Numerical Deembedding Technique for Planar Discontinuities with Periodically Nonuniform Feed Lines Sheng Sun, Lei Zhu School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798 Received 4 September 2007; accepted 2 December 2007

ABSTRACT: A numerical short-open calibration technique is applied to deembedding of planar discontinuities with periodically perturbed nonuniform feed lines in the full-wave method of moments (MoM) algorithm. Different from the other deembedding techniques that are based on the assumption of uniform feed lines, this proposed technique exhibits an unparalleled capability on modeling of planar circuits with nonuniform feed lines. To demonstrate this feature, the open and gap discontinuities are modeled under periodically nonuniform feed configuration, and the effective per-unit-length transmission parameters of slow-wave and electromagnetic bandgap structures are extracted from the full-wave MoM simulation. Two periodically nonuniform microstrip-line resonator circuits are modeled and C 2008 Wiley Periodicals, Inc. Int J RF and then confirmed by EM simulators and measurement. V Microwave CAE 00: 000–000, 2008.

Keywords: periodically nonuniform feed line; planar discontinuity; deembedding; short-open calibration technique; method of moments

modeling, as an individual two-port error box. There is no requirement to constitute a uniform feed line beforehand with known transmission parameters, i.e., phase constant and characteristic impedance, as required in many of other numerical deembedding techniques [5–9]. Of many available MoM algorithms, two distinctive deembedding techniques [5, 6] were developed and utilized for deembedding of planar discontinuities with uniform feed lines. In [5], standing-wave field amplitudes were calculated at the three different points along the uniform feed line to determine scattering matrix of the core circuit block. In [6], doubledelay technique was proposed to extract the port discontinuity as a purely shunt admittance. Shortly, a generalized short-open calibration based on the evenand odd-excitation of a single through line of length 2L was developed in [1]. Importantly, this deembedding technique does not restrict itself to any particu-

I. INTRODUCTION The numerical short-open calibration technique is initially proposed in [1–3] for modeling and deembedding of planar discontinuities with uniform coupled and uncoupled feed lines. To meet the prompt demands in advanced high-frequency and high-density integrated circuits with complicated configuration, this technique has been recently utilized in [4] to characterize microstrip line discontinuities with nonuniform feed lines. As an unparalleled feature, this deembedding technique allows one to consider the whole feed line section, ranged between the impressed source and the discontinuity under Correspondence to: S. Sun; e-mail: [email protected] DOI 10.1002/mmce.20312 Published online in Wiley InterScience (www.interscience. wiley.com). C 2008 Wiley Periodicals, Inc. V

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Figure 1. Short-open calibration technique for numerical deembedding of a planar two-port microstrip discontinuity with periodically nonuniform feed lines. (a) Layout configuration. (b) Equivalent network representation.

lar type of port discontinuity and is valid for either unbounded or shielded case [1, 8]. On the other hand, there has been an increasing desire in the recent years to investigate the transmission performances of electromagnetic bandgap (EBG) and metamaterials structures with periodically nonuniform configurations and develop their constituted microwave circuits with promising features [10, 11]. Because of the restriction of uniform feed lines in many deembedding procedures, all the available fullwave simulators or software cannot be directly applied to characterization of any planar discontinuity or circuit that is driven by periodically nonuniform feed lines, e.g., EBG- or metamaterial-based feed lines. The main objective of this work is to apply this unique short-open calibration technique to handle this class of challenging problems with periodically perturbed nonuniform feed lines.

II. MICROSTRIP DISCONTINUITIES WITH PERIODICALLY NONUNIFORM FEED LINES Figures 1a and 1b show the layout and equivalent circuit network for modeling and deembedding of a two-port microstrip discontinuity as the device under test (DUT) with two dissimilar feed lines. The calibration can be simply carried out by virtue of the perfect short- and open-circuited standards [1–4], formed in the MoM platform.

A. Open End Discontinuity Figure 2a illustrates a microstrip open end discontinuity with a nonuniform feed line that periodically varies in strip width. Figure 2b shows the relevant equivalent circuit network arranged for numerical deembedding of an equivalent open-ended fringing capacitance in the MoM algorithm. The whole periodically nonuniform feed line section is considered to be a single error box [X], as shown in Figure 2b. After calibrating out this error box relying on microstrip-line short- and open-circuited standards [1–4], the fringing capacitance of such an open end can be effectively deembedded. Figure 2c describes the normalized fringing capacitance (Coc) as a function of periodicity under the fixed slit size, i.e., Ws 5 t 5 0.2 mm. The extracted capacitance rises up and gradually converges to a constant fringing capacitance of its corresponding uniform feed-line (Ws 5 Wp) open end as periodicity (T) is sufficiently enlarged. It can be seen that the influence of perturbed strip width primarily depends on the ratio of T/t.

