonator presented here produces field maxima and minima that are not always located along the -plane. Fig. 4 shows the different (minima/maxima) , , and proelectric field planes with normal vectors , , and . The normal duced for each resonant frequency vectors are described using a spherical coordinate system lo. Fig. 5 shows the cated in the center of the cavity at electric field intensity patterns. For this analysis, a full-wave simulation is conducted using HFSS software and the electric

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and Fig. 5(c) shows the electric field intensity. The rotation anare determined for the odd distribution fields gles and and can be calculated based on the cavity dimensions (3) where mm is the total height of the resonant cavity, and mm is the side dimension of the cavity walls. It is worth noting that all three field distributions produce two field maxima located on either side of their respective axis (not and given the offset angles and ). The visible for electric field distribution closely follows the current density of the ring resonator with the current minima aligned with the axis , , and . III. CIRCUIT MODEL

Fig. 4. Rotations of planes with minima and maxima electric field. (a) E at f = f , symmetrical plane with normal N = z ( = 0 and  = 0). (b) E at f > f plane with normal N ( = 345 ,  = 19:65 ). (c) E at f < f with normal vector N ( = 375 ,  = 19:65 ).

An equivalent circuit for triple-mode filters has been presented in [12] and [13] and is shown in Fig. 5. The circuit parameters are calculated in terms of the low-pass equivalent elfor . In this circuit model, the ements , , and angular resonant frequency for each resonator is given by the matrix of each inverter can be standard (4), while the expressed by (5) (4) (5) The synthesis of this filter closely follows the procedure outlined in [14] and [15]. The coupling coefficients from the different resonant modes are calculated by placing a ground termination on the last resonator and calculating or measuring the input phase reflection. A simple calibration of the reference planes at the ports is conducted to terminate the last resonator and calculate the phase response [15]. The inter-resonator cou180 pling coefficients can be extracted from the zeros and poles (0 crossings) of the input phase reflection. The circuit model inverters can then be calculated and related to the circuit model using the following equations:

Fig. 5. Electric field intensity patterns for: (a) E even mode with field minima along plane AA ( = 45 ), (b) E with field minimum along = 75 ), and (c) E with field minimum along plane plane BB ( CC ( = 105 ).

field intensity is analyzed on each of the three resonant frequencies. Fig. 4(a) shows the first electric field distribution with resonant frequency . has an even characteristic with field minimum located along the symmetrical line . Fig. 5(a) shows the field intensity with two field maxima located on each side of the symmetrical axis and along ( and ). the plane with normal vector Fig. 4(b) shows the field plane for at with normal ( , ). The field minimum is vector . The field intensity is located along the line at shown in Fig. 5(b). Fig. 4(c) shows the field plane for . The normal vector is ( , ). , The field minimum is located along the line

(6) (7) (8) As demonstrated in [12], a fully canonical form is achieved and the cross-couwhen both the input/output coupling are introduced in the circuit. The isolation level in pling the out-of-band region of the filter can be used to determine the appropriate level of coupling strength between the source and load ports [16]. At this point, it is worth noting that some topology dimensions have a natural physical constrain. For example, it is clear that the total height of the cavity separating the ground planes is determined by the total thickness of the filter layers. The size of the perturbation via between the loop and square patch is also determined by the substrate thickness

LUGO AND PAPAPOLYMEROU: PLANAR REALIZATION OF TRIPLE-MODE BANDPASS FILTER USING MULTILAYER CONFIGURATION

at the middle layers. Given these constrains, a computer optimization is conducted using only a minimum number of variable parameters. In this case, the variables in consideration are the coupling patch size and the tuning patches with size . The size constrain of the perturbation via forces the coupling to exist within a small variation of negative coefficient values. This unknown coupling value is calculated by matching the variation in the input phase reflection when the inverter is introduced. The negative value inverter causes the final filter response to have the third transmission zero in the upper side of the passband. This result is in good agreement with the known cross-coupling response of the filter, which produces a transmission zero at the upper side under the condition and a lower side transmission zero under the [12]. A slight adjustment in the condition resonant frequencies of the filter is produced when the tuning patches are varied. The fractional bandwidth of the filter is mostly affected by the inter-resonator coupling , which, in turn, are directly proportional to the coupling size .

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Fig. 6. Equivalent-circuit model for a triple-mode filter with fully canonical response.

IV. FILTER DESIGN METHODOLOGY A. Resonant Cavity Here we present a design outline for a triple-mode fully GHz. The goal is to procanonical form filter with duce a third-order response with three transmission zeros. The following procedure can be followed to scale the filter to an arbitrary frequency band. The current design requires an initial and . For calculation of the resonant cavity dimensions simplicity, the cavity is design under the constrains and . This constrain guarantees that the dimensions of the resonant loop remains within the and boundaries of the cavity. Setting GHz in (1) and (2), it occurs when follows that the resonance from mode mm. The different layers of the filters are mm, mm, simply located at heights and mm, where is the total height of the filter or cavity dimension . At this point, the sides of the filter are mm. The center replaced by metal vias with diameter of the via is placed along the cavity wall. This allows some of the standard planar attributes such as integration, system on package (SOP), and compactness of design.

Fig. 7. Simulated response for p = 0:508; 0:635;and 0:762 mm and optimal r = 0:556 mm.

Fig. 8. Simulated response for r = 0:482; 0:556; and 0:635 mm and optimal p = 0:635 mm.

B. Square Loop The square-loop resonator is designed with arm sizes mm. The first resonance split for the square-loop and is produced with introduction of the modes capacitive square patches [7]. The size of these patches affect , and also the coupling between the synchronous resonators gives a degree of freedom, allowing the control of the location of the two loop resonances. This will be one of the variables used for fine tuning and optimization of the final response. Its value is dependant on the fractional bandwidth , and external . quality factor of the filter

C. Input/Output Ports The ports are designed with the standard 90 orthogonal feed. This allows the dual-mode operation of the loop and also creates one of the transmission zero on the lower side of the passband. This effect has been studied in [12] and it is attributed to the inis calcutroduction of the coupling between source and load. lated from the isolation level of the filter, as shown in [16]. The ports are designed as strip lines with characteristic impedance .

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Fig. 9. Construction layers for the fabricated triple-mode filter.

D. Coupling Via and Optimization The coupling via with diameter mm and square patch are introduced to control both the inter-resonance and the mode resonant frequency . couplings This corresponds to the asynchronous middle resonator from the equivalent-circuit model in Fig. 6. A variation in the diameter of and mm the perturbation via shows very little effect on the overall response. This is to be expected given the vertical direction of the electric field (permodes. The size pendicular to the ground plane) of the mm was simply chosen because it is well within the dimensions of the perturbation patch and it is large enough to guarantee an appropriate electric contact to both the patch and resonant loop. The structure is then optimized using the perturbation patches and as variables. By computer optimization, it is found that the optimal triple-mode response with fully canonical form is produced mm and mm. Fig. 7 shows the simwhen response for the various values of ulated and mm and a fixed optimal value of mm. Here, the effect of the perturbation patch can bee seen as it directly affects the split of degenerate modes. The case with mm corresponds to the dual-mode operation where only the loop resonances have been coupled. The case with mm shows stronger coupling for and and, therefore, a larger fractional bandwidth . Fig. 8 shows the response for the resonator with fixed optimal mm and mm. In the and variable size case with mm, the synchronous loop resonances are shifted to higher frequencies, resulting in a passband interrupted by a transmission zero. This transmission zero is . The case produced from the introduced cross-coupling with mm shows the resonant loop frequencies shifted mm, to lower values with respect to the optimal case ( mm). The equivalent circuit in Fig. 5 models the behavior of the structure, and the optimal parameters are found , , , to be , nF pH nF, and pH. The optimized dimensions mm , mm , of the filter are

Fig. 10. Measured (solid line) versus simulated (dashed line).

mm mm

,

mm

, and

.

V. FILTER FABRICATION AND RESULTS The substrate used in this filter was Rogers Duroid with a diand and 0.5 oz electric constant of of copper metallization. Fig. 9 shows the fabricated layers. A substrate thickness of 1.27 mm was used for layers 1 and 4 and a thickness of 0.635 mm was used for the middle layers 2 and mm. The via-holes for the 3. The total filter height cavity have a diameter of 1.27 mm, while the perturbation via has a diameter of 0.381 mm. The boards where individually patterned using standard photolithography, while the vias where completely filled using cooper wires of gauges AWG 17 and 27. An aluminum fixture was fabricated with alignment pins of 1/8 in. The fixture served as the ground planes and as the holder for the subminiature A (SMA) connectors. The measurements were conducted after a short, open, load, and thru (SOLT) standard calibration. Excellent agreement can be seen between the measured and simulated transmission and return responses in Fig. 10. The insertion loss of the filter was 1.1 dB, including the loss of the SMA connectors. This shows an advantage over standard dual-mode planar filters with insertion losses ranging from 2.4 dB [7] to 3.1 dB [17]. The largest effective dimension of the mm is also smaller than the waveguide triple filter mode reported in [18] with a largest dimension of 19.6 mm at

LUGO AND PAPAPOLYMEROU: PLANAR REALIZATION OF TRIPLE-MODE BANDPASS FILTER USING MULTILAYER CONFIGURATION

10 GHz. The return loss was better than 11 dB across the band. This may be further improved by increasing the external coupling of the filter. This can be easily accomplished by enlarging the overlapping area between the input/output ports and the res. onant loop. The measured fractional bandwidth was

VI. CONCLUSION A triple-mode bandpass filter has been designed using a multilayer approach for the first time. The filter topology consists of a square loop resonant pattern located inside a reflector cavity. The walls of the cavity were designed using periodic metal vias that connect the top and bottom ground planes for a overall planar configuration. A filter prototype was fabricated and measured, producing a fully canonical filtering function with three resonant poles and three transmission zeros. The filter yielded an insertion loss of 1.1 dB at the passband centered at 5.8 GHz. The final prototype approaches the performance of waveguide filters in terms of loss, while keeping a compact size profile comparable to standard planar filters.

REFERENCES [1] A. E. Williams and A. E. Atia, “Dual-mode canonical waveguide filters,” IEEE Trans. Microw. Theory Tech., vol. MTT-25, no. 12, pp. 1021–1026, Dec. 1977. [2] S. J. Fiedzinsko, “Dual-mode dielectric loaded cavity filters,” IEEE Trans. Microw. Theory Tech., vol. MTT-30, no. 9, pp. 1311–1316, Sep. 1982. [3] L. H. Chua and D. Mirshekar-Syahkal, “Analysis of dielectric loaded cubical cavity for triplemode filter design,” Proc. Inst. Elect. Eng.—Microw., Antennas, Propag., vol. 151, no. 1, pp. 61–66, Feb. 2004. [4] J. Hattori, H. Wakamatsu, H. Kubo, and Y. Ishikawa, “2 GHz band triple mode dielectric resonator duplexer for digital cellular base station,” in Asia–Pacifc Microw. Conf., 2000, pp. 1315–1318. [5] I. C. Hunter, J. D. Rhodes, and V. Dassonville, “Triple mode dielectric resonator hybrid reflection filters,” Proc. Inst. Elect. Eng.—Microw., Antennas, Propag., vol. 145, pp. 337–343, Aug. 1998. [6] A. C. Kundu and I. Awai, “Control of attenuation pole frequency of a dual-mode microstrip ring resonator bandpass filter,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 6, pp. 1113–1117, Jun. 2001. [7] A. Görür, “Description of coupling between degenerate modes of a dual-mode microstrip loop resonator using a novel perturbation arrangement and its dual-mode bandpass filter applications,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 2, pp. 671–677, Feb. 2004. [8] J.-S. Hong, “Coupling of asynchronously tuned coupled microwave resonators,” Proc. Inst. Elect. Eng.—Microw., Antennas, Propag., vol. 147, no. 5, pp. 354–358, Oct. 2000. [9] X. D. Huang and C. H. Cheng, “A novel coplanar-waveguide bandpass filter using a dual-mode square-ring resonator,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 1, pp. 13–15, Jan. 2006. [10] S. J. Park, K. V. Caekenberghe, and G. M. Rebeiz, “A miniature 2.1-GHz low loss microstrip filter with independent electric and magnetic coupling,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 10, pp. 13–15, Oct. 2004. [11] J. S. Hong and M. J. Lancaster, “Bandpass characteristics of new dualmode microstrip square loop resonators,” Electron. Lett., vol. 25, no. 11, pp. 891–892, May 1995. [12] L. H. Chua and D. M. Syahkal, “Analysis and design of three transmission zeros bandpass filter utilizing triple-mode dielectric loaded cubical cavity,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2003, pp. 937–940.

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[13] ——, “Rigorous analysis of effect of input-output direct coupling on triple-mode third-order Chebyshev bandpass filter,” in Eur. Microw. Conf. Dig., Munich, Germany, Oct. 2003, pp. 179–182. [14] W. Steyn and P. Meyer, “A shorted waveguide-stub coupling mechanism for narrowband multimode coupled resonator filters,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 6, pp. 1622–1625, Jun. 2004. [15] A. E. Atia and H.-W. Yao, “Tuning and measurement of couplings and resonant frequencies for cascaded resonators,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2000, pp. 1637–1640. [16] K. A. Zaki, C. Chen, and A. E. Atia, “A circuit model of probes in dual-mode cavity,” IEEE Trans. Microw. Theory Tech., vol. 36, no. 12, pp. 1740–1746, Dec. 1988. [17] A. Görür, “A novel dual-mode bandpass filter with wide stopband using the properties of microstrip open-loop resonator,” IEEE Microw. Wireless Compon. Lett., vol. 12, no. 10, pp. 386–389, Oct. 2002. [18] G. Lastoria, G. Gerini, M. Guglielmi, and F. Emma, “CAD of triplemode cavities in rectangular waveguide,” IEEE Microw. Guided Wave Lett., vol. 8, no. 10, pp. 339–342, Oct. 1998. Cesar Lugo (S’01–A’02–M’06) received the B.S. degree and M.S. degree in electrical and computer engineering from the Georgia Institute of Technology, Atlanta, in 2002 and 2003, respectively, and is currently working toward the Ph.D. degree in electrical engineering at the Georgia Institute of Technology. He has developed several synthesis and design techniques for reconfigurable RF/millimeter-wave components such as filters, antennas, couplers, phase shifters, and impedance tuners. He has authored or coauthored over 15 scientific papers in peer-reviewed journals and conferences. His research interests include hybrid semiplanar design of microwave components and adaptive algorithms for electromagnetic simulation.

John Papapolymerou (S’90–M’99–SM’04) received the B.S.E.E. degree from the National Technical University of Athens, Athens, Greece, in 1993, and the M.S.E.E. and Ph.D. degrees from The University of Michigan at Ann Arbor, in 1994 and 1999, respectively. From 1999 to 2001, he was a faculty member with the Department of Electrical and Computer Engineering, University of Arizona, Tucson. During the summers of 2000 and 2003, he was a Visiting Professor with The University of Limoges, Limoges, France. From 2001 to 2005, he was an Assistant Professor with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, where he is currently an Associate Professor. He has authored or coauthored over 120 publications in peer-reviewed journals and conferences. His research interests include the implementation of micromachining techniques and microelectromechanical systems (MEMS) devices in microwave, millimeter-wave, and terahertz circuits and the development of both passive and active planar circuits on semiconductor (Si/SiGe, GaAs) and organic substrates [liquid-crystal polymer (LCP), low-temperature co-fired ceramic (LTCC)] for system-on-a-chip (SOC)/SOP RF front ends. Dr. Papapolymerou currently serves as the vice-chair for Commission D of the U.S. National Committee of URSI and as an associate editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. During 2004, he was the chair of the IEEE Microwave Theory and Techniques (MTT)/Antennas and Propagation (AP) Atlanta Chapter. He was the recipient of the 2004 Army Research Office (ARO) Young Investigator Award, the 2002 National Science Foundation (NSF) CAREER award, the Best Paper Award presented at the 3rd IEEE International Conference on Microwave and Millimeter-Wave Technology (ICMMT2002), Beijing, China, and the 1997 Outstanding Graduate Student Instructional Assistant Award presented by the American Society for Engineering Education (ASEE), The University of Michigan at Ann Arbor Chapter. His student was also the recipient of the Best Student Paper Award presented at the 2004 IEEE Topical Meeting on Silicon Monolithic Integrated Circuits in RF Systems, Atlanta, GA.

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Design of a Ten-Way Conical Transmission Line Power Combiner Dirk I. L. de Villiers, Student Member, IEEE, Pieter W. van der Walt, Senior Member, IEEE, and Petrie Meyer, Member, IEEE

Abstract—Axially symmetric power combiners, such as radial line and conical line combiners, are very effective in combining the output signals from a large number of power amplifiers over a wide band with low losses. The main problem with radial lines is the behavior of the characteristic impedance against radial distance, which makes design of radial combiners difficult and normally optimization based. In this paper, a step-by-step design procedure is presented for the design of a conical line combiner. The design strategy relies on the transverse electromagnetic properties of the conical line to eliminate the need for complex full-wave optimization in the design process. Circuit models are instead employed and optimized to achieve a wide matched bandwidth. A ten-way prototype was developed at -band, which displayed more than an octave matched bandwidth with low insertion loss. Index Terms—Combiners, conical combiners, conical transmission lines, -way splitters, passive components, radial combiners.

I. INTRODUCTION XIALLY symmetric power combiners, such as radial and conical types, are used extensively in microwave solidstate power amplification systems [1], [2]. They offer a number of advantages over the corporate or chain-type combiners, es, the most pecially for a large number of combining ports important being lower loss and smaller size due to the minimization of path lengths, and improved amplitude and phase balance due to the symmetry. In radial combiners, the energy travels in the radial direction between a central port and axially symmetric peripheral ports within a cylindrical parallel-plate transmission line (radial line). Radial line combiners have been thoroughly investigated, and computer algorithms and design strategies for their analysis have been developed. These design strategies include design using simplified circuit models of the structure [3], electromagnetic (EM) field analysis of the structure [4], and combinations of these two methods [5], [6]. Many radial combiners have been constructed to operate at microwave frequencies. These include 30-way combiners built at 12.5 GHz, demonstrating a 25% bandwidth with a return loss of around 13 dB [6], [7], a 20-way combiner built at 14 GHz, demonstrating a 57% bandwidth with a 17-dB return loss [4], and eight- and 16-way

A

Manuscript received June 30, 2006; revised October 11, 2006. This work was supported by Reutech Radar Systems (Pty) Ltd. The authors are with the Department of Electrical and Electronic Engineering, University of Stellenbosch, Stellenbosch 7600, South Africa (e-mail: [email protected]; [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.2006.890065

combiners built at 10 GHz, demonstrating 20-dB return-loss bandwidths of 33% and 25%, respectively [5]. The combiners in [5] use a conical line in the transition between the central coaxial line and radial line. The main problem with radial lines is the behavior of the characteristic impedance against radial distance [8], [9], which limits performance and makes design of radial combiners difficult and normally optimization based. A conical power combiner is very similar to a radial combiner, except that a conical transmission line is used. Conical transmission lines have the significant advantage that they support a transverse electromagnetic (TEM) mode and, therefore, have a constant characteristic transmission line impedance against radial distance. This greatly simplifies the design and modeling of the structure for broadband applications. It also allows for a simple broadband coaxial-to-conical line transition [10]. The precision construction of the conical transmission line has historically been more difficult than the construction of a simple parallel-plate radial line, however, modern computer numerical control (CNC) lathes machine the conical structure effortlessly. Conical combiners have been built at -band, -band, an demonstrating a 15% bandwidth [11], [12] and, at eight-diode conical combiner was constructed using GaAs IMPATT diodes to generate 17.9 W of output power at 14.6 GHz [13]. No information is, however, given on the design of these structures, and very little is given on performance. In this paper, a design approach is presented for much improved conical line power combiners. Due to the TEM nature of the lines, the approach can use standard TEM theory, combined with a smooth transition design from [10], and stepped impedance coaxial matching networks to achieve bandwidths of up to 74%. The combiner is fed by ten axially symmetric peripheral coaxial lines connected to probes, which couple energy magnetically into a conical transmission line that terminates, at its apex, in a coaxial air line. Stepped impedance matching networks are incorporated into the central and peripheral feeding coaxial lines. The design is simple to execute and, in contrast to radial line designs, requires only one field analysis, together with two optimization procedures using circuit models. For improved performance, the back-short distance of the conical line can be optimally found by a parameter sweep using a field solver such as Computer Simulation Technology (CST)’s Microwave Studio (CST MWS). The constructed combiner has more than an octave bandwidth from 6.5 to 14.1 GHz, with the measured return loss better than 14.5 dB across the band.

0018-9480/$25.00 © 2007 IEEE

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Fig. 3. Construction of coaxial-to-conical line constant impedance transition profile. Fig. 1. Basic configuration of the unmatched conical power combiner.

A. Central Transition Design 50- input ports, the In an -way combiner fed by for impedance of the conical line would normally be parallel ports, or 5 for a the line to be matched to the ten-way combiner. The characteristic impedance of a conical transmission line operating in the TEM mode is found as [8] (1) where the characteristic impedance of free space has been . The characteristic impedance of approximated to an air dielectric coaxial line is given by (2) Fig. 2. Numbering of the ports of a ten-way combiner.

II. COMBINING STRUCTURE A 2-D cross section of the basic structure of the conical transmission line power combiner is shown in Fig. 1, and a sketch showing the port numbering is shown in Fig. 2. The combiner consists of three sections, i.e.: 1) the central coaxial to conical transition; 2) the conical transmission line; and 3) the input port probes. The central coaxial line has an air and , respectively. dielectric and inner and outer radii of The conical line is also air filled and is defined by the angles and . Probes that feed the input coaxial lines are placed at a distance from the axis. A short circuit is placed in the conical line a distance from the center of the peripheral ports. The desired mode within the combining structure is the dominant TEM mode, which has perfect axial symmetry and no circumferential variations. The TEM electric field has a -directed component only, and the magnetic field has a -directed component only. Higher order modes also have axial symmetry, but their polarity change circumferentially every angle (where is the mode number). In order to prevent the excitation of higher modes, which would cause amplitude and phase imbalance between individual peripheral ports, the conical transmission line must be fed symmetrically.

. Equation (1) reduces to (2) when , One of the advantages of using a conical line is the simple broadband coaxial-to-conical transition possible. A profile that connects the 5- coaxial line to the 5- conical line can be defined with two circles that produce a curved conical transmission line of constant impedance from point to point [10]. A side view of this profile is shown in Fig. 3. The details for the design and proof of the constant impedance characteristic of the profile can be found in [10]. This transition produces very low reflections and has much higher peak power-handling capability than the simple transition in Fig. 1. Simulations of the simple and curved transitions and a comparison of the reflection and peak electric field for an input power of 0.05 W is shown in Fig. 4. Using an air-dielectric breakdown field strength of 2.9 MV/m, from Fig. 4, the maximum predicted power-handling capability of the smooth transition is approximately three times higher than for the sharp edged transition. These results should be treated cautiously because of the uncertainty in predicting fields near sharp corners. B. Conical Transmission Line and Peripheral Port Probe Design The conical transmission line in the combining structure is defined by the two angles, i.e., and . Since the impedance of the line is determined by a ratio of these angles, as shown in (1), any one of the angles can be chosen freely (in this case, ).

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Fig. 5. Field simulation model of the combining structure. The model uses magnetic walls to utilize the symmetry of the structure.

Fig. 6. Field simulation results of the combining structure central output port reflection coefficient.

Using the symmetry of the structure, only a quarter of the structure is analyzed. The final value is found as mm. The field simulation model used for the simulation is shown in Fig. 5, and the predicted reflection coefficient at the central port is shown in Fig. 6. Fig. 4. Simulated comparison of a simple (sharp corner) reactive conical/coaxial transition and a curved (rounded corner) constant impedance transition [10]. (a) Reflection. (b) Electric-field strength in the transition at 10 GHz with an input power of 0.05 W and R = 3:22 mm and R = 3:5 mm.