B. Gap Discontinuity A microstrip gap discontinuity with uniform and/or periodically nonuniform feed lines may be used as one of the indispensable elements in the characterization of various periodically nonuniform microstrip resonator circuits. Let us consider a gap structure with two identical uniform (Ws 5 Wf 5 0.6 mm) or periodically nonuniform (Ws 5 0.2 mm and Wf 5 0.6

International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce

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microstrip resonator circuits with miniaturized size and self-contained upper-stopband. Figure 4b depicts its equivalent circuit network with two unequal selfcapacitances, Cp1 and Cp2, and a mutual capacitance, Cg. Figure 4c shows the extracted capacitances with (solid-line) and without (dotted-line) periodically loaded slits. As the coupling space (S) is increased, the self- and mutual-capacitances have similar growing trend with those of the symmetrical gap while Cp2 exceeds Cp1 due to its wide strip width at the open end.

III. PERIODICALLY NONUNIFORM MICROSTRIP LINES Now, let us move to characterize and deembed periodically nonuniform microstrip line with varied feed lines, as shown in Figure 5. They are investigated herein using our proposed deembedding technique to extract their two effective per-unit-length transmission parameters under the feeding of uniform or peri-

Figure 2. Microstrip open end discontinuity with periodically nonuniform feed line. (a) 3D Physical view. (b) Equivalent circuit network. (c) Extracted open end capacitance versus periodicity (T).

mm) feed lines as illustrated in Figure 3a. Its equivalent circuit model in Figure 3b is composed of a mutual-capacitance (Cg) and two equal self-capacitances (Cp) in a p-network. Figure 3c shows the extracted mutual-/self-capacitances, normalized by strip width (Wf), with respect to coupling space (S) under er 5 1.0, 2.5, and 10.8. At a fixed low frequency of 2.0 GHz, these two capacitances with periodically nonuniform feed lines are found to be slightly lower than those with uniform feed lines. As the coupling space is enlarged, Cg decreases to zero while Cp increases toward a constant value. In comparison to the traditional gap discontinuity, the current gap structure has the decreased even-mode capacitance (Ceven 5 2Cp) and increased odd-mode capacitance (Codd 5 2Cg 1 Cp) as the gap distance is shortened. Next, an asymmetrical gap discontinuity as shown in Figure 4a is investigated as a coupling element in the modeling of various periodically nonuniform

Figure 3. Microstrip gap discontinuity with uniform or periodically nonuniform feed lines. (a) Physical layout. (b) Equivalent circuit network. (c) Extracted mutual-/selfcapacitances versus gap spacing (S).

International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce

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Figure 5. Physical layout for deembedding of periodically nonuniform microstrip line with finite periodic cells (N: the number of periodic cells, T: the periodicity). (a) Uniform feed lines. (b) Periodically nonuniform feed lines. (c) Equivalent circuit network.

Figure 4. Microstrip gap discontinuity with dissimilar uniform and periodically nonuniform feed lines. (a) Physical layout. (b) Equivalent circuit network. (c) Extracted mutual-/self-capacitances versus gap spacing (S).

odically nonuniform microstrip lines. As discussed in the earlier section, the two-port structures in Figures 5a and 5b can be commonly classified into two distinctive parts, feed-line related error boxes and core effective uniform nonuniform transmission line section, as illustrated in Figure 5c. As detailed in [1–4, 12], two error boxes can be consistently characterized via perfect short- and open-circuited standards in the MoM algorithm. As they are calibrated out, the Y-matrix or ABCD-matrix of the central line section with a length of Lp can be simply deembedded. Thus, their effective frequency-dispersive characteristic impedance (Zp) and propagation constant (cp) can be explicitly obtained in terms of four elements of ABCD-matrix, i.e., Ap, Bp, Cp, and Dp, such that sffiffiffiffiffiffi Bp Zp ¼ ReðZp Þ þ j ImðZp Þ ¼ Cp

ð1Þ

Figure 6. Extracted effective per-unit-length transmission parameters of periodically nonuniform microstrip line. (a) Normalized phase constant. (b) Characteristic impedance.