Note that smaller values of would result in a more gradual transition, but with a reduced circumference. To keep the structure as compact as possible, with a large enough circumference to accommodate the ten input ports, is chosen as 90 . can then be calculated from (1) as for a 5- line. The position of the peripheral ports is determined by calculating the circumference needed to accommodate ten subminia(in this case, ture A (SMA) connectors at a certain radius mm). Extendable posts, inserted from the bottom of the structure in Fig. 1, are used as the peripheral probes that are connected to the center pins of the feeding coaxial transmission lines formed by SMA connectors. From [14], larger diameter posts give a wider matched bandwidth with a maximum diameter limited to the SMA dielectric. The widest commercially available post has mm. To determine , a parameter sweep is done with a field solver to give the minimum of at a center frequency of 10 GHz.

III. FEEDING TRANSMISSION LINE MATCHING NETWORKS The only impedance matching section required in the structure is the 5–50- central coaxial line section. However, overall matching bandwidth can be further improved by employing stepped impedance matching networks at each of the peripheral input ports as well. These networks are easily modeled analytically and can, therefore, be optimized very quickly and efficiently using circuit simulation programs such as Applied Wave Research Inc.’s Microwave Office (MWO). A schematic representation of the model to be optimized is shown in Fig. 7. The -parameter block is generated by a field simulation of the combining structure and is not optimized. Only the transmission line sections are optimized. The central output port is on the left-hand side of Fig. 7 and the peripheral input ports are on the right-hand side. A. Peripheral Input Matching Network Description A simple way to construct the transmission lines feeding the input ports would be to use extended dielectric SMA connectors inserted into holes in the metal structure. By removing some of the dielectric, as shown in Fig. 8, two short transmission

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Fig. 7. Schematic representation of circuit to be optimized in MWO. The S -parameter block was generated with a field simulation of the combining structure and needs not be optimized.

Fig. 8. Sketch of an extended dielectric SMA connector inserted into the top of the metal structure to form two short coaxial transmission lines.

lines can be formed with different impedances. The bottom sechas a 50- impedance like the normal SMA tion of length has a variable connector, but the second section of length impedance. The impedance can be calculated as (3) with (4) is the relative permittivity of the dielectric (Teflon: ). The effective relative permittivity of the partially filled coaxial line is found by solving Laplace’s equation within the boundary conditions imposed by the structure, and then calculating the static capacitance, as shown in the Appendix. The variables for optimization are, therefore, the lengths of the two transmission lines ( and ), bearing in mind that the total length of available connectors is constant, and the impedance of the second length of line ( ), which can be varied from 50 to approximately 65 in order not to completely remove the dielectric, which would make insertion into the hole in the metal structure impossible. B. Central Output Matching Network Description The output transmission line matching network is a stepped impedance coaxial airline. The outer conductor diameter is chosen as 7 mm to be compatible with a precision -type connector. The diameter of the inner conductor is stepped to obtain the desired impedance levels. Two sections of lengths and

Fig. 9. Sketch of the stepped impedance coaxial airline feeding the combining conical line.

are used. The optimization variables are, therefore, the lengths of the two sections ( and ), and their impedances, which are dependent on the inner radii ( and ) [see (2)]. Fig. 9 shows the configuration. The inner conductor mm of the -type connector is also shown in this figure. The discontinuities caused by the steps in the inner conductor can be modeled as capacitances to ground [15]. These capacitances are included in the model of Fig. 7 as the lumped-element . capacitors The final values obtained for the parameters are mm, mm, mm, mm, mm, mm, and mm. IV. SUMMARY OF THE STEP-BY-STEP DESIGN PROCEDURE The full design process can be summarized as follows. 1) The combining structure is designed and the exact response of the structure is determined with a full-wave solver. a) Determine the impedance of the conical line from the . number of ports as b) Construct the central transition between conical and coaxial lines, as described in [10], using and calculating from (1). where the input ports are c) Determine the radius placed based on the width of the connectors, the spacing between the connectors, and the number of as small as possible to connectors used. Keep reduce higher order modes.

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Fig. 10. 2-D side view of the conical combiner structure showing all connectors, feeding probes, and fastening screws.

Fig. 11. Simulated and measured reflection coefficient at the central output port.

d) Determine the diameter of the feeding probes. Wider probes give better bandwidth, but the diameter is limited by the outer diameter of the input feeding coaxial lines. e) Determine the length of the back-short using a field can also simulation parameter sweep. (Optional: be used for a slightly detuned performance.) f) Analyze the entire structure with a field solver to get the -parameters at all the ports. 2) Optimize stepped impedance transmission lines on the input and output sides of the combining structure in MWO to achieve a wide matched bandwidth. V. MEASURED RESULTS A 2-D side view of the manufactured structure is shown in Fig. 10 showing all the components of the assembly. The aluminum structure was manufactured with a CNC lathe. The simulated and measured reflection results of the common port of the constructed combiner is shown in Fig. 11. A matched bandwidth of 74% is achieved with a maximum return loss of 14.7 dB from 6.5 to 14.1 GHz. The difference between the

Fig. 12. Measured transmission coefficients in the operating band of the combiner (S , n = 2; 3; . . . ; 11). (a) Amplitude. (b) Phase.

simulated and measured results is caused by the difficulty in accurately simulating and manufacturing a large structure with very narrow gaps. The gap in the central coaxial-to-conical transition is only 0.28 mm compared to the diameter of the structure, which is 65.2 mm. The measured transmission characteristics of the combiner is shown in Fig. 12. A maximum amplitude imbalance of 1 dB and a phase imbalance of 5 is observed in the 8–12-GHz band, and an amplitude imbalance of 1.5 dB and a phase imbalance of 10 is observed in the 6–14-GHz band. The simulated and measured isolation characteristics of the combiner are shown in Fig. 13 where good agreement between the results is demonstrated. It is noted that the worst isolation is between ports 2 and 7 (the ports located at the opposite sides of the combiner), which is consistent with the remarks made in mode when the [4]. This is caused by the excitation of the structure is driven in an unsymmetrical fashion. As reported in [4], these values are consistent with reactive radial combiners. No attempt was made to improve the isolation of the combiner,

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Fig. 14. Measured insertion loss in the operating band of the combiner.

optimization of the entire structure is normally necessary to obtain wideband operation. A ten-way conical combiner was constructed, which demonstrates excellent wideband performance, as well as low losses. The structure is very compact and lightweight and is, therefore, ideally suited for space and airborne applications. APPENDIX EFFECTIVE DIELECTRIC CONSTANT OF A PARTIALLY FILLED COAXIAL LINE

Fig. 13. Simulated and measured isolation in the operating band of the combiner. (a) Simulated. (b) Measured.

The effective permittivity of a partially filled coaxial transmission line, such as the one shown in Fig. 8, can be obtained by solving the static capacitance of the line and comparing it to the static capacitance of a normal coaxial line filled with dielectric. The potential distribution in a partially filled coaxial line is found by solving Laplace’s equation in cylindrical coordinates and applying the proper boundary conditions. This is straightforward, and the potential is given by [9] as (5)

but it is known that the isolation will improve when the number of ports is increased [5]. Fig. 14 shows the total insertion-loss characteristic of the combiner. The maximum loss in the operating band is 1.1 dB. The loss includes the effects of the SMA transitions, as well as an SMA to -type transition used in the measurement setup.

(6) The capacitance follows from integrating the surface charge density per unit length over the inner electrode to find

VI. CONCLUSION A simple technique has been presented for designing conical transmission line power combiners. The technique is general and may be applied to the design of similar -way conical line combiners. Due to the TEM properties of conical transmission lines, the design technique calls for very little full-wave model optimization, but instead relies on optimization of analytic transmission line models to achieve a wide operating bandwidth. This is in contrast to the design of radial line combiners, which do not support a pure TEM transmission mode, where full-wave model

(7) with the capacitance given by (8)

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This can be rewritten in a form resembling the capacitance of a normal coaxial line given by [16] (9) as (10) The effective permittivity can be found from (9) and (10) as (11) with the characteristic impedance following as (12)

ACKNOWLEDGMENT The authors would like to thank Reutech Radar Systems (Pty) Ltd., Stellenbosch, South Africa, for the financial support of this project, and Applied Wave Research Inc., El Segundo, CA, and Computer Simulation Technology, Darmstadt, Germany, for the use of software licenses. The authors would also like to thank U. Buttner and W. Croukamp, both with the Department of Electrical and Electronic Engineering, University of Stellenbosch, Stellenbosch, South Africa, for valuable help during construction of the prototype. REFERENCES [1] K. J. Russel, “Microwave power combining techniques,” IEEE Trans. Microw. Theory Tech., vol. MTT-27, no. 5, pp. 472–478, May 1979. [2] K. Chang and S. Cheng, “Millimeter-wave power-combining techniques,” IEEE Trans. Microw. Theory Tech., vol. MTT-31, no. 2, pp. 91–107, Feb. 1983. [3] G. W. Swift and D. I. Stones, “A comprehensive design technique for the radial wave power combiner,” in IEEE MTT-S Int. Microw. Symp. Dig., May 1988, pp. 279–281. [4] M. E. Bialkowski and V. P. Waris, “Electromagnetic model of a planar radial-waveguide divider/combiner incorporating probes,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 6/7, pp. 1126–1134, Jun./Jul. 1993. [5] S. Nogi, F. Okazaki, and K. Fukui, “A broadband conical-radial wave power divider/combiner,” in Asia–Pacific Microw. Conf., Dec. 1994, pp. 507–510. [6] A. E. Fathy, S. Lee, and D. Kalokitis, “A simplified design approach for radial power combiners,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 247–255, Jan. 2006. [7] E. Belohoubek, R. Brown, H. Johnson, A. Fathy, D. Bechtle, D. Kalokitis, and E. Mykietyn, “30-way radial power combiner for miniature GaAs FET power amplifiers,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1986, pp. 515–518. [8] N. Marcuvitz, Waveguide Handbook. London, U.K.: Peregrinus, 1986.

[9] S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, 3rd ed. New York: Wiley, 1994. [10] P. W. van der Walt, “A novel matched conical line to coaxial line transition,” in Proc. South African Commun. Signal Process. Symp., Sep. 1998, pp. 431–434. [11] K. J. Russel and R. S. Harp, “Broadband diode power-combining techniques,” Air Force Avion. Lab., Wright-Patterson Air Force Base, Wright-Patterson AFB, OH, Interim Tech. Rep. 1, Mar. 1978. [12] R. S. Harp and K. J. Russel, “Conical power combiner,” U.S. Patent 4 188 590, Feb. 1980. [13] J. P. Quine, J. G. McMullen, and D. D. Khandelwal, “ -band IMPATT amplifiers and power combiners,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1978, pp. 346–348. [14] R. Collin, Field Theory of Guided Waves, 2nd ed. New York: IEEE Press, 1991. [15] P. Somlo, “The computation of coaxial line step capacitances,” IEEE Trans. Microw. Theory Tech., vol. MTT-15, no. 1, pp. 48–53, Jan. 1967. [16] D. Pozar, Microwave Engineering, 2nd ed. New York: Wiley, 1998.

Ku

Dirk I. L. de Villiers (S’05) was born in Langebaan, South Africa, on October 13, 1982. He received the B.Eng degree in electrical and electronic engineering from the University of Stellenbosch, Stellenbosch, South Africa, in 2004, and is currently working toward the Ph.D. degree in electrical and electronic engineering at the University of Stellenbosch. His current research interests include passive devices, high-power microwave amplifiers, and theoretical electromagnetics.

Pieter W. van der Walt (SM’80) was born in Germiston, South Africa, on April 12, 1947. He received the B.Sc., B. Eng., M. Eng., and Ph.D. degrees in electronic engineering from the University of Stellenbosch, Stellenbosch, South Africa, in 1970, 1973, and 1982, respectively. In 1971, he joined the Department of Electrical and Electronic Engineering, University of Stellenbosch, and served as Dean from 1993 to 2002. Upon his retirement from the university, he joined Reutech Radar Systems (Pty) Ltd., Stellenbosch, South Africa, as Technology Executive responsible for RF technology. His main research interests include network synthesis and linear and nonlinear circuit design. Prof. van der Walt is a senior member of the South African Institute of Electrical Engineers (SAIEE), and is currently chairman of the IEEE Antennas and Propagation (AP)/Microwave Theory and Techniques (MTT) Chapter of the South Africa section.

Petrie Meyer (S’87–M’88) was born in Bellville, South Africa, in 1965. He received the B.Eng., M.Eng., and Ph.D. degrees from the University of Stellenbosch, Stellenbosch, South Africa, in 1986, 1988, and 1995, respectively. In 1998, he joined the Faculty of Engineering, University of Stellenbosch, where he is currently a Professor of microwave engineering. His current research interests include EM analysis, passive devices, and mathematical modeling techniques.

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Five-Level Waveguide Correlation Unit for Astrophysical Polarimetric Measurements Giuseppe Virone, Riccardo Tascone, Member, IEEE, Massimo Baralis, Augusto Olivieri, Oscar Antonio Peverini, and Renato Orta, Senior Member, IEEE

Abstract—Recent astrophysical measurements require an accurate evaluation of the correlation product between very weakly correlated signals. Waveguide correlation units represent a very appealing solution for these applications because of their stability and low-loss features. A new waveguide correlation unit architecture is discussed in this paper. It extends over five levels in the -plane and integrates hybrid couplers, filters, a phase shifter, matched loads and various matched transitions. All the aspects concerning the analysis, design, and manufacturing procedures are described, focusing on the reduction of the spurious correlation -band prototype has terms (systematic errors). An aluminum been designed and experimentally tested. The device exhibits spurious correlation parameters of the order of 30 dB, in a 10% frequency band, in good agreement with the predicted data. Thanks to this feature, the presented architecture is suitable for high-performance correlation polarimeters, where very high sensitivity to the relevant signal, as well as very high rejection of both the unpolarized component of the radiation under test and the instrument noise, are required. Index Terms—Correlation units, directional couplers, filters, hybrid phase discriminators, phase shifters (PSs), polarimeters, polarization measurements, radiometers, Stokes parameters, waveguide components, waveguide networks.

and are the spectral distribuof the sky emission, where tions of the electric-field Cartesian components, and the symbol denotes the spectral average in the operating frequency band

These measurements are accomplished by a set of correlation polarimeters [4], [5] that cover the - to -band frequency range. In each polarimeter, the radiation under test is collected by a corrugated horn working in dual circular polarization. A waveguide polarizer [6] then converts the circular polarizations into two linear ones and an orthomode transducer [7] separates them into two different rectangular waveguides [8]. Therefore, at the orthomode transducer outputs, under ideal conditions, the available signals are

(2)

I. INTRODUCTION

T

HE astrophysical community has demonstrated great interest in the construction of multifrequency polarization maps of the galactic synchrotron emission and of the faint cosmic microwave background polarization. Since these signals, hereinafter called sky polarization, are several orders of magnitude smaller than the unpolarized radiation, instruments with very high sensitivity, stability, and rejection of the unpolarized component are needed. In this framework, the Sky Polarization Observatory project (SPOrt) [1] and the Baloon-borne radiometers for the Sky Polarization Observations project (Bar-SPOrt) [2] aim at the measurement of the and Stokes parameters [3]

(1)

Manuscript received August 22, 2006; revised November 2, 2006. This work was supported by the Italian Space Agency (ASI) under the framework of the Sky Polarization Observatory (SPOrt) project (selected by the European Space Agency). The authors are with the Istituto di Elettronica ed Ingegneria dell’Informazione e delle Telecomunicazioni, I-10129 Turin, Italy (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.890066

By substituting (2) in (1), one obtains that both the and Stokes parameters can be simultaneously evaluated as the real and imaginary parts of the correlation product between the signals and (i.e., the two components of the incoming radiation in the circular polarization basis)

(3) One of the most important features of the SPOrt and Bar-SPOrt polarimeters is that the correlation product is analogically performed at the antenna frequency using a passive device called the waveguide correlation unit. Thus, the typical problems of heterodyne receivers [9], where the correlation product is evaluated after a down-conversion stage, are avoided. This paper deals with a new waveguide correlation unit architecture, where all the required passive components are integrated and arranged to obtain a very compact device, as well as to provide a symmetric structure. In this way, the manufacturing procedure is simplified so that the corresponding systematic errors are reduced. Inside the unit, the correlation product is carried out by four hybrid directional couplers and a phase shifter (PS). Two input waveguide filters and two additional directional couplers are used to define the operating

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Fig. 1. Scheme of the five-level waveguide correlation unit. Apart from the correlator itself, which is composed of the four hybrid directional couplers DC , with k = 1; . . . ; 4, and the 90 PS, two input filters F and F and two 9-dB directional couplers DC and DC are also integrated. The meanings of the various arrows are explained in the text.

frequency band and to monitor the total power level of the incoming signals, respectively. Eight transitions to standard waveguides and four matched loads are also integrated. The configuration extends along five levels using - and -plane discontinuities to form the various components. For this reason, it has been called the five-level waveguide correlation unit (European Patent 05019553.6). The scheme of the proposed device and the corresponding specifications are discussed in Section II, where an accurate treatment of the systematic errors is reported, with particular emphasis on the rejection of the unpolarized radiation. The architecture and component layout are described in detail in Section III. Section IV reports the analysis and synthesis -band prototype. The cortechniques used to design the responding predicted data for the latter are compared to the experimental results in Section V. II. SCHEME AND WORKING PRINCIPLE The scheme of the waveguide correlation unit is shown in Fig. 1. The correlator itself consists of four hybrid directional , with ) and a 90 PS. The same circouplers ( cuit has been used in [10] for radar applications and in [11] for direct-conversion receivers. From the beam-forming network perspective, it can be considered as a 4 4 Butler matrix for which a waveguide implementation is described in [12]. A similar architecture has also been proposed in [13], as an alternative to network analyzer measurements, and a substrate-integrated-waveguide implementation has been described in a recent study [14]. Apart from the correlator itself, the scheme of Fig. 1 contains and ) with the purpose of defining two waveguide filters ( the instrument frequency band, and two 9-dB directional couand ) for monitoring the signal level in the plers ( two channels. With reference to Fig. 1, the two input signals and are and , respectively. Subsequently, a fraction is filtered by and to supply the power monitoring drawn off by

and , respectively. The remaining signals ( outputs and ) are equally split into the two branches. Since a high isolation level between the two branches is required, two hybrid and ) are used instead of two directional couplers ( power splitters [14]. In the upper branch (called the -branch), and are combined by the hybrid coupler the signals to produce the output signals and . As a result, the latter and , respectively. are ideally proportional to underIn the lower branch (called the -branch), the signal phase shift with respect to . and are then goes a combined by the hybrid coupler . Therefore, in the ideal and are proportional to case, the output signals and , respectively. with (as well as The detection of the outputs and ) is performed by square-law diodes. Subsequently, the converted signals are sent to two differential instrumentation amplifiers. Hence, after a low-pass filtering, the obtained signals (apart from proportionality factors) are

(4) By substituting the ideal expressions of the outputs in (4), it and are proporresults that the low-frequency outputs tional to the real and imaginary parts of the correlation product , respectively,

(5) From (5), it should be noted that if the signals and correspond to the outputs of an antenna system with dual circular poand will be proportional larization, as in (2), the outputs to the and Stokes parameters of the incoming radiation [see (3)]. and It should be noted that, in the ideal case, the signals do not depend on the quantities and because, in (5), they are eliminated by cancellation. This is no longer true in

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the real case, where a perfect cancellation does not occur, owing to the nonideal behavior of the various components. Therefore, and are affected by the values of both the quantities and . For this reason, even in the case of uncorrelated , a spurious output signal (proincoming signals and ) will be present. portional to This behavior is extremely critical in sky polarization measurements, where the levels of and are related to both the unpolarized component of the incoming radiation and to the instrument noise referred at the antenna section. These quantities, which can also be described in terms of the Stokes parameters and

(6) are several orders of magnitude higher than and . As a consequence, the waveguide correlation unit has to be designed to and , as much reduce the spurious output signal, due to as possible. In other words, it has to provide a very high rejection of the and Stokes parameters of the incoming radiation. This rejection requirement can be expressed in terms of the (with and ) scattering parameters of the circuit depicted in Fig. 1. By substituting the correin (4), one obtains sponding

(7) Assuming that and have a constant spectral distribution in the operating frequency band and exploiting (3) and (6), the following matrix relationship can be introduced: (8) where

Fig. 2. Exploded view of the five-level waveguide correlation unit. The eight standard-waveguide connection flanges, the two covers [(a) and (e)], the two plates [(b) and (d)] containing the waveguides, and the central coupling layer (c) are visible.

is close to 0 dB, so that , , and , all the elements of the matrix approach zero. Therefore, a high level of rejection of the and parameters can be obtained by designing the hybrid couplers and with a very low tracking error. In order to obtain a rejection level of 30 dB (SPOrt and Bar-SPOrt requirements), the average value (in the frequency band) of the tracking error has to be less than 0.02 dB. Equation (9) also shows that two cross-coupling terms and , which correspond to a rotation error in the – -plane, exist. Assuming that the above-mentioned requirement on the coupler tracking error is met, it is simple to and approach zero when the transfer prove that and are almost identical, functions corresponding to with . For this reason, the i.e., correlation unit has to be designed to assure good symmetry between the two input signal paths even though significant manufacturing uncertainties are present. III. FIVE-LEVEL ARCHITECTURE

(9)

(10) From these equations, it is possible to observe that (as previand are ously stated) in the real case, the parameters through the affected by the other Stokes parameters and matrix . However, if the tracking (ratio between the transmission coefficients of the direct and coupled port) of and

The direct implementation of the circuit scheme of Fig. 1 involves a cross-like geometrical configuration, which does not guarantee good symmetry between the two channels. In order to overcome this problem, and according to the previously defined requirements (see Section II), a new architecture has been conceived. The configuration is depicted in Fig. 2. It is a five-layer structure where the rectangular waveguides are obtained within two plates (b) and (d) of constant thickness. These plates are separated by a 0.1-mm-thick metal sheet (c) (the thickness of this layer can be the same for all the operating frequency bands) and are closed by two covers (a) and (e) where the eight input/output waveguide flanges are placed. The components layout is shown in Fig. 3. The directional couplers (dashed line contoured) are obtained by coupling two parallel waveguide lengths. This task is accomplished by means

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Fig. 3. Layout of the five-level waveguide correlation unit. From the top: left waveguide plate (b), central coupling layer (c), and right waveguide plate (d). Input/output ports: solid line. Filters: dotted line. Directional couplers: dashed line. PS: dashed–dotted line.

of -plane rectangular apertures in the central layer (c) (narrow wall couplers) [15], [16], with the same height as the waveguide height. The PS (dashed–dotted line contoured) is a cascade of -plane stubs, which are made up of rectangular holes in the waveguide plate (d) and the central layer (c). In order to maintain a high level of integration, the two waveguide filters (dotted line contoured) are obtained by a cascade of -plane irises. The matched loads (i.e., pyramidal absorbers) connected to the diwith are also integrated. rectional couplers The various components are connected together with different types of matched transitions, forming a snake-like geometry, which expands on five levels in the -plane. With reference to Fig. 1, the white arrows represent direct (straight) connections, whereas the dashed arrows indicate the 180 -plane bends, which are composed of three sections of uniform bend (piecewise uniform bends) [17]. The shaded arrows correspond to the input/output L-shaped junctions, which are -plane 90 abrupt bends, matched with -plane steps. Although the dimensions of the internal waveguides were chosen in order to minimize the dispersion effects of the directional couplers, thanks to the L-shaped junctions, the external connections use standard rectangular waveguides (highlighted with solid-line circles in Fig. 3). -plane piecewise uniform As depicted in Fig. 3, two 90 bends (after the input L junctions) and two -plane shaped and ) are L junctions (before the matched loads of

present in order to align the two input ports and to reduce the overall dimensions, respectively. If the scheme of Fig. 1 is divided into four quadrants as indicated, one can observe that the components that belong to the I and III quadrants are realized on the left waveguide plate (b) of Fig. 3, whereas those that belong to the II and IV quadrants are realized on the right one (d). An important feature of the current architecture is that all the mechanical parts have only thru-holes. Hence, they can be manufactured by wire spark erosion, which is one of the most accurate manufacturing techniques. Moreover, as can be noted from Fig. 3, the two waveguide plates (b) and (d) are identical (apart from a 180 rotation). Consequently, they can be simultaneously manufactured by overlapping the two pieces. This expedient leads to a high symmetry of the overall device, regardless of the manufacturing uncertainties. In this way, and are achievable (see low levels of the parameters Section II), otherwise the electrical symmetry between the two plates (b) and (d) would be critical because of the two integrated filters. Finally, it should be noted that the two waveguide plates (b) and (d) can be manufactured before the central coupling layer (c) so that the directional couplers can be designed on the basis of the actual measured waveguide broad wall dimensions and central metal sheet thickness. In this way, very low tracking errors can be obtained, leading to low levels of the elements.