International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce

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Figure 7. Physical layout and equivalent circuit network for deembedding of periodically nonuniform EBG microstrip line.

coshðcp Lp Þ ¼ ap þ jbp ¼

Ap þ D p 2

ð2Þ

For a lossless periodic microstrip line, no energy is consumed along the direction of propagation in its passband such that Im(Zp) and ap/k0 always become zero. Figure 6 depicts calculated characteristic impedance and phase constant of a periodically nonuniform microstrip line with a periodicity of T 5 0.6 mm. As compared with the uniform case, the periodically nonuniform microstrip line achieves an enlarged normalized phase constant (bp/k0) and characteristic impedance (Zp). In addition, the results related to periodically nonuniform feed line are very close to those with uniform feed line. It implies that as the periodic cell number is sufficiently large, the external feed line configuration seems to hardly affect the guided-wave characteristics in the middle periodic section. As a validation, the model with uniform feed line is processed in Sonnet EM simulator again. The obtained results are plotted together in Figure 6 and have a good agreement with the other two cases. As the periodicity (T) is increased to the extent that the total length Lp 5 NT is multiples of a halfwavelength in length, the periodically nonuniform microstrip line may operate under the domain of its attenuation in certain frequency bands. In these stopbands or EBG [13], both cp and Zp become complex with sharp and quick frequency-dependent variation. Figures 7a and 7b show the layout and equivalent circuit arrangement for deembedding of such an EBG structure with periodically nonuniform feed lines. Since the periodic unit cells in the feed line section are exactly the same as those in the EBG line under modeling, only one unit-cell needs to be considered in the core EBG section. Thus, the total length of Lp 5 NT can be significantly shortened to T, where N 5

1. This procedure in deembedding of EBG structure not only largely reduces the CPU time and memory, but also avoids the spurious resonances that usually happened at the upper frequencies, where the feed

Figure 8. Extracted effective per-unit-length transmission parameters of periodically nonuniform EBG microstrip line. (a) Normalized propagation constant. (b) Characteristic impedance.

International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce

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Figure 9. Periodically nonuniform microstrip line resonator circuit capacitively excited by two uniform feed lines. (a) Physical layout. (b) Equivalent circuit representation.

line length is close to quarter-, half-, or full-wavelength. Figures 8a and 8b depict the extracted effective per-unit-length parameters of the periodic EBG transmission line in Figure 7, i.e., cp and Zp, under two different gap spacing of t 5 0.2 and 0.6 mm. At low frequencies, these EBG structures operate in their lowpass bands, where Im(Zp) and ap/k0 are negligibly small close to zero while bp/k0 and Re(Zp) rise up as a function of frequency. As the frequency increases to 11.1 or 13.1 GHz, two distinct stopbands or EBGs emerge with nonzero Im(Zp) and ap/k0 for two cases. In these stopbands, the attenuation constant (ap/k0) comes out, rises up/down, and disappears. In the same stopbands, characteristic impedance is dominated by its imaginary part and its value drops off with the frequency. These exhibited EBG performances are found exactly the same as those reported in [13]. They will be used in the next section to constitute an EBG-embedded periodically nonuniform microstrip line resonator circuit with good harmonic suppression and miniaturized size

5 0.6 mm and its layout is illustrated in Figure 9a with reference to Figure 6. It has a finite cell number of N 5 16. Its relevant equivalent network topology is depicted in Figure 9b for modeling of the whole resonator circuit in terms of a simple transmission line network. Figure 10 represents the derived S21magnitude of this periodically nonuniform line resonator circuit. The first four resonant frequencies are quasi equally distributed in the frequency range of 4.0 to 18.0 GHz, i.e., and occur at 4.46, 8.80, 13.00, and 17.04 GHz, where the effective length of such a line resonator line with L 5 NT is approximately equal to k/2, k, 3k/2, and 2k, respectively. Simulated results are well confirmed with those from the Sonnet EM [14], Agilent Momentum [15] simulators, and measurement. Emphatically, our results are obtained

IV. PERIODICALLY NONUNIFORM MICROSTRIP RESONATORS To validate the deembedded circuit and transmission line parameters under periodically nonuniform feed lines, two periodically nonuniform line resonator circuits, capacitively driven via coupling gaps, are constructed and characterized. The first resonator is made up of a periodically nonuniform slow-wave microstrip line section with a small periodicity of T

Figure 10. Comparison among predicted and measured frequency responses of S21-magnitude of periodically nonuniform microstrip line resonator circuit. (MoM: method of moments; SOC: short-open calibration).