VIRONE et al.: FIVE-LEVEL WAVEGUIDE CORRELATION UNIT FOR ASTROPHYSICAL POLARIMETRIC MEASUREMENTS

IV. ANALYSIS AND DESIGN -band fiveOn the basis of the described architecture, a level waveguide correlation unit has been designed with the aid of several analysis and synthesis tools. Apart from the piecewise uniform bends (see [17]), the analysis of the various components was carried out by applying the method of moments with weighted Gegenbauer polynomials as basis functions to represent the aperture field distribution with the right edge conditions [18]. The projections of the basis functions onto the modal sets, for each kind of discontinuity, were evaluated in the spectral domain by exploiting the analytical knowledge of their Fourier transform. The material losses were taken into account by introducing the relevant impedance boundary conditions in the moment method application [19]. In this way, the loss phenomena were accurately modeled, also including the losses produced by the evanescent fields excited by the discontinuities. Finally, the scattering matrix of the full network was obtained by cascading the generalized scattering matrices of the various components [20]. The 13-cavity filters were designed using the synthesis technique presented in [21]. It is mainly based on a distributed-parameter-circuit model of the device and on the identification of an abstract system to account for losses, frequency dispersion, and higher order mode interaction phenomena. A similar procedure was used to design the various transitions since they can be considered as filtering structures where the dimensions of the input and output ports are assigned and the most important feature is not the out-of-band rejection, but the matching. As far as the directional couplers are concerned, the synthesis was carried out by describing the device in the sum and difference mode basis and applying the waveguide polarizer design technique reported in [6]. The desired coupling level and the return loss/directivity can, in fact, be obtained by simultaneously controlling the differential phase shift between the transmission coefficients of these two modes (e.g., 90 for a 3-dB coupling level and 41.5 for a 9-dB coupling level) and their reflection coefficients. The hybrid directional couplers were designed with eight apertures in order to satisfy the severe requirement on the tracking error reported in Section II; four apertures were instead used for the 9-dB coupler. A similar strategy was also adopted for the design of the ninestub PS. V. EXPERIMENTAL RESULTS -band silverHere, the experimental results concerning a plated aluminum prototype are compared to the simulations. The spark-erosion manufactured device is shown in Figs. 4 and 5. The input/output waveguides are WR28. The torque of the mounting screws (M4) is 1.5 Nm. The pyramidal absorbers are made of ECCOSORB MF-190. First, the 8 8 scattering matrix of the prototype was measured by means of a two-port vector network analyzer (VNA). Fig. 6 shows the transmission coefficients from input ports and to the power monitoring ports and . Both parameters approximately exhibit a 9.4-dB level in the 10% passband centered at 32 GHz (in the lossless case, it would have

Ka

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Fig. 4. Front view of the -band five-level waveguide correlation unit prototype. The dimensions are 257 mm 82 mm.

2

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Fig. 5. Five layers of the -band five-level waveguide correlation unit prototype. The two covers, the two plates containing the waveguides, and the central metal sheet.

been 9.1 dB) and a high out-of-band rejection, which is in excellent agreement with the simulated results (an effective resiscm was used in the simulations). tivity value of 4.5 With the same measurement setup, return losses of the order of 30 dB and isolations better than 35 dB were found at the input/ output ports. The input port reflection coefficients are shown in Fig. 7. Although some slight discrepancies between the measurements and simulations are present, due to manufacturing uncertainties, the frequency responses of the two channels are very similar. and , respecFigs. 8 and 9 show the ratios tively. These quantities, which are the most important ones of the correlation unit (since they are related to the rejection of the and Stokes parameters) are in good agreement with the predicted data. It should be noted that, as appears from Fig. 1, and correspond to the tracking of the couand , respectively. and were specifplers ically designed to exhibit a cubic-like frequency behavior in order to minimize the tracking error (see Section II).

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Fig. 6. Transmission coefficients S and S of the Ka-band prototype. The level is approximately 9.4 dB for both parameters in the 10% frequency band centered at 32 GHz.

Fig. 8. Magnitude of the ratio S =S in the 10% frequency band. This parameter corresponds to the tracking of the coupler DC .

Fig. 7. Reflection coefficients at the input ports of the Ka-band prototype.

Fig. 9. Magnitude of the ratio S =S , in the 10% frequency band. This parameter corresponds to the tracking of the coupler DC .

Thanks to the symmetrical five-level waveguide correlation unit architecture, the various transmission ratios were also measured by exploiting an indirect technique. Each transmission ratio was obtained from two reflection measurements at the corresponding output (or input) ports. In this way, any movement of the connecting cables is avoided because the VNA cables maintain the same position they had during the thru-reflect-line (TRL) calibration procedure. For example, the indirect measurewas carried out by the following: ment of and ; • connecting the VNA to the output ports • performing a reflection measurement with matched loads connected to the input ports and , in order to obtain and (the other ports were the reflection coefficients also terminated on matched loads); • substituting the matched load at input port with a reactive load (e.g., a short circuit) and measuring the corresponding and at ports and , rereflection coefficients spectively.

At this point, since

0

and

can be written as

where is the reflection coefficient of the reactive load (which can be computed does not need to be known), the ratio as

Obviously, the out-of-band behavior cannot be characterized with this technique. Good results were obtained using the previously described method. Figs. 8 and 9 show the excellent agreement between the direct measurements, indirect measurements and predicted

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Fig. 10. Phase of the ratio S =S . This parameter represents the phase equalization of the two channels in the U -branch.

data. The average value of the tracking error (difference from 0 dB) in the operating frequency band is less than 0.02 dB. The phase equalization between the two channels and is another important parameter since it is related to the levels of and (see Section II). Both the measurements and are predicted data that correspond to the phase of reported in Fig. 10. The indirect measurement was performed and and the by connecting the VNA to the input ports reactive/matched loads to port . The measured data are in good agreement with the simulations. The difference of approximately two degrees between the two measurements is due to the essentially depends connecting cables. The phase of on the phase differences between the transmission coefficients and has of the two 13-cavity filters. Since the phase of a variation of approximately 3500 in the passband and their difference is instead of the order of few degrees, it is possible to state that the two filters are practically identical. This result and the high similarity of the two input reflection coefficients (see Fig. 7) are due to both the device architecture and its manufacturing strategy, which leads to a high degree of symmetry between the two input channels. Since the results of Fig. 10 prove phase equalization between the two filters, the error of approximately 1 between the measured (indirect) and the computed phase of the transmission , shown in Fig. 11, should be totally attributed ratio to the manufacturing uncertainties of the PS. As far as the direct measurement is concerned, even in this case, the discrepancies between the two measurements are due to the connecting cables. The previous measurements were elaborated, according to (9) and (10), to obtain the spectral distribution of the elements of the and are reported matrices and . The direct terms in Fig. 12, showing a high out-of-band rejection in excellent agreement with the predicted data. In the passband, the levels and 4.1 dB for are approximately 4.2 dB for (in the lossless case, both levels would have been 3.6 dB). In and are better than Fig. 13, the cross-coupling terms 20 dB. The discrepancy between the predicted and measured data is due to the small phase errors reported in Figs. 10 and 11.

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Fig. 11. Phase of the ratio S =S . This parameter represents the phase equalization of the two channels in the Q-branch.

Fig. 12. Diagonal (direct) terms of the transmission matrix H . The transmisand 4.1 dB for H . sion levels in the passband are 4.2 dB for H

0

0

Fig. 13. Off-diagonal (cross-coupling) terms of the transmission matrix H .

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carried out by very efficient synthesis and analysis techniques, entails that the analogically computed correlation product is only very slightly affected by the uncorrelated components of the input signals. As a result, if the signals entering the unit correspond to the outputs of a dual circularly polarized antenna feed system, a high rejection of the Stokes parameters and of the incoming radiation occurs so that an accurate measurement of and can be performed. -band prototype has been built. The comparison beA tween the measured and theoretical data shows both the accuracy of the adopted analysis tools and the success of the proposed manufacturing solution. The presented results would seem to encourage a future realization of a -band waveguide correlation unit with the same five-level configuration. Fig. 14. Elements of the matrix

Fig. 15. Elements of the matrix

K corresponding to the Q-branch.

K corresponding to the U -branch.

Finally, Figs. 14 and 15 show the spectral distribution of the elements of , which quantify the rejection of the and Stokes parameters (i.e., the spurious correlation terms). and are of the order of 30 dB and and are better than 32 dB, thus fulfilling the specifications. It should be noted that the predicted value of 70 dB for is meaningless from an experimental point-of-view. It was obtained from (10) as the difference between two almost identical . Hence, quantities its sensitivity, with respect to the manufacturing and measuring uncertainties, is extremely high (the same phenomena does not since the PS is present). For this reason, even occur for these measured data can be considered in good agreement with the simulations. VI. CONCLUSION A new waveguide architecture has been conceived after a detailed analysis of the correlation unit systematic errors. The high symmetry of the structure and its particular design, which was

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[15] H. Schmiedel and F. Arndt, “Field theory design of rectangular waveguide multiple-slot narrow-wall couplers,” IEEE Trans. Microw. Theory Tech., vol. MTT-34, no. 7, pp. 791–798, Jul. 1986. [16] T. Shen, Y. Rong, and K. A. Zaki, “Full-wave optimum design of millimeter-wave rectangular waveguide narrow-wall branch couplers,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 13–19, 1999, vol. 4, pp. 1729–1732. [17] G. Virone, R. Tascone, M. Baralis, O. A. Peverini, A. Olivieri, and R. Orta, “Piecewise-uniform bends in rectangular waveguide,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 4, pp. 289–291, Apr. 2005. [18] T. Rozzi, F. Moglie, A. Morini, W. Gulloch, and M. Politi, “Accurate fullband equivalent circuits of inductive posts in rectangular waveguide,” IEEE Trans. Microw. Theory Tech., vol. 40, no. 5, pp. 1000–1009, May 1992. [19] O. A. Peverini, R. Tascone, M. Baralis, G. Virone, and D. Trinchero, “Reduced-order optimized mode-matching CAD of microwave waveguide components,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 311–318, Jan. 2004. [20] T. Itoh, Numerical Techniques For Microwave and Millimeter-Wave Passive Structures. New York: Wiley, 1989. [21] R. Tascone, P. Savi, D. Trinchero, and R. Orta, “Scattering matrix approach for the design of microwave filters,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 3, pp. 423–430, Mar. 2000.

Massimo Baralis was born in Turin, Italy, in 1970. He received the Laurea degree in electronics engineering from the Politecnico di Torino, Turin, Italy, in 2001. Since May 2001, he has been with the Electromagnetics Group, Istituto di Elettronica e di Ingegneria dell’Informazione e delle Telecomunicazioni (IEIIT), Italian National Research Council (CNR), Turin, Italy. His research is focused on the application of numerical methods and measurements techniques and concerns the analysis and design of microwave and millimeter-wave passive components, feed systems, and antennas.

Giuseppe Virone was born in Turin, Italy, in 1977. He received the Electronic Engineering degree (summa cum laude) and Ph.D. degree in electronic and communication engineering from the Politecnico di Torino, Turin, Italy, in 2001 and 2006, respectively. In 2002, he joined the Istituto di Elettronica e di Ingegneria Informatica e delle Telecomunicazioni (IEIIT), Italian National Research Council (CNR), Turin, Italy, as a Research Assistant. He is currently a Researcher with the IEIIT, CNR. His research activities concern the design and numerical analysis of microwave and millimeter-wave passive components for feed systems, antennas, frequency-selective surfaces, compensated dielectric radomes, and industrial applications.

Oscar Antonio Peverini was born in Lisbon, Portugal, in 1972. He received the Laurea degree (summa cum laude) in telecommunications engineering and Ph.D. degree in electronics engineering from the Politecnico di Torino, Turin, Italy, in 1997 and 2001, respectively. From August 1999 to March 2000, he was a Visiting Member with the Applied Physics/Integrated Optics Department, University of Paderborn, Paderborn, Germany. In February 2001, he joined the Istituto di Ricerca sull’Ingegneria delle Telecomunicazioni e dell’Informazione (IRITI), an institute of the Italian National Council (CNR). Since December 2001, he has been a Researcher with the Istituto di Elettronica e di Ingegneria dell’Informazione e delle Telecomunicazioni (IEIIT), a newly established institute of the CNR. He teaches courses on electromagnetic field theory and applied mathematics at the Politecnico di Torino. His research interests include numerical simulation and design of surface acoustic wave (SAW) waveguides and interdigital transducers (IDTs) for integrated acoustooptical devices, of microwave passive components and radiometers for astrophysical observations, and microwave measurements techniques.

Riccardo Tascone (M’02) was born in Genoa, Italy, in 1955. He received the Laurea degree (summa cum laude) in electronic engineering from the Politecnico di Torino, Turin, Italy, in 1980. From 1980 to 1982, he was with the Centro Studi e Laboratori Telecomunicazioni (CSELT), Turin, Italy, where his research mainly dealt with frequency-selective surfaces, waveguide discontinuities, and microwave antennas. In 1982, he joined the Centro Studi Propagatione e Antenne (CESPA), Turin, Italy, of the Italian National Research Council (CNR), where he was initially a Researcher and, since 1991, has been a Senior Scientist (Dirigente di Ricerca). He has been Head of the Applied Electromagnetics Section, Istituto di Ricerca sull’Ingeneria delle Telecomunicazioni e dell’Informazione (IRITI), an institute of the CNR. Since September 2002, he has been with the Istituto di Elettronica e di Ingegneria dell’Informazione e delle Telecomunicazioni (IEIIT), a newly established institute of the CNR. He has held various teaching positions in the area of electromagnetics with the Politecnico di Torino. His current research activities are in the areas of microwave antennas, dielectric radomes, frequency-selective surfaces, radar cross section, waveguide discontinuities, microwave filters, multiplexers, optical passive devices, and radiometers for astrophysical observations.

Augusto Olivieri was born in Courmayeur (AO), Italy, in 1942. He received the Diploma degree in telecommunication from the Istituto A. Avogadro di Torino, Turin, Italy, in 1963. From 1964 to 1967, he was with Poste Telecomunicazioni e Telegrafi (PTT). From 1967 to 1971, he was a Laboratory Technician with the Department of Electronics, Politecnico di Torino. In 1971, he joined the Centro Studi Propagatione e Antenne, Turin (CESPA), Italian National Research Council (CNR). He is currently with the Istituto di Elettronica e di Ingegneria dell’Informazione e delle Telecomunicazioni (IEIIT), Turin, Italy. His primary interests cover a range of areas of telecommunication, radio propagation, antennas, measurement of microwave components, and instrumentation for advanced astrophysical observations.

Renato Orta (M’92–SM’99) received the Laurea degree in electronics engineering from the Politecnico di Torino, Turin, Italy, in 1974. Since 1974, he has been a member of the Department of Electronics, Politecnico di Torino, initially as an Assistant Professor, then as an Associate Professor and, since 1999, as a Full Professor. In 1985, he was Research Fellow with the European Space Research and Technology Center (ESTEC-ESA), Noordwijk, The Netherlands. In 1998, he was Visiting Professor (CLUSTER chair) with the Technical University of Eindhoven, Eindhoven, The Netherlands. He currently teaches courses on electromagnetic-field theory and optical components. His research interests include microwave and optical components, radiation and scattering of electromagnetic and elastic waves, and numerical techniques.

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Optical Summation of RF Signals Mourad Chtioui, Alexandre Marceaux, Alain Enard, Frédéric Cariou, Corinne Dernazaretian, Daniéle Carpentier, and Mohand Achouche

Abstract—We present two concepts dedicated to the optical summation of RF signals for radar applications. The first technique relies on a photodiode simultaneously illuminated on its top and back sides. We experimentally demonstrate the summation of two signals covering the whole 25-GHz photodiode bandwidth with amplitude and phase errors below 1 dB and 4 , respectively. The heterodyne beating noise, which occurs in this type of summation device, is also discussed. The second technique is based on a traveling-wave detector array. We experimentally demonstrates the summation of four signals in the whole 25-GHz photodiodes bandwidth. However, the “summation bandwidth” is limited to 10 GHz due to amplitude and phase errors. Index Terms—Dual illuminated photodiode (DI-PD), electrooptical measurements, heterodyne beating noise, optical summation, traveling-wave detector array (TWDA).

I. INTRODUCTION UTURE generation phased-array antennas will open new prospects for multiple simultaneous independent beams performing different functions and sharing a common antenna aperture leading to an increased system affordability. However, this is accompanied by increased technological requirements such as wide instantaneous bandwidth, reduced size and weight, and high isolation from both electromagnetic interference and crosstalk between module or subarray feeds. Photonics technologies will provide an interconnect solution for both future airborne antennas and ground based radars. Over the last three decades, optical architectures have been widely studied, including highly linear optical links [1], [2], optical beamforming networks (OBFNs) often associated with true time delay (TTD) beam steering [3]–[5], allowing for wide instantaneous bandwidth operation without beam squinting, and optical summation [6]. The summation function is a key building block of future active antenna systems. Basically, it consists in adding in phase input RF signals coming from radiating elements, and previously weighted and phase shifted, to a single output RF signal. With microwave technologies, the number of channels is limited by the size and weight of microwave combiners. It has been shown recently that optical technologies can overcome these limitations [6]. In this case, the summation function is realized

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Manuscript received March 10, 2006; revised September 25, 2006. M. Chtioui, A. Enard, F. Cariou, C. Dernazaretian, D. Carpentier, and M. Achouche are with the Alcatel-Thales III–V Laboratory, 91460 Marcoussis, France (e-mail: [email protected]; [email protected]). A. Marceaux was with the Alcatel-Thales III–V Laboratory, 91460 Marcoussis, France. He is now with the RF Distribution Department, Thales Air Defence, 91470 Limours, France. 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.2006.889321

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Fig. 1. Two main methods for the optical summation of RF signals: (a) using a single photodiode and beams concentrators or (b) using distributed photodetectors on a high-impedance line.

at the photodetector stage. On the basis of the several solutions that have been studied, we can define two categories. The first one consists of concentrating the optical beams in the optical domain and to illuminate a single photodetector, as shown in Fig. 1(a). The beam concentration can be performed by different techniques such as optical couplers, wavelength multiplexers, or a multichannel concentrator (fiber or glass based). The second technique consists of distributing several photodiodes along a high-impedance transmission line, as shown in Fig. 1(b). In this case, the summation function is performed in the RF domain. The final choice of the summation method and technique is dependent upon the required performances of the system, the architecture of the optical beamformer, and the level of the summation stage in the whole architecture. In this paper, we report on the study of two techniques performing the summation of RF signals. This paper is arranged as follows. Section II presents the first technique based on a dual illuminated photodiode (DI-PD), and Section III presents the second technique based on a traveling wave detector array (TWDA). In Sections II and III, we first briefly describe the principle of the solution and show the experimental evidence of the summation function. We then focus on the limiting factors of the technique. Our conclusions are reported in Section IV. We emphasis here the fact that all the measurements presented in this paper have been performed under the same conditions to facilitate the comparison between the two techniques.

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Fig. 2. Schematic of a DI-PD. Fig. 3. Experimental setup for summation measurements using a DI-PD.

II. SUMMATION USING A DI-PD B. Experimental Setup for the Summation A. Utility and Principle In the case of a summation technique based on a beam concentrator and a single photodiode, different technologies can apply, each with its own advantages and drawbacks. We have recently shown that using a standard optical coupler can lead to heterodyne beating noise, which degrades the final performances of the system [7]. This is one of the most critical parameters of the summation. An alternative to avoid this phenomenon consists of using a wavelength multiplexer device [6], which is often implemented with a multibeam-forming architecture [4]. Another alternative, without a careful selection of the wavelength of each laser (which increases the overall cost of the system), consists of geometrically separating the optical beams on the photodetector by using a multicore fiber or integrated waveguides [6]. The main difficulty in this approach resides in the tradeoff existing between the bandwidth of the system and the number of signals that can be summed. On one hand, the size of the sensitive area of the photodiode is fixed for a given aimed cutoff frequency (capacitance). On the other hand, a minimal distance separating each optical guide must be respected in order to avoid crosstalk between the channels. We propose a simple way to multiply by two the number of input signals without enlarging the active area of the photodiode. It consists of illuminating the photodetector both on the top side (P side) and the back side (N side), as shown in Fig. 2. The so-called DI-PD has a conventional p-i-n structure with an undoped InGaAs photoabsorption layer. Only two conditions are necessary to ensure the system is running properly. First, the substrate must be sufficiently thinned to avoid beams divergence and lateral crosstalk. Second, an antireflection coating must be deposited on both sides of the device. When two modulated optical signals (1 and 2) simultaneously illuminate the photodiode’s top and back sides (as shown in Fig. 2), the two optical beams are absorbed separately, and the generated RF signals are coherently added (the component’s dimension being negligible compared to the RF wavelength). In addition, the two optical beams interfere in the absorption layer and generate a heterodyne beating current (regarded as a noise) at a frequency equal to the difference of the frequencies that correspond to the wavelength of the lasers.

Fig. 3 shows the experimental setup for summation measurements on an InGaAs/InP DI-PD available in the laboratory. The 20- m-diameter photodiode exhibits a responsivity of 0.7 A/W and a bandwidth of 25 GHz. We use an electrical network analyzer whose port 1 is connected to the RF input of an electrooptical modulator. The modulated optical beam is then divided into two parts by a 3-dB coupler with each output connected to an optical fiber. The two optical fibers are placed parallel to the test bench plane and polished at a 50 angle to couple the light in the photodiode. One of the fibers is coupled to the top side of the photodiode and the second to its back side. The two optical paths have approximately the same length to ensure that the signals arrive in phase on the active area of the detector. On one of the two paths, we insert an adjustable optical delay line to finely tune the phase difference. The photodiode then converts the two optical signals into an electric sum signal, which is collected by a coplanar probe connected on the network analyzer port 2. -parameter represents the optical to elecThe measured trical frequency response of the whole optical link modulator photodiode) [8]. By using a calibrated electrooptical modulator, we can correct the measured parameters, to obtain the optical to electrical frequency response of the photodiode. The optical delay line is adjusted to obtain a phase difference between the two signals equal to zero at low frequencies. In this situation, the signal measured at the photodiode’s output corresponds to the sum of the two optical signals. Separated signal measurements, as well as their sum, are shown in Fig. 4. We clearly obtain an increase of 6 dB for two equal signals in phase. We can also adjust the optical time delay , to obtain between the two signals (with any phase difference , where is the microwave frequency). For ps, this phase difference is of 180 instance, when at 10 GHz. Thus, we obtain an extinction of the two signals arriving in opposition of phase at this frequency (see Fig. 4). The same phenomenon occurs at 30 GHz, etc. Previous measurements enable us to check the good summation functionality of the device, but do not allow us to quantify the operation’s quality. Indeed, because of the p-i-n photodiode intrinsic properties, the conversion of the optical input signal

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Fig. 4. Measurements of the sum and the extinction of two signals using a DI-PD.

into an electric signal occurs differently according to the illumination side. This leads to slight phase and amplitude differences between the signals (see Section II-D). Moreover, beams interference in the absorption layer generates an heterodyne beating noise at the frequency corresponding to the difference of frequencies of the two optical signals. This noise adds to the noise of the optic links and deteriorates its performances. In the following, we will quantify this noise and these errors in amplitude and phase. We will then discuss the optimization of a DI-PD dedicated to the summation of RF signals. C. Heterodyne Beating Noise To better understand the behavior of the heterodyne beating noise, we consider a simple model based on the interferences’ study of two contrapropagative plane waves in the photoabsorption layer. We suppose that the two optical wavelengths are sufficiently close to generate a beating current in the frequency range of interest, and we assume that the P and N doped layers are transparent at this wavelength. The expression of the photons flow crossing the absorption layer in the -direction (see Fig. 2) is as follows:

(1) where represents the absorbing layer thickness, denotes is the wave-vector the absorption coefficient, and is proamplitude and the frequency of laser 1 (laser 2). portional to the optical power of the first (second) plane wave is proportional to . This photons flow is exand pressed as the sum of three terms. The first two terms correspond to the stationary part of the flow arising from the P and N sides, respectively. This stationary flow represents the signal’s useful part. The third term represents an interference traveling . The ampliwave whose group velocity is , tude of this oscillating term is proportional to which accounts for the attenuation of the two initial electromagnetic waves during their propagation through the absorption layer while interfering. We obtain the expression of the carrier generation rate from the derivative of the photons flow in the absorption layer

Fig. 5. Measurement and simulation of the heterodyne beating noise versus heterodyne frequency in a DI-PD (central wavelength 1 = 1:55 m and absorption layer thickness d = 1:3 m).

, and we inject the obtained rate in the differential equations of a classical “drift-diffusion” model governing the carriers and currents densities variations in the harmonic regime [9]. After solving these equations and integrating the currents densities, we obtain the term of heterodyne beating current. To check the acuity of our model, we measured this heterodyne beating current in the DI-PD. The two optical signals come from a distributed feedback (DFB) laser with a fixed wavelength and a tunable laser ( ) with an external tunable cavity (TUand, consequently, NICS). The variation of the wavelength the variation of the heterodyne beating frequency is obtained by changing the laser cavity length. Thus, we sweep the field of and that of negapositive heterodyne frequencies for . The resulting RF power is meative frequencies for sured thanks to a power meter and normalized to the power corresponding to the photodiode dc photocurrent flowing in a 50load. The resulting experimental points for a fixed wavelength m are plotted together with the simulation results in Fig. 5. The asymmetry of the heterodyne beating level as a function of frequency is due to the difference between the electrons and holes saturated velocities ( and , respectively). Indeed, the amplitude of the beating current flowing in the device varies according to whether the interference traveling wave is moving towards the P side (i.e., in the holes drift direction) or towards the N side (i.e., in the electrons drift direction). As a result, the curve’s shape primarily depends on the absorption layer thickness d (1.3 m for the used photodiodes), and on the carrier velocity values. By slightly adjusting the carrier velocity values, which are not precisely known, a good matching between measurements and modeling is obtained. The fitted and cm/s. velocity values are Since the heterodyne beating current is generated by an interference traveling wave [third term of (1)], minimum and maximum values are reached for particular values of the thickness and for a fixed heterodyne frequency. The maximum (minimum) beating values are reached for thicknesses equal to odd multiple (even multiple) of a quarter-wavelength in the absorption layer. In the case of a thickness equal to an odd multiple of a quarter wavelength (this is not the case of the used DI-PD for the

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Fig. 6. Decrease of the heterodyne beating noise in terms of the absorption layer width (d) (simulated at a zero heterodyne frequency: 1 = 2 = 1:55 m).

previous measurements in Fig. 5), the heterodyne beating level versus the heterodyne frequency presents an absolute maximum at a zero heterodyne frequency. Fig. 6 shows the heterodyne GHz. We observe level as a function of the thickness for that the beating level reaches maximum and minimum values. for As expected, the maxima’ amplitude varies as high values. The beating signal being considered as a noise, it is thus necessary to reduce its level by using photodiodes with m for a beating noise a thick absorption layer (e.g., lower than 30 dB). Another solution could consist of seeking an absorption layer thickness corresponding to the minima on Fig. 6 (even multiple of a quarter-wavelength). This is fortum , where itously the case for the used photodiodes the beating noise remains lower than 39 dB. Unfortunately, in this case, an acute precision is required on the absorbing layer thickness to actually reach such a minima (an error of 5% on causes an increase of the beating noise amplitude of more than 25 dB). Moreover, the minima on Fig. 6 are only observed for a given wavelength so that any change on the incoming wavelength would also cause an increase on the beating noise. Therefore, to ensure a beating noise lower than from 40 to 50 dB, it is essential to design photodiodes with a thick absorbing layer.

D. Amplitude and Phase Errors In a surface illuminated p-i-n photodiode, the incoming light amplitude decreases exponentially in the propagation direction. As a result, the photogenerated carrier concentration is larger at the illumination side. The electrons (holes) then drift towards the N P layer under the electric field influence with a velocity equal to the carrier saturation velocity. The electron saturation velocity being higher than the hole saturation velocity, the electric output signal amplitude and phase differ slightly according to whether the photodiode is top or back side illuminated [10]. In the case of signals coming from both sides, this implies errors in amplitude and phase on the signals’ sum. In particular, the phase errors are crucial for radar applications since it results in an antenna beam deviation and, thus, an error in the target localization. The maximum tolerable error in phase is approximately 6 [11]. Based on the experimental setup of Fig. 3, we have measured the signal’s amplitude and phase corresponding to the photodiode’s sides. The deduced amplitude and phase errors are

Fig. 7. Measurements and simulations of: (a) amplitude and (b) phase errors in a DI-PD.

plotted on Fig. 7 together with the simulation results. Within the whole 25-GHz photodiode’s bandwidth, the amplitude error remains below 1 dB [see Fig. 7(a)], and the phase error does not exceed 4 [see Fig. 7(b)]. A good agreement with the simulation results is observed, validating our model. Fig. 8(a) shows the simulated amplitude errors and Fig. 8(b) shows the simulated phase errors for an absorbing layer thickness of 2 m (triangle), 4.5 m (circle), and 7.5 m (square), which correspond to an aimed heterodyne beating level lower than 30, 40, and 50 dB, respectively. In each case, the corresponding 3-dB cutoff frequency is also indicated [12]. We notice that, unlike the heterodyne beating noise, these errors increase significantly with the absorption layer thickness, leading to a compromise between the heterodyne beating noise and errors in amplitude and phase. III. SUMMATION USING A TWDA The second method we have considered for realizing the optical summation consists of distributing several photodetectors along an RF propagation line to obtain a TWDA. The propagation line geometry and the whole architecture are optimized so that the RF signals generated at each detector propagate without reflections and add coherently at the RF line extremity [13]. Previous studies on TWDA have essentially focused on high saturation power and high efficiency aspects [13]–[16]. To our knowledge, no previous discussions have been reported concerning the amplitude and phase errors, which are key parameters for the summation application.

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Fig. 9. Schematic of traveling-wave photodetector array including optical time delays and a Z load impedance.

Fig. 8. (a) Amplitude and (b) phase errors simulations in a DI-PD for different absorption layer thicknesses d. The 3-dB cutoff frequency is also indicated in each case.

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A. Principle The simplest approach to combine the currents issuing from several photodetectors is to connect the detectors in parallel. Unfortunately, in this case, the detectors’ capacitances are added and the circuit RC time constant is increased proportionally to the number of photodiodes, degrading the system’s bandwidth. As an alternative, the capacitance of each detector can be embedded in the TWDA arrangement, as shown in Fig. 9. The photodetectors are interconnected by inductances , thus forming an artificial transmission line with a characteristic impedance of , being the individual photodetectors capacitance. The RF signal issuing from each photodetector, ideally, propagates without reflection towards the artificial line load impedance, extremities. One extremity is connected to a the other represents the output access of the summation device. Moreover, to sum the generated RF signals coherently at the summation device output access, it is necessary to deliver the optical signals with a time delay between adjacent detectors equal to the microwave signal propagation delay along the part of the artificial transmission line between the two detectors. Fig. 9 illustrates this principle. This time delay can be carried out by optical fibers [13] or optical waveguides [15] upstream from the detectors. The signals are then added in a constructive way in a large frequency range. We have designed such a TWDA thanks to a modeling tool based on a transfer matrix [14].

Fig. 10. Studied architectures for the realization of TWDA prototypes. (a) Wired version. (b) Flip-chip version.

B. Characteristics of the Artificial Transmission Line The adaptation inductances are generally carried out using short lengths of high-impedance transmission lines, where the inductive character overrides the capacitive one. Since our photodiodes have coplanar access, we use a high-impedance coplanar line (gold lines on the InP substrate). For the electrical design, we have used a quasi-static model similar to that in [16] to determine the dimensions of the coplanar line. We have studied two different architectures. The first one, called the “wired version,” consists of connecting a bar of four topside illuminated photodiodes to a high-impedance coplanar line with wire bonding, as shown in Fig. 10(a). The second method, called the “flip-chip version,” consists of equipping a high-impedance coplanar line with gold stud bumps and of mounting four backside illuminated photodiodes by flip-chip using a thermocompression process, as shown in Fig. 10(b). The latter architecture has the advantage of avoiding the parasitic inductances added by bonding wires, which limit the system bandwidth. We have fabricated and measured prototypes of summation devices in both versions by using the same photodiodes as in the DI-PD. The obtained performances are reported below. Using a vector network analyzer, we have measured the various prototypes -parameters. In Fig. 11(a), we present the

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Fig. 12. (a) Characteristic impedance and (b) attenuation constant calculated from the measured S -parameters. Fig. 11. Measurements of the: (a) S 11- and (b) versions of artificial transmission lines (TWDA).

S 21-parameters for the two

measured -parameter module, which corresponds to the return losses coefficient. This parameter represents the device’s . In the matching to the reference impedance -parameter remains flip-chip version case, the measured lower than 15 dB up to 25 GHz, while it exceeds this level, beyond 18 GHz, for the wired version. -parameter In Fig. 11(b), we also present the measured module, which corresponds to the transmission coefficient. This parameter represents the transmitted signal attenuation by both return and linear propagation losses (due to dissipative effects) along the artificial transmission line. This figure shows that the flip-chip version also has a better transmission performance. -parameter gives us a The 3-dB cutoff frequency of the rough estimation of the device’s bandwidth (23 GHz for the flip-chip version and 18 GHz for the wired version). The performances degradation observed on the -parameters is caused by the various parasitic elements, in particular, the photodiodes’ series resistance and the wire-bonding inductances. Indeed, these elements degrade the artificial transmission-line characteristic impedance and increase the transmission losses. We have calculated this characteristic impedance , as well as the linear propagation losses (or attenuation constant) from the measured -parameters. The obtained results are shown in Fig. 12. We notice that the flip-chip version presents performances largely better than those of the wired version. This is particularly true for the characteristic impedance, which is [see Fig. 12(a)]. Moreover, the backside illumivery close to nated photodiodes used to make the flip-chip version prototypes

present a series resistance lower than that of the topside illuminated photodiodes used for the wired version, which explains why the linear losses are definitely weaker for the flip-chip version [see Fig. 12(b)]. Thereafter, we will expose only the results concerning the flip-chip version since this device presents the best performances. C. Experimental Setup for the Summation Measurements The measurement setup used to highlight the summation function on the flip-chip device is presented in Fig. 13. It is very similar to that used for the summation based on a DI-PD. The modulated optical signal is equally distributed on the four ways of the summation device using a coupler 1 towards 4. Optical time delays of values , , and are, respectively, added on the paths D2, D3, and D4 using optical delay lines [the paths are noted so that D1 represents the diode nearest to the load impedance and D4 represents the diode nearest to represents the the output measurement access (Fig. 13)]. microwave travel time along the artificial transmission line between two adjacent photodiodes (approximately 7 ps). The RF signals arriving on an optical carrier are then converted into electric signals in each photodiode. These electric signals are propagated along the artificial line and are added in phase to obtain the signal sum at the output access. This signal is thereafter transmitted to the network analyzer port 2 via the -parameter coplanar probe. Again, we correct the measured by removing the contribution of the calibrated modulator to obtain the optical to electrical response of the summation device. In Fig. 14(a), we present the module of the flip-chip device optical to electrical frequency response by illuminating only one photodiode at the time. These measurements show that the shape

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Fig. 13. Experimental setup for summation measurements using a four-element TWDA (the paths are noted so that D1 represents the diode nearest to the load impedance and D4 represents the diode nearest to the output measurement access).

Fig. 15. Measurements of: (a) amplitude and (b) phase errors in a four-element TWDA (flip-chip version).

load resistance side. Fig. 14(b) shows the coherent summation result of signals issued from several illumination configurations. As expected, we observe a signal increase of 6 dB when successively illuminating one, two, and four photodiodes. As in the DI-PD summation case, we can obtain an extinction of two signals at any given frequency by controlling the optical time delay so that the signals arrive in opposition of phase at this frequency. In Fig. 14(b), we present the extinction of the signals resulting from D1 and D2 at 10 GHz. Again, the previous measurements allow us to check the good summation functionality of the device, but not to quantify this operation’s quality. D. Amplitude and Phase Errors Fig. 14. Frequency response of the four-element summation module (TWDA): (a) illuminating one photodetector at the time and (b) illuminating many photodetectors at the same time.

of the frequency response is not the same depending on the illuminated photodetector (different electrical paths). Indeed, the signals from the photodiodes closest to the load impedance and, thus, the furthest from the output access, are mainly attenuated along their transmission path by the propagation losses, as shown on the curve of the D1 photodiode. On the other hand, the signals resulting from the photodiodes closest to the output access (in particular, D4 and D3) present resonances due to the multiple reflections at the interfaces with the photodiodes at the

Indeed, because signals resulting from different photodiodes cover different paths, they do not reach the summation device output with the same amplitude and phase, resulting in errors in amplitude and phase of the signal sum. Fig. 15(a) shows the amplitude errors measurements for a four-elements TWDA. The errors remain lower than 1 dB for frequencies inferior to 10 GHz. Beyond 10 GHz, the errors increase dramatically. Fig. 15(b) shows the phase errors measurement between the different photodiodes. The measured errors remain acceptable between two adjacent photodiodes up to a frequency of approximately 20 GHz. However, between a nonadjacent photodiodes, this error is no longer acceptable beyond 12 GHz, which represent, with the amplitude error, the principal limitation of this device.

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TABLE I VALUES OF THE ABSORBING LAYER THICKNESS (d), THE 3-dB CUTOFF FREQUENCY, THE p-i-n PHOTODIODE MAXIMUM DIAMETER, AND THE AMPLITUDE AND PHASE ERRORS FOR SOME AIMED LEVELS OF HETERODYNE BEATING NOISE

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at the interfaces line/photodiodes. This phenomenon increases as the distance between the photodiodes is increased. From this point-of-view, a monolithic realization of this type of component would make it possible to reduce the influence of these various parasitic elements and would increase the “summation bandwidth” (up to more than 20 GHz) with an increased number of channels (up to 8, 16, and more). In addition, a monolithic integration of a parallel optical feed using different lengths of integrated waveguides [15] appears to be a reliable solution for the practical implementation of the small optical time delays (a few picoseconds) necessary for the velocity matching of the distributed photodiodes.

IV. SUMMARY AND DISCUSSION We have studied two concepts dedicated to the optical summation of RF signals for radar applications. The first technique relies on a photodiode simultaneously illuminated on its top and back sides. We have experimentally demonstrated the summation of two signals in the whole bandwidth of a 25-GHz photodiode with amplitude and phase errors inferior to 1 dB and 4 , respectively. We have shown that when designing a DI-PD, a compromise arises on the choice of the absorbing layer thickness . Indeed, when increasing , the heterodyne beating noise decreases, while the amplitude and phase errors increase. This is illustrated in Table I, where the thickness , 3 dB cutoff frequency, amplitude and phase errors, and photodiode maximum diameter are given for aimed heterodyne beating level of 30, 40, and 50 dB. We clearly observe that a strong requirement on the heterodyne beating noise level leads to a very thick absorbing layer accompanied by high amplitude and phase errors. In addition, the photodiode diameter should be kept as wide as possible to ensure that a sufficient channels number can be summed. All these constraints orientate this concept towards narrowband applications for which the amplitude and phase errors can be minimized, even in the case of a thick absorbing layer. For instance, by using two eightchannels multicore fibers, similar to the one described in [6], a 16-channel summation device based on a DI-PD technique can achieve a 6-GHz bandwidth with a heterodyne beating noise lower than 40 dB, and phase and amplitude errors, respectively, lower than 10 and 2 dB. The second technique is based on a TWDA. We have designed and measured two prototypes by distributing four photodiodes along a coplanar transmission line. The first prototype uses wire bonding, while the second one uses flip-chip mounting. We have measured the two prototype -parameters and concluded that the matching and losses are better in the case of the flip-chip prototype. We have then experimentally demonstrated the summation of four signals in the whole 25-GHz photodiodes bandwidth. However, the “summation bandwidth” is limited to 10 GHz due to amplitude and phase errors. Although this concept is intrinsically compatible with large bandwidth applications, the amplitude and phase errors are the principal limitations. These errors are partly generated by the various parasitic and dispersive elements, in particular, wire bonding, photodiodes’ series resistance, etc. Moreover for the realized artiof a photodiode is ficial transmission line, the capacitance lumped contrary to a real transmission line where the capacitance is distributed along the line causing multiple reflections

ACKNOWLEDGMENT The authors would like to thank F. Poingt, Alcatel-Thales III–V Laboratory, Marcoussis, France, P. Berdaguer, Alcatel-Thales III–V Laboratory, C. Jany, Alcatel-Thales III–V Laboratory, and M. Riet, Alcatel-Thales III–V Laboratory, for their help in the technological process, F. Blache, Alcatel-Thales III–V Laboratory, for fruitful discussions, and N. Vodjdani, Agence Nationale De La Recherche, Paris, France, for initiating this project. REFERENCES [1] Edward, I. Ackerman, and C. H. Cox, “RF fiber-optic link performance,” IEEE Micro, pp. 50–58, Dec. 2001. [2] K. Garénaux, V. Quet, T. Merlet, and O. Maas, “Demonstration of 0 dB gain reactively matched optical links for very high purity signal distribution in S -band ground based radar system,” in IEEE Int. Microw. Photon. Top. Meeting, Sep. 10–12, 2003, pp. 43–46. [3] J. J. Lee, R. Y. Loo, S. Livingston, V. I. Jones, J. B. Lewis, H. W. Yen, G. L. Tangonan, and M. Wechsberg, “Photonic wideband array antennas,” IEEE Trans. Antennas Propag., vol. 43, no. 9, pp. 966–982, Sep. 1995. [4] S. Blanc, M. Alouini, K. Garenaux, M. Queguinier, and T. Merlet, “Optical multibeamforming network based on WDM and dispersion fiber in receive mode,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 402–411, Jan. 2006. [5] L. A. Bui, A. Mitchell, K. Ghorbani, T. H. Chio, S. Mansoori, and E. R. Lopez, “Wide-band photonically phased array antenna using vector sum phase shifting approach,” IEEE Trans. Antennas Propag., vol. 53, no. 11, pp. 3589–3596, Nov. 2005. [6] T. Merlet, S. Formont, D. Dolfi, S. Tonda-Goldstein, N. Vodjdani, G. Auvray, S. Blanc, C. Fourdain, Y. Canal, and J. Chazelas, “Photonics for RF signals processing in radar systems,” in IEEE Int. Microw. Photon. Top. Meeting, Oct. 4–6, 2004, pp. 305–308. [7] C. Thibon, F. Dross, A. Marceaux, and N. Vodjdani, “Discussion on RIN improvement using a standard coupler,” IEEE Photon. Technol. Lett., vol. 17, no. 6, pp. 1283–1285, Jun. 2005. [8] P. D. Hale and D. F. Williams, “Calibrated measurement of optoelectronic frequency response,” IEEE. Trans. Microw. Theory Tech., vol. 51, no. 4, pp. 1422–1429, Apr. 2003. [9] G. Lucovsky, R. F. Schwarz, and R. B. Emmons, “Transit time considerations in p-i-n diodes,” J. Appl. Phys., vol. 35, no. 3, 1964. [10] J. Gowar, Optical Communication Systems, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1993, pp. 458–458. [11] P. J. Matthews, “Practical photonic beamforming,” in IEEE Int. Microw. Photon. Top. Meeting, Melbourne, Australia, Nov. 1999, pp. 271–274. [12] K. Kato, S. Hata, K. Kawano, and A. Kozen, “Design of ultrawideband, high-sensitivity p-i-n photodetectors,” IEICE Trans. Electron., vol. E76-C, pp. 214–221, 1993. [13] C. L. Goldsmith, G. A. Magel, and R. J. Baca, “Principles and performance of traveling-wave photodetector arrays,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 8, pp. 1342–1350, Aug. 1997. [14] L. Y. Lin, M. C. Wu, T. Itoh, T. A. Vang, R. E. Muller, D. L. Sivco, and Y. Cho, “High-power high-speed photodetectors-design, analysis, and experimental demonstration,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 8, pp. 1320–1331, Aug. 1997.

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[15] S. Murthy, T. Jung, T. Chau, M. C. Wu, D. L. Sivco, and A. Y. Cho, “A novel monolithic distributed traveling-wave photodetector with parallel optical feed,” IEEE Photon. Technol. Lett., vol. 12, no. 6, pp. 681–683, Jun. 2000. [16] Y. Hirota, T. Ishibashi, and H. Ito, “1.55-m wavelength periodic traveling-wave photodetector fabricated using unitraveling-carrier photodiode structures,” J. Lightw. Technol., vol. 19, no. 11, pp. 1751–1758, Nov. 2001. Mourad Chtioui was born in M’saken, Tunisia, in 1982. He received the Engineer degree in electrical engineering from the Ecole Superieure d’Electricite (SUPELEC), Gif-Sur-Yvette, France, in 2005, and is currently working toward the Ph.D. degree at the Alcatel-Thales III–V Laboratory, Marcoussis, France. His research interests are design, fabrication, and characterization for both optical summation devices for radar applications and high saturation-current photodiodes for microwave photonic links applications at 1.55 m.

Alexandre Marceaux was born in 1975. He received the Micro-optoelectronic and Materials Engineer degree and Ph.D. degree in optoelectronics from the Institut National des Sciences Appliquées (INSA), Rennes, France, in 1998 and 2001, respectively. From 1998 to 2001, he was with the Laboratoire de Physique du Solide (LPS), INSA, where he was engaged in research on ultrafast Fe-doped InP multiquantum-well (MQW) saturable absorber microcavities for all-optical 2R regeneration. In 2001, he joined Thales Research and Technology. In 2005, he joined the Alcatel-Thales III-V Laboratory, Marcoussis, France, as Research Staff Member, where he was in charge of the development of photodetectors for microwave photonic applications. In 2006, he joined the RF Distribution Department, Thales Air Defence, Limours, France. He has authored several papers in referred journals and conference proceedings. His current activities include optical distribution for radars signals, packaging of opto-electronic devices, and optical sensors.

Alain Enard, photograph and biography not available at time of publication.

Frédéric Cariou, photograph and biography not available at time of publication.

Corinne Dernazaretian, photograph and biography not available at time of publication.

Daniéle Carpentiera, photograph and biography not available at time of publication.

Mohand Achouche received the Master degree in material sciences and Ph.D. degree from the University of Paris VII, Paris, France, in 1993 and 1996, respectively. His Ph.D. research conducted at France Telecom Research Centre (CNET), Bagneux, France, concerned technology of InP HEMTs. In 1997, he joined the Ferdinand-Braün Institute, Berlin, Germany, where he was involved with GaAs HBTs for mobile communications. In 2000, he joined Alcatel Research and Innovation, Marcoussis, France, where he was initially involved with 40-Gb/s waveguide photodiodes. Since 2004, he has been in charge of a research team involved with high-speed photodiodes for optical fiber telecommunication and microwave photonic links.

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A Fully Electronic System for the Time Magnification of Ultra-Wideband Signals Joshua D. Schwartz, Student Member, IEEE, José Azaña, Member, IEEE, and David V. Plant, Senior Member, IEEE

Abstract—We present the first experimental demonstration of a fully electronic system for the temporal magnification of signals in the ultra-wideband regime. The system employs a broadband analog multiplier and uses chirped electromagnetic-bandgap structures in microstrip technology to provide the required signal dispersion. The demonstrated system achieves a time-magnification factor of five in operation on a 0.6-ns time-windowed input signal with up to 8-GHz bandwidth. We discuss the advantages and limitations of this technique in comparison to recent demonstrations involving optical components. Index Terms—Analog multipliers, frequency conversion, microstrip filters, signal detection, signal processing.

I. INTRODUCTION

T

HE increasing attractiveness of ultra-wideband (UWB) communication, in the context of both short-range wireless and long-range radar and imaging systems, has spurred research towards updating conventional signal-processing systems to accommodate the characteristically large bandwidths of UWB pulses, which span several gigahertz. These bandwidths generally surpass the capabilities of current instrumentation for analog-to-digital conversion (ADC) and arbitrary waveform generation (AWG), both of which represent key signal-processing tools in the development of UWB systems [1]–[5]. Temporal imaging or “time-stretch” systems, in which arbitrarily shaped time-limited input signals can be distortionlessly magnified, compressed, or reversed in the time domain, are capable of extending the operating bandwidths of existing ADC and AWG systems by adding a bandwidth-conversion step, as illustrated in Fig. 1. Temporally magnifying (i.e., stretching) an analog UWB input signal prior to ADC would permit the sampling of the signal to be performed at a relaxed rate without losing information [1], [2]. Similarly, in the context of AWG, introducing a time-compression block at the system output would compress the generated waveforms to bandwidths of interest for UWB communication beyond the capabilities of current AWG implementations, which are limited at 2 GHz [4], [5]. Manuscript received August 8, 2006; revised October 24, 2006. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada. J. D. Schwartz and D. V. Plant are with the Photonics Systems Group, Department of Electrical and Computer Engineering, McGill University, Montréal, QC, Canada H3A 2A7 (e-mail: [email protected]). J. Azaña is with the Institut National de la Recherche Scientifique-Energie, Matériaux et Télécommunications, Montréal, QC, Canada H5A 1K6 (e-mail: [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.2006.890069

Recent demonstrations of time-stretching systems intended for the aforementioned microwave applications have almost exclusively invoked the optical domain [1]–[5]. This presents a problem for microwave systems from an integration standpoint, as signal conversion to and from the optical domain necessitates the cumbersome element of electrooptic modulation, and the referenced works require costly mode-locked or supercontinuum laser sources. The decision to perform time-stretching operations in the optical domain was motivated in part by a perceived lack of a sufficiently broadband low-loss dispersive element in the electrical domain such as that which is readily available in optical fiber [2]. In our recent study [6], we proposed a fully electronic implementation for a time-magnification system by using chirped electromagnetic-bandgap (EBG) structures in microstrip technology as a source of broadband dispersion. Here, we present the first experimental demonstration of a working 5 time-magnification system for signals with bandwidths of several gigahertz. We also discuss the capabilities and limitations of fully electronic time-stretch systems, as compared to their optically implemented counterparts, and we investigate timing and system resolution issues. This paper is organized as follows. In Section II, we provide a very brief review of the theory of time-stretch systems before discussing, in Section III, the measured performance of our time-magnification system. Section IV details some of the challenges and limitations associated with purely electronic timestretch systems, paying particular attention to how they compare to electrooptic implementations.

II. BACKGROUND The theory of time stretching originated in the work of Caputi [7] and can be understood in the context of an elegant space–time duality between the paraxial diffraction of a propagating wave in space and the narrowband dispersion of a propagating wave in time, which share a common mathematical framework [8]–[10]. As a result of this duality, the traditional spatial imaging system consisting of a thin lens between two diffractive media (such as air) has an analog in the time domain, which consists of a “time lens” between two dispersive media. The function of the “time lens” is to emulate in the time domain the effect that a thin curved lens surface has in space on a propagating wave: namely, quadratic phase modulation. Whereas a space-lens introduces into a -propagating wave a phase term quadratically related to spatial – coordinates, a “time lens” introduces into a propagating wave a quadratic-phase term in the time domain. This can be achieved in electronics

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Fig. 1. Conceptual illustration of the application of time stretching. Time magnification facilitates ADC by reducing the frequency and bandwidth of the input signal (top), while time compression enables high-frequency AWG by frequency upshifting the generated waveform (bottom).

Fig. 2. Schematic representation of a time-stretch system. Reference linear frequency sweep is shown generated passively. An LPF is employed to select the difference-frequency product of the mixing operation.

by mixing a signal against a linear frequency ramp, which is straightforward to implement using an analog multiplier. A schematic depiction of time stretching is presented in , bandwidth Fig. 2. An input signal of carrier frequency , and constrained to time-window is passed through a dispersive element with linear group delay of slope (nanoseconds/gigahertz). Given a value of satisfying the condition (1) this dispersion is sufficient to perform a frequency-to-time mapping of the signal, ordering the spectral components [11]. The dispersed signal is then difference-frequency1 mixed against a linear frequency sweep of slope , shown generated in Fig. 2 by a passive process: applying dispersion to an impulse. The re, a new sulting product has a reduced central frequency , and the bandwidth is group-delay slope 1In

this discussion, we neglect the sum-frequency components of the mixing operation, as these can be easily filtered out. It is possible to perform time stretching using the sum frequencies instead, and the decision rests upon the available operation bandwidth of the output dispersive network or on the choice of a magnifying or compressing system [12].

Fig. 3. Frequency-time representation of dispersed input signal and reference impulse (frequency sweep), as well as difference-frequency mixer output.

reduced in this step in proportion to the ratio while the signal remains the same duration in time. A frequency-time representation of this mixing process in illustrated in Fig. 3, valid if (1) is respected. If this signal is then compressed (i.e., dis), the original input signal persed with group-delay slope

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TABLE I COMPARISON OF COMMONLY CHIRPED STRUCTURES

can be recovered from the signal envelope, stretched temporally by the magnification factor and reduced in amplitude by the same factor, respecting conservation of energy (assuming lossless mixing and dispersion operations). For , magnification is achieved, while indicates time compression and has the effect of time reversing the signal. It should be noted that the output signal is effectively an amplitude modulation on a chirped carrier corresponding to the difference-frequency band created in the mixing process, and must still be envelope detected somehow to recover signal information. III. EXPERIMENTAL DEMONSTRATION The authors proposed and simulated an electronic time-magnification system in [6] and here present the first experimental demonstration of such a system intended for use at bandwidths of interest to UWB communication systems. We have found that the dispersive requirements of time-stretch systems can be met in the electrical domain by using chirped EBG structures, which were recently introduced in planar microstrip technology [13]. A reflective bandgap can be created in a microstrip line by “wiggling” the strip width in a sinusoidal fashion to induce Bragg-like local reflections, where the propagating wave is coupled to the same, but counter-propagating mode, and is reflected. Introducing a linear frequency chirp into the structure is accomplished by changing the period of the strip-width modulation along the length of the device, thus reflecting different frequencies at different lengths along the microstrip, yielding a linear group-delay slope (dispersion). These devices are similar to chirped fiber Bragg gratings in the optical domain, and chirped surface acoustic wave (SAW) structures common to radar, although they are far less lossy and easily achieve greater bandwidths than their SAW counterparts, as summarized in Table I. A photograph of an etched 28-cm chirped EBG microstrip designed for operation between 2–10 GHz is shown in Fig. 4. This microstrip was meandered to fit into a 10-cm etching process, though the radius of curvature and the strip spacing were designed to exhibit negligible radiative and cross-coupling effects. To generate the reference frequency sweep, we employed a passive-generation scheme, as proposed in Caputi’s original system, and in a manner similar to the recent optical time-stretch system schemes using dispersed ultra-short optical pulses [1]–[3]. Actively generating a frequency ramp spanning up to 8 GHz at a sweep rate of several gigahertz/nanosecond is prohibitively difficult, and there exists, to the authors’ knowledge, no suitable voltage-controlled oscillator for this purpose that is commercially available. A Picosecond Pulse

Fig. 4. From [6]. Photograph of the 28-cm chirped EBG microstrip used in the demonstration. Microstrip is gold metallized on an alumina substrate and is meandered to fit a 10-cm etching process.

Labs model 3600 impulse generator was used to create a train of 7.5-V 70-ps impulses to provide useful bandwidth out to 10 GHz. These impulses were dispersed using a chirped EBG microstrip, resulting in a linear frequency sweep. Ideally, this sweep would exhibit a flat-amplitude response since the goal is to have a difference-frequency mixing product that contains only those amplitude features belonging to the RF input signal. Unfortunately, a dispersed impulse will feature amplitude rolloff associated with its spectral content with diminished amplitude towards the higher frequencies. However, since this rolloff represents a priori information, its effect on signal measurements can be compensated for in post-process as needed. Our demonstration system employs three chirped microstrip EBG designs (wherever dispersion is indicated in Fig. 1). We selected Coorstek’s ADS-96R substrate for these microstrips (1.27-mm thickness, , GHz). The EBGs were designed to have group-delay slopes ns/GHz, ns/GHz, and ns/GHz for the input dispersive network, reference frequency-sweep generator, and output compressing network, respectively, where we note the change in sign of the group-delay slope of the third (compressive) EBG structure. The system magnification factor was designed to be 5, while the slope required of the reference frequency sweep was determined according to [7] as (2) The input and reference-sweep microstrips both operated in the band of 2–10 GHz and were 28 cm in length, while the output compressive network was designed for operation from 1–2.5 GHz and was 38 cm in order to have enough periods at

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Fig. 5. Simulated (dashed line) and measured (solid line) S 11 and group delay (reflected port) of the photographed microstrip of Fig. 4. The group-delay slope is approximately 0.5 ns/GHz. Loss is somewhat higher in measurement (2–3 dB) due to end connectorization impedance mismatching, which also worsens the measured group-delay ripple by contributing to long-path resonances.

0

Fig. 6. MATLAB illustration of the “101” input signal centered at 5 GHz (dashed line) and the impulse (solid line) after they have undergone dispersion in chirped EBG microstrips. These are the inputs to the multiplier; their timing with respect to each other determines the output frequency difference.

these low frequencies to induce reflection. The microstrips were designed according to the empirical formula proposed in [13]

(3) This expression yields an impedance modulation centered around 50 —in our case, this impedance varied from 35 to 65 at the widest and narrowest points of the strip, respectively. The microstrip is of length , local period “ ” at the center of , and a chirp , which is related to the structure the target group-delay slope (nanoseconds/gigahertz) by (4) is the effective permittivity of the microstrip. We Here, employed in (3) an asymmetric Gaussian apodization window to help flatten the response and increase the reflectivity of the EBG for the longer lossier round-trips of the lower frequencies. We refer the interested reader to [11] and [13] for further details concerning this procedure. Once fabricated, the microstrips were mounted on aluminum baseplates and end connectorized to subminiature A (SMA) connectors. Each microstrip was 50- terminated on the far end. The microstrips were then paired with an appropriate commercially available 6-dB directional coupler covering their bandwidths in order to circulate the reflected dispersed signals. and group-delay A sample of chirped EBG microstrip response are presented in Fig. 5, both from simulation using Agilent’s Momentum software (a method-of-moments tool for full-wave analysis), and measured using Agilent’s 8720ES 20-GHz vector network analyzer. This microstrip, chosen for use in the reference-frequency generation step, was designed ns/GHz. Chirped for a group-delay slope of microstrip EBGs suffer from group-delay ripple (as do their optical counterparts) due to undesired multiple internal reflections (long-path Fabry–Perot resonances). Apodization techniques can be employed to help mitigate this and other effects, which contribute to overall aberration in the imaging system [14]. Our measured results feature a more pronounced ripple than

Fig. 7. 3-bit “101” test input at 5 Gb/s (top), shown from measurement before carrier modulation) and time-magnified outputs from simulation of the microstrips (middle) and from measurement (bottom). Outputs are normalized absolute values and have envelopes highlighted for visualization purposes.

simulations suggested, most likely due to the addition of reflections resulting from impedance mismatch at the soldered SMA end connectors; however, a measured group-delay slope of 0.506 ns/GHz was verified by smoothing this ripple using the averaging functions of the vector network analyzer. The role of the “time lens” in our system was played by an analog multiplier (e.g., a Gilbert-cell architecture with predistortion circuitry), which can have very broad bandwidths through judicious choice of impedance-matching schemes [15], [16]. A differential multiplier topology was chosen to minimize noise effects and improve the conversion gain performance. The analog multiplier was fabricated in an available 0.5- m SiGe BiCMOS technology and was simulated using the Cadence software suite. The design used a 3.3-V supply and simple resistive networks for broadband impedance matching since power consumption was not a concern for this demonstration. We targeted a conversion gain of approximately 8 dB in the mixing process and employed an additional external amplifier at the output stage, which compensated for the anticipated

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Fig. 8. Depiction of three parallel channels forming a free-running time-magnification front-end step for ADC.

losses associated with directional couplings and the time-magnification process itself. Input signals were brought on-chip using high-frequency probes from Cascade Microtech’s Infinity series, while a difference-frequency single-ended output of 1–2.5 GHz was passed off-chip via wirebond through the test board. The sum frequencies generated were inherently filtered, as they were attenuated by the test board, which effectively acted a low-pass filter (LPF) since the sum frequencies spanned 4.5–20 GHz. For test inputs, the authors chose to work with differential digital bit-signaling for the ease with which windowing could be applied by strings of zeros, and since this is complicit with our system’s resolution limitations (see Section IV). For example, a simple 3-bit sequence “101” at 5 Gb/s and 2-V amplitude, generated differentially by an Anritsu MP1763B pulse pattern generator (PPG), fit within the target time window of 0.6 ns and was upconverted on a 5-GHz carrier using off-shelf diode mixers. Differential bit sequences were padded with zeros out to a 64-bit length and were sent through identical microstrip dispersion elements and couplers. The signal input and reference impulse, after dispersion in their respective chirped EBGs, were received by the analog multiplier. The impulse was triggered by a 1/64 clock output from the PPG and coarse timing adjustments were made by observing the multiplier output as different bits in the 64-bit stream were toggled to locate a nonzero product. In this fashion, the temporal alignment of the signal under test and the reference frequency sweep at the multiplier input could be verified. An illustration of the post-dispersion signals being received by the multiplier is presented from measurement in Fig 6. These dispersed signals represent the spectral content of the RF input signal and impulse, ordered in the time domain, as described in [11] and subject to the 2–10-GHz bandpassing of the EBG microstrips. The amplitude of the dispersed reference impulse (solid trace) is nearly flat within the bandwidth of the input signal and, thus, post-processing was deemed unnecessary. The multiplier product was compressed by the final EBG microstrip and a one-shot measurement was taken using a Tektronix CSA8000 scope with a 20-GHz electrical module. The result is illustrated in Fig. 7 in comparison to the (pre-carrier) “101” RF input, and also results predicted in [6] using the simulated microstrip -parameters and ideal multiplication. No post-

processing has been applied to the measured signal, and the envelope is shown by interpolation (in MATLAB) for illustrative purposes. A time-scale factor of 5 has been applied to the time axis of the simulated and measured outputs to facilitate comparison. The “101” bit pattern is clearly distinguishable. Outputs have been normalized for clarity; the actual measured output amplitude was 140 mV at its peak after coupling/microstrip losses, time-magnification loss (a factor of 5) and the multiplier conversion gain of the system. It is clear that some distortion has taken place, which can be attributed to microstrip group-delay ripple (which shows up as an amplitude ripple) and also the frequency responses of the analog multiplier, directional couplers, and reference impulse, which are not flat. IV. DISCUSSION Here, we will discuss several challenges associated with implementing this system entirely in the electrical domain, particularly when contrasted with optical implementations. A. Integrated Systems Optical time-magnification systems have demonstrated time-apertures of approximately 2 ns [2], though because of (1) and because chirp cannot be increased indefinitely without paying a cost in the EBG reflectivity, it is clear that achieving larger time windows or even continuous operation would require segmenting and interleaving the input signal through parallel channels of operation [2]. Fig. 8 depicts what this would look like for a three-channel magnification system for an ADC application. This parallelization would be highly inconvenient in an optically enabled system, as it would require multiple electrooptic modulators and mode-locked lasers, which is cost prohibitive. By contrast, an electrical implementation requires several chirped microstrip EBGs, which are fairly long, but, being planar, can be stacked to conserve space. The authors suspect that a stripline-oriented EBG would be particularly convenient in a multiple metal-layer process. Parallelizing the system would also require several multiplier circuits and signal interleaving, which is straightforward and represents little overhead. One impulse generator, representing the most costly component, could still be employed by increasing its repetition rate and interleaving the impulses into each channel.

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B. Time-Bandwidth Products and System Resolution The input temporal aperture of a time-stretch system is intimately related to its bandwidth, and an appropriate figure-of-merit for a time-stretch system is the time-bandwidth product, defined as the product of the system’s temporal aperture and its RF bandwidth, because of the implicit tradeoff between these two elements [2]. For the demonstrated system, 5, this product is modest: approximately 0.6 ns 8 GHz whereas demonstrations in the optical domain have been shown to achieve products as high as 60 [17]. Furthermore, the resolution of our demonstrated system is particularly limited because the electrical carrier frequency is so close to the bandwidth of the baseband RF signal. Systems employing photonic components benefit from very high resolutions since optical carrier frequencies are much higher than RF signal bandwidths. The electrical system demonstrated here cannot resolve finer than approximately 0.1-ns features in a 0.6-ns window (permitting six resolution elements; hence, the authors’ choice of digital bit-signaling for this demonstration). However, system resolution can be improved by choosing higher RF carrier frequencies. A carrier frequency of 15 GHz would result in 18 resolution elements assuming the same 0.6-ns window used here. This is made challenging because losses incurred in microstrips scale , and an upper limit between 15–20 GHz is anticipated with before attenuation becomes a problem. Using an EBG with negative chirp values (reflecting the higher lossier frequencies first) can help mitigate loss effects. The authors wish to point out that the choice of frequencies in this demonstration was limited by available technologies, which are not state-of-the-art (e.g., 0.5- m SiGe), computer simulation-time constraints, and available impulse generation equipment. Electronic time-stretch systems may achieve better resolution and time-bandwidth figures exceeding ten with broader-bandwidth multipliers and longer EBGs designed for higher carrier frequencies. In an estimate of what could be reasonably be expected from the state-of-the-art, we note that analog multipliers with bandwidths of 23 GHz have been reported [16], and that the reference frequency sweep can be extended to larger bandwidths by using narrower impulses, with some commercially available impulse-forming networks now capable of forming 15-ps impulses. The authors believe the most limiting factor of the electronic implementation is that EBG microstrips with higher group-delay slopes and bandwidths would require, in general, more length. For example, targeting a 2-ns window of an RF signal of 8-GHz bandwidth (a time-bandwidth product of 16) would require a group-delay slope of greater than 1 ns/GHz and a corresponding microstrip length of approximately 48 cm according to (1) and the design equations laid out in [11]. Since raising the effective permittivity would mitigate this problem, the authors again anticipate the role that stripline EBG structures may have to play since they use true rather than effective permittivity. Stripline structures also offer true TEM-mode propagation (as opposed to quasi-TEM in microstrip), which is highly advantageous for building broadband directional couplers [18] and, thus, the authors suspect these couplers could be integrated directly into the EBG device design.

Fig. 9. Temporal misalignment with the time-aperture results in degraded quality of signal envelope. A 150-ps misalignment outside of the 600-ps aperture results in degradation of detected bit amplitude by over 50%.

Fig. 10. Measured system output for a 6-Gb/s “10101” bit pattern of 0.82 ns. This bit pattern exceeds the rated 0.6-ns window length and one bit is noticeably attenuated and broadened.

C. Timing Issues It is important in a time-stretch system that the dispersed input signal arrives at the mixer input at the same time as the reference frequency sweep to obtain the correct difference frequency. If part of the input signal is time shifted away from the ideal triggering case depicted in Fig. 3, two problems occur, which are: 1) there may be no component of the reference frequency sweep at the mixer input or 2) the mixer product will be frequency shifted and, thus, fall outside the passband of the subsequent dispersive device. By keeping the reference pulse at a fixed time while advancing the input bit sequence in time, the demonstration system (0.6-ns window, “101”’ sequence) tolerates approximately 150 ps of temporal misalignment outside this window before one of the input “1” bits becomes approximately 50% attenuated, as shown in Fig. 9 using MATLAB to process the measured -parameter data of the chirped EBGs. Similarly, we demonstrate in Fig. 10 a measured oscilloscope response to a 6-Gb/s bit pattern of “10101,” which is 0.82 ns in length and, thus, is larger than the system’s temporal aperture. As a result, one of the bits is “defocused” in this temporal imaging system, emerging broader and attenuated. Time-stretch systems are also

SCHWARTZ et al.: FULLY ELECTRONIC SYSTEM FOR TIME MAGNIFICATION OF UWB SIGNALS

subject to aberrations due to departures from ideal quadratic phase behaviors, a detailed discussion of which can be found in [19]. D. Other Points of Comparison Another point of comparison with optical systems is output signal detection, which must be achieved by extracting the envelope of a chirped-carrier signal. For systems employing signal conversion to the optical domain, this step is trivial due to high optical carrier frequencies (in terahertz) and the mere act of photodetection is sufficient to recover the desired signal envelope. This is not the case for an electrical time-stretch system in which a peak-detection or down-conversion scheme must be employed to recover the amplitude features of the signal on a one-shot basis. It is also interesting to note the ability of optical time-stretch systems to entirely remove the first dispersive element of the system, and they can recast the magnification factor in terms of two dispersive elements related by an inequality rather than a precise equality, sometimes referred to as “simplified” temporal imaging [1], [20]. Electrical systems may not be able to benefit in this regard since one requirement of this simplified implementation is that the bandwidth at the output of the time lens should be much greater than the input bandwidth, and this would likely present a challenge for electronic systems in the context of practical analog multiplier design for UWB systems where input bandwidths are already large. V. CONCLUSION In this paper, we have reported on the first demonstration of a fully electronic implementation of a time-magnification system for bandwidths in the range of UWB systems. This has significant implications for improving the effective bandwidth of ADCs to accommodate the spectrum of UWB communication systems. The key development enabling this system is the chirped EBG structure, which is a broadband source of strong dispersion. We have demonstrated 5 magnification of a 0.6-ns signal window with 8-GHz RF bandwidth. A system employing longer EBG structures, using current impulse generation equipment with the state-of-the-art in analog multiplier design, would be capable of achieving time-bandwidth products in excess of ten and would be comparable to what has been demonstrated using electrooptic modulation schemes, but without the costly optical components and with promising integration opportunities. The authors will shortly follow up this study with an experimental demonstration of an electronic time-compression system to demonstrate the potential application of this technique for AWG applications in UWB communication technologies. ACKNOWLEDGMENT The authors gratefully acknowledge the assistance of M. Guttman, N. Kheder, and M. Nikolic, all with McGill University, Montréal, QC, Canada, for their role in the design and layout of the analog multiplier, as well as Dr. M. El-Gamal, McGill University, for his assistance providing access to test equipment.

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REFERENCES [1] F. Coppinger, A. S. Bhushan, and B. Jalali, “Photonic time-stretch and its application to analog-to-digital conversion,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 7, pp. 1309–1314, Jul. 1999. [2] Y. Han and B. Jalali, “Photonic time-stretched analog-to-digital converter: fundamental concepts and practical considerations,” J. Lightw. Technol., vol. 21, no. 12, pp. 3085–3103, Dec. 2003. [3] F. Coppinger, A. S. Bhushan, and B. Jalali, “Time magnification of electrical signals using chirped optical pulses,” Electron. Lett., vol. 34, no. 4, pp. 399–400, Feb. 1998. [4] J. Azaña, N. K. Berger, B. Levit, and B. Fischer, “Broadband arbitrary waveform generation based on microwave frequency upshifting in optical fibers,” J. Lightw. Technol., vol. 24, no. 7, pp. 2663–2675, Jul. 2006. [5] J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photon. Technol. Lett., vol. 15, no. 4, pp. 581–583, Apr. 2003. [6] J. Schwartz, J. Azaña, and D. V. Plant, “A fully electronic time-stretch system,” presented at the 12th Int. Antenna Technol. Appl. Electromagn. Symp., 2006. [7] W. J. Caputi, “Stretch: A time-transformation technique,” IEEE Trans. Aerosp. Electron. Syst., vol. 7, no. 2, pp. 269–278, Mar. 1971. [8] B. Kolner, “Space–time duality and the theory of temporal imaging,” IEEE J. Quantum Electron., vol. 30, no. 8, pp. 1951–1963, Aug. 1994. [9] P. Naulleau and E. Leith, “Stretch, time lenses, and incoherent time imaging,” Appl. Opt., vol. 34, no. 20, pp. 4119–4128, Jul. 1995. [10] B. Kolner and M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett., vol. 14, no. 12, pp. 630–632, Jun. 1989. [11] M. A. G. Laso, T. Lopetegi, M. J. Erro, D. Benito, M. J. Garde, M. A. Muriel, M. Sorolla, and M. Gugliemi, “Real-time spectral analysis in microstrip technology,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 705–717, Mar. 2003. [12] C. Bennett and B. Kolner, “Principles of parametric temporal imaging—Part I: System configurations,” IEEE J. Quantum Electron., vol. 36, no. 4, pp. 430–437, Apr. 2000. [13] M. A. G. Laso, T. Lopetegi, M. J. Erro, D. Benito, M. J. Garde, M. A. Muriel, M. Sorolla, and M. Guglielmi, “Chirped delay lines in microstrip technology,” IEEE Microw. Wireless Compon. Lett., vol. 11, no. 12, pp. 486–488, Dec. 2001. [14] M. J. Erro, M. A. G. Laso, T. Lopetegi, M. J. Garde, D. Benito, and M. Sorolla, “A comparison of the performance of different tapers in continuous microstrip electromagnetic crystals,” Microw. Opt. Technol. Lett., vol. 36, no. 1, pp. 37–40, Jan. 2003. [15] B. Tzeng, C. Lien, H. Wang, Y. Wang, P. Chao, and C. Cheng, “A 1-17-GHz InGaP–GaAs HBT MMIC analog multiplier and mixer with broadband input-matching networks,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 11, pp. 2564–2568, Nov. 2002. [16] M. D. Tsai, C. S. Lin, C. H. Wang, C. H. Lien, and H. Wang, “A 0.1-23-GHz SiGe BiCMOS analog multiplier and mixer based on attenuation-compensation technique,” in IEEE Radio Freq. Integr. Circuits Symp., 2004, pp. 417–420. [17] Y. Han, O. Boyraz, and B. Jalali, “Ultrawide-band photonic time-stretch A/D converter employing phase diversity,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 4, pp. 1404–1408, Apr. 2005. [18] D. Pozar, Microwave Engineering, 3rd ed. Hoboken, NJ: Wiley, 2005. [19] C. Bennett and B. Kolner, “Aberrations in temporal imaging,” IEEE J. Quantum Electron., vol. 37, no. 1, pp. 20–32, Jan. 2001. [20] J. Azaña, N. K. Berger, B. Levit, and B. Fischer, “Simplified temporal imaging systems for optical waveforms,” IEEE Photon. Technol. Lett., vol. 17, no. 1, pp. 94–96, Jan. 2005.

Joshua D. Schwartz (S’01) received the B.Eng in electrical engineering (with honors) from McGill University, Montréal, QC, Canada, in 2003, and is currently working toward the Ph.D. degree at McGill University. He is currently with the Photonics Systems Group, McGill University. He has authored or coauthored over 12 publications and conference presentations. His research areas of interest include real-time microwave signal processing, EBG structures, and metamaterials. Mr. Schwartz was the recipient of a 2003 British Association Medal as the top graduating student of his B.Eng program.

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José Azaña (M’03) received the Ingeniero de Telecomunicación and Ph.D. degrees in optical signal processing from the Universidad Politécnica de Madrid (UPM), Madrid, Spain, in 1997 and 2001, respectively. He recently joined the Institut National de la Recherche Scientifique (INRS)-Energie, Montréal, QC, Canada, where he is currently a Research Professor with the Ultrafast Optical Processing (UOP) Group. His research has resulted in over 50 publications in scientific and engineering journals. Dr. Azaña is a member of the Optical Society of America (OSA). He has been recognized with several distinctions in both Spain and Canada.

David V. Plant (S’86–M’89–SM’05) received the Ph.D. degree in electrical engineering from Brown University, Providence, RI, in 1989. Since 1993, he has been a Professor and member of the Photonic Systems Group, Department of Electrical and Computer Engineering, McGill University, Montréal, QC, Canada. In January, 2006, he became the Associate Dean of Research and Graduate Education. He is also a James McGill Professor (Tier I Canada Research Chair institutional equivalent). Dr. Plant is an IEEE Distinguished Lecturer (2005–2007). He is a Fellow of the Optical Society of America. He is a member of Sigma Xi. He was the recipient of the 2006 IEEE Canada R. A. Fessenden Medal.

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Optimization and Implementation of Impedance-Matched True-Time-Delay Phase Shifters on Quartz Substrate Balaji Lakshminarayanan, Member, IEEE, and Thomas M. Weller, Senior Member, IEEE

Abstract—In this paper, design equations for true-time-delay microelectromechanical (MEM) phase shifters comprised of impedance-matched slow-wave unit cells are presented. The input to these equations requires only the specification of the maximum frequency of operation ( max ) and substrate dielectric constant ( ). The basis of the design methodology is to maximize the phase shift versus insertion loss performance. Experimental data for three 1-bit devices that are 11.44-mm long ( max = 12 GHz), 4.6-mm long ( max = 50 GHz), and 3.12-mm long ( max = 110 GHz) shows a maximum phase deviation of 3% compared to predicted performance, with 11 less than 19 dB from 1 to 110 GHz. The worst case insertion loss is 0.9 dB for max = 12 GHz, 1.16 dB for max = 50 GHz, and 2.65 dB for max = 110 GHz. The MEM beams are actuated using high-resistance SiCr bias lines with typical actuation voltage around 30–45 V. Index Terms—Microelectromechanical systems (MEMS), phase shifter, RF MEMS, true time delay (TTD).

I. INTRODUCTION

O

VER THE past eight years, considerable improvements have been made towards the development of low loss and broad true-time-delay (TTD) microelectromechanical (MEM) phase shifters. The phase shifter topology that is typically used for broadband applications is the distributed MEM transmission line (DMTL). The DMTL, initially reported by Barker and Rebeiz [1], usually consists of a uniform length of high-impedance coplanar waveguide (CPW) transmission line that is loaded by periodic placement of discrete MEM capacitors. The increase in the distributed capacitance in the down-state provides a differwith respect to the phase in the upstate. ential phase shift The DMTL design has been demonstrated from - to -band [2]–[9]. These MEM devices are often designed such that the is less than 10 dB for the two phase states. Based on the criteria, an optimal footprint for the CPW and MEM design was derived in [1].

Manuscript received January 18, 2005; revised October 15, 2006. This work was supported by the National Science Foundation under Grant ECS-9875235. This work was supported in part by the University of South Florida Center for Ocean Technology. B. Lakshminarayanan is with the Electrical and Computer Engineering Department, University of California at San Diego, La Jolla, CA 92093 USA (e-mail: [email protected]). T. M. Weller is with the Electrical Engineering Department, University of South Florida, Tampa, FL 33625 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.890076

Recently, 1- and 4-bit phase shifters based on cascaded slow-wave unit-cells section were presented in [10]. The design per unit achieves high return loss in both states, a large length, and phase shift per decibels that is comparable to or better than previously reported performance. The goal of this paper is to provide a methodology for the design of a slow-wave unit cell and calculate an optimal physdB is maximized. ical layout such that the figure-of-merit First, closed-form expressions for the MEM capacitances and the spacing are derived assuming a quasi-TEM model of a transmission line. The input to the derivation is the maximum operand the dielectric constant of the subating frequency per millimeter for the unit strate. Using these expressions, cell is derived. A closed-form expression for the conductor loss dB calculation. (from conformal mapping) is used in the To simplify the design process, the parasitics of the MEM devices and the discontinuity effects are not taken into account. Using the methodology outlined herein, designs with three different values (12, 50, and 110 GHz) were fabricated, and measured data indicate agreement to within 10% of the predata. The designs chosen for fabrication do not repredicted sent the optimum solutions, based on the presented optimization method, due to limitations in the fabrication facilities. II. MODELING AND OPTIMIZATION OF AN IMPEDANCE-MATCHED UNIT CELL A. Design Equations The model for a slow-wave unit cell in its two operating states, excluding the parasitics due to the MEM bridges and the discontinuities, is shown in Fig. 1. In the normal state, the model is comprised of a transmission line of length , with a capac. In the slow-wave itor to ground due to the shunt bridge state, the model is comprised of two shunt capacitors due to , separated by the transmisthe ground-plane bridges sion line of length (which is equal to the total length through the ground-plane slot). The impedance and Bragg’s frequency for each state of the unit cell is given by (1) and (2) as follows [7]:

0018-9480/$25.00 © 2007 IEEE

(1)

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(4)

where

Fig. 1. Ideal model for the slow-wave unit cell. (a) In the normal state. (b) In the slow-wave state. This model does not take into account the parasitics due to the bridge and the discontinuities. The total length through the slot (s ) is equal to 2S + S .

(2) and are the per unit length inductance and cawhere and are the per pacitance in the normal state, while unit length inductance and capacitance in the slow-wave state. and are the capacitance and spacing between the shunt bridges. and are the capacitance and spacing between and are the effective characthe ground-plane bridges. teristic impedance in the normal and slow-wave states. The per unit length capacitance and inductance in each state are given by (3) [1]

is the complete elliptic integral of the first kind, , , is the width of the center conductor, is the ground-to-ground spacing in the normal state, and is the substrate thickness. The variables and are replaced with and for calculating the impedance in the slow-wave state. The required input parameters for the design of a slow-wave or the maximum unit cell are the Bragg’s frequency and the substrate dielectric confrequency of operation . The Bragg’s frequency is set to 2.6 to maintain stant a linear phase shift in both states [7]. From these specifications, in the slow-wave the required ground-to-ground spacing state is calculated using (5). To minimize radiation loss, the ground-to-ground spacing in a uniform CPW line is typically . However, for the unit cell used maintained less than herein, the phase velocity-matching condition for radiation to occur is not satisfied [14], [15]. Therefore, a larger value of can be chosen while maintaining single-mode operation. is set to 65 , which translates to In this study, at . The total length along the midpoint through the slot (6), where is in the slow-wave state is equal to is the length of the slot in the north–south direction and the length of the slot in the east–west direction (Fig. 1). Since the overall length of the unit cell cannot exceed set by the . However, this spacing is further Bragg’s frequency, required to accommodate the reduced by a factor of is set to 80 m. two ground-plane pedestals. In this study, The approximate signal path through the midpoint in the slot is calculated using (7) as follows: (5) (6) (7)

(3) in which is the effective dielectric constant of the transmission line and is the free-space velocity. Since the unit cell is and can be related to the constructed using a CPW line, physical line parameters using conformal mapping, as given in (4) [11] as follows:

Given the impedance-match requirement in both states , the only unknown variable in (7) is the separation . This separation is derived using (1) and (2) and has the form and the bridge capacitance given in (8). The spacing of the shunt beam is then calculated by rearranging (1) and is given in (8) as follows:

(8) It is seen from (8) that is inversely proportional to and the effective dielectric constant of the substrate. For m, , and assuming a spacing, quartz substrate, m, m, and m for

LAKSHMINARAYANAN AND WELLER: OPTIMIZATION AND IMPLEMENTATION OF IMPEDANCE-MATCHED TTD PHASE SHIFTERS

Fig. 2. C versus center conductor width W (with G = 300 m) for quartz . substrate. f is set to 2:6f

Fig. 3. Maximum value of f = 2 :6 f .

G

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versus substrate dielectric constant assuming

and GHz, respectively. Fig. 2 shows as a function of assuming and m. The impedance-matched condition is applied to the slowwave state resulting in a closed-form equation for the ground, as given in (9), as follows: plane bridge capacitance (9) Using (1)–(9), the phase constants in each state ( , ) and are derived in (10) and (11) as follows: the net phase shift

(10)

(11) , mm versus CPW footprint ( , , , For a given and ) is calculated using (4)–(11). However, the variable in (7) varies as [see Fig. 1(b)]. Therefore, the maximum value of is limited through (8) and is shown in Fig. 3 on quartz versus dielectric constant. The maximum value for and GHz is 1704, 306, and 160 m, with respectively. Using (4)–(11), the phase shift per millimeter versus the confor the slow-wave unit cell has been calductor width culated assuming a quartz substrate. Fig. 4 shows the calcumm for and GHz. CPW widths lated and m were assumed in generating these curves - and -GHz designs. Smaller widths of for the and m were used to minimize radiation for the -band design. The center conductor width for each case is

Fig. 4. Calculated phase shift per millimeter versus the center conductor width (W ) on a quartz substrate. The three data pairs pertain to f = 12; 50; and 110 GHz with W = 100 m and G , as shown in Fig. 3.

m, while is maintained at the maximum value shown in Fig. 3. and , It is seen from Fig. 4 that for a given becomes larger as the center conductor width decreases (high impedance). This is because a larger loading capacitance is needed to load the line to 50 and, therefore, the bridge capacitance has a larger effect on the phase velocity . Furthermore, two different impedance values with different total CPW widths result in the same phase shift. For example, with GHz, mm with 85 in m and 96 in the the slow-wave state m . The same is obtained normal state m and for 98 in the slow-wave state 116 in the normal state m . mm increases with deFig. 4 also demonstrates that m creasing . For example, in the case of GHz , the phase shift increases from 94 mm to 118 mm as decreases from 500 to 300 m. However, one of the implications of reducing is that the pull-in voltage required to actuate the shunt bridge increases. For a 1- m-thick Au plated bridge suspended 1.5 m above the CPW line with

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TABLE I COMPARISON OF TOTAL LOSS VERSUS RADIATION LOSS AT f = 110 GHz. THE CONTRIBUTION OF RADIATION LOSS TO THE TOTAL LOSS IS SHOWN IN PARENTHESES

Fig. 5. Calculated phase shift per millimeter versus the center conductor width (W ) on a quartz substrate for G = 300 m and W = 100 m. The CPW width G is fixed at the maximum value, as calculated in Fig. 3.

residual stress Mpa, , width m, and a GPa; V for a 300- m-long Young’s modulus V for a 200- m-long bridge. Therefore, bridge, while for pull-in voltages 30 V, is designed to be 300 m. An exception to this guideline is used at -band, in which case is set 200 m to reduce radiation loss. mm versus the conductor width for Fig. 5 shows is set to the the slow-wave unit cell on quartz. In this figure, maximum allowable value, as calculated in Fig. 3, with m and m. It is seen from this figure that the phase shift increases with . This is due to the fact that a smaller loading capacitance is needed to load the line to 50 . From Figs. 2–5, it can be seen that, for a given spacing and , mm increases as increases (low impedance in normal state), while mm increases as decreases (high impedance in slow-wave state). B. Optimization One of the first reported optimization methods pertaining to nonlinear distributed loaded lines was based on the work of Rodwell et al. [14]. In [14], footprints for minimum insertion loss were presented. The distributed line analysis was extended to optimize for the best phase shift by Nagra and York [15]. Barker and Rebeiz [13] applied a different method to a DMTL in which the MEM device was optimized to provide maximum phase shift for the minimum amount of insertion loss. In this paper, a method similar to [13] is applied to the slow-wave unit dB is obtained. cell so that maximum The insertion loss for a design on quartz is dominated by conductor loss. An analytical expression for the conductor loss per has been calculated using conformal mapping unit length [11] and is reproduced in (12) as follows:

(12)

where is the metal thickness, is the surface resistance, and is the conductivity of the metal. Dielectric and radiation losses are also present on these circuits. However, the dielectric loss calculated using a closed-form equation is negligible ( 1.5% of the total loss) up to 110 GHz [13]. Furthermore, the radiation loss is only significant ( 10% of the total loss) for width m and frequency 75 GHz (Table I). data for a 1-cmTo verify the accuracy of (12), measured long uniform CPW line on 500- m-thick quartz is compared with the predicted loss. Fig. 6(a) shows the measured and calculated loss versus frequency for different CPW line dimensions . It is seen from this figure that (12) underestimates the measured loss, as evidenced by the correction factor (ranging from 1.1 to 1.54) required to match the measured data. Furthermore, comparison of the measured and predicted loss as a function of total CPW width at 50 GHz shows that a single multiplication factor cannot be found to match the measured data Fig. 6(b). Since a single correction factor cannot be found, (12) is used without any multiplication factor and will predict higher dB than measured data. Therefore, (11) and (12) are only ) that provide trends for used to predict the widths ( and dB and not precise values. maximum The bridge resistance is calculated using (13) as follows:

(13) and are the shunt and ground plane bridge resiswhere tance, respectively. The loss due to the resistance of the bridge is 1% at GHz, 7% at GHz, and 23% at GHz These values are typical of a MEM bridge with a similar footprint [13]. The calculations were made for the slow-wave state, which typically has the greater loss when compared with the normal state, due to the longer signal path and interaction with an additional beam. Apart from the conductor loss, the unit cell also has loss due to contact resistance of the bridge, conductor roughness, and leakage via the SiCr bias lines. These effects are difficult to calculate and are not accounted for in the optimization procedure. dB is found by dividing the phase The figure-of-merit shift per unit cell by the loss per unit cell and is plotted in Fig. 7 for a quartz substrate. In this figure, the widths and vary independently and the corresponding dB is set to is indicated by the contour lines. The parameter is set to 300 m in light of the pull-in voltage consideration. 2 length of MEM beam for the GHz 600 m

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Fig. 6. (a) Measured and calculated CPW line loss versus frequency on quartz. (b) Comparison between measured and predicted loss at 50 GHz versus CPW pitch S m .

( = 300 )

design, and to the maximum allowable values of 306 and 160 m for the 50- and 110-GHz designs, respectively. The and is indicated on the top characteristic impedance and right-hand side of the plots. The maximum dB calcuand is listed in Table II. The 0 contour lated for a given in Fig. 7(b) and (c) is due to the fact that a high value of (in the normal state) is required to load the line to 50 and required in the slow-wave state. offsets the low dB values as indicated in Table II Although such high are theoretically possible, fabrication limitations restrict what can be achieved in practice. For example, on a quartz substrate GHz, the maximum dB dB with requires a shunt capacitance of 1.4 fF. Achieving such small capacitance is difficult due to fringing effects. Assuming around 8 fF, the optimization a minimum achievable m and procedure described above shows that m are good choices for the MEM bridge design. dB to These constraints reduce the maximum achievable

1 = 12

Fig. 7. =dB contours versus center conductor widths (W and W ). GHz. (b) f GHz. (c) f GHz. The (a) f characteristic impedance Z and Z corresponding to the width is shown on top/right axis. The “star” correspond to the conductor width (W and W ) used in verification.

= 50

= 110

600 dB on quartz. As the design equations do not use the correction factor for conductor loss and do not account for parasitics, measured values will be lower than 600 dB.

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TABLE II CALCULATED CPW WIDTHS (W=G AND W =G IN MICROMETERS) FOR MAXIMUM =dB

1

TABLE III CHARACTERISTICS OF THE THREE UNIT-CELL DESIGNS USED IN THIS VERIFICATION. PREDICTED AND MEASURED =dB IS ALSO LISTED

1

Fig. 9. Comparison of measured and calculated unit cell ; ; and GHz). (f

= 12 50

110

1 for three designs

Fig. 10. Measured S and S for the three unit-cell designs. The data for GHz is shown in inset. f

= 12

Fig. 8. Fabricated unit cells and accurate models for the unit cell in the normal and slow-wave states from [12].

C. Unit-Cell Implementation In order to verify the calculated dimensions that yield maximum dB, three slow-wave unit-cell designs were fabricated on a 500- m-thick quartz substrate. The three designs correspond to and GHz. The details pertaining to the fabrication process are presented in [10]. Measurements were performed using a Wiltron 360B vector network analyzer (VNA) up to 50 GHz. -band measurements were performed using a WR-10 waveguide module in conjunction with the Wiltron 360B. The bridge capacitances ( and ) and spacing ( and ) were calculated using (1)–(12), and

are listed in Table III. A photograph of the fabricated devices is shown in Fig. 8. The comparison between measured and calculated is shown in Fig. 9. The agreement is within 10% through 110 GHz. The discrepancy between the data sets may be because the calculation does not account for additional inductance due to current bending at the junctions. The worst case and over both states are shown in Fig. 10. is 20 dB through 110 GHz, demonstrating that the effective impedance is maintained close to 50 . The worst case measured is greater than 0.1 dB for the - and -GHz designs, and greater than 0.34 dB for the GHz design; these results translate to 371 dB at 12 GHz, 348 dB at 50 GHz, and 148 dB at 110 GHz. These values are lower than the predicted values in Fig. 7; however, the contact resistance, conductor roughness, and signal leakage via SiCr bias lines contribute additional loss and reduce the dB. Simply scaling the 50-GHz measured figure-of-merit by a conductor loss correction factor of 1.5 (Fig. 6) brings the measurement and prediction into close agreement.

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Fig. 11. Fabricated 1-bit phase shifters.

Fig. 13. Comparison between measured and equivalent-circuit model for the 1-bit =dB data.

1

Fig. 12. Comparison between measured and equivalent circuit model for the 1-bit  data.

1

TABLE IV MEASURED PERFORMANCE OF THE THREE PHASE SHIFTER DESIGNS AT CORRESPONDING f

III. 1-bit PHASE SHIFTER PERFORMANCE 1-bit phase shifters were constructed by cascading 13 unit GHz, ten unit cells for cells corresponding to GHz, and 12 unit cells for GHz (Fig. 11). In order to predict the performance of the phase shifters accurately, a semilumped model for the unit cell [12] was extracted from the unit cell measured results presented above. In this model, the parasitics associated with the MEM bridges and the current bending at the junctions are taken into account. The transmission line loss is modeled as a resistor and incorporates behavior to represent skin-depth loss. Fig. 12 shows the data. The agreement comparison of measured and modeled between the measured data and the model for the three designs is within 3% through 110 GHz. The worst case insertion loss is approximately 0.9 dB for GHz, 1.16 dB for GHz, and 2.65 dB for GHz (Table IV). The measured dB at 12,

Fig. 14. Comparison between measured and equivalent-circuit model for 1-bit S data in both the states. (a) f GHz. (b) f GHz. GHz. (c) f

= 110

= 12

= 50

50, and 110 GHz is 429 dB, 358 dB, and 150 dB, respectively (Fig. 13). The higher insertion loss at -band may be due to higher signal leakage in the SiCr. The measured

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(Fig. 14) is below 19 dB for both states from 1 to 110 GHz, indicating a good impedance match for all these designs. IV. SUMMARY In this paper, equations pertaining to the design of a slow-wave microelectromechanical systems (MEMS) true-time-delay phase shifter are presented. The inputs to these and equations are the maximum frequency of operation . To simplify the design the substrate dielectric constant process, the parasitics of the MEM bridges and certain discontinuities are not taken into account. It is shown that a low effective impedance in the normal state and a high effective impedance per section. in the slow-wave state result in the maximum Furthermore, the unit cell was optimized to provide maximum phase shift for the minimum amount of insertion loss. Optimal ) under these conditions center conductor widths ( and and GHz) on quartz are for three designs ( calculated. These designs were measured and verified to be data through 110 GHz. within 10% of the predicted 1-bit phase shifters were also developed in order to further validate the design optimization procedure. A semilumped model [12] was used to accurately predict the performance of the unit cell. The agreement between the measured data and the is below model is good through 110 GHz. The measured 19 dB for both states, and the measured and simulated are within 3%. The worst case insertion loss is approximately GHz, 1.16 dB for GHz, and 0.9 dB for GHz. 2.65 dB for REFERENCES [1] S. Barker and G. Rebeiz, “Distributed MEMS true-time delay phase shifters and wideband switches,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 11, pp. 1881–1889, Nov. 1998. [2] J. S. Hayden, A. Malczewski, J. Kleber, C. L. Goldsmith, and G. M. Rebeiz, “2- and 4-bit DC–18 GHz microstrip MEMS distributed phase shifters,” in IEEE MTT-S Int. Microw. Symp. Dig., Phoenix, AZ, May 2001, pp. 219–221. [3] Y. Liu, A. Borgioli, A. S. Nagra, and R. A. York, “ -band 3-bit lowloss distributed MEMS phase shifter,” IEEE Microw. Guided Wave Lett., vol. 10, no. 10, pp. 415–417, Oct. 2000. [4] H. Kim, S. Lee, J. Kim, J. Park, Y. Kim, and Y. Kwon, “A -band CPS distributed analog MEMS phase shifter,” in IEEE MTT-S Int. Microw. Symp. Dig., Philadelphia, PA, Jun. 2003, pp. 1481–1484. [5] B. Lakshminarayanan and T. Weller, “Distributed MEMS phase shifters on silicon using tapered impedance unit cells,” in IEEE MTT-S Int. Microw. Symp. Dig., Seattle, WA, Jun. 2002, pp. 1237–1240. [6] H. T. Kim, J. H. Park, S. Lee, S. Kim, J. M. Kim, Y. K. Kim, and Y. Kwon, “ -band 2-b and 4-b low-loss and low-voltage distributed MEMS digital phase shifter using metal–air–metal capacitors,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2918–2923, Dec. 2002.

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[7] J. S. Hayden and G. Rebeiz, “Very low-loss distributed -band and -band MEMS phase shifters using metal-air-metal capacitors,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 1, pp. 309–314, Jan. 2003. [8] J. Hung, L. Dussopt, and G. Rebeiz, “Distributed 2- and 3-bit -band MEMS phase shifters on glass substrates,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 2, pp. 600–606, Feb. 2004. [9] J. Perruisseau-Carrier, R. Fritschi, P. Crespo-Valero, and A. K. Skrivervik, “Modeling of periodic distributed MEMS-application to the design of variable true-time delay lines,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 600–606, Jan. 2006. [10] B. Lakshminarayanan and T. Weller, “MEMS phase shifters using cascaded slow-wave structures for improved impedance matching and/or phase shift,” in IEEE MTT-S Int. Microw. Symp. Dig., 2004, vol. 2, pp. 725–728. [11] K. C. Gupta, R. Garg, I. Bahl, and P. Bhartia, Microstrip Lines and Slotlines. Norwood, MA: Artech House, 1996. [12] B. Lakshminarayanan and T. Weller, “Design and modeling of 4-bit slow-wave MEMS phase shifters,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 120–127, Jan. 2006. [13] S. Barker and G. Rebeiz, “Optimization of distributed MEMS transmission-line phase shifters— -band and -band designs,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 11, pp. 1956–1966, Nov. 2000. [14] M. J. W. Rodwell, M. Kamegawa, R. Yu, M. Case, E. Carman, and K. S. Gilboney, “GaAs nonlinear transmission lines for picosecond pulse generation and millimeter-wave sampling,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 7, pp. 1194–1204, Jul. 1991. [15] A. S. Nagra and R. A. York, “Distributed analog phase shifter with low insertion loss,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 9, pp. 1705–1711, Jul. 1999.

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Balaji Lakshminarayanan (S’00–M’03) received the B.E. (with honors) degree from the Birla Institute of Technology and Science (BITS), Pilani, India, in 1995, and the M.S. and Ph.D. degrees in electrical engineering from the University of South Florida, Tampa, in 1999, and 2005, respectively. He currently conducts post-doctoral research with the University of California at San Diego, La Jolla. His research interests are in the area of RF-MEMS techniques for microwaves, application of micromachining for millimeter-wave circuits, and electromagnetic modeling of very large scale integration (VLSI) and microwave circuits.

Thomas M. Weller (S’92–M’95–SM’98) received the B.S., M.S., and Ph.D. degrees in electrical engineering from The University of Michigan at Ann Arbor, in 1988, 1991, and 1995, respectively. He is currently an Associate Professor with the Electrical Engineering Department, University of South Florida, Tampa. In 2001, he cofounded Modelithics Inc. He has authored or coauthored over 80 papers. Dr. Weller was a corecipient of the 1996 Microwave Prize presented by the IEEE Microwave Theory and Techniques Society (IEEE MTT-S). He was a recipient of a National Science Foundation (NSF) 1999 CAREER Award, and the IEEE MTT-S 2005 Outstanding Young Engineer Award.

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An Exposure System for Long-Term and Large-Scale Animal Bioassay of 1.5-GHz Digital Cellular Phones Kanako Wake, Member, IEEE, Atsushi Mukoyama, Soichi Watanabe, Member, IEEE, Yukio Yamanaka, Toru Uno, Senior Member, IEEE, and Masao Taki, Member, IEEE

Abstract—We developed an exposure system for an animal bioassay to evaluate possible health effects by the exposure with 1.5-GHz-band personal digital cellular system used in Japanese telecommunications. For large-scale and long-term study, i.e., 500 rats were employed for two years, the exposure system was designed to simultaneously expose 100 rats and to easily maintain. The specific absorption rate (SAR) inside the rat was evaluated with numerical simulation using anatomical rat models and was validated by comparing it with experimental dosimetry using phantom models. Variation and uncertainty of the SAR in the rat for each exposure was also statistically investigated using logged data, which had been acquired in the exposure system during the animal bioassay. We confirmed that the desired exposure (a SAR averaged over a brain of 2 W/kg for the high-dose group and 0.67 W/kg for the low-dose group and a SAR average over a whole body of less than 0.4 W/kg) had been fairly achieved during the exposure period of two years. A relatively large variation of the SAR was found in the male high-dose group. Index Terms—Bioassay, cellular phone, electromagnetic field (EMF), exposure, rat, specific absorption rate (SAR).

I. INTRODUCTION HERE IS a great concern by the general public related to the possible health effects of exposure to electromagnetic fields (EMFs) from cellular phones. The World Health Organization (WHO) established the International EMF Project to assess scientific evidence on the possible effects EMF might have. Although no adverse effects on health have been scientifically confirmed from exposure to low-level RF fields for extended periods, there has been little information available to assess whatever health risks there may be from exposure to pulsed or modulated RF fields such as those used in telecommunications. Therefore, the WHO has recommended further biological studies of

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Manuscript received April 21, 2006; revised August 24, 2006. This work was supported in part by the Committee to Promote Research on the Possible Biological Effect of Electromagnetic Field, Ministry of International Affairs and Communications. K. Wake, S. Watanabe, and Y. Yamanaka are with the National Institute of Information and Communications Technology, Tokyo 184-8795, Japan (e-mail: [email protected]; [email protected]). A. Mukoyama is with Sony Ericsson Mobile Communications Japan, Tokyo 108-0075, Japan. T. Uno is with the Department of Electrical and Electronic Engineering, Tokyo University of Agriculture and Technology, Tokyo 184-8588, Japan. M. Taki is with the Department of Electrical and Electronic Engineering, Tokyo Metropolitan University, Tokyo 192-0397, Japan. 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.2006.889149

good quality, especially large-scale, standard two-year animal bioassays [1]. To date, several animal bioassays have been performed to evaluate the effects that the EMF from cellular phones have on health.1 The purpose of this study was to develop an exposure system for a large-scale and long-term animal bioassay and analyze the specific absorption rate (SAR) inside animals. In our experiments, Fisher 344 rats were exposed to EMF for the personal digital cellular (PDC) system used in Japanese telecommunications. The minimal requirements for the exposure setup for bio-experiments have been summarized by Kuster and Shönborn [2]. The following lists six additional requirements necessary for our exposure system to simulate the biological effects of using cellular phones: 1) to concentrate the EMF exposure to the target tissue or organ, i.e., brain; 2) to deal with a significant number of animals (large scale); 3) to maintain the experimental period over the life span of animals (long term); 4) to quantitatively evaluate and accurately control the exposure; 5) to reduce the stress of the animals in the exposure setups; 6) to avoid EMF leakage from the exposure setups. For a study of brain tumor carcinogenicity using rats, over 200 animals are necessary and the experimental period needs to be at least two years. Several setups have been developed to expose rodents to EMF for animal bioassays [5]–[8]. Some exposure system were designed for whole-body exposure using radial waveguides [3], [4]. There are also several setups for brain localized exposure [5]–[8]. Although an exposure system using a small loop antenna [8] has advantages in localizing the exposure, these types of systems need an antenna for each animal and are, therefore, not suitable for large-scale experiments. In some systems, on the other hand, animals are radially displaced around a resonant wire antenna [5]–[7] or radial waveguide exposure systems. Burkhardt et al. and Swicord et al. used a half-wavelength dipole antenna [5], [6], and Watanabe et al. used a quarter-wavelength monopole antenna [7] Localization of the exposure with these systems is not better than with the former ones, although many animals can be simultaneously exposed. One reason for the low performance of the localized exposure is because of the high SAR around the lower part of the head, i.e., the chin, which is not the target tissue, i.e., the brain [7]. 1[Online].

Available: http://www.who.int/peh-emf/research/database/en/

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We developed an exposure system [9] for large-scale and long-term animal bioassays based on previous monopole exposure system. We improved localization of exposure by locating the antenna feed point on the top plane, where the target tissue, i.e., the brain of the rats, was closer to the feed point. While we previously reported preliminary SAR characteristics [9], more accurate and carefully validated investigations into SAR characteristics with improved anatomical rat models and antenna models are discussed in this paper. Variations and uncertainty in the rat SAR throughout the two-year period were also statistically investigated using logged data, which had been acquired in the exposure system during the animal bioassay [10]. Fig. 1. Antenna input power determined for animal bioassay based on the preliminary calculation of SAR for various sized rats from [9].

II. EXPOSURE CONDITIONS AND SYSTEM B. Exposure System A. Exposure Conditions The rats were exposed to a signal used in a PDC system. PDC is based on the technology of time division multiple access (TDMA) and is a unique digital modulation system used in Japanese wireless telecommunications. The PDC signal is characterized by a burst of 20 ms, a duty cycle of 1/3, and a carrier frequency of 800 MHz or 1.5-GHz band. A 1.439-GHz signal (the up-link frequency of the 1.5-GHz band) was used in this animal exposure system. The rats were divided into five groups, i.e., two cage-controls, sham, low-, and high-exposure groups in the animal bioassay [10]. Rats, except in the complete cage-control group, received maternal administration of ethylnitrosourea before exposure. Each group is composed of 50 males and 50 females. Males and females were exposed separately. The brain-average SARs were set to 2 and 0.67 W/kg for the high- and low-exposure groups, respectively, if the wholebody average SAR was less than 0.4 W/kg. The target value of 2 W/kg is chosen to test the limited value of the local SAR. To test not only one exposure condition, we chose the lower level of 0.67 W/kg, which is one-third of the target value of 2 W/kg. The temporal peak SAR of the lower exposure condition is 2 W/kg because the duty cycle of the PDC signal is one-third. We limited the whole-body average SAR to less than 0.4 W/kg to neglect the thermal effects of whole-body temperature elevation. We, therefore, reduced the antenna input power so that the whole-body average SAR did not exceed 0.4 W/kg even if the brain-average SAR did not achieve the target value. From our preliminary calculations of the rat SAR, we evaluated its dependence on the rat weight [9]. We then determined the target values of the antenna input power for various rat weights, as shown in Fig. 1. The brain-average SAR of rats under 140 g was lower than 2 W/kg because the whole-body average SAR exceeded 0.4 W/kg if the brain-average SAR was 2 W/kg. This corresponds to approximately the first two and five weeks from the start of exposure for male and female rats, respectively. The rats in the sham, low-, and high-exposure groups were placed in the exposure systems for 90 min/day, five days/week, for two years.

A 1.439-GHz PDC signal (the up-link frequency of the 1.5-GHz band) was generated with a signal generator (E4438C, Agilent, Tokyo, Japan), divided into ten with a divider (PD1920-12, R&K, Tokyo, Japan), amplified with amplifiers (A1920-3838-R, R&K), and delivered to ten exposure boxes in the system [see Fig. 2(a)]. Both forward and reverse power were monitored with a power meter (NRT-Z43, Rhode&Schwarz, Munich, Germany) for each antenna to assure that the radiated power would stay constant. The antenna input power was maintained within 5% of the target value by controlling the gain of amplifier with a PC. The date and time for the start and end of each EMF exposure, forward, and reverse powers, temperature in the boxes, as well as the weight of the rats, were recorded on the PC. Fig. 2(b) and (c) shows a photograph and schematic view of an exposure box. The exposure box was 90 90 60 cm , and the inside, except for the metal ceiling, was covered with an electromagnetic absorber (UP-10, Tohoku Chemical Industries, Tokyo, Japan) whose attenuation at the 1.5-GHz band is more than 26 dB. The feed point for the quarter-wavelength monopole antenna was located at the center of the ceiling, which was regarded as the ground plane. Ten rats were fixed inside a plastic holder much like a carrousel. The distance between the antenna and each rat’s nose was 3 cm and the distance between the ceiling and each rat’s back was 0.5 cm. During the exposure period of two years, the test animals grew rapidly. The weight of rats increased from under 100 g to over 400 g, as shown in Fig. 3. There was also a difference between males and females. Plastic holders were prepared for rats with four different sizes, i.e., with inner diameters of 4.0, 4.7, 5.5, and 6.2 cm. Fresh air was ventilated to reduce stress in the rats caused by the constraints. The temperature of the air was controlled at 22 C 2 C and the air-flow rate where rats can stick out their nose was approximately 1 m min. Since ten rats were placed in each exposure box, the system could, therefore, simultaneously expose 100 rats to the EMF signal. The system was repeatedly used three times to handle 300 rats in the sham, low-, and high-exposure groups in a day. We rotated the schedule for each subgroup consisting of ten rats with the exposure boxes and the exposure time throughout a day. Therefore, the frequency that each subgroup was exposed to in

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Fig. 4. Experimental view and X-ray CT image of the rat. (a) Rat in plastic tube. (b) X-ray CT image of rat.

III. DOSIMETRY A. Methods The SAR is in proportion to the square of the electric field as follows: (1) where , , and denote the electric field (rms), conductivity, and density of tissues, respectively. The SAR can also be evaluated by temperature elevation due to exposure as follows:

(2)

Fig. 2. Exposure system and box. A 1.439-GHz PDC signal is delivered to ten exposure boxes. Ten rats are simultaneously exposed to EMF with one exposure box. (a) Diagram of exposure system. (b) Photograph of exposure box. (c) Schematic view of the exposure box.

where denotes the temperature elevation and denotes the specific heat capacity. There are two techniques to evaluate the SAR inside an animal. The first is numerical analysis and the other is experimental dosimetry. The finite-difference time-domain (FDTD) method [11] is widely used for numerical analysis because it can deal with heterogeneity in biological beings with millimeter resolution. However, the resolution of millimeter order is sometimes not enough to precisely model an antenna. On the other hand, a practical antenna and exposure setup can be used in experimental dosimetry. An elevated temperature in (homogeneous) phantoms due to exposure can be measured noninvasively with a thermographic camera [12]. Experimental dosimetry often has limitations in modeling biological beings when homogeneous phantoms are used for practical reasons. The internal temperature is measured only in a certain section of the phantom. Numerical dosimetry with a proprietary FDTD code was used to evaluate the SAR inside rats in this study. We also validated the calculation results by comparing them with the experimental results. B. Models

Fig. 3. Growth curve of rats. (a) Females. (b) Males.

each box and during each period of a day were scheduled to be the same.

The complex structures of biological beings greatly influence the SAR distributions. We developed anatomical rat models based on X-ray computer tomography (CT) images of the rats fixed in the same plastic holders used in the animal bioassay. Fig. 4(a) and (b) shows a rat in a plastic holder and an X-ray CT image of it. Rats gain weight and change their shape during long-term exposure (Fig. 3). Although scaled models have often been used for modeling different-sized ones, the ratio

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Fig. 6. Calculated and measured input impedance of antenna. (a) Model. (b) Input impedance.

Fig. 5. Numerical rat models. (a) 126 g. (b) 263 g. (c) 359 g.

TABLE I TISSUES OF NUMERICAL RAT MODELS AND THEIR DIELECTRIC PROPERTIES AT 1.439 GHz

Fig. 7. Rat phantoms. Phantoms are bisected to allow thermographic temperature rise measurement in the sagittal section.

of growth can vary with each body part (e.g., head, legs, and body) or the shape of the animals can vary with age. We, therefore, developed three different sized rat models (126, 263, and 359 g). We also created other-sized models (76–430 g) by numerical scaling based only on weight information to evaluate the smooth relationship between the weight of the rats and the SAR. The models in Fig. 5 have a spatial resolution of 1 mm and consist of eight types of tissues (bone, brain, cerebrospinal fluid, eye, fat, muscle, nerve, and skin). Thought the cell size of 1 mm is not enough to express the thickness of thin tissue, i.e., skin, the influence on the whole-body and the brain-averaged SAR was thought to be small and neglected. Table I lists the dielectric properties of tissues at 1.439 GHz [13] employed in the FDTD analysis. The anatomy of the rat model, the arrangement of the ten rat models, and the antenna model were improved from those used in the previous study [9]. Fig. 6(a) shows the antenna model used in the analysis. The antenna was modeled with 1-mm resolution and fed with a onecell-width gap. The antenna input impedance without rats is plotted in Fig. 6(b). The calculated results fairly agree with measured data, especially the real part of the input impedance, which has dominant effects on antenna input power. The ceiling of the exposure box was assumed to be a perfect conductor, and second-order approximations of Mur’s boundary condition [14] were employed for the other walls covered with

the absorber. The plastic holders were not modeled in our FDTD calculation. We analyzed only a quarter of the calculation region using symmetrical conditions to save computational resources. We assumed ten rats were exposed in an equal arrangement. There are two-and-half rats in the quarter region of the system with symmetrical conditions. The SAR was evaluated only with one complete rat model. Homogeneous rat phantoms with the same shape as the numerical rat models were used for the experimental dosimetry. The ingredients for the phantom material for the 1.5-GHz band have been discussed elsewhere [15]. The phantoms could be bisected to allow thermographic temperature rise measurement in the sagittal section, as shown in Fig. 7. The dielectric properties of the phantoms were measured with the Network Analyzer (8753E, Hewlett-Packard, Tokyo, Japan) and probe (85070B, Hewlett-Packard). Their relative permittivity was approximately 58.14 and conductivity was 1.17 S/m. The specific heat capacity of phantoms was 3.77 J/g/K. The elevation in temperature due to high-power EMF exposure in the section of the phantom was measured with a thermographic camera (TVS-8100MK II, Nippon Avionics, Tokyo, Japan). C. SAR Analysis First, to validate our calculations, we compared SAR distributions from numerical and experimental dosimetry. Fig. 8(a) and (b) shows the calculated SAR distribution in a

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Fig. 9. SAR distributions in the sagittal section of heterogeneous rat models. SAR is normalized against their maximum value. (a) 126 g. (b) 263 g. (c) 359 g.

Fig. 8. SAR in homogeneous model (359 g) evaluated with calculation and experiment. (a) Calculated SAR distribution in sagittal section. (b) Measured temperature distribution in sagittal section. (c) Calculated and measured SAR along rat head for antenna input power of 1 W.

homogeneous rat model and the measured temperature distribution in a homogeneous phantom, respectively. Electrical , S/m) were properties of the phantom ( used for the calculation with homogeneous model. The temperature distribution was measured after 60-s exposure with antenna input power of 30 W. Both distributions are similar. We can see from Fig. 8(c) that the SARs through the brain also quantitatively agree well with each other. These results confirm the reliability of the numerical analysis we did. Calculated SAR distributions in the sagittal sections of heterogeneous rat models are shown in Fig. 9. The SAR is normalized against their maximum value. We can see that the exposure is localized within the rat’s head. Fig. 10 plots the whole-body average SAR and brain-average SAR as a function of the rat’s weight with an antenna input power of 1 W. These results were derived from FDTD simulation. Approximate curves for both whole-body average SAR and brain-average SAR were determined with fourth-order polynomial function. Dotted lines show the results of the 126-, 263-, and 359-g model actually derived from X-ray CT images. Other results were derived by numerical scaling based on weight information. Due to improvements to the rat and antenna models, the evaluated SAR values are slightly different from the preliminary calculations [9].

Fig. 10. Whole-body and brain-average SAR in several different size rats with antenna input power of 1 W. Lines are approximated curves with fourth-order polynomial functions.

IV. STATISTICAL EVALUATION OF SAR FOR TWO-YEAR ANIMAL BIOASSAY The date, time, antenna input power, and temperature inside each exposure box were recorded on a PC every 5 min in the animal bioassay [10]. The weights of the rats were also measured and recorded periodically (Fig. 3). Fig. 11(a) and (b) shows examples of logged data for antenna input power and temperature of the female high-dose group on a certain day. Five exposure boxes were used to expose five subgroups consisting of ten rats in the female high-dose group. We can see that both the antenna input power and the temperature were fairly constant during the exposure period of 90 min. The power variations during the exposure period were smaller than 2%. The fluctuations in temperature inside each exposure box were within 0.3 C during the exposure period. However, the average temperature inside each box differed because of the displacement of the exposure boxes. To avoid the influence of temperature differences among the exposure boxes, the frequency that a subgroup was placed in each of ten exposure boxes was the same, as described above. The changes in the rat’s weight during two years are summarized in Fig. 3. It should be noted that the rats in the sham, low-, and high-exposure groups gained less weight than the controlled groups because of handling and the constraints, although there were no significant differences between the sham, low-,

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Fig. 12. Evaluated SAR for each exposed group in animal bioassay. Error bars show the variation of SAR calculate with (8). (a) Whole-body average. (b) Brain average.

culated in same way. We assumed the average and the variaand ) tion of power absorption for unit input power ( are estimated by (5) and (6) using an approximated curve with ], which relates the a fourth-order polynomial function [ rat’s weight with power absorption for unit input power (Fig. 10) Fig. 11. Example of log data for animal bioassay. (a) Antenna input power. (b) Temperature.

(5) (6)

and high-exposure groups. Variation of weights among 50 rats in each group was larger in older and bigger size rats. The SARs in rats are in proportion to antenna input power and power absorption for unit input power. The power absorption for unit input power is related to each rat’s weight, as shown in Fig. 10. Therefore, the fluctuation of antenna input power and the variation of the rat’s weight in each group influence the SAR rat experienced with each exposure in animal bioassay. Other possible parameters such as the movements of rats, which can also influence the SAR in animal bioassay, were not considered in this study. With the logged data of antenna input power, the average for a certain exposure is defined as power (3) is the number of data and is the data, where is evaluated with and the variation of power (4) The average and the variation of weight ( and ) among 50 rats in each exposure group on a certain day are also cal-

The average SAR of each dose group for a certain exposure was defined as (7) using average antenna input power and average power . We used (8) to estimate absorption for unit input power the variation of the SAR for an exposure to each dose group as follows: (8) The assumption was made that the fluctuation of the antenna input power and the variations of power absorption for as a consequence of the variation of rat’s unit input power are independent of each other. weight The evaluated whole-body average SAR and brain-average SAR for each dose group during the animal bioassay are plotted in Fig. 12. The average SAR in each exposure was shown together with an error bar for SAR variation calculated with (8). We can see that the whole-body average SARs are lower than 0.4 W/kg for all groups throughout the experimental period of two years. We can, therefore, neglect the significant thermal effects of the whole body in our animal bioassay. Although the brain-average SAR was relatively smaller than the target value (2 W/kg for the high-dose group and 0.67 W/kg for the

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TABLE II WHOLE-BODY AND BRAIN-AVERAGE SAR (IN WATTS/KILOGRAM) AVERAGED FOR TWO YEARS

Values in () were the average and maximum variations in the SAR due to weight differences and power fluctuations during two years.

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TABLE III AVERAGE TEMPERATURE OF EACH GROUP FOR TWO YEARS

males was over 10%. This large variation is mainly due to the first term in the square root of (8). When the size of the rat is big, variation of weights among each group becomes large. Fig. 13 plots the average temperature of the air ventilated into the exposure box for each group during the animal bioassay, and the mean values are listed in Table III. The temperature was measured with the platinum thermo-resistance of 0.5 C resolution. There were no differences between the sham, low-, and high-dose groups or between the male and female groups. V. CONCLUSIONS We have developed an exposure system for the large-scale and long-term animal bioassay that could expose 100 rats simultaneously to a 1.5-GHz-band PDC signal, and it was easy to maintain. We evaluated detailed SAR characteristics using anatomical rat models with various weights and validated our calculations by comparing them with the data obtained by experimental dosimetry. The variation in the SAR is evaluated statistically using the logged data acquired during the two-year animal bioassay. We have confirmed that desired exposure was fairly achieved during most of the experiment. The uncertainty in the SAR was also evaluated and found to be relatively high for the male group. REFERENCES

Fig. 13. Average temperature inside exposure box for each group in animal bioassay. (a) Females. (b) Males.

low-dose group) in the early period, desired exposures were nearly achieved during the remainder of the two years. Table II lists the average SAR values for each group throughout the experimental period of two years. The average value of the brain-average SAR is approximately 20% and 14% lower than the target value for female and male rats, respectively. These differences are mainly due to the error in the preliminary calculations using relatively coarse rats and antenna models [9]. The uncertainty in the SAR due to power fluctuations and weight variations for each dose group were also summarized for a two-year period. The average and maximum variation of the SAR were listed in () in Table II. The whole-body and brainaverage SAR can be largely affected by the size or weight of the rats. The maximum variation of the brain-average SAR for

[1] “Technical report WHO fact sheets 181,” World Health Org., Geneva, Switzerland, 1998. [2] N. Kuster and F. Shönborn, “Recommended minimal requirements and development guidelines for exposure setups of bio-experiments addressing the health risk concern of wireless communications,” Bioelectromagnetics, vol. 21, pp. 508–524, 2000. [3] V. W. Hansen, A. K. Bitz, and J. R. Streckert, “RF exposure of biological systems in radial waveguides,” IEEE Trans. Electromagn. Compat., vol. 41, no. 4, pp. 487–493, Nov. 1999. [4] Q. Balzano, C. Chou, R. Cicchetti, A. Faraone, and R. Y. Tay, “An efficient RF exposure system with precise SAR estimation for in vivo animal studies at 900 MHz,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 11, pp. 2040–2049, Nov. 2000. [5] M. Burkhardt, U. Spinelli, and N. Kuster, “Exposure setup to test effects of wireless communications systems on the CNS,” Health Phys., vol. 73, no. 5, pp. 770–778, 1997. [6] M. Swicord, J. Morrissey, D. Zakharia, M. Ballen, and Q. Balzano, “Dosimetry in mice exposed to 1.6 GHz microwaves in a carrousel irradiator,” Bioelectromagnetics, vol. 20, pp. 42–47, 1999. [7] S. Watanabe, M. Taki, and Y. Yamanaka, “A microwave exposure setup for the heads of Sprague–Dawley rats,” in XXXVIth Int. Union Radio Sci. Gen. Assembly, 1999, p. 863. [8] C. K. Chou, K. W. Chan, J. A. McDougall, and A. W. Guy, “Development of a rat head exposure system for simulating human exposure to RF fields from handheld wireless telephones,” Bioelectromagnetics, vol. 20, no. 4, pp. 75–92, 1999. [9] S. Watanabe, A. Mukoyama, K. Wake, Y. Yamanaka, T. Uno, and M. Taki, “Microwave exposure setup for a long-term in vivo study,” in Proc. Int. IEEE AP-S Symp., 2000, vol. 1, pp. 225–228.

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[10] T. Shirai, M. Kawabe, T. Ichihara, O. Fujiwara, M. Taki, S. Watanabe, K. Wake, Y. Yamanaka, K. Imaida, M. Asamoto, and S. Tamano, “Chronic exposure to a 1.439 GHz electromagnetic field used for cellular phones dose not promote -ethylnitrosourea induced central nervous system tumors in F344 rats,” Bioelectromagnetics, vol. 26, pp. 59–68, 2005. [11] A. Taflove, Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method. Norwood, MA: Artech House, 1998. [12] A. W. Guy, “Analysis of electromagnetic fields induced in biological tissues by thermographic studies on equivalent phantom models,” IEEE Trans. Microw. Theory Tech., vol. MTT-19, no. 2, pp. 205–214, Feb. 1968. [13] C. Gabriel, “Compilation of the dielectric properties of body tissues at RF and microwave frequencies,” Brooks Air Force Base, Brooks AFB, TX, Tech. Rep. AL/OE-TR-1996-0037, 1996. [14] G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat., vol. EMC-23, no. 4, pp. 377–382, Nov. 1981. [15] Y. Okano, K. Ito, I. Ida, and M. Takahashi, “The SAR evaluation method by a combination of thermographic experiments and biological tissue-equivalent phantoms,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 11, pp. 2094–2103, Nov. 2000.

N

Kanako Wake (M’05) received the B.E., M.E., and D.E. degrees in electrical engineering from Tokyo Metropolitan University, Tokyo, Japan, in 1995, 1997, and 2000, respectively. She is currently with the National Institute of Information and Communications Technology (NICT), Tokyo Japan, where she is involved with research on biomedical electromagnetic compatibility. Dr. Wake is a member of the Institute of Electronics, Information and Communication Engineers (IEICE), Japan, the Institute of Electrical Engineers (IEE), Japan, and the Bioelectromagnetics Society. She was the recipient of the 1999 International Scientific Radio Union (URSI) Young Scientist Award.

Atsushi Mukouyama was born in Saitama, Japan, in 1977. He received the B.E. degree from the Tokyo University of Agriculture and Technology, Tokyo, Japan in 2000. He is an Engineer with Sony Ericsson Mobile Communications Japan, Tokyo, Japan. His research interests include small antennas for mobile phone.

Soichi Watanabe (S’93–M’96) received the B.E., M.E., and D.E. degrees in electrical engineering from Tokyo Metropolitan University, Tokyo, Japan, in 1991, 1993, and 1996, respectively. He is currently with the National Institute of Information and Communications Technology (NICT), Tokyo, Japan. His main interest is research on biomedical electromagnetic compatibility. Dr. Watanabe is a member of the Institute of Electronics, Information and Communication Engineers (IEICE), Japan, the Institute of Electrical Engineers (IEE), Japan, the Institute of Electrical and Electronics Engineering, and the Bioelectromagnetics Society. He was the recipient of several awards including the 1996 International Scientific Radio Union (URSI) Young Scientist Award and 1997 Best Paper Award presented by the IEICE.

Yukio Yamanaka was born in Yamaguchi Prefecture, Japan, on March 25, 1958. He received the B.S. and the M.S. degrees in electrical engineering from Nagoya University, Nagoya, Japan, in 1980 and 1983, respectively. In 1983, he joined the Radio Research Laboratory, Ministry of Posts and Telecommunications [now the National Institute of Information and Communication Technology (NICT)], Tokyo, Japan, where he has been engaged in the study of statistical characteristics of man-made noise and electromagnetic compatibility (EMC) measurements. He is currently the Group Leader of the EMC Group, NICT.

Toru Uno (M’85–SM’02) received the B.S.E.E. degree from the Tokyo University of Agriculture and Technology (TUAT), Tokyo, Japan, in 1980, and the M.S. and Ph.D. degrees in electrical engineering from Tohoku University, Sendai, Japan, in 1982 and 1985, respectively. In 1985, he became a Research Associate with the Department of Electrical Engineering, Tohoku University, and then an Associate Professor from 1991 to 1994. He is currently a Professor with the Department of Electrical and Electronic Engineering, TUAT. From August 1998 to May 1999, he was on leave from the TUAT as a Visiting Scholar with the Electrical Engineering Department, Pennsylvania State University. He has authored two books regarding the FDTD method for electromagnetics and antennas. He was an Associate Editor for the IEICE Transactions on Communications (2000–2005). His research interests include the electromagnetic inverse problem, computational electromagnetics, medical and subsurface radar imagings, and EMC. Dr. Uno is a member of the Institute of Electronics, Information and Communication Engineers (IEICE), Japan, American Geophysical Union (AGU), Applied Computational Electromagnetic Society (ACES), the Japan Society for Simulation Technology, and the Japan Society of Archaeological Prospection. He has served as a secretary of the Technical Group on Antennas and Propagation of the Institute of Electronics, IEICE (1999–2000). He also served as a vice-chair of the IEEE Antennas and Propagation Society (AP-S) Japan Chapter (2003–2004) and as a chair (2005–2006). He was the recipient of the Young Scientist Award and the Distinguished Contributions Award presented by the IEICE.

Masao Taki (M’02) was born in Tokyo, Japan, in 1953. He received the B.E., M.E., and Ph.D. degrees from the University of Tokyo, Tokyo, Japan, in 1976, 1978, and 1981, respectively. In 1981, he joined the Metropolitan University, Tokyo, Japan, where he is currently a Professor with the Department of Electrical and Electronic Engineering. He has been engaged in research on the compatibility of EMFs with the human body. Dr. Taki is a member of the Bioelectromagnetic Society, the Institute of Electronics, Information and Communication Engineers (IEICE), Japan, the Institute of Electrical Engineers (IEE), Japan, the IEEE Microwave Theory and Techniques Society (IEEE MTT-S), and the Japan Health Physics Society. Since 1995, he has been a member of the International Commission on Non-Ionizing Radiation Protection (ICNIRP). He is the chairman of the Japanese National Committee for International Electrotechnical Commission (IEC) TC106 and the chairman of the Japanese National Committee for URSI-K.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 2, FEBRUARY 2007

351

Letters Corrections to “Efficient Implementations of the Crank–Nicolson Scheme for the Finite-Difference Time-Domain Method Guilin Sun and C. W. Trueman There is a misprint in [1, eq. (19)]. The correct equation is as follows:

tan2 !12 t ( 1 )2 2 1x sin2 21y sin2 21z sin 2 = 12 + 12 + 12 (19a) tan2 !12 t 1 + ( 1 )6 sin2 21x sin2 21y sin2 21z 12 12 12 ( 1 )2 2 1x sin2 21y sin2 21y sin 2 = 12 + 12 + 12 2 1x sin2 21y + ( 1 )2 sin 1 22 12 1 x 1z 2 2 + ( 1 )2 sin 1 22 sin 1 22 sin2 21y sin2 21z 2 +( 1 ) 1 2 (19b) 12 c

Charles H. Cox III, Edward I. Ackerman, Gary E. Betts, and Joelle L. Prince In [1, Fig. 7], there was an error in the vertical axis. The correct label is “Dynamic Range (dB in 500 MHz),” as shown here in Fig. 1. The references for each point in the figure are as listed in [1].

t

x

y

c

c

Corrections to “Limits on the Performance of RF-Over-Fiber Links and Their Impact on Device Design”

t

x

t

x

c

t

c

t

c

t

z

y

y

y

x

y

x

z

y

z

z

:

Fig. 1. Analog link third-order spurious-free dynamic range versus frequency in a 500-MHz noise bandwidth.

REFERENCES [1] G. Sun and C. W. Trueman, “Efficient implementations of the Crank–Nicolson scheme for the finite-difference time-domain method,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 5, pp. 2275–2284, May 2006.

Manuscript received October 20, 2006; revised November 15, 2006. G. Sun is with the Department of Electrical and Computer Engineering, McGill University, Montreal, QC, Canada H3A 2A7 (e-mail: [email protected]). C. W. Trueman is with the Department of Electrical and Computer Engineering, Concordia University, Montreal, QC, Canada H3G 1M8. Digital Object Identifier 10.1109/TMTT.2006.889340

REFERENCES [1] C. H. Cox III, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the performance of RF-over-fiber links and their impact on device design,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 2, pp. 906–920, Feb. 2006.

Manuscript received October 16, 2006; revised October 19, 2006. The authors are with Photonic Systems Inc., Billerica, MA 01821 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.890072

0018-9480/$25.00 © 2007 IEEE

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 2, FEBRUARY 2007

351

Letters Corrections to “Efficient Implementations of the Crank–Nicolson Scheme for the Finite-Difference Time-Domain Method Guilin Sun and C. W. Trueman There is a misprint in [1, eq. (19)]. The correct equation is as follows:

tan2 !12 t ( 1 )2 2 1x sin2 21y sin2 21z sin 2 = 12 + 12 + 12 (19a) tan2 !12 t 1 + ( 1 )6 sin2 21x sin2 21y sin2 21z 12 12 12 ( 1 )2 2 1x sin2 21y sin2 21y sin 2 = 12 + 12 + 12 2 1x sin2 21y + ( 1 )2 sin 1 22 12 1 x 1z 2 2 + ( 1 )2 sin 1 22 sin 1 22 sin2 21y sin2 21z 2 +( 1 ) 1 2 (19b) 12 c

Charles H. Cox III, Edward I. Ackerman, Gary E. Betts, and Joelle L. Prince In [1, Fig. 7], there was an error in the vertical axis. The correct label is “Dynamic Range (dB in 500 MHz),” as shown here in Fig. 1. The references for each point in the figure are as listed in [1].

t

x

y

c

c

Corrections to “Limits on the Performance of RF-Over-Fiber Links and Their Impact on Device Design”

t

x

t

x

c

t

c

t

c

t

z

y

y

y

x

y

x

z

y

z

z

:

Fig. 1. Analog link third-order spurious-free dynamic range versus frequency in a 500-MHz noise bandwidth.

REFERENCES [1] G. Sun and C. W. Trueman, “Efficient implementations of the Crank–Nicolson scheme for the finite-difference time-domain method,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 5, pp. 2275–2284, May 2006.

Manuscript received October 20, 2006; revised November 15, 2006. G. Sun is with the Department of Electrical and Computer Engineering, McGill University, Montreal, QC, Canada H3A 2A7 (e-mail: [email protected]). C. W. Trueman is with the Department of Electrical and Computer Engineering, Concordia University, Montreal, QC, Canada H3G 1M8. Digital Object Identifier 10.1109/TMTT.2006.889340

REFERENCES [1] C. H. Cox III, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the performance of RF-over-fiber links and their impact on device design,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 2, pp. 906–920, Feb. 2006.

Manuscript received October 16, 2006; revised October 19, 2006. The authors are with Photonic Systems Inc., Billerica, MA 01821 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.890072

0018-9480/$25.00 © 2007 IEEE

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A. Hanke E. Hankui G. Hanson Z. Hao H. Happy A. R. Harish L. Harle L. D. Haro F. J. Harris H. Harris M. Harris P. Harrison R. G. Harrison O. Hartin H. Hashemi K. Hashimoto O. Hashimoto J. Haslett S. Hay J. Hayashi L. Hayden T. Heath J. Heaton M. P. Heijden G. Heiter J. Helszajn R. Henderson D. Heo P. Herczfeld H. Hernandez J. J. Herren K. Herrick F. Herzel J. S. Hesthaven K. Hettak P. Heydari T. S. Hie M. Hieda A. Higgins A. Hirata J. Hirokawa T. Hirvonen J. P. Hof K. Hoffmann R. Hoffmann M. Hoft E. Holzman J. S. Hong S. Hong W. Hong A. Hoorfar K. Horiguchi Y. Horii T. S. Horng J. Horton J. Hoversten H. Howe H. M. Hsu H. T. Hsu J. P. Hsu P. Hsu C. W. Hsue M. Z. Hualiang C. W. Huang F. Huang G. W. Huang J. Huang T. W. Huang W. Huei M. Huemer H. T. Hui J. A. Huisman A. Hung C. M. Hung J. J. Hung I. Hunter M. Hussein E. Hutchcraft B. Huyart J. C. Hwang J. N. Hwang R. B. Hwang M. Hélier Y. Iida S. Iitaka P. Ikonen K. Ikossi M. M. Ilic A. Inoue T. Ishikawa T. Ishizaki S. Islam Y. Isota M. Ito N. Itoh T. Itoh Y. Itoh T. Ivanov D. Iverson M. Iwamoto Y. Iyama D. Jablonski D. Jachowski R. Jackson R. W. Jackson A. Jacob M. Jacob S. Jacobsen D. Jaeger B. Jagannathan V. Jamnejad V. Jandhyala M. Janezic M. Jankovic R. A. Jaoude B. Jarry P. Jarry J. B. Jarvis A. Jastrzebski B. Jemison W. Jemison S. K. Jeng A. Jenkins Y. H. Jeong A. Jerng T. Jerse P. Jia X. Jiang B. Jim J. G. Jiménez J. M. Jin J. Joe R. Johnk L. Jonathan J. Joubert E. J. Jr N. C. Jr R. Judaschke J. Juntunen D. Junxiong T. Kaho M. Kahrs T. Kaiser S. Kalenitchenko V. Kalinin T. Kalkur Y. Kamimura H. Kanai S. Kanamaluru H. Kanaya K. Kanaya

Digital Object Identifier 10.1109/TMTT.2007.892055

S. Kang P. Kangaslahtii V. S. Kaper B. Karasik N. Karmakar A. Karwowski T. Kashiwa L. Katehi H. Kato K. Katoh A. Katz R. Kaul R. Kaunisto T. Kawai K. Kawakami A. Kawalec T. Kawanishi S. Kawasaki H. Kayano M. Kazimierczuk R. Keam S. Kee L. C. Kempel P. Kenington A. Kerr A. Khalil A. Khanifar A. Khanna F. Kharabi R. Khazaka J. Kiang J. F. Kiang Y. W. Kiang B. Kim C. S. Kim D. I. Kim H. Kim H. T. Kim I. Kim J. H. Kim J. P. Kim M. Kim W. Kim S. Kimura N. Kinayman A. Kirilenko V. Kisel M. Kishihara A. Kishk T. Kitamura K. I. Kitayama T. Kitazawa T. Kitoh M. Kivikoski G. Kiziltas D. M. Klymyshyn R. Knochel L. Knockaert Y. Kogami T. Kolding B. Kolundzija J. Komiak G. Kompa A. Konczykowska H. Kondoh Y. Konishi B. Kopp K. Kornegay T. Kosmanis P. Kosmas Y. Kotsuka A. Kozyrev N. Kriplani K. Krishnamurthy V. Krishnamurthy C. Krowne V. Krozer J. Krupka W. Kruppa D. Kryger R. S. Kshetrimayum H. Ku H. Kubo A. Kucar A. Kucharski W. B. Kuhn T. Kuki A. Kumar M. Kumar C. Kuo J. T. Kuo H. Kurebayashi K. Kuroda D. Kuylenstierna M. Kuzuhara Y. Kwon G. Kyriacou P. Lampariello M. Lancaster L. Langley U. Langmann Z. Lao G. Lapin L. Larson J. Laskar M. Latrach C. L. Lau A. Lauer J. P. Laurent D. Lautru P. Lavrador G. Lazzi B. H. Lee C. H. Lee D. Y. Lee J. Lee J. F. Lee J. H. Lee J. W. Lee R. Lee S. Lee S. G. Lee S. T. Lee S. Y. Lee T. Lee T. C. Lee D. M. Leenaerts Z. Lei G. Leizerovich Y. C. Leong R. Leoni P. Leuchtmann G. Leuzzi A. Leven B. Levitas R. Levy G. I. Lewis H. J. Li L. W. Li X. Li Y. Li H. X. Lian C. K. Liao M. Liberti E. Lier L. Ligthart S. T. Lim E. Limiti C. Lin F. Lin H. H. Lin

J. Lin K. Y. Lin T. H. Lin W. Lin Y. S. Lin E. Lind L. Lind L. F. Lind D. Linkhart P. Linnér D. Linton A. Lipparini D. Lippens V. Litvinov A. S. Liu C. Liu J. Liu J. C. Liu Q. H. Liu S. I. Liu T. Liu T. P. Liu O. Llopis D. Lo J. LoVetri N. Lopez Z. Lou M. Lourdiane G. Lovat D. Lovelace H. C. Lu K. Lu L. H. Lu S. S. Lu Y. Lu V. Lubecke S. Lucyszyn R. Luebbers N. Luhmann A. Lukanen M. Lukic A. D. Lustrac J. F. Luy C. Lyons G. Lyons G. C. M H. Ma J. G. Ma Z. Ma P. Maagt S. Maas G. Macchiarella P. Macchiarella J. Machac M. Madihian A. Madjar V. Madrangeas A. Maestrini G. Magerl S. L. Mageur A. A. Mahmoud S. Mahmoud F. Maiwald A. H. Majedi M. Makimoto S. Makino J. Malherbe G. Manara R. Manas G. Manes T. Maniwa R. Mansour D. Manstretta J. Mao S. G. Mao A. Margomenos R. Marques G. Marrocco J. Martel J. Martens J. Marti G. Martin E. Martinez K. Maruhashi J. E. Marzo H. Masallaei N. Masatoshi D. Masotti G. D. Massa B. Matinpour T. Matsui A. Matsushima S. Matsuzawa H. Matt G. Matthaei L. Maurer J. Mayock J. Mazierska S. Mazumder G. Mazzarella K. McCarthy G. McDonald R. McMillan D. McNamara D. McQuiddy F. Medina C. Melanie A. Á. Melcon F. Mena C. C. Meng H. K. Meng W. Menzel F. Mesa A. C. Metaxas R. Metaxas P. Meyer E. Michielssen A. Mickelson D. Miller P. Miller B. W. Min R. Minasian J. D. Mingo J. Mink B. Minnis F. Miranda D. Mirshekar C. Mishra S. Mitilineos R. Mittra K. Miyaguchi M. Miyakawa H. Miyamoto R. Miyamoto M. Miyashita M. Miyazaki K. Mizuno S. Mizushina J. Modelski W. V. Moer S. Mohammadi H. Moheb J. Mondal M. Mongiardo P. Monteiro C. Monzon A. D. Morcillo J. Morente T. Morf D. R. Morgan M. Morgan

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