International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce

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Figure 11. Periodically nonuniform EBG microstrip line resonator circuit capacitively excited by two uniform feed lines. (a) Physical layout. (b) Equivalent circuit representation.

via equivalent cascaded networks, thus the whole simulation only took a few seconds. Overall, these four sets of results are found in very reasonable agreement with each other over the plotted frequency range. Some small and visible discrepancies among them are mainly caused by different deembedding/calibration procedures and different numbers/configurations of selected cells in numerical implementation. In particular, such a difference becomes more and more significant as the |S21| is reduced much lower than 220 dB in the upper frequency range. Basically, it is attributed to approximation of the deembedding procedures used in those commercial tools. The circuit model parameters of a port discontinuity in MoM become increasingly frequency-dependent as the frequency increases, so the lumped port model is no longer accurate. On the other hand, the characteristic impedance of a microstrip line in an inhomogeneous substrate becomes inconsistent under the three usual definitions at high frequencies, i.e., the voltage-current, power-current, and power-voltage definitions. However, as emphasized and exhibited in our previous works, e.g. [1, 4 and 13], the short-open calibration or SOC procedure considers each of the periodically nonuniform feed lines as a two-port error box, in which ABCD matrix is numerically derived in the consistent MoM algorithm. Further, the above EBG microstrip line can be utilized to make up a periodically nonuniform line resonator circuit with harmonic-suppression. Figures 11a and 11b represent the relevant layout and equivalent circuit model. The EBG unit cell is herein chosen as t 5 0.2 mm with reference to Figure 8. Based on the

similar equivalent network as illustrated in Figure 11b, this EBG-embedded resonator with N 5 3 is characterized via three different numerical approaches and the simulated S21-magnitudes are plotted as in Figure 12. The first three resonant frequencies now occur at 4.80, 8.80, and 10.80 GHz. The distance between adjacent resonant frequencies is gradually reduced from 4.00 to 2.00 GHz. It is primarily caused by the fact that the phase constant (bp/k0) increases with frequency as shown in Figure 8a. Because of the existed bandgap beyond 11.1 GHz, there is no visible fourth resonant frequency in the plotted range until 20.0 GHz. In other words, the fourth transmission pole is faded within the bandstop of such an EBG structure. Finally, such a periodically

Figure 12. Comparison among predicted and measured frequency responses of S21-magnitude of a periodically nonuniform EBG microstrip line resonator circuit. (MoM: method of moments; SOC: short-open calibration).

International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce

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nonuniform resonator circuit is fabricated and its measured result is plotted in Figure 12 against three simulated graphs for experimental verification.

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V. CONCLUSIONS

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In this work, the short-open calibration technique has been applied to model and deembed a few different planar microstrip discontinuities with periodically nonuniform feed lines in the full-wave MoM algorithm. In this aspect, microstrip open and gap discontinuities as well as slow-wave and EBG microstrip lines have been thoroughly investigated. Extracted circuit model parameters are utilized to efficiently characterize a periodically nonuniform microstrip resonator circuit on the basis of simple transmission line theory. The predicted results are evidently validated by those from two commercial MoM simulators and the measurement of a fabricated resonator circuit. This unique deembedding technique in the full-wave MoM not only provides an efficient approach to modeling and designing of practical planar circuits with a number of circuit elements, but also analyzes a variety of high-density and microwave integrated circuits via cascaded subcircuit blocks.

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1. L. Zhu and K. Wu, Unified equivalent circuit model of planar discontinuities suitable for field theory-based CAD and optimization of M(H)MICs, IEEE Trans Microwave Theory Tech 47 (1999), 1589–1602. 2. M. Farina and T. Rozzi, A short-open deembedding technique for method-of-moments-based electromagnetic analyses, IEEE Trans Microwave Theory Tech 49 (2001), 624–628. 3. V.I. Okhmatovski, J. Morsey, and A.C. Cangellaris, On deembedding of port discontinuities in full-wave

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CAD models of multiport circuits, IEEE Trans Microwave Theory Tech 51 (2003), 2355–2365. S. Sun and L. Zhu, Short-open calibration technique for field theory-based parametric extraction of planar discontinuities with nonuniform feed lines, IEEE MTT-S Int Microwave Symp (2007), 273–276. D.C. Chang and J.X. Zheng, Electromagnetic modeling of passive circuit elements in MMIC, IEEE Trans Microwave Theory Tech 40 (1992), 1741–1747. J.C. Rautio, A de-embedding algorithm for electromagnetics, Int J Microwave Millimeter Wave Comput Aided Eng 1 (1991), 282–287. J.C. Rautio and R.F. Harrington, An electromagnetic time-harmonic analysis of shielded microstrip circuits, IEEE Trans Microwave Theory Tech 35 (1987), 726– 730. J.C. Rautio and V.I. Okhmatovski, Unification of double-delay and SOC electromagnetic deembedding, IEEE Trans Microwave Theory Tech 53 (2005), 2892–2898. M.P. Spowart and E.F. Kuester, An orthogonalitybased deembedding technique for microstrip networks, IEEE Trans Microwave Theory Tech 53 (2005), 938– 945. F. Yang, R. Coccioli, Y. Qian, and T. Itoh, Planar PBG structures: Basic properties and applications, IEICE Trans Electron E83-C (2000), 687–696. G.V. Eleftheriades, Enabling RF/microwave devices using negative-refractive-index transmission-line (NRITL) metamaterials, IEEE Antennas Propag Mag 49 (2007), 34–51. S. Sun and L. Zhu, Guided-wave characteristics of periodically nonuniform coupled microstrip lines: Even and odd modes, IEEE Trans Microwave Theory Tech 53 (2005), 1221–1227. L. Zhu, Guided-wave characteristics of periodic coplanar waveguides with inductive loading-unit-length transmission parameters, IEEE Trans Microwave Theory Tech 51 (2003), 2133–2138. Sonnet EM Suite, Sonnet Software Inc., New York, 2004. Advanced Design System (ADS) 2005a, Agilent Technol., Palo Alto, CA, 2005.

BIOGRAPHIES Sheng Sun received the B.Eng. degree in information engineering from the Xi’an Jiaotong University, Xi’an, China, in 2001, and the Ph.D. degree in microwave engineering from the Nanyang Technological University, Singapore, in 2006. From 2005 to 2006, he was a Research Fellow with the Integrated Circuits and Systems Laboratory, Institute of Micro-

electronics, Singapore. Since 2006, he has been a Research Fellow with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. His current research interests include microwave circuits, numerical electromagnetic modeling and deembedding techniques, microwave measurement, and millimeter-wave integrated circuits. Dr. Sun was the recipient of the Young Scientist Travel Grant (YSTG) presented at the 2004 International Symposium on Antennas and Propagation (ISAP’04), Sendai, Japan.

International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce

Numerical Short-Open Calibration Technique Lei Zhu received the B.Eng. and M.Eng. degrees in radio engineering from the Nanjing Institute of Technology (now: Southeast University), Nanjing, China, in 1985 and 1988, respectively, and the Ph.D. Eng. degree in electronic engineering from the University of Electro-Communications, Tokyo, Japan, in 1993. From 1993 to 1996, he was a Research Engineer with the Matsushita-Kotobuki Electronics Industries, Ltd., Tokyo, Japan. From 1996 to 2000, he was a Research Fellow with the Ecole Polytechnique de Montreal, University of Montreal, Quebec, Canada. Since July 2000, he has been an Associate Professor with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. His

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research interests include planar filters, planar periodic structures, planar antennas, numerical electromagnetic modeling and deembedding techniques. So far, he has published more than 150 papers in peer reviewed journals and conference proceedings, including 17 in IEEE T-MTT and 26 in IEEE MWCL/MGWL. Dr. Zhu received the Asia-Pacific Microwave Prize Award in 1997, the Silver Award of Excellent Invention from the Matsushita-Kotobuki Electronics Industries, Ltd., Japan, in 1996, and the First-Order Achievement Award in Science and Technology from the National Education Committee, China, in 1993. He served as an Associate Editor for the IEICE Transactions on Electronics during 2003–2005 and has been serving as an Associate Editor for IEEE Microwave and Wireless Components Letters since October 2006. He has been a member of the IEEE MTT-S Technical Committee 1 on Computer-Aided Design since June 2006.

International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce