IEEE MTT-V055-I07 (2007-07) [55, 07 ed.]

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JULY 2007

VOLUME 55

NUMBER 7

IETMAB

(ISSN 0018-9480)

PAPERS

Linear and Nonlinear Device Modeling SiGe HBT’s Small-Signal Pi Modeling .. ......... ........ ......... ......... . T.-R. Yang, J. M.-L. Tsai, C.-L. Ho, and R. Hu Wideband Nonlinear Response of High-Temperature Superconducting Thin Films From Transmission-Line Measurements . ......... ........ ......... ......... ........ ......... ......... ....... J. Mateu, J. C. Booth, and B. H. Moeckly

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Smart Antennas, Phased Arrays, and Radars Ultra-Wideband Multifunctional Communications/Radar System ........ ..... G. N. Saddik, R. S. Singh, and E. R. Brown A Quadrature Radar Topology With Tx Leakage Canceller for 24-GHz Radar Applications .... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ...... C.-Y. Kim, J.-G. Kim, and S. Hong

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Active Circuits, Semiconductor Devices, and Integrated Circuits Design of Ultra-Low-Voltage RF Frontends With Complementary Current-Reused Architectures ...... ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ....... H.-H. Hsieh and L.-H. Lu Electrical Backplane Equalization Using Programmable Analog Zeros and Folded Active Inductors .. ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ . J. Chen, F. Saibi, J. Lin, and K. Azadet Monolithic Integration of a Folded Dipole Antenna With a 24-GHz Receiver in SiGe HBT Technology ........ ......... .. .. ........ ......... ......... .. E. Öjefors, E. Sönmez, S. Chartier, P. Lindberg, C. Schick, A. Rydberg, and H. Schumacher A 4-bit CMOS Phase Shifter Using Distributed Active Switches ......... ........ ......... ........ D.-W. Kang and S. Hong Field Analysis and Guides Waves A New Brillouin Dispersion Diagram for 1-D Periodic Printed Structures ...... ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ..... P. Baccarelli, S. Paulotto, D. R. Jackson, and A. A. Oliner A Nonlinear Finite-Element Leaky-Waveguide Solver .. ......... ....... ... ........ ...... P. C. Allilomes and G. A. Kyriacou Effects of Losses on the Current Spectrum of a Printed-Circuit Line ..... ........ .. J. Bernal, F. Mesa, and D. R. Jackson

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(Contents Continued on Back Cover)

(Contents Continued from Front Cover) CAD Algorithms and Numerical Techniques Microwave Circuit Design by Means of Direct Decomposition in the Finite-Element Method .. ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... .... V. de la Rubia and J. Zapata

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Filters and Multiplexers Bandwidth-Compensation Method for Miniaturized Parallel Coupled-Line Filters ...... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ .... S.-S. Myoung, Y. Lee, and J.-G. Yook Miniaturized Dual-Mode Ring Bandpass Filters With Patterned Ground Plane . ...... R.-J. Mao, X.-H. Tang, and F. Xiao

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Packaging, Interconnects, MCMs, Hybrids, and Passive Circuit Elements Design and High Performance of a Micromachined -Band Rectangular Coaxial Cable ........ ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... M. J. Lancaster, J. Zhou, M. Ke, Y. Wang, and K. Jiang

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Instrumentation and Measurement Techniques A Swept-Frequency Measurement of Complex Permittivity and Complex Permeability of a Columnar Specimen Inserted in a Rectangular Waveguide ... ......... ......... ........ ......... ......... ........ ......... ......... ........ ..... A. Nishikata Phase and Amplitude Noise Analysis in Microwave Oscillators Using Nodal Harmonic Balance ....... ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ... S. Sancho, A. Suárez, and F. Ramirez

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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.2007.903659

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 7, JULY 2007

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SiGe HBT’s Small-Signal Pi Modeling Tian-Ren Yang, Julius Ming-Lin Tsai, Chih-Long Ho, and Robert Hu

Abstract—This paper presents the derivation procedure used in determining the parameters in SiGe HBT’s small-signal model where the Pi circuit configuration is employed. For both the transistor’s external base–collector capacitor and its base spreading resistor, new close-form expressions have been derived. Comparisons with existing approaches vindicate the feasibility and effectiveness of our formulations. With the impact of the lossy substrate effectively modeled and the frequency dependency of the transconductance properly addressed, this proposed extraction approach demonstrates accurate results up to 30 GHz with different bias conditions. Index Terms—Base spreading resistor, HBT, Pi model, SiGe. Fig. 1. HBT’s small-signal Pi model where the substrate network , C , and R . Transconductance G is set to consists of C =(1 + j! ). G e

I. INTRODUCTION

F

OR THE small-signal modeling of an HBT, either Tee or Pi circuit configuration can be used [1]–[5]. Though the Tee circuit reflects the device-physics aspect of this transistor, the Pi circuit in general provides better insight into designing circuits [6], [7] and, thus, will be explored in this paper. As shown in Fig. 1, the SiGe HBT’s Pi model consists of the intrinsic transistor, which is enclosed by the dotted box, the , the lossy substrate network , base spreading resistor , and [8]–[10], the external parasitic capacitor , , , and , and the base, emitter, and collector resistors and . Two time the input and output pad capacitors constants and are added on to the transconductance to account for its magnitude and phase frequency dependency [5], [11]–[15]. Output impedance of this voltage-induced current source is assumed infinite. As is well known, one challenge in SiGe HBT’s small-signal Pi modeling comes from the presence , whose location between and the intrinsic transistor of and , so far, makes the reliable derivation of both largely by way of additional test structures or numerically [3], [16]–[19]. In this paper, proper close-form expressions for these two parameters have been worked out and will be compared with existing approaches [20]–[22]. Fig. 2 shows the HBT under test, which is fabricated using a commercial 0.35- m SiGe–BiCMOS process and has bulk recm for the substrate. The base poly resistance sistivity of 8 is 200 square, while the silicided base poly for inter-connection has a much lower resistance of a few square. The emitter

Fig. 2. SiGe HBT under test. (a) Photograph. (b) Schematic of the transistor where the emitter is connected to ground, the base is the input, and the collector is the output.

layout consists of four fingers, with each 5.1- m long. Twoport short, open, load, and thru (SOLT) calibration is performed using 100- m Cascade probes on the ceramic substrate provided by the same vendor. With the Agilent network analyzer’s output power set to 10 dBm, losses due to the additional cables and bias-Ts’ will pull the power level down by 0.2 dB/GHz. DC bias for this transistor comes from HP4142B modular dc source/monitor. The 30-GHz upper frequency is mainly determined by the available frequency range of the coaxial cables used in the measurement. II. HBT SMALL-SIGNAL PI MODELING A. Determination of

Manuscript received October 28, 2006; revised January 28, 2007. The work of R. Hu was supported by the National Science Council, R.O.C., under Contract NSC 95-2221-E-009-315. T.-R. Yang was with the Department of Electronics Engineering, National Chiao Tung University, Hsin-Chu, Taiwan 300, R.O.C., and also with VIA Technologies, Taipei, Taiwan 100, R.O.C. He is now serving in the R.O.C. Army. J. M.-L. Tsai and C.-L. Ho are with VIA Technologies, Taipei, Taiwan 100, R.O.C. R. Hu is with the Department of Electronics Engineering, National Chiao Tung University, Hsin-Chu, Taiwan 300, R.O.C. (e-mail: shuihushuihu@yahoo. com). Digital Object Identifier 10.1109/TMTT.2007.900214

,

,

,

, and

Fig. 3 shows the flowchart in determining the transistor’s small-signal parameters. To find out the parasitics of the input and output pads, an open-pad test structure is designed where the transistor itself has been removed. Frequency-independent fF, capacitance can, therefore, be obtained as fF. The measured cross-coupling capacitance between input and output is three orders less and can be neglected. Using an appropriate deembedding procedure, these capacitors , , can be removed from the transistor’s model. Since

0018-9480/$25.00 © 2007 IEEE

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Fig. 3. Flowchart for the determination of the transistor’s small-signal parameters.

Fig. 5. Reverse-biased HBT for the determination of substrate network. versus (a) Schematic. (b) Measured and simulated substrate admittance Y frequency. The solid curves are the measured results; the dashed curves are the simulated ones with C = 21:4 fF, C = 57:5 fF, and R = 128 .

the reverse-biased intrinsic transistor resembles two separate caand are conpacitors, as shown in Fig. 5(a). As far as cerned, port 1 on the left of the schematic can be connected to ground and the signal is injected into port 2 on the right. If is much smaller than the impedance of the series cirfrom port 2 cuit, then most of the current passing through will flow down the branch rather than the branch. as open circuit, we then have By treating this , where (1) Fig. 4. Saturated HBT for the determination of R , R , and R . (a) Schematic. (b) By extrapolating the measured resistance to those corresponding to infinite base current, we have R = 7:5 , R = 4:1 , and R = 4:3 . The solid curves are measured at 2 GHz; the dashed ones are at 5 GHz.

and are beneath the first-layer metal, they are beyond the reach of a short-circuit test structure, but can be determined by forcing the transistor into saturation [23]. By setting the current flowing out of the collector to be half of the base current, we slowly increase the base current and voltage, from 1 and 18.8 mV, respectively, to 11 and 95 mV, respectively. Since this saturated intrinsic transistor can now be modeled as two conducting diodes, as shown in Fig. 4(a), a Tee circuit configuration, especially at low frequency, emerges. By extrapolating the measured resistance to that corresponding to infinite , and base current, we have the frequency-independent , equal to 7.5, 4.1, and 4.3 , respectively, as illustrated in Fig. 4(b). Here, the solid curves are those corresponding to 2 GHz, and the dashed curves are for 5 GHz. The impacts of and on the transistor’s Pi model are ready to be both , however, will be temporarily retained for removed now; the determination in Section II-B of the substrate network. B. Determination of Substrate Network Though mathematically the substrate network can be decided when the transistor is in saturation, the small in-parallel renders the derived substrate parameter values highly susceptible to measurement uncertainties. Reliable results can be obV, tained by reverse-biasing the transistor. With A, V, and A,

Since the complex-number can provide only two constraints at each frequency point, analytical solutions (of genuinely frequency independent) for the three parameters constructing the substrate network cannot be obtained; rather, numerical algorithms need to be used. As

(2)

and

(3) Least squares fit over the whole frequency range then gives fF, fF, and , respectively. Fig. 5(b) shows the admittance of the substrate network. Here, the solid curves are the measured results; the overlapping dashed curves are their simulated counterparts. Of course, other similar formulations can also be employed for the derivation of , , and [9], [10]. If only series [24], or parallel, circuit is used to model the substrate network, close-form analytical expressions can indeed be written, but the resulting parameters will be highly frequency dependent and, thus, are not useful. C. Determination of With both the substrate network and readily removed from the schematic of the reverse-biased transistor, analytical

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Fig. 6. Reverse-biased HBT where the substrate network and R in the previous schematic have been deembedded for the purpose of determining C . (a) Schematic. (b) Measured C versus frequency.

Fig. 7. Parameter values of the reverse-biased transistor with different base voltages. (a) R versus V where the circle markers are the derived results. (b) C , C , and C versus V .

solutions for the remaining circuit components, as shown in and are known where Fig. 6(a), can be obtained once

(4) By defining

as (5) Fig. 8. Measured and simulated S -parameters of the reverse-biased HBT. The solid curves are the measured results; the overlapping dashed curves are their simulated counterparts.

we have

(6) If the admittance matrix of the , then designated as

,

, and

sub-circuit is

(7) and, thus, (8) The measured results are fF, , fF, and as shown in Fig. 6(b), fF. The three other expressions derived using (7) also give the same value. If we change the base voltage while keeping the collector node grounded, different reverse-biased parameter values can be oband are intained, as illustrated in Fig. 7, where both dependent of the base voltage in this reverse-biased condition. on the transistor’s Pi model can be reNow, the impact of moved, i.e., deembedded.

In the conventional approach using IC-CAP, detailed layout information, dc capacitance measurement, and numerical fine (and tuning have to be employed for the determination of ). Besides, it postulates that has to be independent of of a frequency. Recently, an analytical formulation for normal-biased transistor has been proposed [20]. However, the extensive use of least squares fits makes it to a large degree at each freresemble a numerical approach. In our case, quency point can be directly calculated and compared. Indeed, , sometimes more than as demonstrated in Section II-D on one analytical solutions can be written for a certain parameter; however, not all of them render the same result over the intended frequency range. Figs. 8 and 9 show the -parameters of the reverse-biased transistor. Here, the solid curves are the measured results; the overlapping dashed curves are their simulated counterparts. Bias condition and the component values used in the simulation are tabulated in Table I.

D. Determination of Normal-Biased Now, with the bias of the transistor being set at mA (current density 1.13 mA m ),

V, V, and

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Fig. 11. S and 1=Y curves used in deriving R . (a) S used for the extrapolation of (16) from 0.1 to 30 GHz. (b) 1=Y used for the extrapolation of (17) from 0.5 to 30 GHz. Both curves move clockwise as frequency increases. Fig. 9. Measured (solid) and simulated (dashed) S -parameters of the reversebiased HBT on the Smith chart. TABLE I REVERSE-BIASED TRANSISTOR

On the other hand, if the transconductance can be treated as in terms of and a real number, we can also express [21], i.e.,

(12)

with (13) and

(14) Fig. 10. R and the normal-biased intrinsic transistor. (a) Schematic. (b) Measured R where curve 1 is derived using (11), curve 2 is using (12), and curve 3 is using the algorithm suggested in [20].

A, the normal-biased , as shown in Fig. 10(a), and are known. As can be determined when both

needs to be assumed Here, the current flowing through branch. Fig. 10(b) displays much larger than that on the where curve 1 comes from our proposed (11), the derived the upper bound curve 2 is from (12), and the somehow lower bound curve 3 is using the algorithm suggested in [20]. The discrepancy between these three curves instigates a further numerical survey as follows. As suggested in the IC-CAP user’s manual, by assuming the and transconductance to be a real number

(9) i.e., (15)

(10) then the input impedance (with 50expressed as

therefore,

output loading) can be

(16) (11)

contour on the Smith Extrapolation of the corresponding of 25 , as illustrate chart to infinite frequency results in a to be much larger in Fig. 11(a). Alternatively, by assuming

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than the impedance of at high enough frequency, the incan be obtained by extrapolating the curve to tended the -axis [22], as (17) is around 25 . Both numerical apIn Fig. 11(b), this proaches, therefore, confirm our analytical method. , though the numerical apRegarding the derivation of proaches render correct results, they are incapable of revealing and, thus, the frequency dependence (or independence) of cannot be viewed as wideband modeling in this respect. Strictly exists only at infinite frequency. On speaking, a valid the other hand, since the frequency variable is not used in the three discussed analytical (and, thus, wideband) methods, at every frequency point can be independently obtained and over compared. Among the three, (12) has a near-constant the widest bandwidth; however, the 6- offset relative to all the other discussed approaches limits its application. While the one suggested in [20] gives valid results for frequency from for 15 to 30 GHz, our proposed (11) can have satisfying frequency from down below 10 to 30 GHz and, thus, is the most preferred. is directly derived at the intended Furthermore, since this bias point, rather than adopted from values using other bias conditions, the current crowding effect [25]–[28], even if exists, will not affect the validity of our proposed formulation, and since the simulated -parameters agree with their measured counterparts, as demonstrated in Section II-E, it is just fine using a single , rather than a more complicated sub-circuit [29], in modeling this part of the transistor.

Fig. 12. Measured and simulated transconductance. (a) Magnitude of the transconductance where the solid curve is the measured result, i.e., Y Y ; the overlapping dashed curve is its simulated counterpart with G = 123:8 mS,  = 1:5 ps, and  = 1:2 ps. (b) Phase of the transconductance where the solid curve is the measured result; the overlapping dashed curve 1 is its simulated counterpart; dashed curve 2 is the simulated phase with only  , but no  ; dashed curve 3 is with  only.

0

nonzero in the transconductance, the conventional sion needs to be revised. As

(19) where the leakage currents flowing through and are assumed negligible at high frequency in the approximation, we then have

(20)

E. Determination of Normal-Biased Intrinsic Parameters With deembedded, parameters of the normal-biased intrinsic transistor can, therefore, be determined as fF, , and fF. With the transconductance defined as [5], [11]–[13] (18) is the angular frequency, we then have mS, ps, and ps. In Fig. 12(a), the solid curve is the magnitude of the measured and the overlapping dashed curve is its simulated counterpart. Apparently, is needed for explaining this magnitude frequency dependency. In Fig. 12(b), , the overlapping the solid curve is the phase of the measured dashed curve 1 is its simulated counterpart with both and employed. If we retain the time constant while omitting , or retain , but omitting , shown as dashed curves 2 and 3, respectively, phase discrepancy can be observed. Knowing the intrinsic transistor’s parameters now enables the calculation of the cutoff frequency , which is the frequency is equal to 1. With where the magnitude of the current gain where

expres-

and (21) Fig. 13 shows the curves in logarithmic and linear scales. In Fig. 13(a), solid curve 1 corresponds to the total transistor, solid curve 2 is with the intrinsic transistor only, the dashed straight lines are their high-frequency linear approximations in this logarithmic scale. Fig. 13(b) has the same results, but expressed in is 42 GHz when the total transistor is emlinear scale. Thus, ployed, as by extrapolating the logarithmic curve to the -axis, equal to 56 GHz for the intrinsic resistor, which and we have agrees with the 54.1 GHz calculated using (20); it is 58.4 GHz using (21), also a valid approximation. Likewise, by defining the gain as [30]

(22)

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Fig. 13. Magnitude of h versus frequency. (a) In logarithmic scale where the solid curve 1 is with the total transistor; solid curve 2 is with the intrinsic transistor only. The two dashed curves are their high-frequency linear approximation for deriving f . (b) In linear scale.

Fig. 16. Measured (solid line) and simulated (dotted line) S -parameters of the normal-biased HBT on the Smith chart.

TABLE II NORMAL-BIASED TRANSISTOR

Fig. 14. Magnitude of gain U versus frequency. (a) In logarithmic scale where the two overlapping solid curves are the measured and simulated results with the total transistor; the dashed curve is the high-frequency linear approximation . (b) In linear scale. for deriving f

Fig. 17. Measured and simulated S -parameters of the HBT biased at V = 2 V, I = 6:6 mA, V = 0:95 V, and I = 65:7 A. The solid curves are the measured results; the overlapping dashed curves are their simulated counterparts. The error  is 0.38%.

as shown in Figs. 15 and 16. Here, the solid curves are the measured results; the overlapping dashed curves are their simulated counterparts. If we define the error as Fig. 15. Measured and simulated S -parameters of the normal-biased HBT. The solid curves are the measured results; the overlapping dashed curves are the simulated ones. The error  defined in (23) is 0.13%.

(23) where is the stability factor, the maximum oscillation fre, which is the frequency at which is equal to 1, quency can be easily obtained. Both the measured -parameters and the GHz, as model-based -parameters give the same shown in Fig. 14. Accuracy of the HBT’s small-signal Pi modeling can be verified by comparing the measured and simulated -parameters,

where the summation is over the frequency points from 0.1 to 30 GHz. The calculated error is 0.13%. The bias condition and parameter values used in the simulation are tabulated in Table II. Applying the same collector and base voltages, another transistor of the same size on the same wafer is measured, with

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TABLE III TRANSISTOR BIASED AT DIFFERENT BASE VOLTAGES

Fig. 18. Measured and simulated S -parameters of the HBT biased at V = 2 V, I = 2:8 mA, V = 0:90 V, and I = 19:4 A. The solid curves are the measured results; the overlapping dashed curves are their simulated counterparts. The error  is 0.27%.

and then 0.73 mA, with the corresponding current density being 1.08, 0.46, and 0.12 mA m , respectively. The simulated results in these cases agree with their respective measured counterparts, as shown in Figs. 18 and 19. On the other hand, if is changed from 0.95 to 1.0 V, will increase from 6.6 to 11.8 mA (current density 1.93 mA m ) with slightly improved gain response at low frequency; the simulated results still follow the and under measured ones, as shown in Fig. 20. Both these different base voltages are tabulated in Table III. III. CONCLUSION

Fig. 19. Measured and simulated S -parameters of the HBT biased at V = 2 V, I = 0:73 mA, V = 0:85 V, and I = 4:6 A. The solid curves are the measured results; the overlapping dashed curves are their simulated counterparts. The error  is 0.86%.

In this paper, new procedures for deriving the SiGe HBT’s small-signal Pi modeling have been developed. For the external and the base spreading resistor base–collector capacitor , reliable analytical solutions have been proposed and compared with other methods. The lossy substrate effect has also been appropriately modeled. Agreements between the measured and simulated results in each derivation step thus vindicates the accuracy and efficiency of our new modeling approach. In addition to the intended normal-biased condition, this proposed approach shows satisfying results for different biasing conditions. Therefore, circuits designed using the HBT’s small-signal Pi model can be accurately analyzed. In the future, we plan to extend this modeling work to address each parameter’s nonlinear effect, thus facilitating the design of mixers. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers of this TRANSACTIONS for suggestions and encouragement. REFERENCES

Fig. 20. Measured and simulated S -parameters of the HBT biased at V = 2 V, I = 11:8 mA, V = 1:0 V, and I = 124 A. The solid curves are the measured results; the overlapping dashed curves are their simulated counterparts. The error  is 0.55%.

consistent results shown in Fig. 17. If is changed from 0.95 to 0.9 V and then 0.85 V, will decrease from 6.6 to 2.8 mA

[1] U. Basaran, N. Wieser, G. Feiler, and M. Berroth, “Small-signal and high-frequency noise modeling of SiGe HBTs,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 3, pp. 919–928, Mar. 2005. [2] B. Li and S. Prasad, “Basic expressions and approximations in smallsignal parameter extraction for HBT’s,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 5, pp. 534–539, May 1999. [3] B. Li, S. Prasad, L. W. Yang, and S. C. Wang, “A semianalytical parameter-extraction procedure for HBT equivalent circuit,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 10, pp. 1427–1435, Oct. 1998. [4] D. A. Teeter and W. R. Curtice, “Comparison of hybrid Pi and Tee HBT circuit topologies and their relationship to large signal modeling,” in IEEE MTT-S Int. Microw. Symp. Dig., Denver, CO, Jun. 1997, vol. 2, pp. 375–378. [5] A. Schuppen, U. Erben, A. Gruhle, H. Kibbel, H. Schumacher, and U. Konig, “Enhanced SiGe heterojunction bipolar transistors with 160 ,” in Int. Electron Device Meeting Tech. Dig., 1995, pp. GHz-f 743–746. [6] R. Hu, “Wide-band matched LNA design using transistor’s intrinsic gate–drain capacitor,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 3, pp. 1277–1286, Mar. 2006. [7] R. Hu and T. H. Sang, “On-wafer noise parameter measurement using wideband frequency-variation method,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 7, pp. 2398–2402, Jul. 2005.

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[8] M. Pfost, H. Rein, and T. Holzwarth, “Modeling substrate effects in the design of high-speed Si-bipolar IC’s,” IEEE J. Solid-State Circuits, vol. 31, no. 10, pp. 1493–1501, Oct. 1996. [9] U. Basaran and M. Barroth, “An accurate method to determine the substrate network elements and base resistance,” in Proc. IEEE Bipolar/ BiCMOS Circuits Technol. Meeting, Sep. 2003, pp. 93–96. [10] H. Y. Chen, K. M. Chen, G. W. Huang, and C. H. Chang, “An improved parameter extraction method of SiGe HBTs’ substrate network,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 6, pp. 321–323, Jun. 2006. [11] J. A. Seitchik, A. Chatterjee, and P. Yang, “An accurate bipolar model for large-signal transient and AC applications,” in Int. Electron Devices Meeting Tech. Dig., 1987, pp. 244–247. [12] M. Reisch, High-Frequency Bipolar Transistor. Berlin, Germany: Springer-Verlag, 2003, ch. 3. [13] M. Schröter and T. Y. Lee, “Physics-based minority charge and transit time modeling for bipolar transistors,” IEEE Trans. Electron Device, vol. 46, no. 2, pp. 288–300, Feb. 1999. [14] M. Kahn, S. Blayac, M. Riet, P. Berdaguer, V. Dhalluin, F. Alexandre, and J. Godin, “Measurement of base and collector transit times in thinbase InGaAs/InP HBT,” IEEE Electron Device Lett., vol. 24, no. 7, pp. 430–432, Jul. 2003. [15] M. Malorny, M. Schroter, D. Celi, and D. Berger, “An improved method for determining the transit time of Si/SiGe bipolar transistors,” in Proc. IEEE Bipolar/BiCMOS Circuits Technol. Meeting, Sep. 2003, pp. 229–232. [16] D. Costa, W. U. Liu, and J. S. Harris, “Direct extraction of the AlGaAs/GaAs heterojunction bipolar transistor small-signal equivalent circuit,” IEEE Trans. Electron Devices, vol. 38, no. 9, pp. 2018–2024, Sep. 1991. [17] B. Ardouin, T. Zimmer, H. Mnif, and P. Fouillat, “Direct method for bipolar base–emitter and base–collector capacitance splitting using high frequency measurements,” in Proc. IEEE Bipolar/BiCMOS Circuits Technol. Meeting, Oct. 2001, pp. 114–117. [18] W. Sansen and R. G. Mayer, “Characterization and measurement of the base and emitter resistances of bipolar transistors,” IEEE J. Solid-State Circuits, vol. SSC-7, no. 12, pp. 492–498, Dec. 1972. [19] T. Fuse and Y. Sasaki, “An analysis of small-signal and large-signal base resistances for submicrometer BJT’s,” IEEE Trans. Electron Devices, vol. 42, no. 3, pp. 534–539, Mar. 1995. [20] L. Degachi and F. M. Ghannouchi, “Systematic and rigorous extraction method of HBT small-signal model parameters,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 2, pp. 682–688, Feb. 2006. [21] D. W. Wu and D. L. Miller, “Unique determination of AlGaAs/GaAs HBT’s small-signal equivalent circuit parameters,” in IEEE 15th GaAs Symp. Tech. Dig., San Jose, CA, Oct. 1993, pp. 259–262. [22] W. J. Kloosterman, J. C. J. Paasschen, and D. B. M. Klaassen, “Improved extraction of base and emitter resistance from small signal high frequency admittance measurements,” in Proc. IEEE Biploar/BiCMOS Circuits Technol. Meeting, Sep. 1999, pp. 93–96. [23] S. A. Maas and D. Tait, “Parameter-extraction method for heterojunction bipolar transistors,” IEEE Microw. Guided Wave Lett., vol. 2, no. 12, pp. 502–504, Dec. 1992. [24] S. D. Harker, R. J. Havens, J. C. J. Paasschens, D. Szmyd, L. F. Tiemeijer, and E. F. Weagel, “An S -parameter technique for substrate resistance characterization of RF bipolar transistor,” in Proc. IEEE Bipolar/BiCMOS Circuits Technol. Meeting, Sep. 2000, pp. 176–179. [25] J. E. Larry and R. L. Anderson, “Effective base resistance of bipolar transistors,” IEEE Trans. Electron Devices, vol. ED-32, no. 11, pp. 2503–2505, Nov. 1985. [26] J. S. Yuan, J. J. Liou, and W. R. Eisenstadt, “A physics-based current-dependent base resistance model for advanced bipolar transistors,” IEEE Trans. Electron Devices, vol. 35, no. 7, pp. 1055–1062, Jul. 1988. [27] M. Schröter, “Modeling of the low-frequency base resistance of single base contact bipolar transistors,” IEEE Trans. Electron Devices, vol. 39, no. 8, pp. 1966–1968, Aug. 1992. [28] V. Fournier, J. Dangla, and C. Dubon-Chevallier, “Investigation of emitter current crowding effect in heterojunction bipolar transistors,” Electron. Lett., vol. 29, no. 20, pp. 1799–1800, Sep. 1993. [29] W. B. Tang, C. M. Wang, and Y. M. Hsin, “A new extraction technique for the complete small-signal equivalent-circuit model of InGaP/GaAs HBT including base contact impedance and AC current crowding effect,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 10, pp. 3641–3647, Oct. 2006.

[30] M. S. Gupta, “Power gain in feedback amplifiers, a classic revisited,” IEEE Trans. Microw. Theory Tech., vol. 40, no. 5, pp. 864–879, May 1992.

Tian-Ren Yang was born in Taiwan, R.O.C., in 1982. He received the B.S.E.E. and M.S.E.E. degrees from National Chiao Tung University, Hsin-Chu, Taiwan, R.O.C., in 2004 and 2006, respectively. From 2005 to 2006, he was involved with high-frequency measurements and HBT’s device modeling with VIA Technologies, Taipei, Taiwan, R.O.C. He is currently serving in the R.O.C. Army. His research interests include device modeling, RF and microwave circuit design, and applied electromagnetics.

Julius Ming-Lin Tsai was born in Taipei, Taiwan, R.O.C., in June 1976. He received the B.S. and Ph.D. degrees in power mechanical engineering from National Tsing Hua University, Hsin-Chu, Taiwan, R.O.C., in 1995 and 2004, respectively. From June 2003 to April 2004, he was a Visiting Scholar with Carnegie–Mellon University, Pittsburgh, PA, where he was involved with low-noise CMOS accelerometers. He is currently with VIA Technologies, Taipei, Taiwan, R.O.C., where he in involved with electrostatic discharge (ESD) devices, high-frequency small-signal modeling, and communication system measurements. Dr. Tsai was the recipient of a scholarship presented by the National Science Council of Taiwan, R.O.C.

Chih-Long (Tony) Ho received the B.S. degree in electronics from Fu-Jen Catholic University, Taiwan, R.O.C., in 1988, and the M.S. degree in communication engineering from National Chiao Tung University, Hsin-Chu, Taiwan, R.O.C., in 1996. From 1988 to 1999, he was a Microwave Engineer with the Chung-Shan Institute of Science and Technology (CSIST), where he was engaged in the design of microwave module systems. In 1999, he joined Agilent Technologies as a Technical Consultant. Since 2002, he has been the Manager of the Technology Development Department, VIA Technologies, Taipei, Taiwan, R.O.C., where he is actively involved in both the device characterization and RF transceiver integrated-circuit manufacturing system. His research interest is RF and microwave integrated circuits for wireless communications.

Robert Hu received the B.S.E.E. degree from National Taiwan University, Taipei, Taiwan, R.O.C., in 1990, and the Ph.D. degree from The University of Michigan at Ann Arbor, in 2003. From 1996 to 1999, he was with Academia Sinica, Taipei, Taiwan, R.O.C., where he was involved with the millimeter-wave receivers. In 1999 and 2003, he was with the California Institute of Technology, Pasadena, where he was involved with millimeter-wave wideband receivers. He is currently with the Department of Electronics Engineering, National Chiao Tung University, Hsin-Chu, Taiwan, R.O.C. His research interests include microwave and millimeter-wave electronics.

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Wideband Nonlinear Response of High-Temperature Superconducting Thin Films From Transmission-Line Measurements Jordi Mateu, Member, IEEE, James C. Booth, and Brian H. Moeckly

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Abstract—We report on a technique for extracting an accurate in superconducting value of the nonlinear inductance transmission lines. This novel technique assesses the frequency dependence of the transmission line’s nonlinear response. A wideband nonlinear measurement system was used to simultaneously measure the third-order spurious signals at 1 2, 2 1, 1 2, 2 1, 1 , and 2 frequencies. Measurements for different values of the fundamental frequencies 1 and 2 allow us to study the spurious signal generation from 1 to 21 GHz. We demonstrate this technique by measuring coplanar waveguide several superconducting YBa2 Cu3 O7 transmission line geometries patterned in a single chip at 80 K. The results show a linear frequency dependence of the nonlinear response, indicating a dominant contribution of the nonlinear . inductance over the nonlinear resistance The experimentally obtained nonlinear inductances are then used to determine device-independent measures of the linearity of the thin-film material in order to provide the foundation for modeling the nonlinear response of specific devices.

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Index Terms—Nonlinear response, superconductors, thru-reflect-line (TRL) calibration, transmission-line measurements, wideband nonlinear measurement system.

I. INTRODUCTION

S

UPERCONDUCTING technology enables us to implement high-performance passive microwave devices and systems [1]. In particular, the extremely high quality factor of superconducting filters considerably improves the selectivity and sensitivity of a wireless communication base-station receiver [2], [3]. However, the inherently nonlinear response of superconducting materials may limit some of their potential applications [4]. This nonlinear response takes various forms, from compression and saturation to generation of harmonics and intermodulation (IM) products [5]. The analysis of the performance of Manuscript received November 29, 2006; revised March 20, 2007. This work was supported in part under the Fulbright Program, by the Spanish Ministry of Education and Science, under Contract RYC-2005-001125, under Project TEC2006-13248-C04-02/TCM, and by the U.S. Government. J. Mateu was with the National Institute of Standards and Technology, Boulder, CO 80305 USA. He is now with the Department of Signal Theory and Communications, Universitat Politècnica de Catalunya and Centre Technologic de Telecomunicacions de Catalunya, Barcelona, 08034, Spain (e-mail: [email protected]). J. C. Booth is with the National Institute of Standards and Technology, Boulder, CO 80305 USA (e-mail: [email protected]). B. H. Moeckly is with Superconductor Technologies Inc., Santa Barbara, CA 93111 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.2007.900212

Fig. 1. Equivalent circuit model of the dz elemental segment of a transmission line. R , L , C and G are the (linear) resistance, inductance, capacitance, and conductance per unit length, respectively. L i and R i are the current dependence contribution on the inductance and resistance, respectively [8].

1 ()

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superconducting devices taking nonlinear effects into account is a very important key to accelerate the incorporation of superconducting devices in microwave applications and has been extensively studied during the past several years. These studies include theoretical models of the physical phenomenon [6], [7], circuit models and analysis [8], and measurements [9]–[12]. Common to [5]–[12] is the distributed origin of the nonlinear response in superconducting devices due to the current dependence on the superconducting penetration depth. In the case of a quasi-TEM transmission line, these nonlinear effects are accounted for each incremental segment of the transmission line and a nonlinear as a nonlinear distributed inductance , as shown in Fig. 1 [8]. distributed resistance Accurate determination of the nonlinear circuit parameters and ) is important for improving our under( standing of the fundamental nonlinear response in superconducting thin-film materials [5] in order to create accurate circuit models of superconducting devices [8], [13], [14] and to develop strategies to reduce the nonlinear response at the circuit and component level [15]–[17]. The most commonly used technique to characterize the superconducting nonlinear response consists of scalar measurements of IM products in resonators and filters [12] or third-harmonic measurements in transmission lines [11] at a single frequency point (or several points close in frequency). These measurements are then used to extract the nonlinear circuit parameters and ). However, due to the lack of information ( on the phase of the measured spurious signals, which are rarely measured [18], and having measured the spurious signal at only a single frequency, these measurements may be normally modeled by either a nonlinear inductance or a nonlinear resistance [8], whose accuracy depends on the measured value at a single frequency. The technique used in this study consists of simultaneously , , measuring the third-order spurious signals at , , , and frequencies, generated in a

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broadband transmission line. To do this, we applied the wideband nonlinear measurement system reported in [19]. In [19], we used the frequency spacing between the two-tone signal and to primarily control the time varying of the incident signal and extract the time dependence of the dielectric nonlinear response. Here, the frequencies of the two-tone incident and were set to produce spurious signals from 1 signal to 21 GHz, and from the relationships between the measured spurious signals, we assess the frequency dependence of the superconducting nonlinear response. The resulting frequency dependence is used to distinguish between an origin from the nonlinear inductance or nonlinear resistance. Furthermore, this frequency dependence can be used to obtain a more accurate value of the nonlinear inductance than measuring a spurious signal at only a single frequency. Here, we demonstrate this technique by measuring several superconducting coplanar waveguide (CPW) transmission line geometries, patterned on a single chip, at 80 K. The extracted circuit parameters of Fig. 1 are then used to obtain device-independent parameters that can be used as a figure of merit for quantifying the nonlinear response of the film and, therefore, predicting the nonlinear response in real devices such as resonators and filters. Besides, this allows us to discern between the local [6] and nonlocal [7] origin of the superconducting nonlinear response at this temperature. II. NONLINEAR EFFECTS IN HIGH-TEMPERATURE SUPERCONDUCTING TRANSMISSION LINES Many measurements on superconducting devices suggest that nonlinear effects arise from a quadratic current dependence of the nonlinear inductance and nonlinear resistance [11], [12]. and , where is the That is, and are the RF current flowing through the line, and coefficients setting the strength of the nonlinear effects. Considering this quadratic nonlinear dependence in a low loss and perfectly matched transmission line, the nonlinear inductance and the nonlinear resistance may be related to the observable spurious signals by analyzing the equivalent circuit of Fig. 1. Hence, the power of third-order IM products and the third harmonics at the output of the superconducting transmission line are, respectively, [8] (1)

TABLE I TEST STRUCTURES

linear term . Note that where each spurious signal occurs.

refers to the frequency

III. TEST STRUCTURES We used a superconducting YBa Cu O (YBCO) sample of 400-nm thickness deposited on a 0.5-mm-thick LaAlO (LAO) substrate with K. The sample is patterned with three different sets of CPW structures. The width of the center conductor ( ) and the gaps between the center conductor and ground planes ( ) are summarized in Table I. We refer to each set as C11, C22, and C55, respectively. The three sets have lines , , , of several lengths, i.e., , and mm. The detailed dimensions in Table I were measured after patterning. Accurate dimensions of the structures under test are needed to relate the measured circuit parameters to the material properties, to be discussed later. Among the transmission lines, each set contains a short circuit and a resistor with identical cross sections. Identical structures were also fabricated from gold film of thickness 0.5 m deposited on a bare LAO substrate, which are used as a reference calibration set. IV. MEASUREMENTS We performed on-wafer measurements at 80 K on our YBCO CPW structures using a cryogenic microwave probe station. Directly from measurements in transmission lines, we obtained the distributed circuit parameters corresponding to the equivalent circuit model for an incremental length of transmission line of Fig. 1. From a quasi-TEM mode propagating through the line, the properties of the superconductor are primarily related to the and resistance inductance per unit length. In order to characterize and , we need first to extract the linear distributed parameters of the su, , and ). Meaperconducting transmission line ( , surements for obtaining the linear distributed parameters and and ) were the nonlinear distributed parameters ( performed separately.

and (2) and . Here, and where are, respectively, the characteristic impedance and the length is the frequency of the measured of the line. The value of , , , or ) and IM product ( is the frequency of the measured third harmonic ( or ), where and are the frequencies of the input signals. In this analysis, the powers of the two fundamental signals are set to be equal to . From (1) and (2), it is clear that by measuring the scalar power of the spurious signals, we can extract only the value of the non-

A. Broadband Frequency Response We started by obtaining the complex propagation constant up to 25 GHz using multiline thru-reflect-line (TRL) [20] calibration techniques. These techniques were applied to our test structures and to a reference calibration set (calset) of CPW transmission lines with a nondispersive and low-loss substrate. The calset consists of gold lines with embedded lumped resistors. Measurements of these resistors were used to obtain the distributed capacitances of the calset of transmission lines [21], which were used to obtain the characteristic impedance of the calset transmission lines [22]. We then determined the characteristic impedances of the test structures using the calibration

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TABLE II THIRD HARMONIC AND IM PRODUCT MEASUREMENTS

Fig. 2. Measured inductances per unit length for three cross-sectional transmission lines from 0.5 to 25 GHz.

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Fig. 3. Diagram of the wideband measurement setup. f and f represent the sources. A, IS, LPF and 2-PS represent amplifiers, isolators, low-pass filters, and two-port power splitters, respectively. 2-PC and 3-PC are the two- and three-port power combiners, respectively, and finally, SA indicates the spectrum analyzer. The reference branches contain a phase shifter and variable attenuator.

comparison method [23]. From the complex propagation constant and the characteristic impedance of the line, we directly obtained the linear distributed parameters of the line , , , [24]. These measurements were performed for a power and delivered by the network analyzer of 5 dBm, which ensured the linear response of the measured superconducting transmission line. We are mainly interested in , as this is related to the penetration depth of the superconducting thin film [1]. Fig. 2 shows the measured inductance per unit length of the test transmission lines. The inductance is flat versus frequency due to the frequency-independent superconducting penetration depth. B. Nonlinear Response According to (1) and (2), measurements of the IM products and third harmonics are related to the nonlinear distributed parameters and . To extract their contribution to the nonlinear response, we determined the wideband nonlinear response of our superconducting transmission lines by simultane, , ously measuring all third-order products ( , , , and frequencies) when the transand mission lines were fed with two fundamental signals at frequencies. The wideband measurement system to perform these measurements is shown in Fig. 3. A detailed description of this setup is given in [19]. The measurements performed with this measurement system are summarized in Table II. Each column in Table II shows the

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Fig. 4. Measured spurious signals at f f (squares), f f (circles), and f (diamonds) generated in a CPW transmission line of 21-m width of center conductor and 42.85-m gap as a function of the incident power. The length of the lines is 11.25 mm. Inset: measures of Fig. 4 scaled by K and the frequencies where the spurious signals occur.

fundamental frequencies and corresponding to the experiments reported in this study. We used fundamental signals within the relatively narrow frequency range of 4–7 GHz to generate IMD and third harmonic signals over the range from 1 to 21 GHz, giving a wideband characterization of the nonlinear response. Our simple picture of the material dependent nonlinear response in superconducting transmission lines (see Section II) demands a simple relationship between the different IM signals , , , and ) and third harmonics ( ( and ). Although it has been implicitly assumed in previous studies that these simple relationships hold, until now it has not been verified experimentally. Fig. 4 shows the measured spurious signals as a function of the power of the fundamental tones (the power of the two incident signals are kept balanced), on a log–log scale, for the of the C22 set, when MHz. For longest line the sake of clarity, Fig. 4 shows spurious signals only at fre(in circles), (in squares), and quencies (in diamonds). Note that the spurious signals follow a slope of 3, which is consistent with the quadratic nonlinear dependence assumed in Section II [8]. The inset of Fig. 4 shows the measured nonlinear signals when scaled by the simple geometrical relationships that connect the different manifestations of nonand of (1) and (2)], as well as by the linear response [ . The collapse squared frequencies of the spurious signals of the data onto a single line demonstrates the equivalence of different manifestations of nonlinear response in the same device and indicates a dominant contribution of the nonlinear inductive . This term over the nonlinear resistive term result is important because these simple relationships form the

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TABLE III MEASURED NONLINEAR INDUCTANCE

VI. NONLINEAR DEVICE-INDEPENDENT MEASURE

Fig. 5. Measured nonlinear term as a function frequency for the structures under tests. Circles, squares, and diamonds correspond to C11, C22, and C55, respectively.

basis of our circuit models, and also because it implies that the material nonlinear properties of the thin film can be extracted from measurements of any third-order spurious signal.

V. NONLINEAR INDUCTANCE Although from the data represented in Fig. 4 we interpreted a dominant contribution of the nonlinear inductance over the nonlength), linear resistance (at least for the C22 structure with here we corroborate this by extracting the nonlinear term for each experiment detailed in Table II. The nonlinear terms and , which generate spurious signals over a wide range of frequencies, are at the frequencies of the fundamental signals. Therefore, one does not expect to observe any and in these meafrequency dependence of the terms surements; rather the frequency dependence shown in (1) and (2) is the frequency dependence of the generated spurious signals. Fig. 5 depicts the experimentally extracted nonlinear term as a function of frequency . The nonlinear terms corresponding to C55, C22, and C11 CPW structures are, respectively, represented by diamonds, squares, and circles. Note that the narrower the center conductor of the line, the higher its measured nonlinear term. From Fig. 5, we also see that the nonlinear term of our three sets of structures can be fit by a linear frequency dependence, indicated in Fig. 5 by dashed lines. This corroborates the dominance of the nonlinear inductance over the nonlinear resistance. Moreover, the slope of these linear fittings may be used to and obtain an accurate value of the nonlinear inductance to reduce the uncertainties occurring. The nonlinear inductance coefficients obtained using the procedure above are detailed in Table III for each set of CPW transmission lines. Note that these values provide a unified description of all third-order spurious signals generated in a superconducting transmission line. In Table III, we also indicate the variance of the uncertainties on the extracted nonlinear term.

Having extracted values of the nonlinear distributed inducfor each set of CPW transmission lines tance coefficient (patterned on the same superconducting thin film), here we proceed to obtain device-independent parameters characterizing the nonlinear response of the superconducting material. As briefly mentioned in Section I, the nonlinear effects in superconductors are caused by a current dependence of the penetration depth. Consequently, the nonlinear inductance must be related to the variation of the penetration depth [1]. There are two main approaches for modeling the variation of the superconducting penetration depth. The approach of Dahm and Scalapino [6] makes use of the local theory in which the variation of the penetration depth is modeled as a function of current density distribution in the cross section of the super, conducting transmission line as where is the penetration depth for and is the term quantifying the current density dependence, assuming a local theory. An alternative approach [7] derives from nonlocal electrodynamics, in which the variation of the penetration depth depends on the average current density, as , where is the term quantifying the current density dependence assuming nonlocal theory. Both theories result in a quadratic current density dependence of the penetration depth. However, both theories arrive at different expressions of the nonlinear inductance coefficient. From the local theory [6], the nonlinear inductance coefficient can be written as (3) where and and are, respectively, the current density and area of the cross section. The nonlocal theory calculates the nonlinear inductance coefficient as [7] (4) where . The terms and can be seen as nonlinear geometrical factors, which depend on the current distribution over the cross section (defined by ), which, in turn, depends on the penetration depth . and , respectively, from (3) and (4), we Isolating see that their uncertainties depend on the uncertainty of the non, the linear part of the penetralinear inductance coefficient tion depth , and the determination of the current density over has been the cross section of the line. The uncertainty of previously shown in this paper. A. Penetration Depth Here, we extract the linear part of the penetration depth from the measured (linear) distributed inductance , shown in

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Fig. 8. (a) Device-independent nonlinear term using local theory [6] [see (3)]. (b) Device-independent nonlinear term using nonlocal theory [7] [see (4)]. Both extract from the measured nonlinear term.

Fig. 6. Inductance per unit length L for the measured structures as a function of the superconducting penetration depth  .

Fig. 7. (a) Nonlinear geometric factor 0 scaled by the superconducting penetration depth squared. (b) Nonlinear geometric factor 0 scaled by the superconducting penetration depth. Both as a function of the penetration depths for the measured structures.

Fig. 2, using the procedure described in [25]. This technique [25] makes use of the Sheen method [26] to obtain the disand . In tributed parameters in a superconducting strip Fig. 6, we show the value of as a function of the penetration depth for C11, C22, and C55. From Fig. 6 and using the meaof Fig. 2, we obtain a superconducting penetration sured depth of 500 nm at 80 K with less than 5% deviation. This shows consistency in the measurements small deviation on and numerical methods used for modeling. We attribute this deviation to uncertainties resulting from the measurements, as can be seen in Fig. 2. B. Nonlinear Penetration Depth To assess how the accuracy in the determination of the penand , etration depth affects the extracted value of Fig. 7(a) and (b) plots the terms and , respectively, as a function of the penetration depth . These results show very different behavior between local and nonlocal assumptions. For the local assumption [see Fig. 7(a)], the variaversus depends on the structure and shows the tion of biggest deviation for the narrowest lines (C11). Using a value of penetration depth of 500 nm with a 5% of uncertainty would rewith 6% of uncertainty. For sult in an estimated value of the nonlocal assumption [see Fig. 7(b)], the variation of

does not depend on the structure, and would result in versus . The results a 5% uncertainty in the determination of in Fig. 7 may have important implications on strategies for material improvement to reduce nonlinear effects [27], which are, however, beyond of the scope of this paper. Finally, from the values of the nonlinear inductance coeffi, the linear penetration depth , and the nonlinear cient and/or , we extract the device-ingeometrical factors dependent measures of nonlinearities for both theories, local and , reand nonlocal. Fig. 8(a) and (b) depict spectively, as a function of frequency, directly from the measured nonlinear term. While the data of Fig. 8(a) almost collapse onto a single line, the data of Fig. 8(b) give different slope for C55, C22, and C11. These results clearly show that the local theory [6] gives more consistent results, at least at 80 K. Using is directly related to the superconducting local theory, pair-breaking current [6], the value of which is consistent with that measured in the same sample [11], [28] and other samples [12]. Our wideband measurements, therefore, yield more accurate values of the pair-breaking current than does a single-frequency measurement [28]. Full assessment of the superconducting nonlinear response would require nonlinear measurements as a function of temperature [11], [12]. However, measurements at lower temperatures would result in very weak spurious signals, being difficult to measure in a transmission line configuration [5], with the measurement setup of Fig. 3. VII. CONCLUSION The technique presented in this study has provides an accurate characterization of the nonlinear properties of superconducting materials by measuring the wideband nonlinear spurious signal response in superconducting transmission lines. This technique also requires characterization of the linear properties. By using this technique, we have also demonstrated the unified description of the wideband nonlinear response of the superconducting material. We started from the accurate extracted nonlinear inductance coefficients, addressed the question of local versus nonlocal theory, and concluded that local theory provides more consistent results. Furthermore, these accurate nonlinear inductance coefficients were used to obtain device-independent measures of the linearity of the superconductor material itself.

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REFERENCES [1] M. J. Lancaster, Passive Microwave Device Applications of High Temperature Superconductors. Cambridge, U.K.: Cambridge Univ. Press, 1997. [2] B. A. Willemsen, HTS Wireless Applications, ser. NATO Sci., Ser. E: Appl. Sci., H. Weinstock and M. Nisenoff, Eds.. Norwell, MA: Kluwer, 2002, vol. 375, ch. 15 Microw. Supercond., pp. 387–416. [3] M. I. Salkola and D. J. Scalapino, “Benefits of superconducting technology to wireless CDMA networks,” IEEE Trans. Veh. Technol., vol. 55, no. 3, pp. 943–955, May 2006. [4] M. I. Salkola, “Nonlinear characteristics of a superconducting receiver,” Appl. Phys. Lett., vol. 88, pp. 012501-1–012501-3, 2006. [5] J. C. Booth, K. Leong, S. A. Schima, C. Collado, J. Mateu, and J. M. O’Callaghan, “Unified description of nonlinear effects in high temperature superconductor microwave devices,” J. Supercond., vol. 19, no. 7–8, pp. 531–540, Nov. 2006. [6] T. Dahm and D. J. Scalapino, “Theory of intermodulation in superconducting microstrip resonator,” J. Appl. Phys., vol. 81, no. 4, pp. 2002–2012, 1997. [7] D. Agassi and D. E. Oates, “Nonlinear Meissner effect in a high-temperature superconductor,” Phys. Rev. B, Condens. Matter, vol. 72, pp. 014538–014538, 2005. [8] C. Collado, J. Mateu, and J. M. O’Callaghan, “Analysis and simulation of the effects of distributed nonlinearities in microwave superconducting devices,” IEEE Trans. Appl. Supercond., vol. 15, no. 1, pp. 26–39, Mar. 2005. [9] J. C. Booth, J. Bell, D. Rudman, L. Vale, and R. Ono, “Geometry dependence of nonlinear effects in high temperature superconducting transmission lines at microwave frequencies,” J. Appl. Phys., vol. 86, no. 2, pp. 1020–1020, 1999. [10] B. Willemsen, K. Kihlstrom, and T. Dahm, “Unusual power dependence of two-tone intermodulation in high-T c superconducting microwave resonators,” Appl. Phys. Lett., vol. 4, no. 5, pp. 753–755, 1999. [11] K. Leong, J. C. Booth, and S. A. Schima, “A current-density scale for characterizing nonlinearity in high-T c superconductors,” IEEE Trans. Appl. Supercond., vol. 15, no. 2, pp. 3608–3611, Jun. 2005. [12] D. E. Oates, S.-H. Park, D. Agassi, G. Koren, and K. Irmaier, “Temperature dependence of intermodulation distortion in YBCO: Understanding nonlinearity,” IEEE Trans. Appl. Supercond., vol. 15, no. 2, pp. 3589–3564, Jun. 2005. [13] M. I. Salkola, “Intermodulation response of superconducting filters,” J. Appl. Phys., vol. 98, pp. 023907–023907, Jul. 2005. [14] T. Dahm and D. J. Scalapino, “Analysis and optimization of intermodulation in high-T c superconducting microwave filter design,” IEEE Trans. Appl. Supercond., vol. 8, no. 4, pp. 149–157, Dec. 1998. [15] D. Seron, C. Collado, J. Mateu, and J. M. O’Callaghan, “Analysis and simulation of distributed nonlinearities in ferroelectrics and superconductors for microwave applications,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 3, pp. 1154–1160, Mar. 2006. [16] J. Mateu, J. C. Booth, C. Collado, and J. M. O’Callaghan, “Intermodulation distortion in coupled-resonator filters with nonuniform distributed properties: Use in HTS IMD compensation,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 4, pp. 616–624, Apr. 2007. [17] J. Mateu, J. C. Booth, and B. H. Moeckly, “Nonlinear response of combined superconductor/ferroelectric devices: First experimental step,” IEEE Trans. Appl. Supercond., Jul. 2007, to be published. [18] J. C. Booth, K. Leong, S. A. Schima, A. Jargon, D. C. Degroot, and R. Schwall, “Phase-sensitive measurements of nonlinearity in high-temperature superconductor thin film,” IEEE Trans. Appl. Supercond., vol. 15, no. 2, pp. 1000–1003, Jun. 2005. [19] J. Mateu, J. C. Booth, and S. A. Schima, “Characterization of the nonlinear response in ferroelectric thin-film transmission lines,” IEEE Trans. Microw. Theory Tech., submitted for publication. [20] R. Marks, “A multiline method of network analyzer calibrations,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 7, pp. 1205–1215, Jul. 1991. [21] D. F. Williams and R. B. Marks, “Transmission line capacitance measurements,” IEEE Microw. Guided Wave Lett., vol. 1, no. 9, pp. 243–245, Sep. 1991. [22] R. B. Marks and D. F. Williams, “Characteristic impedance determination using propagation constant measurement,” IEEE Microw. Guided Wave Lett., vol. 1, no. 6, pp. 141–143, Jun. 1991. [23] D. F. WilliamsR. B. Marks, “Accurate transmission line characterization,” IEEE Microw. Guided Wave Lett., vol. 3, no. 8, pp. 247–249, Aug. 1993. [24] M. D. Janezic, D. F. Williams, V. Blaschke, A. Karamcheti, and C. S. Chang, “Permittivity characterization of low-k thin films from transmission- line measurements,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 1, pp. 132–136, Jan. 2003.

[25] K. Leong, J. C. Booth, and J. H. Classen, “Determination of the superconducting penetration depth from coplanar-waveguide measurements,” J. Supercond., to be published. [26] D. M. Sheen, S. M. Ali, D. E. Oates, R. S. Withers, and J. A. Kong, “Current distribution, resistance, and inductance for superconducting strip transmission lines,” IEEE Trans. Appl. Supercond., vol. 1, no. 2, pp. 108–118, Jun. 1991. [27] B. M. Andersen, J. C. Booth, and P. J. Hirschfield, “Disorder effects on the intrinsic nonlinear current density in YBaCuO,” J. Appl. Phys., submitted for publication. [28] J. Mateu, J. C. Booth, and B. H. Moeckly, “Frequency dependence of the nonlinear response of YBCO transmission line,” Appl. Phys. Lett., vol. 90, pp. 012512-1–012512-3, 2007. Jordi Mateu (M’03) was born in Llardecans, Spain, in 1975. He received the Telecommunication Engineering and Ph.D. degrees from the Universitat Politècnica de Catalunya (UPC), Barcelona, Spain, in 1999 and 2003, respectively. Since October 2006, he has been Research Fellow with the Department of Signal Theory and Communications, UPC. From May to August 2001, he was Visiting Researcher with Superconductor Technologies Inc., Santa Barbara, CA. From October 2002 to August 2005, he was Research Associate with the Centre Technologic de Telecomunicacions de Catalunya (CTTC), Catalonia, Spain. Since September 2004, he has held several Guest Researcher appointments with the National Institute of Standards an Technology (NIST), Boulder, CO, where he was a Fulbright Research Fellow from September 2005 to October 2006. In July 2006, he was a Visiting Researcher with the Massachusetts Institute of Technology (MIT) Lincoln Laboratory. From September 2003 to August 2005, he was a Part-Time Assistant Professor with the Universitat Autònoma de Barcelona. His primary research interests include microwave devices and system and characterization and modeling of new electronic materials including ferroelectrics, magnetoelectric, and superconductors. Dr. Mateu was the recipient of the 2004 Prize for the best doctoral thesis in fundamental and basic technologies for information and communications presented by the Colegio Oficial Ingenieros de Telecomunicación (COIT) and the Asociación Española de Ingenieros de Telecomunicación (AEIT). He was also the recipient of a Fulbright Research Fellowship, an Occasional Lecturer Award for visiting MIT, and a Ramón y Cajal Contract.

James C. Booth received the B.A. degree in physics from the University of Virginia, Blacksburg, in 1989, and the Ph.D. degree in physics from the University of Maryland at College Park, in 1996. His doctoral dissertation concerned novel measurements of the frequency-dependent microwave surface impedance of cuprate thin-film superconductors. Since 1996, he has been a Physicist with the National Institute of Standards and Technology (NIST), Boulder, CO, originally as a National Research Council (NRC) Post-Doctoral Research Associate (1996–1998) and currently as a Staff Scientist. His research with NIST is focused on exploring the microwave properties of new electronic materials and devices including ferroelectric, magneto-electric, and superconducting thin films, as well as developing experimental platforms integrating microfluidic and microelectronic components for RF and microwave frequency characterization of liquid and biological samples.

Brian H. Moeckly received the B.S. degrees (both with distinction) in electrical engineering and physics from Iowa State University, Ames, in 1987, and the M.S. and Ph.D. degrees in physics from Cornell University, Ithaca, NY, in 1990 and 1994, respectively. From 1994 to 1996, he was member of Technical Staff, from 2000 to 2002, he was Supervisor of Materials Technology, and from 2002 to 2003, he was Manager of Fabrication and Materials Technology with Conductus Inc. From 2003 to 2004, he was Manager of Materials Science and Process Development with Superconducting Technologies Inc. (STI), Santa Barbara, CA. He is currently Director of Materials Research and Development with STI. He has authored or coauthored over 50 peer-reviewed papers and has given over 20 invited presentations. He holds two patents with five pending.

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Ultra-Wideband Multifunctional Communications/Radar System George N. Saddik, Student Member, IEEE, Rahul S. Singh, Member, IEEE, and Elliott R. Brown, Fellow, IEEE

Abstract—We have designed, simulated, fabricated, and tested an ultra-wideband (UWB) multifunctional communication and radar system utilizing a single shared transmitting antenna aperture. Two surface acoustic wave bandpass chirp filters were used to modulate the radar and communications pulses, generating linear frequency modulation waveforms with opposite slope factors. The system operates at a center frequency of 750 MHz with 500 MHz of instantaneous bandwidth. The measured range resolution is 63 cm (25 in) using targets with a radar cross section of 2.7 m2 . The probability of detection was measured to be 99%, and the probability of false alarm was 7% with the communication and radar systems operating simultaneously. The bit error rate for simultaneous communication at 1 Mb/s, and radar at 150 kHz 3. Our pulse repetition frequency and 1.5-ns pulsewidth is 2 UWB multifunctional system demonstrates the ability to simultaneously interrogate the environment and communicate through a shared transmitting antenna aperture, while realizing a simple system architecture with low output power and not employing time-division multiplexing. Index Terms—Chirp, communications, radar, RF.

I. INTRODUCTION

T

ODAY, multifunctional systems are used in our daily life from personal digital assistants (PDAs) to cell phones. Among the advantages of having a multifunctional system are low cost and reduced size. The military has taken advantage of multifunctional systems by developing broadband RF apertures that are capable of simultaneously operating communication, radar and electronic warfare [1]–[4], and multifunctional unattended ground sensor networks that can be optimized depending on location [5], [6]. To achieve simultaneous operation, we have proposed [7], [8] using linear frequency modulated (LFM) signals of opposite slopes for the communications and radar transmitted pulses. The quasi-orthogonality of opposite slope LFM signals is being exploited to allow for simultaneous communication and radar operation through a shared antenna aperture. The quasi-orthogonal concept has been adapted from orthogonal signal design in communication theory. The opposite slope expanded LFM radar and communication signals, combined and Manuscript received September 8, 2006; revised March 23, 2007. This work was supported by the U.S. Army Research Office under Multiuniversity Research Initiative Grant “Multifunctional, Adaptive Radio, Radar and Sensors.” The authors are with the Electrical and Computer Engineering Department, University of California at Santa Barbara, Santa Barbara, CA 93016 USA (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.2007.900343

transmitted through a shared aperture, are not completely orthogonal, but are near orthogonal and, hence, affect each other in their respective receivers. The combined radar and communications signals are inputs to the matched chirp filter in the appropriate radar and communications receivers; the signal matched to the filter is compressed and the unmatched signal is further expanded, reduced in amplitude, and is seen as additional noise at the output of the matched filter. If the signals were completely orthogonal, then the unmatched signals would not be seen at the output. In Section III, we simulate this behavior and show that the two signals are not completely orthogonal to each other, and quantify in terms of signal-to-noise ratio (SNR) the quasi-orthogonal term. While traditionally LFM signals have been used for radar to improve range resolution while maintaining adequate average transmitting power [9], in 1962 Winkler proposed the use of LFM signals for analog communications [10]. Soon thereafter, digital communication applications were proposed and implemented using LFM signals [11]–[16]. Recently, an indoor chirp spread-spectrum wireless communication system [17], [18] was demonstrated using chirped -differential quadrature -DQPSK) to achieve high data rates. phase-shift keying ( The primary advantage found in using chirp waveforms was the mitigation of multipath fading without the use of complicated signal processing [19]. Another advantage is using passive surface acoustic wave (SAW) chirp filters, as done in [17], [18] and in this study; they bring much greater hardware simplicity and reduction in power in the transmitter and receiver designs. Simulations were conducted previously to establish feasibility [7], and the pursuit of hardware realization was undertaken. This paper reports on the successful design, implementation, and demonstration of simultaneous communications and radar operation using the quasi-orthogonality of the upand down-chirp waveforms, and without the constraint of time-division multiplexing. II. ULTRA-WIDEBAND (UWB) SYSTEM OVERVIEW A. System Architecture The communication and radar transmitters use LFM signals that are identical in frequency range, but have opposite chirp slopes: positive (up-chirp) and negative (down-chirp), respectively (Fig. 1). Both the expander and compressor chirp filters used in our implementation have down-chirp slopes; to achieve the up-chirp, the spectrum was inverted using a mixer MHz either after the expander with filter or before the compressor filter followed by a low-pass Bessel filter to suppress the upper sideband [20]. The center frequency of the chirp filters is 750 MHz with a bandwidth of

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Fig. 1. Multifunctional communications/radar system block diagram. The transmitter is a down-chirped gated continuous wave (CW) radar signal that is power combined with an up-chirped BPSK communication signal and transmitted through an RHCP helical antenna. The radar echo is received by an LHCP helical antenna that is pulse compressed through a matched filter (up-chirp filter). The communication signal is received by an RHCP helical antenna that is pulse compressed using a down-chirp filter.

500 MHz, chirp duration of 500 ns, chirp rate of 1 GHz/ s, time bandwidth product of 250, and a fractional bandwidth of 0.66, calculated using the UWB definition [21]. UWB signals are further classified based upon the form of signal used: step functions with very short rise times, a very narrow pulse (i.e., impulse), and a single cycle sinusoid pulse. Another classification for UWB signals are those with a bandwidth inversely proportional to the pulsewidth [21]. Additionally, the Federal Communications Commission (FCC) designated frequency bands below 960 MHz for ground penetrating radar, and from 3.1 to 10.6 GHz for other applications [22], such as communication. UWB signals are uniquely different from narrowband signals in that they do not suffer from Rayleigh fading as narrowband systems do, but rather multipath dispersion [23]. Secondly, UWB is known for its penetration capabilities. Thirdly, it can create images with less clutter. The second and third properties are independent of the system architecture; however, the first property is dependent on the system architecture [21]. For these reasons, UWB signals have found applications in military and commercial radar, as well as both military and commercial communication systems.

The obtained compressive filter has an advantage for the communication side of the system; it has a raised cosine window, which provides sidelobe suppression, for reduction of intersymbol interference (ISI). However, this advantage comes at the expense of a broadening of the received compressed pulse, which, in terms of radar operation, degrades the is measured on the expansive and range resolution. When compressive filter, it shows an insertion loss of approximately 35 dB. Although this is a major disadvantage of the current SAW chirp filter, the high center frequency, the large bandwidth, and time-bandwidth product of 250 (processing gain of 24 dB) more than compensates for the insertion loss penalty. The communication system uses a binary phase-shift keying (BPSK) modulation scheme at a data rate of 1 Mb/s. The modulated data signal is injected into a single-pole single-throw (SPST) switch, which reduces the bit time from 1 s to 10 ns. The reason for the bit time reduction is to make use of the expansive and compressive properties of the chirp filters, the bit width must have 1–2 cycles of the 750-MHz center frequency of the chirp filters. This puts a limit on the minimum and maximum width that a communication bit can have: 1.3 and 2.6 ns,

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respectively. Having a bit width greater than the 2.6-ns limit reduces the ability of the chirp filter to expand and compress the bit. The modulated and bit reduced data signal is fed into the expander down-chirp filter, which is followed by a mixer to invert the spectrum. The radar signal is a single sinusoid pulsed waveform with a 750-MHz carrier, pulse duration of 1.5 ns, and a pulse repetition frequency (PRF) of 150 kHz. The waveform is amplified and fed into the down-chirp filter followed by a gain stage. The two opposite chirp signals, radar and communication, are then simply combined through a Wilkinson power combiner, amplified ( 27-dBm peak output power), and transmitted through a single wideband shared antenna aperture. The radar and communication receivers employ separate antennas for reception. The radar receive antenna is located adjacent to the radar transmitter antenna (bistatic configuration), and the communication receiver antenna is located several meters down range from the transmitter. The first stage of the communication and radar receiver is a gain stage followed by a chirp filter (compressor) having the appropriate chirp slope. The output from the communication compression filter is connected to gain stage and followed by a carrier recovery circuit. The recovered carrier is used as the local oscillator (LO) into a down-conversion mixer, which demodulates the RF signal to baseband. The down converted signal is then directed into a clock and data recovery (CDR) circuit. The output from the radar compression filter is fed into a gain stage, which is coupled to a square law detector and the output is displayed on the Agilent 54846B Infiniium digital sampling oscilloscope.

The measured range above is significantly outside the near-field range (i.e., in the far field), thus permitting the use of Friis’s equation. The free-space link loss was calculated to be 87.9 dB using

B. Antenna Design and Link Analysis Two right-handed circularly polarized (RHCP) and one lefthanded circularly polarized (LHCP) antennas were designed for the system [24]. The helical antenna was chosen for its ease of construction, large fractional bandwidth (from 1.7 to 1) [25], and circular polarization. The designed antenna medium directivity was 10.7 dBi and the fractional bandwidth was matched to that of the chirp filters, with 7.5 dB of return loss across the operating band with reference to 50 . Finally, circular polarization is beneficial due to the inherent immunity to multiple reflections and multipath. With the antennas and system architecture chosen, a simple analysis was performed to evaluate the expected performance using Friis’s radar equation. The maximum range was calculated to be 19.2 m using the radar equation

(2) where is the free-space wavelength, is the RCS, and is the maximum range. The receiver RF gain was calculated to be 40.5 dB. The receiver bandwidth is 500 MHz, and the calculated noise figure is 2 dB. The minimum detectable signal was calcuof 10 dB. lated to be 75 dB with an III. SIMULATIONS AND ANALYSIS To better understand the simultaneous operation of the communications and the radar systems, an analysis and simulation was performed on the communications receiver. The objective was to quantify the interference of the radar signal on the communications signal through the matched filter and the quasi-orthogonality between the two signals. The simulation was performed in MATLAB by filtering a communication and radar signal through the communication matched down-chirp filter. The analysis was performed to verify the results and to provide a closed-form expression for each signal through the communication’s matched filter. The communications and radar signals are given as (3) (4) where ( MHz ns) is the modulation rate, ( MHz) is the ns) is the pulsewidth. center frequency of operation, and ( The unitary energy communications matched filter is (5) where ( ) is the time-bandwidth product. The time-domain output signals through the matched filter were computed by multiplying the signals and the filter in the frequency domain and then taking the inverse Fourier transform [26]. The resulting matched filter signals for the communications and radar signals as inputs is then found to be (6) (7)

(1) ( 27 dBm) is the power transmitted, and where (10.7 dB) are the gain of the transmitting and receiving antennas, respectively, (40 cm) is the free-space wavelength, (2.7 m ) is the radar cross section (RCS) for the targets used in (10 dB) is the minimum detectable signal. Section IV, and The measured maximum range with an output SNR of 10 dB at the receiver is 15.2 m, which is within close agreement of the calculated maximum range with the same SNR requirement.

(8) is the expected compression waveform of an up-chirp is the output of a down-chirp signal waveform, while through a down-chirp compressor. Using MATLAB to numer, it is found to be approximately 2 and is ically evaluate treated as a constant to evaluate the performance of the system.

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Fig. 2. Spectrum and phase plots of simulated signals. (a) Communications up-chirp in the frequency domain. (b) Communications (up) and radar (down) chirp phases.

From the expression, it is observed that the output is not compressed, but is rather a down-chirp signal that is twice as long as the input with the modulation rate halved, and the overall power halved. This leads to being able to state a peak power communication SNR for our system due to signal interference from the radar signal (9) The transmitted communications up-chirp signal (3) and radar down-chirp signal (4) were convolved with the communication down-chirp matched filter (5) in MATLAB. This simulated the output of the communications receiver stimulated by a matched communication and unmatched radar input waveforms. The input waveforms are plotted in Fig. 2, and the simulated output waveforms of the communication receiver is plotted in Fig. 3. The simulated output plots (solid line) and the evaluated analytic (diamonds) expression (6)–(8) are plotted together to show agreement between the simulated results and analytical expression. Finally, the output plots show for the input communications signal that the peak output voltage of (15.8 V), and the communications receiver is equal to for the input radar signal, that the peak output voltage of the V (the input signals communications receiver is equal to ). The peak voltage outputs support the above power were 1 SNR expression (9). IV. EXPERIMENTAL RESULTS

Fig. 3. Simulated and analytical expression (6)–(8) results are displayed via the solid line and diamonds, respectively. (a) Expected pulse compression output of the matched filter (down-chirp) to the input communication signal (up-chirp). (b) Input radar signal (down-chirp) being further expanded by the communication matched filter (down-chirp), i.e., the signal being expanded twice.

Fig. 4. Measured range resolution of two targets placed 10 m from the transmitting and receiving antennas.

A. Radar Results The radar experimental results were collected in a large open parking lot where the interference from other objects was minimized. The targets used in our experiment were two rectangular

sheets of metal measuring 45 cm RCS of 2.7 m .

40 cm with a corresponding

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Fig. 5. Measured BER versus PRF in log–log scale with data rate of 2.5 Mb/s and radar pulse 1.5 ns, and exponential curve fit.

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1) The transmitter output was connected directly to the receiver input with a 60-dB attenuator, the same attenuation as expected through 30 m of free space. 2) The carrier signal was hardwired from the transmitter to the receiver down-conversion mixer as the LO. 3) The data clock was hardwired to the receiver re-timing circuit. The BER was measured by comparing 0.1 Mbits of transmitted binary data to the received data. The BER for the communications system only (radar was off) was measured to be less than 10 at 1 Mb/s, and with both systems operating simultanewith the data rate at 1 Mb/s, and ously, the BER was the radar operating at a PRF of 150 kHz. Under the same conditions, the BER was measured while sweeping the PRF of the radar from 100 to 1000 kHz. Fig. 5 shows on a log–log scale an exponential increase in the BER as the PRF is increased. To further illustrate the simultaneous operation of radar and communications, an eye diagram of the communications waveform (with the radar off) at 1 Mb/s and an eye diagram with the radar on with a PRF of 150 kHz are given in Fig. 6. As seen from Fig. 6(a) and (b), the eye diagram closes in the vertical and horizontal directions by 6 and 4.7 dB, respectively, which is within close agreement of the simulated results discussed above. V. CONCLUSION

Fig. 6. Measured communications eye diagram with: (a) the radar off and (b) the radar on. (a) Communication system operating independently has very little noise, which is attributed to the electronics. (b) In comparison, a significant rise in the noise level due to the radar operating simultaneously, but still maintaining two distinct voltage levels (i.e., the eye is still open).

As an example of the radar operation (Fig. 4), the two targets were placed approximately 10 m away from the system and separated from each other in range by 63 cm. The probability of detection was measured to be 99% with the communication system off using a logic analyzer. With the communication system operating at a data rate of 1 Mb/s, the probability of detection was 99%, and the probability of false alarm was 7%. B. Communications Results To achieve benchmark results for the system, the bit error rate (BER) of the system was experimentally measured under the following conditions.

A multifunctional UWB communication and radar system sharing the same antenna aperture, with a simple architecture, and low power has been implemented and tested. The simulations and implemented system have demonstrated the feasibility of simultaneous operation of radar and communications using a shared transmitting antenna aperture. The range resolution was demonstrated at 10 m to be 63 cm and the probability of detection and probability of false alarm were 99% and 7%, respecat a data tively. The best case BER was measured to be rate of 1 Mb/s, with the radar system on simultaneously and a radar pulsewidth and PRF of 1.5 ns and 150 kHz, respectively. In future systems, to improve system performance such as maximum range, range resolution, and the BER, several design changes to the system hardware and signal processing could be made. The first significant change to improve system operation would be to increase the time-bandwidth product by increasing the bandwidth and/or the dispersion time in the chirp filter; this would increase the SNR for the system, as shown in (9). One approach to increasing the time-bandwidth product is to design a SAW chirp filter at a higher frequency with larger bandwidth and dispersion time than the chirp filters used in the system. The authors have simulated in [8] a SAW filter using AlN-on-SiC at a 10-GHz frequency and bandwidth near 7.5 GHz. We believe that fabricating a SAW filter at 10 GHz and a bandwidth near 7.5 GHz is feasible. It has been shown in the past that SAW filters in the 10-GHz range on lithium niobate can be fabricated using electron beam lithography, proper transducer design, and electrode material [27]. SAW filters utilizing AlN-on-SiC have also been demonstrated to operate in the 30-GHz range [28]. Thus, with the current advancement in electron beam lithography, the high quality of III–V material available at the University of California at Santa Barbara (UCSB), and the proper design of the transducer, an AlN-on-SiC SAW filter in the 10-GHz

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range and bandwidth near 7.5 GHz is feasible. Further, we believe that by scaling up higher in frequency ( 40 GHz), the insertion loss of the SAW filter will significantly increase since, as the metal electrodes are reduced in thickness and width, losses will arise due to skin effects. In addition, the system operation can be improved by using UWB antennas specifically designed for the receiver and transmitter, thus improving the phase linearity of the system, directivity, and gain. This would improve the dispersion and mismatch loss with respect to the antenna used in the system, in turn improving the maximum range, range resolution, and BER. Finally, improving the various system components, such as the low-noise amplifier (LNA) sensitivity in the receiver, and increasing the transmit power would further improve the system performance.

ACKNOWLEDGMENT The authors wish to thank R. Bernardo, Integrated Circuit Systems Inc., Worcester, MA, for his generous donation of two expansive and two compressive matched SAW chirp filters. The authors additionally wish to thank Dr. D. Palmer, U.S. Army Research Office AMSRD-ARL-RO-EL, Research Triangle Park, NC, for program management.

REFERENCES [1] P. K. Hughes and J. Y. Choe, “Overview of advanced multifunction RF systems (AMRFS),” in Proc. IEEE Int. Phased Array Syst. Technol. Conf., May 2000, pp. 21–24. [2] P. Antonki, R. Bonneau, R. Brown, S. Ertan, V. Vannicola, D. Weiner, and M. Wicks, “Bistatic radar denial/embedded communications via waveform diversity,” in Proc. IEEE Radar Conf., May 2001, pp. 41–45. [3] J. H. Wehling, “Multifunction millimeter-wave systems for armored vehicle application,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 3, pp. 1020–1025, Mar. 2005. [4] G. C. Tavik, C. L. Hilterbrick, J. B. Evins, J. J. Alter, J. G. Crnkovich, Jr., J. W. de Graaf, W. Habicht, II, G. P. Hrin, S. A. Lessin, D. C. Wu, and S. M. Hagewood, “The advanced multifunction RF concept,” IEEE Trans Microw. Theory Tech., vol. 53, no. 3, pt. 2, pp. 1009–1020, Mar. 2005. [5] R. A. Burne, A. L. Buczak, V. R. Jamalabad, I. Kadar, and E. R. Eadan, “Self-organizing cooperative sensor network for remote surveillance,” in Proc. SPIE, Jan. 1999, vol. 3577, pp. 124–134. [6] M. Tubaishat and S. Madria, “Sensor networks : An overview,” IEEE Potentials, vol. 22, no. 2, pp. 20–23, May 2003. [7] M. Roberton and E. R. Brown, “Integrated radar and communication based on chirped spread-spectrum techniques,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2003, vol. 1, pp. 611–614. [8] G. N. Saddik and E. R. Brown, “Towards a multifunctional LFM-waveform communication/radar system: Single-crystal AlN-on-SiC SAW filters for X -band,” in GOMAC Tech. Conf., Las Vegas, NV, Apr. 2005, pp. 399–402. [9] C. E. Cook and M. Bernfeld, Radar Signals—An Introduction to Theory and Application, 1st ed. New York: Academic, 1967. [10] M. R. Winkler, “Chirp signals for communications,” in IEEE WESCON Conv. Rec., 1962. [11] G. W. Barnes, D. Hirst, and D. J. James, “Chirp modulation system in aeronautical satellites,” in Proc. 87th AGARD Avion. in Spacecraft Conf., 1971, pp. 30.1–30.10. [12] G. W. Judd and V. H. Estrick, “Applications of surface acoustic wave (SAW) filters—An overview,” in Proc. Soc. Photo-Opt. Instrum. Eng., Jul. 1980, vol. 239, pp. 220–235. [13] W. Hirt and S. Pasupathy, “Continuous phase chirp (CPC) signals for binary data communication—Part I: Coherent detection and Part II: Non-coherent detection,” IEEE Trans. Commun., vol. COM-29, no. 6, pp. 836–856, Jun. 1981.

[14] G. F. Gott and J. P. Newsome, “H.F. data transmission using chirp signals,” Proc. Inst. Elect. Eng., vol. 118, pp. 1162–1166, Sep. 1971. [15] A. J. Berni and W. D. Gregg, “On the utility of chirp modulation for digital signaling,” IEEE Trans. Commun., vol. COM-21, no. 6, pp. 748–751, Jun. 1973. [16] J. Burnsweig and J. Wooldridge, “Ranging and data transmission using digital encoded FM—‘Chirp’ surface acoustic wave filters,” IEEE Trans. Microw. Theory Tech., vol. MTT-21, no. 4, pp. 272–279, Apr. 1973. [17] A. Springer, M. Huemer, L. Reindl, C. C. W. Ruppel, A. Pohl, F. Seifert, W. Gugler, and R. Weigel, “A robust ultra-broad band wireless communication system using SAW chirped delay lines,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 12, pp. 2213–2218, Dec. 1998. [18] A. Springer, W. Gugler, M. Huemer, R. Keller, and R. Weigel, “A wireless spread spectrum communication system using SAW chirped delay lines,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 4, pp. 754–760, Apr. 2001. [19] W. Gugler, A. Springer, and R. Weigel, “A robust SAW-based chirp – =4 DQPSK system for indoor applications,” in Proc. IEEE Int. Commun. Conf., New Orleans, LA, 2000, pp. 773–777. [20] D. K. Barton, C. E. Cook, and P. H. Hamilton, Radar Evaluation Handbook, 1st ed. Boston, MA: Artech House, 1991. [21] J. D. Taylor, Ultra-Wideband RADAR Technology, 1st ed. Boca Raton, FL: CRC, 2000. [22] J. R. Andrews, “UWB signal sources, antennas and propagation,” in IEEE Wireless Commun. Technol. Top. Conf., Oct. 15–17, 2003, pp. 439–440. [23] K. Siwiak, “The potential of ultra wideband communications,” in 12th Int. AP-S Conf., Mar. 2003, vol. 1, pp. 225–228. [24] J. D. Kraus, Antennas, 2nd ed. New York: McGraw-Hill, 1988. [25] J. D. Kraus, “Helical beam antennas for wideband applications,” Proc. IRE, vol. 36, no. 10, pp. 1236–1242, Oct. 1948. [26] C. E. Cook, “Pulse compression—Key to more efficient transmission,” Proc. IRE, vol. 48, no. 3, pp. 310–316, Mar. 1960. [27] K. Yamanouchi, “Generation, propagation, and attenuation of 10 GHzrange SAW in LiNbO3,” in IEEE Ultrason. Symp., 1998, pp. 57–62. [28] Y. Takagaki, T. Hesjedal, O. Brandt, and K. H. Ploog, “Surface-acoustic-wave transducers for the extremely-high-frequency range using AlN/SiC(0001),” Semiconduct. Sci. Technol., vol. 19, pp. 256–259, 2004.

George N. Saddik (S’97) received the Bachelor of Science degree in electrical engineering from California State Polytechnic University, Pomona, in 1998, and is currently working toward the M.S. and Ph.D. degrees at the University of California at Santa Barbara (UCSB). He was a Product Engineer with the WatkinsJohnson Company, Palo Alto, CA, where he was involved with microwave transceivers, and was later an Application Engineer involved with RF semiconductors for commercial applications. His research interests include design and fabrication of FBAR filters in AlN and BST and their integration with GaN-based HEMT technology, and their application to communication systems.

Rahul S. Singh (M’06) received the B.Scs. degree in electrical engineering and mathematics from Southern Methodist University, Dallas, TX, in 1997, and the M.S. and Ph.D. degrees in electrical engineering from the University of California at Los Angeles (UCLA), in 1999 and 2005, respectively. He is currently a Post-Doctoral Researcher with the Electrical and Computer Engineering Department, University of California at Santa Barbara, where he conducts research in RF systems and integration, biomedical ultrasound including both hard tissues (dental) and soft tissues, terahertz imaging, and gigasonics. His research interests also include piezoelectric materials (perovskites and nitrides), transducer design, analog and digital signal processing, radar applications, and system research.

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Elliott R. Brown (M’92–SM’97–F’00) received the Ph.D. degree in applied physics from the California Institute of Technology, Pasadena, in 1985 He is currently a Professor of electrical and computer engineering with the Electrical and Computer Engineering Department, University of California at Santa Barbara (UCSB), where he teaches courses in solid-state engineering, RF sensors, and terahertz science. His research concerns the terahertz field in several areas, including ultra-low-noise rectifiers, photomixing sources, the terahertz phenomenology of biomaterials, and terahertz remote sensor and imager design and simulation.

Other areas of his research include multifunctional RF electronics and systems and biomedical ultrasonic imaging in hard tissue, particularly imaging in human teeth in collaboration with the University of California at Los Angeles (UCLA) Dental School. Prior to UCSB, he was a Professor of electrical engineering with UCLA. Prior to that, he was a Program Manager with the Electronics Technology Office, Defense Advanced Research Projects Agency (DARPA), Arlington, VA. He performed his post-doctoral research with the Lincoln Laboratory, Massachusetts Institute of Technology (MIT). Dr. Brown is a member of the American Physical Society. He was the recipient of a 1998 Award for Outstanding Achievement presented by the U.S. Office of the Secretary of Defense.

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A Quadrature Radar Topology With Tx Leakage Canceller for 24-GHz Radar Applications Choul-Young Kim, Student Member, IEEE, Jeong-Geun Kim, and Songcheol Hong, Member, IEEE

Abstract—A quadrature radar topology with a Tx leakage canceller is presented. The canceller is composed of four branch-line hybrid couplers, a 90 delay line, and a Wilkinson combiner. It has high Tx-to-Rx isolation and wide bandwidth characteristics. The isolation is 35.27 dB at 24 GHz and does not drop below 30 dB over a 2-GHz bandwidth. Given that the isolation of the branch-line hybrid coupler is 21.5 dB, the enhancement of the isolation is expected to be 13.77 dB. A quadrature Doppler radar front-end with the canceller was implemented. The noise level in the mixer output is lower by 9 dB compared with that of a conventional quadrature radar. This radar can measure speed as low as 0.76 mm/s, which correspond to a 0.12-Hz Doppler shift. It is also able to detect signs of the Doppler shift from a moving target due to its quadrature topology. Index Terms—Quadrature radar, radar sensor, velocity measurement.

I. INTRODUCTION ICROWAVE AND millimeter-wave radars have garnered much attention in many applications such as speed-detection sensors, collision-avoidance sensors, traffic control sensors, health-care sensors, and human motion detectors [1]. For these applications, high performance, small size, and low cost are crucial for the radars [2], [3]. Continuous wave (CW) radars are based on very simple structures, allowing small size with low-cost solutions. In CW radar systems such as frequency modulated continuous wave (FMCW) radars or Doppler radars, a high Tx-to-Rx isolation is required to improve sensitivity [4]. A system with two antennas is likely to have high Tx-to-Rx isolation; however, the system size is two times larger than a system with a single antenna. A system with a single antenna must be implemented with a circulator or a coupler following the antenna. As circulators, and especially couplers, show low isolation characteristics, especially at millimeter-wave frequencies, there are significant Tx leakages from the transmitter to receiver. A reflected power canceller (RPC) has been proposed in order to achieve good Tx-to-Rx isolation in a single antenna system [5], [6]. Recently, an RPC based on digital signal processing (DSP) was reported

M

Manuscript received December 7, 2006; revised March 16, 2007. This work was supported in part by the Ministry of Information and Communication and the Institute for Information Technology Advancement under the Information Technology Leading Research and Development Support Project. C.-Y. Kim and S. Hong are with the School of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea (e-mail: [email protected]). J.-G. Kim was with the School of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea. He is now with the Electrical and Computer Engineering Department, University of California at San Diego, La Jolla, CA 92037 USA. Digital Object Identifier 10.1109/TMTT.2007.900316

[7]. This RPC shows good Tx-to-Rx isolation performance up to -band frequencies. Another approach to reduce Tx leakage, termed “balanced radar topology,” has been proposed [8]. However, in this approach, when a signal is transmitted, half of the transmitting signal power is dissipated at the terminating resistor of the coupler. In addition, half of the received signal power is dissipated. Quadrature receivers can detect the speed and direction of a target and can also measure the displacement [9], [10]. Even human vital signals are detected very well with quadrature mixing [11], [12]. The receivers receive Doppler signals with much higher signal-to-noise ratios [13]. In [14], an approach was proposed wherein the Tx leakage is cancelled using Lange couplers and a Wilkinson combiner in a conventional CW radar with a single antenna. The Tx leakage canceller has a higher level of Tx-to-Rx isolation compared to that of a single Lange coupler and also has wideband characteristics. In this paper, a Tx leakage canceller is implemented using a branch-line hybrid coupler as a quadrature coupler and the quadrature radar topology is also presented. In spite of the narrowband characteristics of the branch-line hybrid coupler, the 30-dB isolation bandwidth of this canceller is over 2 GHz due to the canceller topology. Quadrature radar topology with the proposed canceller can reduce power losses in the transmit and receive paths. In contrast with the balance radar topology [8], there is no power loss in the transmit path. A quadrature radar without a Tx leakage canceller was also fabricated for comparison with the quadrature radar with the Tx leakage canceller. The noise level of the mixer output 9 dB lower as a result of the improved Tx-to-Rx isolation relative to that of a conventional quadrature radar. This type of radar can measure speeds as low as 0.76 mm/s, which corresponds to a 0.12-Hz Doppler shift. It is also able to detect signs of the Doppler shift due to its quadrature topology. II. PRINCIPLE OF THE QUADRATURE RADAR TOPOLOGY WITH Tx LEAKAGE CANCELLERS Fig. 1 presents block diagrams of the quadrature radar without the Tx leakage canceller, the quadrature radar with the proposed Tx leakage canceller, and the quadrature radar with a different Tx leakage canceller implementation involving a quadrature coupler instead of a Wilkinson combiner along with a 90 delay line. The quadrature radar shown in Fig. 1(a) consists of an antenna, a signal source, low-noise amplifiers (LNAs), a Wilkinson divider, two mixers, two low-pass filters (LPFs), and two quadrature couplers. One coupler is used to separate the Tx signal from the Rx signal and the other coupler is used to generate the quadrature signal for quadrature mixing. The output signal from the signal source is transferred to the antenna through the quadrature coupler and radiated through the

0018-9480/$25.00 © 2007 IEEE

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Fig. 2. Output noise in the passive mixer.

figure of the receiver. Second, the baseband noise level in the mixer output can increase, as shown in Fig. 2. The transmitter noise sideband is transferred to the receiver; therefore, the sensitivity of the receiver decreases. The dc offset is mainly generated due to mixing of the local oscillator (LO) signal and the Tx leakage. Flicker noise in the output is generated, as induced dc current from the dc offset exists in the passive mixer [15]. This also degrades the sensitivity of the receiver. If the Tx leakage signal can be suppressed, the sensitivity of the receiver will improve. B. Operation of the Tx Leakage Canceller The Tx leakage canceller is composed of four quadrature couplers, a 90 delay line, and a Wilkinson combiner. The transmitting signal is given by (1) Fig. 1. Block diagrams of: (a) quadrature radar without the Tx leakage canceller, (b) quadrature radar with the proposed Tx leakage canceller, and (c) quadrature radar with an alternate implementation of the Tx leakage canceller using a quadrature coupler instead of a Wilkinson combiner along with a 90 delay line.

antenna. The radiated signal is reflected from the target object and captured with the same antenna. The received signal is coupled to an LNA by the quadrature coupler and then down- converted to the baseband by the two mixers. Finally, unwanted signals such as harmonic signals are removed via the LPF. There are significant Tx leakage signals in the isolation port of the quadrature coupler. The proposed quadrature radar shown in Fig. 1(b) has a Tx leakage canceller instead of two quadrature couplers. The Tx leakage canceller has three functions, i.e.: 1) separating the Tx signal and Rx signal; 2) generating the quadrature signal for quadrature mixing; and 3) improving the Tx-to-Rx isolation. A quadrature radar with a different Tx leakage canceller implementation where a quadrature coupler is used instead of a Wilkinson combiner along with a 90 delay line is shown in Fig. 1(c).

where is the transmitting signal, is the amplitude of the transmitted signal, and and are the transmitted frequency and the phase of the transmitted signal, respectively. The operation of the Tx leakage canceller is as follows. 1) Tx Leakage: A large Tx leakage signal appears at the isolation port of a quadrature coupler due to the low isolation property of the quadrature coupler. One port of the quadrature coupler (coupler 3) is connected to a 90 delay line. Thus, the two signals in the two input ports of the Wilkinson combiner are shown as (2) (3) where and are the input signals of the and are the amplitude Wilkinson combiner, of and the amplitude mismatch of , respectively, and and are the phase of and the phase mismatch of , respectively. The leakage signals in the output port of the Wilkinson combiner are given as follows:

A. Tx Leakage Problems In a system with a single antenna, there is a large amount of Tx leakage from the transmitter to the receiver. Several problems result from this significant Tx leakage. First, the leakage can saturate the receiver components and decrease the noise

(4) where (5)

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If the amplitude and phase mismatch are very small, these two signals are antiphased and equal in amplitude. Therefore, when they are combined with the Wilkinson combiner, Tx leakage signals are cancelled out, as they are much smaller than in the case where only one coupler is used. 2) Transmitting Signal: The transmitting signal from the signal source shown in (1) is split into two signals of equal amplitude and a 90 phase difference through the quadrature coupler (coupler 1). These split signals are coupled to the two input ports of the quadrature coupler (coupler 4) through the quadrature couplers (coupler 2, coupler 3) with 3 dB less power. These two signals are then combined by the quadrature coupler (coupler 4) as follows: (6) and are the input signal of the antenna where and the phase of that of the signal, respectively. Half of the transmitting signal power is delivered to the mixers as the LO signal, and half is transmitted to the antenna. Therefore, ideally, there is no power loss in the transmitting path. 3) Receiving Signal: The received signal reflected from the target is described as (7) where is the received signal, is the amplitude of the received signal, and and are the Doppler frequency and the phase of the received signal, respectively. This signal is split into two signals that have a phase difference of 90 and are equal in amplitude at the outputs of the quadrature coupler (coupler 4). Half of the input signal of the quadrature coupler (coupler 3) is coupled to the quadrature coupler (coupler 1) and the other half is fed into a 90 delay line. Following this, the phase of the signal in the input port of a 90 delay line is shifted 90 by a 90 delay line. As a result, this signal is in-phase with the output signal of the quadrature coupler (coupler 2). Thus, the signal is combined by the Wilkinson combiner as follows: (8) and are the output signal of the Wilkinson where combiner and the phase of that signal, respectively. Half of the received signal from the antenna is delivered to the mixers, and the other half is lost since it is delivered to the port of the transmitting source. C. Operation of the Quadrature Radar With the Tx Leakage Canceller The Tx leakage canceller can provide each mixer with quadrature signals. The output signals of the quadrature mixer are described as (9) (10)

Fig. 3. 2.5-D EM simulation results of the Lange coupler [14] and the branchline hybrid coupler.

where and are the output signals of the LPFs followed by the quadrature mixers. Accordingly, it is possible to detect the Doppler signal as well as the sign of this signal. III. Tx LEAKAGE CANCELLERS A. Quadrature Couplers for Tx Leakage Canceller A quadrature coupler, the most important component in the Tx leakage canceller, is a 3-dB directional coupler with a 90 phase difference in the outputs of the through and coupled arms. The performance of a Tx leakage canceller depends on the performance of the quadrature couplers. Using (1) and (2), the isois described as lation of a single coupler dB

(11)

Using (1)–(5) and (11), the isolation of the Tx leakage canceller is described as

dB

(12)

Equations (11) and (12) mean that the isolation performance of a canceller is directly affected by the isolation characteristics of a single coupler . Generally, a Lange coupler or a branch-line hybrid coupler is widely used as a quadrature coupler in microwave circuit design. Lange couplers are the most widely used, as they can be integrated into a monolithic microwave integrated circuit (MMIC). As they show a wideband performance with a small size, a wideband Tx leakage canceller with a compact size can be implemented easily [14]. However, this is difficult to realize on a printed circuit board (PCB). Here, the branch-line hybrid coupler is the ideal quadrature coupler due to convenience of design and implementation on a PCB. A branch-line hybrid coupler generally has narrowband characteristics and requires a comparatively larger area; however, according to our 2.5-D electromagnetic (EM) simulation, it has higher isolation than the Lange coupler, as shown in Fig. 3. Therefore, high isolation performance of the Tx leakage

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Fig. 4. Fabricated Tx leakage canceller using a branch-line hybrid coupler.

canceller using a branch-line hybrid coupler is more readily achieved than in the case of using a Lange coupler. The bandwidth of the Tx leakage canceller using a branch-line hybrid coupler can be improved with the canceller topology. B. Tx Leakage Canceller Using a Branch-Line Hybrid Coupler To demonstrate the topology, a 24-GHz Tx leakage canceller with branch-line hybrid couplers was implemented. A branchline hybrid coupler and microstrip delay line and Wilkinson combiner were designed using the 2.5-D EM simulator of Ansoft Designer. The Tx leakage canceller was fabricated on a RO3003 PCB, which has a dielectric constant of 3 and a thickness of 10 mil. 50 – and 100- high-frequency resistors were utilized. The fabricated Tx leakage canceller is shown in Fig. 4. The performance of the Tx leakage canceller was characterized with an HP8564E spectrum analyzer. First, the output frequency and output power of a MMIC voltage-controlled oscillator (VCO) (HMC533LP4, Hittite) were measured. Second, the output frequency and output power were measured at each output port of the Tx leakage canceller, which has the same VCO. The measured results are shown in Fig. 5. The frequency covers a range of 22.7–24.8 GHz. The transmittances between the signal source port and each LO (I, Q) port are 6.64 and 6.97 dB at 24 GHz, respectively. The achieved transmittance between the signal source port and the antenna port is 4.47 dB at 24 GHz. The achieved isolation between the signal source port and the canceller output port is 35.27 dB at 24 GHz. Given that the isolation of the branch-line hybrid coupler itself is 21.5 dB, an improvement of 13.77 dB in terms of Tx-to-Rx isolation is achieved by the Tx leakage canceller. In the case of the Tx leakage canceller using a Lange coupler, given that the isolation of the Lange coupler itself is 15.1 dB, additional isolation of 13 dB in terms of Tx-to-Rx isolation is achieved by the Tx leakage canceller [14]. Although almost the same isolation improvement is attained for both cancellers, the canceller with the branch-line hybrid coupler has 7-dB higher isolation than that using the Lange coupler. This is due to the

Fig. 5. (a) Measured S -parameter of the VCO to antenna and each mixer input in the Tx leakage canceller. (b) Measured S -parameter of the Tx leakage canceller using a branch-line hybrid coupler.

superior isolation of the branch-line hybrid coupler over the Lange coupler. In spite of the narrowband characteristics of the branch-line hybrid coupler, the 30-dB isolation bandwidth of this canceller is 2 GHz with the canceller topology. IV. FABRICATION OF QUADRATURE RADAR SYSTEM AND MEASUREMENT A quadrature radar with a Tx leakage canceller using a branch-line hybrid coupler was implemented to detect the direction and speed of a moving target. The radar front-end module was fabricated on a RO3003 PCB, as shown in Fig. 6. A 2 1 patch array antenna was designed on the same PCB using Ansoft Designer’s 2.5-D EM simulator. The measured input return loss is shown in Fig. 7. The return loss is 27.4 dB at 24 GHz and the bandwidth ( 10 dB) is approximately 1.5 GHz. Commercially available -band MMICs were used for the VCO (HMC533LP4, Hittite), LNA (HMC517LC4, Hittite), and mixer (HMC292LM3C, Hittite). The baseband outputs of the front-end module were filtered with Stanford Research System model SR560 LNAs using 0.03-Hz high-pass and 10-Hz low-pass analog filters, each with a 20-dB/decade slope and a gain of 26 dB. The preconditioned quadrature output

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Fig. 8. Direction of back and forth, and speed measurement setup.

Fig. 6. Fabricated quadrature radar front-end module.

Fig. 7. Measured input return loss of the 2

2 1 patch array antenna.

signals were converted to digital data with a PCMCIA-type NI DAQ 6024E card, which has a 12-bit input resolution. The data collected with the DAQ card was processed and displayed with a signal processing program developed using LabVIEW 7.0. The measurement setup is shown in Fig. 8. The developed radar sensor was tested while measuring the direction and speed of a moving metal plate (0.09 m ) mounted on a rail that could only move automatically along one axis, back and forth, at a fixed speed. The metal plate was located approximately 3.8 m away from the sensor. Data was received from the sensor and the speed of the moving target was measured. As the metal plate moved slowly, the output signals of the sensor were sampled at a sampling rate of 1000 S/s. The measured results in accordance with back and forth movement of the metal plate are shown in Fig. 9. As shown in Fig. 9, a clean Doppler signal can be obtained with the developed system. Fig. 9(a) shows a case in which the target moves away from the sensor. The phase of the Q-channel signal

Fig. 9. (a) Measured channel responses when the target moves away from the sensor and (b) when the target moves towards the sensor.

is retarded with respect to that of the I-channel. Fig. 9(b) shows a case in which the target moves toward the sensor. The phase of the Q-channel signal is advanced with respect to that of the I-channel. These results coincide with (8) and (9). It is also possible to obtain the target speed from the above results by calculating the Doppler frequency. The Doppler frequency shift is 1.17 Hz. The calculated speed of the target from the Doppler frequency is 7.5 mm/s. Target speeds of 0.75–7.5 mm/s were

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quadrature radar with the Tx leakage canceller were also obtained by FFT, and are shown in Fig. 11. The minimum measurable speed is limited by the corner frequency of the high-pass filter, the flicker noise of the mixer, and the phase noise of the frequency source. Since the reported minimum speed is limited by the mechanical part of the moving target, we predicted the minimum detectable speed of the radar would be much lower than 0.75 mm/s. It is possible to estimate the maximum detection range for the Doppler frequency from Fig. 11 and a radar equation. V. CONCLUSION

Fig. 10. Measured mixer output noise spectral density of quadrature radar with a Tx leakage canceller and without a Tx leakage canceller.

A Tx leakage canceller was implemented with simple passive devices of four branch-line hybrid couplers, a 90 microstrip delay line, and a Wilkinson combiner. The Tx leakage canceller with the branch-line hybrid couplers was fabricated on a PCB and the measured isolation is 35.27 dB at 24 GHz, constituting a 13.77-dB improvement over that of a single branch-line coupler. The canceller with the branch-line hybrid coupler has Tx-to-Rx isolation, 7 dB higher than that attained using a Lange coupler due to the superior isolation characteristics of the branch-line hybrid coupler over the Lange coupler. In spite of the narrowband characteristics of the branch-line hybrid coupler, the 30-dB isolation bandwidth of this canceller has more than 2 GHz due to the canceller topology. This topology was used to implement a quadrature Doppler radar. The noise level in the mixer output is lowered by 9 dB due to the improved Tx-to-Rx isolation relative to that of a conventional quadrature radar without a Tx leakage canceller. This radar module can measure speeds as low as 0.76 mm/s, which corresponds to a 0.12-Hz Doppler shift. It is also able to detect signs of the Doppler shift due to its quadrature topology. With high isolation characteristics and improved performance coupled with its small size, the quadrature radar topology with the proposed Tx leakage canceller can be readily utilized in many radar applications. REFERENCES

Fig. 11. Measured signal spectral density of quadrature radar with the Tx leakage canceller.

also measured with a baseband amp gain (SR 560) of 36 dB at 3 m away. The results are shown in Table I. The direction and speed of the moving target can be detected by the proposed quadrature radar sensor. The quadrature radar without a Tx leakage canceller shown in Fig. 1(a) was also fabricated for comparison with the quadrature radar with the Tx leakage canceller. The mixer output signals of two quadrature radars, as shown in Fig. 10, were measured at a 1000-S/s sampling ratio while the target was stopped. The mixer output noise spectra of both types of quadrature radar are obtained by a fast Fourier transform (FFT). The noise level at the mixer output is lowered by roughly 9 dB due to the improved Tx-to-Rx isolation relative to that of conventional quadrature radar without a Tx leakage canceller. The output spectra of the

[1] H. H. Meinel, “Commercial applications of millimeter waves history, present status, and future trends,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 7, pp. 1639–1653, Jul. 1995. [2] M. E. Russel, C. A. Drubin, A. S. Marinilli, W. G. Woodington, and M. J. Del Checcolo, “Integrated automotive sensors,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 3, pp. 674–677, Mar. 2002. [3] M. Klotz and H. Rohling, “A 24 GHz short-range radar network for automotive application,” in Proc. IEEE Radar Conf., 2001, pp. 115–119. [4] M. I. Skolnik, Introduction to Radar Systems, 2nd ed. New York: McGraw-Hill, 1980. [5] P. D. L. Beasley, A. G. Stove, B. J. Reits, and B. As, “Solving the problems of a single antenna frequency modulated CW radar,” in Proc. IEEE Radar Conf., 1990, pp. 391–395. [6] Q. Jiming, Q. Xinjian, and R. Zhijiu, “Development of a 3 cm band reflected power canceller,” in Proc. IEEE Radar Conf., 2001, pp. 1098–1102. [7] S. Kannangara and M. Faulkner, “Adaptive duplexer for multiband transceiver,” in Proc. IEEE Radio Wireless Conf., 2003, pp. 381–384. [8] J. Kim, S. Ko, S. Jeon, J. Park, and S. Hong, “Balanced topology to cancel Tx leakage in CW radar,” IEEE Microw. Guided Wave Lett., vol. 14, no. 9, pp. 443–445, Sep. 2004. [9] S. Kim and C. Nguyen, “A displacement measurement technique using millimeter-wave interferometry,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 6, pp. 1724–1728, Jun. 2003. [10] S. Kim and C. Nguyen, “On the development of a multifunction millimeter-wave sensor for displacement sensing and low-velocity measurement,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 11, pp. 2503–2512, Nov. 2004.

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[11] D. O. Hogenboom and C. A. Dimarzio, “Quadrature detection of a Doppler signal,” Appl. Opt., vol. 37, no. 13, pp. 2569–2572, May 1998. [12] A. D. Droitcour, O. B. Lubecke, V. M. Lubecke, J. Lin, and G. Kovacs, “Range correlation and I/Q performance benefits in single-chip silicon Doppler radars for noncontact cardiopulmonary monitoring,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 3, pp. 838–848, Mar. 2004. [13] P. Heide, V. Magori, and R. Schwarte, “Coded 24 GHz Doppler radar sensors,” in IEEE MTT-S Int. Microw. Symp. Dig., 1995, pp. 965–968. [14] C. Kim, J. Kim, J. H. Oum, J. Yang, D. Kim, J. Choi, S. Kwon, S. Jeon, J. Park, and S. Hong, “Tx leakage cancellers for 24 GHz and 77 GHz vehicular radar applications,” in IEEE MTT-S Int. Microw. Symp. Dig., 2006, pp. 1402–1405. [15] S. Zhou and M. F. Chang, “A COMS passive mixer with low flicker noise for low power direct-conversion receiver,” IEEE J. Solid-State Circuits, vol. 40, no. 5, pp. 1084–1093, May 2005.

Choul-Young Kim (S’04) received the B.S. degree in electrical engineering from Chungnam National University (CNU), Daejeon, Korea, in 2002, the M.S. degree in electrical engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2004, and is currently working toward the Ph.D. degree at KAIST. His research interests include RF circuits and systems for short-range radar.

Jeong-Geun Kim received the B.S., M.S., and Ph.D. degrees in electrical engineering from the Korea Advanced Institute of Science and Technology (KAIST), Korea, in 1999, 2001 and 2005, respectively. He is currently a Post-Doctoral Research Fellow with the Electrical and Computer Engineering Department, University of California at San Diego (UCSD), La Jolla. His research interests include millimeter-wave integrated circuits and systems for short-range radar and phased-array antenna applications.

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 School 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, Stanford, CA, and Samsung Microwave Semiconductor, Suwon, Korea. 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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Design of Ultra-Low-Voltage RF Frontends With Complementary Current-Reused Architectures Hsieh-Hung Hsieh, Student Member, IEEE, and Liang-Hung Lu, Member, IEEE

Abstract—In this paper, ultra-low-voltage circuit techniques are presented for CMOS RF frontends. By employing a complementary current-reused architecture, the RF building blocks including a low-noise amplifier (LNA) and a single-balanced down-conversion mixer can operate at a reduced supply voltage with microwatt power consumption while maintaining reasonable circuit performance at multigigahertz frequencies. Based on the MOSFET model in moderate and weak inversion, theoretical analysis and design considerations of the proposed circuit techniques are described in detail. Using a standard 0.18- m CMOS process, prototype frontend circuits are implemented at the 5-GHz frequency band for demonstration. From the measurement results, the fully integrated LNA exhibits a gain of 9.2 dB and a noise figure of 4.5 dB at 5 GHz, while the mixer has a conversion gain of 3.2 dB and an IIP3 of 8 dBm. Operated at a supply voltage of 0.6 V, the power consumptions of the LNA and the mixer are 900 and 792 W, respectively. Index Terms—CMOS RF frontends, complementary current-reused topology, down-conversion mixers, low-noise amplifiers (LNAs), moderate inversion, ultra-low power, ultra-low voltage.

dard 0.18- m CMOS process, an ultra-low-voltage LNA and mixer suitable for operations with microwatt power consumption are realized at the 5-GHz frequency band. The behavior of the MOSFETs biased at a reduced overdrive voltage and its impact on the circuit performance of RF frontends are also investigated. Though the fabricated circuits are not targeted at a specific wireless standard, the developed techniques and design guidelines can be easily applied for various short-range wireless applications such as ZigBee, Bluetooth, wideband personal area network (WPAN), and wireless sensor networks. This paper is organized as follows. In Section II, MOSFETs biased at different inversion levels are reviewed, and a transistor model suitable for moderate and weak inversion is introduced. The proposed circuit topologies of the LNA and the down-conversion mixer are described in Sections III and IV, respectively. Section V presents the design and experimental results of the 5-GHz RF frontends. Finally, a conclusion is provided in Section VI. II. MOSFETs AT VARIOUS INVERSION LEVELS

I. INTRODUCTION S THE feature size of MOSFETs continues to shrink, a proportional downscaling in the supply voltage is mandatory to maintain gate–oxide reliability [1]. However, in consideration of the subthreshold leakage and the noise margin required by the digital integrated circuits, the scaling rate of the threshold voltage is relatively slow compared with that of the supply voltage. Consequently, the overdrive voltage of the transistors progressively decreases as the technology advances. It has become an inevitable trend to operate the MOS devices in moderate or weak inversion for certain mixed-signal and RF integrated circuits, motivating the development of low-voltage design techniques exclusively for deep-submicrometer CMOS technologies [2]–[6]. In an RF receiver frontend, the low-noise amplifier (LNA) and the down-conversion mixer are considered the most important building blocks. Typically, these circuits suffer from significant degradation in the RF properties, especially for gain, noise figure, and linearity, as the transistors operate in weak inversion. To overcome the limitations on the supply voltage and the transistor overdrive, a complementary current-reused topology has been proposed for the RF frontend circuits [7], [8]. Using a stan-

A

Manuscript received September 4, 2006; revised April 3, 2007. This work was supported in part by the National Science Council under Grant 94-2220-E002-026 and Grant 94-2220-E-002-009. The authors are with the Graduate Institute of Electronics Engineering and Department of Electrical Engineering, National Taiwan University, Taipei 10617, Taiwan, R.O.C. (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2007.900208

For the design of the RF frontend circuits operating at ultra-low supply voltage and power consumption, the behavior of the MOSFETs with a reduced overdrive voltage is first investigated and modeled. In conventional circuit implementations, the transistors are normally biased in their saturation regions to maximize the transconductance and output resistance. De, the operation of pending on the gate-to-source voltage the MOSFETs is typically classified into three modes: weak, moderate, and strong inversion. Weak inversion indicates that is slightly higher than the threshold voltage the value of . With the insufficient number of carriers in the induced is dominated by the diffusion channel, the drain current component instead of the drift one. Consequently, the device increases, the exhibits exponential I–V characteristics. As drift current becomes more significant. It is often referred to as moderate inversion when the carrier drift reaches a level comparable to the diffusion current. Finally, the drain current is dominated by the carrier drift when a high gate voltage is applied, leading to strong inversion operation with a square-law I–V relationship. In order to perform theoretical analysis for circuit designs, it is desirable to have a more quantitative scale of the inversion level for the MOSFETs. A useful index “inversion coefficient” is defined in [9] and [10] by a normalized drain current, which can be expressed as

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(1)

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Fig. 1. Simulated saturated drain current and the associated inversion coefficient of an n-channel MOSFET with a fixed V .

Note that is the saturation drain current, and are the is the effective channel width and length, respectively, and technology current, which is process dependent, and is defined as (2) where is the weak-inversion slope factor with a typical value is the low-field mobility, is the gate–oxide from 1 to 2, is the thermal voltage. Concapacitance per unit area, and ventionally, the device is considered in weak inversion for and in strong inversion for . For an value ranging from 0.1 to 10, the MOS transistor is operating in moderate inversion. To have a better understanding on the bias conversus for a 0.18- m dition of the transistors, simulated is ilnMOS device under a fixed drain-to-source voltage is obtained from (1) lustrated in Fig. 1, where the values of with a technology current of 0.7 A. According to the definition, it is observed that the transistor operate in weak inversion V and in strong inversion for V. With for between 0.4 and 0.7 V, the transistor is biased in moderate inversion. With various levels of inversion, the I–V characteristics of the MOSFETs are strongly influenced by the bias condition. Typically, distinct expressions of the drain current are required for weak, moderate, and strong inversion to characterize the transistor behavior. For simplicity, a semiempirical expression of the saturation current evolved in [11] and [12] is employed for the analysis and design of the RF frontend circuits in this paper. The saturation current of a MOSFET is given by

Fig. 2. Simulated and the calculated drain currents of a 0.18-m MOSFET operating in the saturation region with a fixed V .

n-channel

Note that (5) has the well-known form of the square-law I–V characteristics. For short-channel devices, the nonideal effects such as velocity saturation, series drain–source resistance, and mobility degradation should also be taken into account for the device modeling. As a result, a modified current expression of (3) is given by [13] (6) where (7) (8) are the mobility reduction Here, the parameters and coefficient and the saturation velocity, respectively. From (7), and the zero-bias drain conductance the transconductance of the device can be derived and are given by

(9) (10)

(3)

In order to verify the device model, the simulated saturation along with the values obtained from current as a function of (6) are plotted in Fig. 2. It is observed that (6) predicts the drain current of the MOSFET operating from weak to strong inversion with sufficient accuracy. Therefore, the derivations in (6)–(10) are employed for the analysis of the RF frontend circuits below.

(4)

III. PROPOSED LNA

with

For a MOS transistor operating in strong inversion, the drain current in (3) can be approximated by (5)

A. Circuit Topology To operate the LNA at reduced supply voltage and power consumption while providing sufficient gain at multigigahertz frequencies, a complementary amplifier with a current-reused

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Fig. 3. Evolution of the proposed LNA. (a) Conventional cascode topology. (b) CMOS amplifier topology. (c) Complementary current-reused topology with two cascaded gain stages. (d) Complementary current-reused topology with three cascaded gain stages.

circuit topology is presented. The evolution of the proposed low-voltage technique is illustrated in Fig. 3. In conventional circuit implementations, a cascode amplifier [14], as shown in Fig. 3(a), is widely used for the LNA designs. With the current-reused feature in the cascode topology, desirable LNA gain is achieved with relatively low current consumption. However, it is not suitable for low-voltage operations due to the stack of nMOS transistors. In order to alleviate the limitations imposed on the supply voltage, an LNA topology, as shown in Fig. 3(b), has been proposed [15]. Due to the use of the CMOS amplifier stages, the required supply voltage is reduced by one transistor overdrive compared with that of the cascode amplifiers. Unfortunately, the gain of the CMOS LNAs is inherently low, especially for low-power operations. For an enhanced gain in ultra-low-power and ultra-low-voltage designs, a complementary current-reused LNA topology with cascaded amplifier stages is proposed, as shown in Fig. 3(c). Note that, under similar bias conditions, the amplifier gain can be further enhanced by increasing the number of cascaded stages in the proposed LNA architecture, as illustrated in Fig. 3(d). However, in practical circuit implementations, the parasitics introduced by the additional gain stages may result in performance degradation. Therefore, an LNA with three cascaded gain stages is employed in this study. The complete circuit schematic of the LNA is shown in Fig. 4, where common-source MOS transistors , , and represent the first, second, and third gain stages, respectively. By connecting the drain inductance of each transistor together with an ac ground provided by the bypass capacitor , the complementary current-reused topology is established. As for the dc bias, the gate of is connected to , while those of and are tied to the ground through resistors. In order to obtain simultaneous power and noise matching for the input stage, the inductive source degeneration with and is adopted. On the other hand, the output matching is provided by and . The networks and between the gain stages are used for inter-stage matching.

Fig. 4. Complete circuit schematic of the proposed LNA suitable for ultra-lowvoltage and ultra-low-power applications.

B. Theoretical Analysis In order to evaluate the circuit performance and to provide useful design guidelines of the proposed LNA for low-voltage and low-power operations, theoretical analysis is performed with respect to the circuit specifications using the MOSFET model, as presented in Section II. 1) Small-Signal Characteristics: As the first active building block in an RF receiver, the LNA is required to provide sufficient gain such that the noise contributed from the following stages can be effectively suppressed. Note that the proposed circuit topology is basically a three-stage amplifier, as illustrated in Fig. 5. To achieve a maximum available gain of the LNA, the of the cascaded stages should effective transconductance be optimized, while power matching is required in between the

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Fig. 5. Circuit schematic of an equivalent three-stage cascaded amplifier with the input, output, and inter-stage matching networks.

stages. Assuming that the losses from the matching network are negligible, the conjugate matching conditions are given by [16]

and gate-to-source capacitance of the MOSFETs, respectively. is given by Generally, the capacitance

(11) (12) (13) (14)

(21) where is a constant with a value of 2/3 for long-channel devices. By combining (9) and (21), the cutoff frequency, which and , is expressed as is defined as the ratio of

In addition to the amplifier gain, the noise figure is also an important design consideration for the impedance matching of the LNA circuits. The matching conditions for minimum noise figures of the cascaded gain stages are given by (15) (16) (17) , , and are the optimum noise source where impedances for the first, second, and third stages, respectively. In typical LNA designs, it is difficult to meet (11)–(17) at the same time. However, due to the use of the source degeneration in the cascaded gain stages, simultaneous power and noise matching can be achieved [17], leading to optimum performance of the LNA. With the matching conditions specified in (11)–(17), the effective transconductance of the gain stages can be derived as (18) (19) (20) where is the operating frequency, is the source and represent the transconductance impedance, and

(22) From (18)–(20), it is clear that the effective transconductance of the gain stages is strongly influenced by the transistor overdrive voltage. Therefore, three cascaded stages are employed in the proposed LNA topology to boost the amplifier gain for ultralow-voltage operations. Note that, from (19)–(20), transconductances of the second and third stages can be effectively enand . However, hanced by reducing the real part of it usually requires transistors with enormous aspect ratios for noise matching, leading to performance degradation, especially at higher frequencies, due to the excessive parasitics from the MOSFETs. Another important specification of the LNA circuits is the impedance matching at the input. The small-signal equivalent circuit of the first gain stage is illustrated in Fig. 6, where is the load impedance. Note that, due to the existence of , the input stage is treated as a bilateral two-port network and . Assuming that the the input impedance is influenced by is relatively small, which is generally overlap capacitance

HSIEH AND LU: DESIGN OF ULTRA-LOW-VOLTAGE RF FRONTENDS WITH COMPLEMENTARY CURRENT-REUSED ARCHITECTURES

Fig. 6. Small-signal equivalent circuit of the input stage.

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Fig. 7. Small-signal equivalent circuit of the LNA input stage with noise current sources.

the case in practical designs, a simplified expression of the input impedance is given by

(23) From (23), the input impedance matching to a 50- system can be achieved by

(30) (31)

(24) (25) 2) Noise Figure: In a typical RF system, the sensitivity of the receiver is determined by the noise figure of the LNA. Due to the use of the source degeneration in the amplifier stages, noise and power matching can be simultaneously achieved. Provided that the gain is sufficiently larger, the noise figure of a cascaded LNA is dominated by the input stage. To have a better understanding on the noise matching, the equivalent circuit of the input stage is and indicate the Thevenin’s depicted in Fig. 7, where equivalent circuit seen from the MOS transistor to the source terminal, and and represent the mean-square values of the gate-induced and channel noise currents, respectively. The expressions of the noise currents are given by (26)

and is a correlation coefficient with a where predicted value of 0.395 [14]. Note that can be expressed as

(32)

and has a value close to unity. From the equivalent circuit in at the gate of is approxFig. 7, the input impedance imated by (33) According to (30) and (33), the simultaneous power and noise is the complex conjugate of matching can be achieved if , resulting in

(27) where and have typical values of 4/3 and 2/3, respectively. is given by Note that (28) (34) is the zero-bias drain conductance. Assuming that where the effects of and are negligible, the two-port noise parameters of the equivalent circuit in Fig. 7 can be derived. , The noise resistance , the optimum source impedance are expressed as [17], [18] and the minimum noise factor follows: (29)

By properly choosing the device parameters to satisfy (34), the while input stage of the LNA exhibits a noise factor of maintaining a maximum power gain and a low input return loss at the operating frequency . To further investigate the influence of the bias conditions , the simulated values along on the minimum noise factor with the calculated ones from (22) and (31) are demonstrated in Fig. 8, which provides useful guidelines for the design of

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Fig. 8. Simulated and calculated minimum noise factor of a 0.18-m n-channel MOSFET with a fixed V .

Fig. 9. Calculated IIP of the gain stage versus gate bias voltage . various values of R

V

for

where the LNAs at various bias conditions. It is obvious that indecreases. For ultra-low-voltage creases as the bias voltage LNAs with the MOSFETs operated in moderate inversion, the obtained from the device model in (22) can be treated as the worst case estimation in practical circuit implementations. 3) Linearity: With the ultra-low supply voltage and power consumption in the LNA design, the small-signal approximation only holds at a reduced input power. As the signal level increases, nonlinear characteristics such as gain compression and intermodulation distortion become significant. Therefore, analysis on the linearity of the proposed LNA is essential for practical applications. Due to the similarity in the circuit topology, the proposed LNA is considered as a three-stage cascaded amplifier to simplify the analysis. By neglecting the dc and higher order harmonic terms in the transfer function, the input thirdof the LNA is given by [14] order intercept point (35) where and are the linear gains of the first and the second and are greater than stage, respectively. Assuming that unity, the input intermodulation distortion of the third stage dominates. Therefore, analysis on the linearity of the third stage is employed to evaluate the overall LNA performance. To investigate the linearity of the amplifier stage, the drain is expressed by a third-order power series as current of

(36) Since the third-order derivative of the drain current with respect to gate voltage is the major contribution to the distortion, it is required to derive the coefficient . Based on the transistor model in (6), is given by [13]

(38) (39) As indicated in [14],

can be approximated by (40)

represents the input resistance of . Since is where of the MOSFET, can actually the transconductance be derived by substituting (9) and (37) into (40) as [13]

(41) versus is depicted in Based on (41), the calculated of the MOSFETs in the amplifiers Fig. 9. It is clear that the generally increases with the gate bias voltage. However, with the source degeneration in the cascaded gain stages, the coefficient in (40) is no longer the transconductance , but the ef, as specified in fective transconductance of the third stage (20). Thus, in the proposed LNA topology, the gain of the third stage can be traded for enhanced amplifier linearity, especially when low-voltage operations are required. 4) Stability Consideration: The stability is another important issue in LNA designs, and can be inspected by the reflection coefficients at the matching networks and terminations. For a general two-port network, in order not to initiate the undesirable oscillation, the necessary and sufficient conditions for the circuit stability are given by [16] (42) (43) (44)

(37)

(45)

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Fig. 10. Evolution of the proposed down-conversion mixer. (a) Conventional Gilbert-cell mixer. (b) Folded cascode topology. (c) Complementary current-reused topology. (d) Complementary current-reused topology with the current-bleeding technique.

where , , , and represent the source, load, input, and output reflection coefficients, respectively. With proper rearrangement, (42)–(45) result in the following two conditions: (46) (47) where (48) Note that (46) and (47) are derived for a single-stage amplifier. For multistage ones, the conditions for unconditional stability should be applied to each one of the gain stages. That is, once each gain stage in the cascaded topology satisfies (46) and (47), the stability of the amplifier is assured. In this particular design, the simplified circuit model in Fig. 5 is adopted to examine the circuit stability. Due to the use of the source degeneration, which is considered a negative feedback at the frequencies of interest, for each one of the gain stages, the proposed amplifier satisfies the unconditionally stable conditions at various process corners. 5) Bias Conditions: Due to the use of the complementary stages for the LNA design, the voltage at the drain of the MOS, as shown in Fig. 4, is considered a quasi-stable dc FETs point under normal bias conditions. Typically, a common-mode feedback (CMFB) is required to provide a stable dc bias at the output. In the proposed topology, the transistors are inevitably operated in moderate or weak inversion due to the reduce supply voltage. The drain currents of the MOSFETs are saturated even smaller than . Thus, by properly selecting with a can endure the aspect ratios of the transistors, the voltage process and supply voltage variations to provide a relatively stable bias for the amplifier even without the CMFB. According to the circuit simulations, a voltage shift less than 50 mV is obwith a 5% variation in the supply voltage for served at various corner analyses.

IV. PROPOSED DOWN-CONVERSION MIXER A. Proposed Topology Fig. 10(a) shows a simplified circuit topology of a Gilbert cell, which is widely used as the down-conversion mixer in RF frontends. The transistor acts as the transconductance stage to convert the RF voltage into a current signal, while and form the communicating stage for frequency translation. With the series-gated topology, the Gilbert cells are not suitable for low-voltage applications. In order to reduce the required supply voltage, a folded cascode mixer [19] is proposed, as shown in Fig. 10(b). Since the dc current reuse between the transconductance and commutating stages no longer exists, the power consumption may increase even with a reduced supply voltage. For low-power and low-voltage applications, the proposed complementary current-reused technique is adopted for the down-conversion mixer as well. A conceptual illustration of the proposed mixer topology is shown in Fig. 10(c), where an nMOS transconductance stage and a pMOS commutating stage and the ground. To enhance the are stacked between the gain of the down-conversion mixer, the current-bleeding technique [20] can also be incorporated into the proposed topology, as shown in Fig. 10(d). The complete circuit schematic of the down-conversion mixer is illustrated in Fig. 11. The RF current generated by is directed to the source of the the transconductance of switching pair through the bypass capacitor , while the frequency translation of the single-balanced mixer is provided . In the mixer design, the by the commutating stage is selected to maximize the conversion gain. load resistance For low-voltage operations, the current-bleeding technique is as the current source. As part adopted by using the resistor of the bias current flows through , excess voltage drop across can be prevented even with large load resistance. As a result, the mixer can operate at a reduced supply voltage while providing sufficient conversion gain for frequency translation. and Note that source degeneration with on-chip inductors is also employed for the transconductance stage to provide

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Fig. 13. Simulated normalized conversion gain (CG) and noise figure (NF) of the proposed mixer as functions of the current bleeding ratio.

Fig. 11. Complete circuit schematic of the proposed down-conversion mixer suitable for ultra-low-voltage and ultra-low-power applications.

Fig. 12. Equivalent circuit model of the proposed mixer for the derivation of the down-conversion gain.

signal amplification and impedance matching at RF frequencies. To have a better understanding of the proposed mixer circuit, detailed analysis with respect to the most important circuit specifications is presented as follows. B. Theoretical Analysis 1) Conversion Gain: The conversion gain of the mixer is evaluated by the equivalent circuit, as shown in Fig. 12. Assuming that the input impedance is matched to 50 and the local oscillator (LO) signals are square waves, the conversion gain is approximately by (49) is the RF frequency, and and are the where and , respectively. The expression transconductances of in (49) can be further simplified as for for

(50) (51)

According to (50) and (51), the conversion gain is independent is sufficiently large, and of the commutating stage when as the transconductance of the commutating decreases with stage becomes too small. Due to the use of the current-bleeding technique, large load can be employed to enhance the conversion gain. resistance , the transconducHowever, as more current flows through tance of the commutating stage diminishes, leading to a degraded conversion gain. as predicted in (51). The optimum conversion gain can be achieved by taking the effects of both and into consideration. With the definition of the currentas the dc current through divided by the bleeding ratio total bias current, the simulated conversion gain versus is presented in Fig. 13. It is observed that the down-conversion mixer exhibits a maximum conversion gain with a current-bleeding ratio of 90% in this particular design. In additional to the conversion gain, the simulated noise figure is also depicted in Fig. 13, indicating an increase in the noise figure as approaches to unity. Therefore, a tradeoff is involved between the conversion gain and the noise figure for circuit implementations. Note that the derivation in (49) is based on the small-signal equivalent circuits of the mixer. Hence, the linearity issue is not taken into account. It is generally true that the current-bleeding technique enhances the down-conversion gain at the cost of the circuit linearity. In order to alleviate such limitations, the transconductance stage of the proposed mixer is biased in the vicinity of the . More detailed derivation and anal“sweet spot” to boost ysis will be presented in Section IV-B.2. 2) Linearity: In a receiver frontend, the linearity and dynamic range are strongly influenced by the down-conversion mixer. To simplify the analysis, the linearity of the mixer is evaluated by the characteristics of the transconductance stage with the assumption that the commutating stage acts as ideal along switches. For a saturated MOSFET, the simulated with the values of and , which are defined in (36), are illusincreases, the transistor operates from trated in Fig. 14. As weak to strong inversion, leading to an increase in the value of . On the other hand, the value of turns from positive to negslightly higher than . ative with a zero-crossing point at of the mixer increases with when It is obvious that the the transistor is operating in strong inversion. For the transistor

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Fig. 16. Simplified illustration of the LO leakage paths to the RF port in the proposed mixer.

IIP

Fig. 14. (a) Simulated and (b) the associated coefficients c and c of the n-channel MOSFET at various gate bias voltages.

Fig. 17. Design flow of the proposed complementary current-reused LNA.

1IIP

Fig. 15. Simulated of the proposed mixer with various values of L for inductive source degeneration.

in the moderate inversion, interestingly, a significant peaking in the magnitude of is observed at the zero-crossing point of , which is typically referring to as the “sweet spot” [13]. Therefore, by properly biasing the transconductance stage of the mixer in the vicinity of the sweet spot, an enhanced can be achieved even with a reduced supply voltage. In the proposed mixer circuit, source degeneration is employed for input matching at the RF frequencies. Fig. 15 shows the simulated of the mixer biased at the sweet spot with deviation in the various values of the source inductance . Based on the simu-

lation results, the peaking at the sweet spot degrades due to the use of the source inductance. Therefore, special care has to be taken in selecting the values of the source degeneration for optimum circuit performance in terms of the input matching, conversion gain, and linearity. 3) Isolation: To reduce the overall power consumption of the RF frontends, low-or zero-IF architectures appear to be particularly well suited for the frequency down-conversion. Consequently, the isolation between the LO and RF ports becomes critically important. Fig. 16 shows a simplified illustration of the LO leakage paths to the RF port in the proposed mixer. In addition to the possible coupling through the gate-to-source capaciof the commutating stage, which is similar to the case tance

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TABLE I CIRCUIT PARAMETERS OF THE LNA

Fig. 19. Measured (solid lines) and simulated (dotted lines) small-signal parameters of the LNA.

Fig. 18. Microphotograph of the fabricated LNA.

in conventional Gilbert-cell mixers, the signal path provided by of the switching pair should also the parasitic capacitance be taken into account. Fortunately, with the current-bleeding is allowed for the mixer detechnique, large load resistance sign. The LO leakage to the RF port through is generally insignificant and can be neglected. Due to the differential operation in terms of the LO signals, the fundamental coupling is theoretically cancelled out at the common-mode nodes, and the LO-to-RF isolation is solely resulted from the mismatch between the MOS devices and the load resistances. With careful selection of the design parameters accompanied by a symmetric layout, port isolation better than 30 dB can be achieved for applications at multigigahertz frequencies. 4) DC Stability: With the complementary current-reused stages, dc stability should be carefully examined in the mixer design. As shown in Fig. 11, the voltage at the drain of the MOSFETs is considered a quasi-stable dc point and is generally susceptible to variations in the fabrication process and the supply voltage. Fortunately, with the current bleeding introduced by , only a small portion of the bias current flows is through the pMOS switching pair. The drain voltage of , leading to good bias stability. From the thus defined by has a deviation less simulation results, the drain voltage

Fig. 20. Two-tone harmonic measurement of the 5-GHz LNA with an input frequency spacing of 20 MHz.

than 70 mV with respect to the supply voltage variation up to 5% for various corner conditions in this particular design. V. CIRCUIT DESIGN AND EXPERIMENTAL RESULTS In order to demonstrate the feasibility of ultra-low-voltage RF frontends with the proposed circuit technique, an LNA and a down-conversion mixer are designed to operate at a supply voltage of 0.6 V for 5-GHz applications. The circuits are implemented in a standard 0.18- m CMOS technology provided by a commercial foundry. The threshold voltages of

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TABLE II PERFORMANCE COMPARISON OF LOW-VOLTAGE AND LOW-POWER LNAS

the nMOS and pMOS transistors are approximately 0.5 V. As for the on-chip passive components, a top AlCu metallization layer of 2- m thickness is available for the inductive elements, while the metal–insulator–metal (MIM) capacitors with oxide intermetal dielectric are also provided. The RF performance of the LNA and the mixer is characterized by on-wafer probing. A. LNA With the circuit schematic, as shown in Fig. 4, the design procedure of the 0.6-V micropower LNA is depicted in Fig. 17. The circuit design starts with the device size and bias point of the cascaded gain stages. For optimum circuit performance in is designed at half terms of gain and linearity, the voltage of the supply voltage, specifying the aspect ratios of gain stages , , and as (52) where and are the mobility of the nMOS and PMOS, respectively. Once the device size and bias point of each stage are determined, the input matching network is accomplished by seand for simultaneous gain and noise lecting the values of matching. On the other hand, the values of the LC networks inand capacitors are designed cluding inductors for power matching in between the gain stages. For optimum circuit performance, iterations may be required in the design procedure. The final circuit parameters of the proposed LNA are tabulated in Table I. Fig. 18 shows a microphotograph of the fabricated LNA with a chip area of 0.86 1.1 mm including the pads. In this design, 3-D spiral inductors utilizing multiple interconnection layers are employed for the source degeneration and the interstage matching to minimize the chip size. Meanwhile, the inductor at the LNA input is realized by a planar structure with a high quality factor for minimum noise figure. Operated at a supply voltage of 0.6 V, the LNA consumes a dc power of 900 W.

Fig. 21. Design flow of the proposed complementary current-reused mixer.

The small-signal characteristics of the fabricated circuit are demonstrated in Fig. 19. Due to the use of the cascaded gain stages, the LNA exhibits a linear gain of 9.2 dB at the center frequency of 5 GHz with a 3-dB bandwidth of 1 GHz. The relatively flat gain response allows wideband applications for frequencies from 4.5 to 5.5 GHz. With the on-chip matching networks, the input and output ports are matched to 50 in the and vicinity of the center frequency. The measured at the center frequency are 12 and 21 dB, respectively. The noise figure of the LNA was measured without external noise matching, exhibiting a minimum value of 4.5 dB at 5 GHz. To and evaluate the large-signal behavior of the LNA,

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TABLE III CIRCUIT PARAMETERS OF THE MIXER

Fig. 23. Measured (solid line) and simulated (dotted line) conversion gain of the mixer versus LO power level.

Fig. 22. Microphotograph of the fabricated mixer.

were characterized by the two-tone harmonic measurement with 20-MHz input frequency spacing. Fig. 20 shows of 27 dBm and of the measured results with 16 dBm at the 5-GHz frequency band. The performance of the fabricated LNA is summarized in Table II along with previously published data [21]–[23] for comparison. It is noted that the proposed LNA exhibits the lowest power consumption and supply voltage while maintaining competitive performance in terms of gain and noise figure at 5 GHz. A widely used figure of merit, gain/power quotient, is also adopted for the performance evaluation. In this design, a gain/power quotient of 10.2 dB/mW is achieved. To the authors’ best knowledge, this is the highest record that has ever been reported in standard CMOS technologies. In this particular design, the LNA performance in terms of the linearity and noise figure are degraded to some extent compared with the conventional circuit implementations. However, for most of short-range communication systems, the physical (PHY) layer specifications are not as stringent [24]. Instead, the fabrication cost and power consumption are major concerns. Therefore, the proposed circuit techniques are well suited for such wireless applications at multigigahertz frequencies. B. Down-Conversion Mixer Fig. 21 shows the simplified design procedure of the ultra-low-voltage down-conversion mixer. The circuit design

Fig. 24. Measured (solid line) and simulated (dotted line) input reflection coefficient of the mixer.

Fig. 25. Two-tone harmonic measurement of the 5-GHz mixer with an input frequency spacing of 20 MHz.

starts with the device size and bias point of the transistors in the transconductance stage. In order to maximize the of the mixer, it is desirable to design the sweet spot at a gate voltage of the supply voltage while maintaining the required input matching at the RF input. Note that the performance of the mixer is strongly influenced by the current-bleeding ratio. A tradeoff has to be made among the conversion gain, noise figure, and linearity for the mixer design. Once the values of

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TABLE IV PERFORMANCE COMPARISON OF LOW-VOLTAGE AND LOW-POWER MIXERS

and are determined, the aspect ratio of commutating stage is chosen such that complete current switching is ensured for the specified LO power level. Similar to the LNA design, iterations may be required to achieve the desirable circuit performance, and the final design values for the 5-GHz down-conversion mixer are summarized in Table III. Fig. 22 shows a microphotograph of the fabricated mixer with a chip area of 0.95 0.7 mm . The RF port of the mixer is with the source degeneration, and the gate matched to 50 voltage of is provided externally via a bias tee. In order to drive the 50- load, the open-source buffer is also employed. To convert the differential signals into a single-ended one, an external power combiner was employed at the IF output. All losses from the adaptors, cables, and power combiners in the measurement setup were calibrated and deembedded in the experimental results. Biased at a reduced supply voltage of 0.6 V, the power consumption of the mixer core is 792 W. With an RF input at 5.2 GHz and an LO frequency of 5.1 GHz, the measured down-conversion gain versus the LO power is illustrated in Fig. 23, indicating a maximum gain of 3.2 dB at an LO power of 2 dBm. The moderate conversion gain of the mixer is mainly due to the limitations on the supply voltage and power consumption. For a fixed IF of 100 MHz, measurement of the down-conversion gain versus the RF frequency was performed, and a 3-dB bandwidth of 1.6 GHz is demonstrated. Due to the use of source degeneration at the RF port, good input matching is achieved. As shown in Fig. 24, the RF return loss in the vicinity of 5 GHz is generally better than 10 dB. In addition, the port-to-port isolations were also characterized, and the obtained LO-to-RF and LO-to-IF isolations are higher than 30 dB. Another important specification in the mixer design is the noise figure. Based on the experimental results, a double-sideband noise figure of 14 dB is achieved at an RF frequency of 5.2 GHz. In order to evaluate the circuit linearity, a two-tone harmonic measurement with a frequency spacing of 20 MHz was carried out. Fig. 25 shows the measured intermodulation distortion versus the input power sweep, indicating a

of 15 dBm and an of 8 dBm at the 5-GHz frequency band. Table IV summarizes of the performance of the proposed mixer along with the results from the previously published data [25]–[27] for comparison. As indicated in Table IV, the designed circuit reveals the lowest power consumption and supply voltage for active mixers while maintaining a reasonable conversion gain at 5-GHz frequency band. VI. CONCLUSION In this paper, complementary current-reused circuit techniques suitable for the design of RF frontends has been demonstrated. Theoretical analysis and design tradeoffs have been presented for the circuit implementations based on a semiempirical MOSFET model. Using a standard 0.18- m CMOS process, an LNA and down-conversion mixer have been designed and fabricated at the 5-GHz frequency band. With the proposed circuit topologies, the RF frontend circuits can operate at a reduced supply voltage of 0.6 V with power consumption less than 1 mW, exhibiting a great potential for applications in ultra-low-power and ultra-low-voltage wireless systems. ACKNOWLEDGMENT The authors would like to thank the National Chip Implementation Center (CIC), Hsinchu, Taiwan, R.O.C., for chip fabrication and National Nano Device Laboratories (NDL), Hsinchu, Taiwan, R.O.C., for chip measurement. The authors would like to express their appreciation to K.-S. Chung, Realtek Semiconductor Corporation, Hsinchu, Taiwan, R.O.C., for valuable discussion. REFERENCES [1] “International technology roadmap for semiconductors,” Semiconduct. Ind. Assoc., 2004 ed. [Online]. Available: http://public.itrs.net/

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[2] H.-H. Hsieh, C.-T. Lu, and L.-H. Lu, “A 0.5-V 1.9-GHz low-power phase-locked loop in 0.18-m CMOS,” presented at the IEEE VLSI Circuits Symp. Jun. 2007. [3] S.-A Yu and P. Kinget, “A 0.65 V 2.5 GHz fractional-N frequency synthesizer in 90 nm CMOS,” in IEEE Int. Solid-State Circuits Conf., Feb. 2007, pp. 304–305. [4] H.-H. Hsieh and L.-H. Lu, “A high-performance CMOS voltage-controlled oscillator for ultra-low-voltage operations,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 3, pp. 467–473, Mar. 2007. [5] N. Stanic, P. Kinget, and Y. Tsividis, “A 0.5 V 900 MHz CMOS receiver front end,” in IEEE VLSI Circuits Symp. Tech. Dig., Jun. 2006, pp. 228–229. [6] S. Chatterjee, Y. Tsividis, and P. Kinget, “0.5-V analog circuit techniques and their application in OTA and filter design,” IEEE J. SolidState Circuits, vol. 40, no. 12, pp. 2373–2387, Dec. 2005. [7] H.-H. Hsieh and L.-H. Lu, “A CMOS 5-GHz micro-power LNA,” in IEEE Radio Freq. Integr. Circuits Symp., Jun. 2005, pp. 31–34. [8] H.-H. Hsieh, K.-S. Chung, and L.-H. Lu, “Ultra-low-voltage mixer and VCO in 0.18-m CMOS,” in IEEE Radio Freq. Integr. Circuits Symp., Jun. 2005, pp. 167–170. [9] D. M. Binkley, M. Bucher, and D. Foty, “Design-oriented characterization of CMOS over the continuum of inversion level and channel length,” in IEEE Int. Electron., Circuits, Syst. Conf., Dec. 2000, pp. 161–164. [10] A.-S. Porret et al., “A low-power low-voltage transceiver architecture suitable for wireless distributed sensors network,” in IEEE Int. Circuits Syst. Symp., May 2000, vol. 1, pp. 56–59. [11] Y. Tsividis, K. Suyama, and K. Vavelidis, “A simple ‘reconciliation’ MOSFET model valid in all regions,” Electron. Lett., vol. 31, no. 6, pp. 506–508, Mar. 1995. [12] Y. Tsividis, Operation and Modeling of the MOS Transistor, 2nd ed. New York: Oxford Univ. Press, 1999. [13] B. Toole et al., “RF circuit implications of moderate inversion enhanced linear region in MOSFETs,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 51, no. 2, pp. 319–328, Feb. 2004. [14] T. H. Lee, The Design of CMOS Radio Frequency Integrated Circuits. Cambridge, U.K.: Cambridge Univ. Press, 1998. [15] T. Taris et al., “A 1-V 2 GHz VLSI CMOS low noise amplifier,” in IEEE Radio Freq. Integr. Circuits Symp., Jun. 2003, pp. 123–126. [16] G. Gonzalez, Microwave Transistor Amplifiers: Analysis and Design, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 1997. [17] T.-K. Nguyen et al., “CMOS low-noise amplifier design optimization techniques,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 5, pp. 1433–1442, May 2004. [18] J. Lu and F. Huang, “Comments on ‘CMOS low-noise amplifier design optimization techniques’,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 7, pp. 3155–3155, Jul. 2006. [19] P. Choi et al., “An experimental coin-sized radio for extremely low power WPAN (IEEE 802.15.4) application at 2.4 GHz,” IEEE J. SolidState Circuits, vol. 38, no. 12, pp. 2258–2268, Dec. 2003. [20] S.-G. Lee and J.-K. Choi, “Current-reuse bleeding mixer,” Electron. Lett., vol. 36, no. 8, pp. 696–697, Apr. 2000. [21] K. Ohsato and T. Yoshimasu, “Internally matched, ultralow dc power consumption CMOS amplifier for L-band personal communications,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 5, pp. 204–206, May 2004.

[22] T. K. K. Tsang and M. N. El-Gamal, “Gain and frequency controllable sub-1 V 5.8 GHz CMOS LNA,” in IEEE Int. Circuits Syst. Symp., May 2002, vol. 4, pp. 795–798. [23] D. Linten et al., “Low-power 5 GHz LNA and VCO in 90 nm RF CMOS,” in IEEE VLSI Circuits Symp., Jun. 2004, pp. 372–375. [24] N.-J. Oh, S.-G. Lee, and J. Ko, “A CMOS 868/915 MHz direct conversion ZigBee single-chip radio,” IEEE Commun. Mag., vol. 43, no. 12, pp. 100–109, Dec. 2005. [25] C. Debono et al., “A 900 MHz, 0.9 V low-power CMOS down-conversion mixer,” in IEEE Custom Integr. Circuit Conf., May 2001, pp. 527–530. [26] V. Vidojkovic et al., “A low-voltage folded-switching mixer in 0.18-m CMOS,” IEEE J. Solid-State Circuits, vol. 40, no. 6, pp. 1259–1264, Jun. 2005. [27] C. Hermann et al., “A 0.6-V 1.6-mW transformer-based 2.5-GHz down-conversion mixer with 5.4-dB gain and 2.8-dBm IIP3 in 0.13-m CMOS,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 2, pp. 488–495, Feb. 2005.

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Hsieh-Hung Hsieh (S’05) was born in Taipei, Taiwan, R.O.C., in 1981. He received the B.S. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 2004, and is currently working toward the Ph.D. degree in electronic engineering at National Taiwan University. His research interests include the development of low-voltage and low-power RF integrated circuits, multiband wireless systems, RF testing, and monolithic microwave integrated circuit (MMIC) designs.

Liang-Hung Lu (M’02) was born in Taipei, Taiwan, R.O.C., in 1968. He received the B.S. and M.S. degrees in electronics engineering from National Chiao-Tung University, Hsinchu, Taiwan, R.O.C., in 1991 and 1993, respectively, and the Ph.D. degree in electrical engineering from The University of Michigan at Ann Arbor, in 2001. During his graduate study, he was involved in SiGe HBT technology and MMIC designs. From 2001 to 2002, he was with IBM, where he was involved with low-power and RF integrated circuits for silicon-on-insulator (SOI) technology. In August 2002, he joined the faculty of the Graduate Institute of Electronics Engineering and the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan, R.O.C., where he is currently an Associate Professor. His research interests include CMOS/BiCMOS RF and mixed-signal integrated-circuit designs.

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Electrical Backplane Equalization Using Programmable Analog Zeros and Folded Active Inductors Jinghong Chen, Member, IEEE, Fadi Saibi, Jenshan Lin, Senior Member, IEEE, and Kamran Azadet, Associate Member, IEEE

Abstract—In this paper, we present a low-power small-area electrical backplane equalizer using programmable analog zeros and folded active inductors. We also present a dc-offset cancellation circuit, which occupies less chip area than the traditional offset cancellation schemes. The equalizer circuit was fabricated in a 1.0-V 90-nm CMOS process. With one zero stage, the equalizer occupies 0.015-mm2 chip area and dissipates 12 mW of power. At 4.25-Gb/s data rate, the equalizer provides 7.8-dB gain boost at the Nyquist frequency. Without the use of any transmitter equalization, the analog zero equalizer demonstrated error-free transmission for pseudorandom-bit-sequence-31 data patterns over 34-in lossy FR4 backplanes. Index Terms—Analog equalization, backplane communication, broadband amplifier, dc-offset cancellation, folded active inductor, inductor shunt peaking, inter-symbol interference (ISI), transceiver.

I. INTRODUCTION LECTRICAL data transmission at multigigabit/second through low-cost copper traces on FR4 boards is a challenging problem [1]–[10]. The copper traces on FR4 boards exhibit frequency-dependent loss, which is caused by skin effect and dielectric loss. Such frequency-dependent losses limit channel bandwidth and causes inter-symbol interference (ISI). Several approaches have been pursued to maintain backplane signal transmission integrity. Passive solutions by using high-quality microwave substrate materials, innovative via-hole techniques, and new connector technology can help to solve the transmission problem. However, such solutions often require the use of costly microwave substrate and special high-bandwidth connectors, while very long trace lengths may still result in unacceptable transmission characteristics [2]. Active solutions may use transmitter pre-emphasis to equalize the channel or to compress the bandwidth using multilevel signaling (e.g., PAM-4) techniques [3], [4]. Transmitter pre-emphasis typically utilizes a single- or two-tap finite impulse filter to combat backplane high-frequency loss. Transmitter pre-emphasis can effectively equalize the channel at lower data rates. However,

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Manuscript received September 12, 2006; revised March 19, 2007. J. Chen is with Analog Devices Inc., Somerset, NJ 08873 USA (e-mail: [email protected]). F. Saibi and K. Azadet are with the LSI Corporation, Allentown, PA 18109 USA (e-mail: [email protected]; [email protected]). J. Lin is with the Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL 32611 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.2007.900342

at higher data rates, the number of taps required in the pre-emphasis filter increases dramatically [3], which results in high power consumption. Electromagnetic interference (EMI) also becomes an issue as the high-frequency signals are emphasized at the transmitter side. Additionally, to be able to adapt a pre-emphasis filter’s transfer function to a particular channel’s characteristics, the transmitter pre-emphasis technique would require a handshaking mechanism between a receiver and transmitter [4]. Multilevel signaling such as PAM-4 alleviates ISI, but has interoperability issues and may suffer from reduced voltage margins that exacerbate crosstalk effects [3]. Another type of equalization is the receiver decision-feedback equalization (DFE). In a DFE, prior data decisions are multiplied by tap weights and subtracted from the received signal, eliminating post-cursor ISI [11], [12]. The DFE equalization does not amplify crosstalk or any other high-frequency noises. The main disadvantage of the DFE, however, is its difficult implementation at higher speeds since, in a DFE, the feedback loop delay should be less than half the baud period. Other problems with DFE are that it cannot cancel precursor ISI and it can only compensate for a fixed time span of the post-cursor ISI. In this paper, we describe a receiver equalization technique that combines a linear analog equalizer with a one-tap lookahead DFE. The analog equalizer incorporates the use of programmable analog high-pass zeros and folded active inductors. The analog zero compensates for the backplane low-pass characteristics by boosting the high-frequency component of the signal. The main advantages of receiver analog equalization are lower power consumption, less silicon area, and most importantly, its capability of performing equalization at higher speeds. Folded active inductor extends the equalizer bandwidth without occupying an excessive chip area, as compared to spiral inductors. The folded active inductor also alleviates the headroom bottleneck of conventional active inductors [13], [14] enabling the equalizer circuit to be implemented under a 1.0-V supply. No on-chip high voltage generation circuit is required. The combination of the analog linear equalizer with the DFE is a tradeoff between power consumption and high-frequency crosstalk enhancement. This paper also presents a dc-offset cancellation scheme, which occupies less chip area than the traditional offset cancellation techniques. Fig. 1 shows a schematic diagram of the receiver front-end including a variable-gain amplifier (VGA) along with the dc-offset cancellation circuit, the analog zero stage, and the DFE. The circuit was fabricated in a 1.0-V 90-nm CMOS process. With one zero stage, the analog high-pass equalizer

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Fig. 3. Programmable degeneration capacitor implemented as a binary weighted capacitor array. Fig. 1. Schematic diagram showing the receiver analog zero high-pass equalizer, the DFE equalizer, and the offset cancellation scheme.

Fig. 2. Schematic view of the analog zero stage with folded active inductor load.

occupies 0.015-mm chip area and dissipates 12 mW of power. At 4.25-Gb/s data rate and when driving a 0.4-pF capacitive load, the analog equalizer provides 7.8-dB gain boost at the baud-rate frequency. Without the use of any transmitter equalization, the analog zero opened the received eye diagram, which was almost closed. At a bit error rate (BER) of 10 , the analog zero with the one-tap look-ahead DFE achieves 84 mV of voltage margin over 34-in lossy FR4 backplane traces. II. EQUALIZATION CIRCUIT A. Analog Zero Equalizer Fig. 2 shows a simplified schematic view of the analog zero high-pass equalization circuit. The analog zero is implemented by a transconductor with source degeneration realized by a and a resistor . parallel connection of a capacitor When not considering the effect of the folded active inductor, the transfer function of the analog zero can be approximately given by (1) where

and , is the low frequency gain of the amplifier, and are the amplifier load resistance and caand pacitance, respectively. The analog zero compensates for the FR4 backplane low-pass characteristics by boosting the high-

frequency component of the signal so that the overall frequency response can be flat within the signal range. The equalizer is time constant of the zero. When the adapted by changing the zero frequency is smaller than the baud-rate frequency, a gain boost at the baud-rate frequency can be achieved. The slope of the gain boost due to the analog zero is 20 dB per decade of frequency and, therefore, a smaller zero yields a larger amount of gain boost. By implementing the degeneration capacitor as a binary weighted capacitor array that is controlled via an time conon-chip microcontroller, as depicted in Fig. 3, the stant of the zero can be adjusted allowing the equalizer to provide tunable frequency boost for a variety of backplane trace while relength or data rate applications. Tuning mains fixed has the advantage of keeping the low-frequency gain of the equalizer to be a constant. Depending on the intended application (e.g., trace length of the backplane or data rate), the equalizer can comprises several such zero stages in cascade to obtain a larger amount of high-frequency boost. When cascading multiple such analog zero stages, the reverse scaling technique [7] can also be employed to avoid significant smallsignal bandwidth reduction. The proposed equalizer that uses the programmable capacitor to control the equalizer parameters can easily achieve a large dynamic range (e.g., high-frequency gain boost) due to the wide control range of the programmable capacitor implemented as a binary capacitor array, whereas the equalizer circuit in [5] with tuning being achieved through a varactor was limited to 10-dB dynamic range because of the limited tunable range of the varactor. It should be noted that, in Fig. 3, one could further combine the binary weighted capacitors with a varactor to allow the equalizer to have both fine and coarse dynamic range tunings. In [6], the zero is created by source degeneration with a fixed capacitor in parallel with a variable resistor. The resistor is implemented by an nMOS transistor and the tuning is performed by controlling the gate voltage of the transistor. While tuning the resistor can provide a large range of control, the low-frequency gain of the equalizer, however, is affected. To achieve a large dynamic range, a large effective resistance must be used, which . In significantly reduces the equalizer low-frequency gain [6], when having a 30-dB dynamic range, the equalizer low-frequency gain is reduced by 20 dB. With a gain attenuation of 20 dB, a high-gain and high-bandwidth amplifier must be used to further amplify the equalized signal to a level sufficient for a reliable operation of the receiver clock and data recovery (CDR) circuit. The high-gain and high-bandwidth amplifier, however, often can consume an excessive amount of power, preventing

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Fig. 5. Folded active inductor load. (a) Schematic. (b) Small-signal model. Fig. 4. Inductor peaking for bandwidth enhancement. (a) Passive inductor. (b) Active inductor. (c) Active inductor with voltage boosting technique.

the tunable resistor approach from being practically used in a networking serializer/deserializer (SerDes) device where low power consumption is a must. To enhance the equalizer bandwidth, an inductor shunt peaking technique was used. Inductor shunt peaking is a well-known technique that moves the pole of an amplifier to a higher frequency by partly tuning out the load capacitance. To achieve an optimum bandwidth extension (e.g., maximally flat frequency response), the inductance can be designed as , and theoretically, the bandwidth can be extended by approximately 70% [13]. As shown in Fig. 4, the inductor load can be implemented by passive spiral inductors [7], conventional active inductors [14], and conventional active inductors with a voltage boosting technique [13], [15]. In [7], a 10-Gb/s backplane equalizer was designed using passive spiral inductors. The passive spiral inductor has a small voltage drop; however, it occupies a large chip area and introduces a significant amount of parasitic capacitance. The equalizer circuit reported in this paper was designed for a 4.25-Gb/s (mostly the CDR SerDes device. The capacitive load load capacitance and the routing capacitance) is approximately of 150 , the 400 fF. Assume an equalizer load resistance inductance required is 3.6 nH. Such spiral inductors (two for each zero stage and 3.6 nH per inductor) occupy a large chip area preventing a compact floor plan. Since a high- factor is not needed in this application, active inductors, on the other hand, can be used. Active inductors are small and operate at where is the unity frequencies up to approximately current-gain frequency of the technology. One of the main drawbacks of the conventional active inductors is their large dc voltage drop caused by the nMOS threshold voltage, which is enlarged by the body effect. The voltage drop, which is approximately 0.6 V in this technology, creates a headroom problem when having a 1.0-V power supply. In [13] and [15], low voltage-drop active inductors with voltage boosting techniques were proposed. However, such techniques require an on-chip high-voltage generation circuit using either a charge-pump [13] or a voltage doubling circuit [15] that make the design more complex and also consume a significant amount of chip area. In this design, the analog zero equalizer employs a folded active inductor shunt peaking technique [16]. Fig. 5(a) shows a schematic view of the folded active inductor. In Fig. 5(a), is the original equalizer load resistance; the resistor and the nMOS transistor form the folded active inductor. From its small-

Fig. 6. Folded active inductor. (a) Impedance versus frequency response. (b) Small-signal equivalent-circuit model.

signal model, shown in Fig. 5(b), the total load impedance can be calculated as (2)

where and are the transconductance and the gate-tosource capacitance of the nMOS transistor. Equation (2) can be simplified to the following expression: (3)

As can be seen from (3), at low frequency, , . By choosing to and at high frequency, , a high-frequency impedance boost can be be larger than achieved. Such a high-frequency impedance boost behaves like an inductor that helps to partly absorb the equalizer load capacversus frequency response is plotted itance. The impedance Fig. 6(a) and a small-signal equivalent-circuit model is plotted in Fig. 6(b). With the folded active inductor load, the equalizer transfer function is then given by

(4) The schematic diagram of the analog zero equalizer with a folded active inductor for bandwidth extension is shown in

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Fig. 8. Schematic view of the analog comparator used for analog zero equalizer offset cancellation.

Fig. 7. Schematic view of the VGA circuit along with the dc-offset cancellation scheme.

Fig. 2. Large voltage can be across the gate and source of the nMOS transistor in the folded active inductor, and it does not cause any headroom problems for the equalizer. The entire circuit can be operated with a 1.0-V supply without the need of an on-chip high-voltage generation circuit. The addition of the folder active inductor can lower the equalizer output common-mode voltage due to the tail current used to bias the nMOS transistor in the folder active inductor. However, such a tail current is usually much smaller than that of the equalizer and its impact on the equalizer output common-mode voltage can often be neglected. For the reported equalizer deis signed for a 4.25-Gb/s SerDes device, shown in Fig. 2, is 500 , and the resistor is chosen chosen as 150 , as 3 k . The 3-bit binary weighted capacitors are chosen as 50, 100, and 200 fF, respectively. The tail current of the equalizer is 4.8 mA, and the tail current of the folder active inductor is 0.4 mA. B. Receiver VGA and Offset Cancellation Fig. 7 shows a schematic diagram of the VGA circuit along with the dc-offset cancellation block. Traditional offset cancellow-pass filter at the equalizer output lation often uses a to extract the dc offset and then feeds it back to the equalizer input for cancellation. Such a technique, however, suppresses not only the offset voltage, but also the low-frequency components of the input signal. In order to achieve a low cutoff frequency, high resistance and capacitance are required, resulting in a large chip area or the need for off-chip components. To overcome this problem, a new on-chip offset cancellation technique was developed. The VGA that drives the analog zero equalizer provides 16 gain steps with a targeted gain range from 6 to 9 dB and a gain resolution of approximately 1.0 dB per step. The gain is adjusted by setting the degeneration resistance implemented as a switched resistance array. The offset cancellation is performed and ) by adjusting the at the VGA input nodes (e.g., resistor. The default common current flowing through the . The adjusting mode voltage is currents are binary weighted and are turned on or off through

with . A total of switching transistors adjusting currents are coupled to the node and one adjusting node. In current (the most significant bit) is coupled to the and mV. Such a setting provides the design, an offset adjustment range of 31 mV. The simulated worst case input referred offset voltage of the receiver is approximately 17 mV. Therefore, such a setting is sufficient to cover dc-offset cancellation through process-voltage-temperature (PVT) varianode provide tions. The current sources connected to the of 1, 2, 4, 8, and 16 mV, voltage drops across resistor which can be used to reduce the actual dc input voltage to the node from the default common mode voltage . The curnode provides a voltage drop of rent source coupled to the , then the 32 mV. If that offset voltage causes node needs to be lowered. Combinainput voltage at the to are tried until the offset is compensated. On tions of , the is the other hand, if the offset causes by 32 mV. After is triggered, the turned on to lower node will be higher than the adjusted dc dc voltage at the node. Signals – are then selectively triglevel at the node until the offset is gered to lower the dc level at the compensated. As shown in Fig. 1, when doing offset calibration, the inputs to the analog equalizer are disconnected by opening switches connected to them to isolate the equalizer from external signal sources. The equalizer outputs are coupled to a comparator whose output is monitored by on-chip digital circuits. The offset is compensated if it is detected that the comparator output makes a transition from “1” to “0” or vice versa : combinations. Fig. 8 shows a schematic for a certain view of the comparator circuit. It should be noted that, in designing the comparator, the comparator itself should not introduce a significant amount of offset. Unlike the equalizer that needs to operate at high frequencies, the comparator only needs to operate at a very low frequency. Long channel length transistors with good matching properties were used to make the offset of the comparator negligible. The proposed offset cancellation scheme avoids the use of large capacitance and resistance, thus providing significant area savings. In addition, the use of only the most significant bit adnode reduces the number of adjusting justing current at the current from ten (if the exact 5-bit adjusting current coupled at node are used at the node) to six, which provides the additional area savings.

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Fig. 9. One-tap look-ahead DFE architecture.

Fig. 10. Schematic view of the latch circuit with offset cancellation.

C. Look-Ahead DFE The analog zero equalizer effectively counteracts ISI due to channel high-frequency loss. However. for some legacy backplanes operated at higher data rates where crosstalk is significant, the analog zero equalizer can amplify the undesired high-frequency crosstalk. To address this issue, our complete receive equalization also incorporates one-tap DFE following the analog zero equalizer. The goal is a tradeoff between the low-power and small-area, but crosstalk enhancing liner analog zero equalizer, and the higher power DFE that does not enhance crosstalk. To achieve an area and power-efficient equalization that limits the crosstalk enhancement while compensating for channel loss, we concluded to use a one-tap DFE to remove most of the ISI from the first trailing position and the analog zero linear equalizer to cancel the remaining ISI from the first trailing position and the ISI from earlier symbols. The one-tap DFE can be mathematically expressed as (5) is the decision made at time , is the analog where zero equalizer output at time , and is the DFE coefficient. The one-tap DFE provides a high-frequency gain boost of and if is set to be the first 20 postcursor coefficient, the ISI from the first preceding symbol can be cancelled. The bottleneck of DFE implementation is that the feedback loop delay should be less than half the baud period. To relax the timing requirement and thus enabling the DFE to reliably work at multigigabit/second in CMOS technology, our receiver employs a one-tap look-ahead DFE architecture that reduces critical timing path to a flip-flop and a 2 : 1 MUX. The look-ahead DFE can be mathematically expressed as if if

(6)

In (6), the analog zero equalizer output is compared to both , and the output is selected by a 2 : 1 MUX using . This is equivalent to the DFE function shown in (5). Fig. 9 shows a block diagram of the one-tap look-ahead DFE circuit. In the implementation, the analog zero equalizer was designed to have a maximum output eye opening of 600 mV, the DFE threshold (e.g., ) was generated around the common-mode voltage of the analog zero equalizer with a range of 150 mV and a step

size of 2.4 mV. The DFE latch shown in Fig. 10 consists of a pair of cross-coupled inverters, three central reset pMOS transistors and two pull-down paths to couple in the input signals. In the latch circuit shown in Fig. 10, DP and DN represent the data input, and RP and RN represent the reference (e.g., DFE threshold) input. The offset of the DFE latch was trimmed by adjusting the tail current of the latch. The offset correction is performed at both the data and the reference signal paths. The input referred offset of the latch is trimmed to be less than 1.6 mV. III. MEASUREMENT RESULTS The analog zero equalizer using programmable analog zeros and a folded active inductor was fabricated using a 1.0-V 90-nm CMOS technology. The equalizer was designed for a 4.25-Gb/s SerDes device. The receiver is terminated at 50 and the incoming signal is coupled to the receiver via on-chip capacitors. A 1.7-nH T-coil inductor is added at the receiver input to compensate for the large capacitance (approximately 300 fF) of the electrostatic-discharge (ESD) device and to provide additional peaking for compensating for the backplane channel loss. The T-coil occupies an area of approximately 90 m 90 m. The core of the analog zero stage occupies 0.015 mm of area and dissipates 12 mW of power. At 4.25-Gb/s data rate, simulation shows that one zero stage ) is sufficient to compensate for loss over a wide range ( response of FR4 trace length. Fig. 11 shows the measured of a 34-in FR4 backplane used in the results presented below. As can be seen from this figure, at the baud-rate frequency (e.g., 2.125 GHz), the backplane has a loss of approximately 10 dB. Fig. 12 shows post-layout simulation results of the gain verse frequency characteristics of the analog zero stage. In the nominal PVT, the low-frequency gain is approximately 1.4 dB and the gain boost at the baud-rate frequency is approximately 7.8 dB, which is slightly less than 10 dB due to a slight underestimate of the routing capacitance at the equalizer output nodes. Since the transconductance of the transistors is dependent on PVT variations, the frequency response in Fig. 12 can vary. In the worst case, the boost at the baud-rate frequency can vary approximately 16%. The curves shown in Fig. 12 were obtained at the nominal PVT corner (e.g., 1.0-V power supply, 35 C temperature, and nominal nMOS and pMOS transistors).

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Fig. 11. Measured S response of a 34-in FR4 backplane trace and the equalized frequency response (e.g., frequency response including the FR4 backplane trace and the analog zero equalizer).

Fig. 13. Normalized equalized and unequalized impulse response.

Fig. 14. Micrograph of the 4.25-Gb/s SerDes device and testing board.

Fig. 12. Gain versus frequency response of the analog zero equalizer. There are a total of eight gain settings.

To meet the stringent production requirement, the analog zero, however, was designed to achieve the maximum peaking at the baud-rate frequency at the worst PVT corner (e.g., 0.95-V supply, 125 C temperature, and slow nMOS and pMOS transistors). In the design, three control bits ( , , and ) are used to program the capacitor shown in Fig. 3, thus there are a total of eight settings for boosting the high-frequency gain. It should be noted that, for higher data rate applications, one can increase the number of binary weighted capacitors and/or the capacitance values to achieve an even larger dynamic range. One can also combine the binary weighted switched capacitors with a varactor to allow the equalizer to have both fine and coarse dynamic-range tunings. Post-layout simulation also indicates that folded active inductor load achieves 27% more bandwidth compared to that of resistor load. The bandwidth extension of the folded active inductor is smaller than that of spiral inductors (practically spiral inductors can achieve approximately 40% gain extension). This is mainly due to the intrinsic capacitance at the drain node (such as the drain to bulk capacitance) of the nMOS transistor in the folded active

Fig. 15. Scoped eye diagrams with 4.25-Gb/s PRBS-31 input pattern. (a) Unequalized. (b) Equalized.

inductor. Such a drain capacitance is directly connected at the equalizer output node and, thus, degrades the bandwidth extension effect. In Fig. 11, we also plot the simulated ac response of the complete system including the FR4 backplane channel and the receiver front-end equalization (excluding the DFE). Effects such as the FR4 channel loss, the packagingand ESD-induced bandwidth reduction, the peaking effect of the T-coil inductor, and the high-frequency gain boost of the analog zero including the VGA are included. The VGA is programmed at approximately 1.0-dB gain setting. As can be seen from this figure, the overall system frequency response is almost flat at frequencies below the baud-rate frequency. The simulated equalized and unequalized system impulse responses are plotted in Fig. 13. A micrograph of the 4.25-Gb/s SerDes device and its testing board is shown Fig. 14. Fig. 15 shows the measured eye diagrams at the analog zero equalizer input

CHEN et al.: ELECTRICAL BACKPLANE EQUALIZATION USING PROGRAMMABLE ANALOG ZEROS AND FOLDED ACTIVE INDUCTORS

Fig. 16. BER versus added dc offset at 4.25-Gb/s data rate.

and output nodes. The eye diagrams were obtained using an on-chip eye-monitoring device. The eye diagram was scoped on-chip with 32 horizontal and 128 vertical steps over 1-UI. The on-chip eye-opening monitor not only reduces the requirement for packaging pins, but also removes the use of 50- testing buffers, which could burn a significant amount of power. The eye diagram was monitored with a pseudorandom bit sequence (PRBS)-31 data pattern at a 4.25-Gb/s data rate and with the response was shown in 34-in FR4 backplane trace whose Fig. 11. As shown in the eye-diagram plot in Fig. 15(a), the loss of the FR4 backplane trace (as well as the losses due to the transmitter output capacitance and the transmit-side packaging parasitic capacitance) results in an eye diagram with no discernible opening at the receiver front-end when transmitter pre-emphasis equalization is not used. Fig. 15(b) shows the eye diagram after the analog zero equalization. The eye diagram is obtained without the use of any transmitter equalization. As can be seen from Fig. 15(b), the analog zero equalizer itself effectively opened the received eye diagram, achieving a vertical eye opening of greater than 97 mV and a horizontal eye opening larger than 0.50 UI. Such an eye opening results in error-free transmissions of PRBS data over 34-in lossy FR4 backplanes with two board connectors. To test the composite equalization with both the analog zero stage and the one-tap look-ahead DFE, a voltage margin test was performed for a range of positive and negative DFE latch threshold voltages. Since the dc offset of the DFE latches is calibrated, the voltage margin is accurate within 1.6 mV. The margin information takes into account the receiver’s equalization, receiver noise, and high-frequency crosstalk from adjacent backplane channels. is As shown in Fig. 16, the voltage margin at a BER of 10 approximately 84 mV for the tested device at a 4.25-Gb/s data rate. IV. CONCLUSIONS This paper has presented a backplane equalization circuit that combines programmable high-pass analog zeros with one-tap look-ahead DFE. The main advantages of the analog zero equalization are lower power consumption, less silicon area, and most

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importantly, its capability of performing equalization at higher data rates. The proposed programmable analog zero can easily achieve a large dynamic range due to the wide control range of the programmable capacitor implemented as a binary weighted capacitor array. The folded active inductor extends the equalizer bandwidth allowing the equalizer to be used at a multigigabit/second data rate. With the folded active inductor shunt peaking technique, the equalizer circuit can be operated with a 1.0-V supply without having headroom problems. The equalizer circuit was implemented in a 1.0-V 90-nm CMOS technology. With one zero stage, the equalizer occupies 0.015-mm chip area and dissipates 12 mW of power. At a 4.25-Gb/s data rate, the analog zero provides 7.8-dB gain boost at the baud-rate frequency. Without the use of any transmitter equalization, the analog zero opened the received eye diagram, which was almost closed. At a BER of 10 , the analog zero with the one-tap look-ahead DFE achieves 84 mV of voltage margin over 34-in lossy FR4 backplane traces. ACKNOWLEDGMENT The authors would like to thank C. Guo, F. Yang, K. Kshonze, J. Anidjar, B. Kapuschinsky, G. Zhang, M. Mobin, and G. Sheets, all with the LSI Corporation, Allentown, PA, for helpful discussions. The authors also would like to thank D. Ryan, LSI Corporation, for performing the chip layout. REFERENCES [1] C. Pelard et al., “Realization of multi-gigabit channel equalization and crosstalk cancellation integrated circuits,” IEEE J. Solid-State Circuits, vol. 39, no. 10, pp. 1659–16689, Oct. 2004. [2] J. H. Sinsky, A. Adamiecki, and M. Duelk, “10-Gb/s electrical backplane transmission using duobinary signaling,” in IEEE MTT-S Int. Microw. Symp. Dig., Apr. 2002, pp. 109–112. [3] J. L. Zerbe et al., “Equalization and clock recovery for a 2.5–10 Gb/s 2-PAM/4-PAM backplane transceiver cell,” IEEE J. Solid-State Circuits, vol. 38, no. 12, pp. 2121–2130, Dec. 2003. [4] J. T. Stonick, G. Y. Weo, J. L. Sonntag, and D. K. Weinlader, “An adaptive PAM-4 5-Gb/s backplane transceiver in 0.25-m CMOS,” IEEE J. Solid-State Circuits, vol. 38, no. 3, pp. 436–443, Mar. 2003. [5] J. S. Choi, M. S. Hwang, and D. K. Jeong, “A 0.18-m CMOS 3.5-Gb/s continuous-time adaptive cable equalizer using enhanced low-frequency gain control method,” IEEE J. Solid-State Circuits, vol. 39, no. 3, pp. 419–425, Mar. 2004. [6] Y. Tomita, M. Kibune, J. Ogawa, W. W. Walker, H. Tamura, and T. Kuroda, “A 10-Gb/s receiver with series equalizer and on-chip ISI monitor in 0.11-m CMOS,” IEEE J. Solid-State Circuits, vol. 40, no. 4, pp. 986–993, Apr. 2005. [7] S. Gondi, J. Lee, D. Takeuchi, and B. Razavi, “A 10-Gb/s CMOS adaptive equalizer for backplane applications,” in IEEE Int. Solid-State Circuits Conf., Feb. 2005, pp. 328–329. [8] J. E. Jaussi, G. Balamurugan, D. R. Johnson, B. Casper, A. Martin, J. Kennedy, N. Shanbhag, and R. Mooney, “An 8-Gb/s source-synchronous I/O link with adaptive receiver equalization, offset cancellation and clock de-skew,” IEEE J. Solid-State Circuits, vol. 40, no. 1, pp. 80–88, Jan. 2005. [9] G. E. Zhang and M. M. Gren, “A 10 Gb/s BICMOS adaptive cable equalizer,” IEEE J. Solid-State Circuits, vol. 40, no. 11, pp. 2132–2140, Nov. 2005. [10] J. H. Chen, G. Sheets, C. Guo, F. Saibi, F. Yang, K. Azadet, J. Lin, and G. Zhang, “Electrical backplane equalization using programmable analog zeros and folded active inductors,” in IEEE 48th Midwest Circuits Syst. Symp., Aug. 7–10, 2005, vol. 2, pp. 1366–1369. [11] K. Krishna, D. A. Yokoyama-Martin, A. Caffee, C. Jones, M. Loikkanen, J. Parker, R. Segelken, J. L. Sonntag, J. Stonick, S. Titus, D. Weinlader, and S. Wolfer, “A multigigabit backplane transceiver core in 0.13 m CMOS with a power-efficient equalization architecture,” IEEE J. Solid-State Circuits, vol. 40, no. 12, pp. 2658–2666, Dec. 2005.

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[12] T. Beukema, M. Sorna, K. Selander, S. Zier, B. L. Ji, P. Murfet, J. Mason, W. Rhee, H. Ainspan, B. Parker, and M. Beakes, “A 6.4-Gb/s CMOS Serdes core with feed-forward ad decision-feedback equalization,” IEEE J. Solid-State Circuits, vol. 40, no. 12, pp. 2633–2645, Dec. 2005. [13] E. Säckinger and W. C. Fischer, “A 3-GHz 32-dB CMOS limiting amplifier for SONET OC-48 receivers,” IEEE J. Solid-State Circuits, vol. 35, no. 12, pp. 1884–1888, Dec. 2000. [14] W. Z. Chen and C. H. Lu, “A 2.5 Gbps CMOS optical receiver analog front-end,” in IEEE Custom Integrated Circuits Conf., May 2002, pp. 359–362. [15] N. Krishnapura, M. Barazande-Pour, Q. Chaudhry, J. Khoury, K. Lakshmikumar, and A. Aggarwal, “A 5-Gb/s NRZ transceiver with adaptive equalization for backplane transmission,” in IEEE Int. Solid-State Circuit Conf., Feb. 2005, pp. 60–61. [16] C. H. Wu, J. W. Liao, and S. I. Liu, “A 1 V 4.2 mW fully integrated 2.5 Gb/s CMOS limiting amplifier using folded active inductors,” in IEEE Int. Circuits Syst. Symp., May 2004, pp. I1044–I1047.

Jinghong Chen (M’06) received the B.S. and M.S. degrees in engineering physics from Tsinghua University, Tsinghua, China, in 1992 and 1994, respectively, the Master of Engineering degree in electrical engineering from the University of Virginia, Charlottesville, in 1997, and the Ph.D. degree in electrical engineering from the University of Illinois at Urbana-Champaign, in 2000. In January 2001, he joined the High-Speed Communication Very Large Scale Integration (VLSI) Research Department, Bell Laboratories, Lucent Technologies, as a Member of Technical Staff. In June 2002, he joined Agere Systems, a spin-off of Lucent Technologies, and was involved with CMOS circuit design for optical and wireless communications. Since October 2006, he has been with Analog Devices Inc., Somerset, NJ, where he is involved with XM satellite radio receivers. His current research involves designing mixed-signal and RF circuits for communication and signal processing systems.

Fadi Saibi received the Ingénieur degree from Ecole Polytechnique, Paris, France, in 1999, and the Ph.D. in electronics and communications from the Ecole Nationale Supérieure des Télécommunications (ENST) (Télécom Paris), Paris, France, in 2005. In 2000, he was a Research Trainee with the High Speed Communications Very Large Scale Integration (VLSI) Research Group, Bell Laboratories, Lucent Technologies, Holmdel, NJ. In 2001, he joined Agere Systems, a spin-off from Lucent Technologies. In April 2007, the LSI Corporation, Allentown, PA, completed its merger with Agere Systems. He has been involved with signal processing for optical and copper wireline communications, multigigabit backplane equalization, and VLSI integration.

Jenshan Lin (S’91–M’94–SM’00) received the B.S. degree from National Chiao Tung University, Hsinchu, Taiwan, R.O.C., in 1987, and the M.S. and

Ph.D. degrees in electrical engineering from the University of California at Los Angeles (UCLA), in 1991 and 1994, respectively. In 1994, he joined AT&T Bell Laboratories (now Bell Laboratories, Lucent Technologies), Murray Hill, NJ, as a Member of Technical Staff, and became the Technical Manager of RF and High Speed Circuit Design Research in 2000. Since joining Bell Laboratories, he has been involved with RF integrated circuits using various technologies for wireless communications. In September 2001, he joined Agere Systems, a spin-off from Lucent Technologies, where he was involved with high-speed CMOS circuit design for optical and backplane communications. In July 2003, he joined the Department of Electrical and Computer Engineering, University of Florida, Gainesville, as an Associate Professor. He has authored or coauthored over 100 technical publications in referred journals and conferences proceedings. He holds five patents. His current research interests include RF system-on-chip integration, high-speed broadband circuits, high-efficiency transmitters, wireless sensors, biomedical applications of microwave and millimeter-wave technologies, and software-configurable radios. Dr. Lin has been active in the IEEE Microwave Theory and Techniques Society (IEEE MTT-S). He is an elected Administrative Committee (AdCom) member serving the term of 2006–2008, and a member of the Wireless Technology Technical Committee. He has served on several conference Steering Committees and Technical Program Committees including the IEEE MTT-S International Microwave Symposium (IMS), Radio Frequency Integrated Circuits (RFIC) Symposium, Radio and Wireless Symposium (RWS), and Wireless and Microwave Technology Conference (WAMICON). He is currently the Technical Program cochair of the 2006 and 2007 RFIC Symposium, and the finance chair of 2007 RWS. He was the recipient of the 1994 UCLA Outstanding Ph.D. Award and the 1997 Eta Kappa Nu Outstanding Young Electrical Engineer Honorable Mention Award. He has been the coauthor/advisor of several IEEE IMS Best Student Paper Awards and advisor of an IEEE MTT-S Undergraduate/Pre-Graduate Scholarship Award.

Kamran Azadet (S’92–A’95) received the Engineering degree from the Ecole Centrale de Lyon, Lyon, France, in 1990, and the Ph.D. degree from Ecole Nationale Supérieure des Télécommunications (ENST), Paris, France, in 1994. From 1990 to 1994, he was a Research Engineer with Matra MHS, Saint Quentin en Yvelines, France, where he was involved in the design of video filters for acquisition systems. In 1994, he joined Bell Laboratories, Holmdel, NJ, where he was involved in the area of color digital CMOS cameras and high-speed transceivers. In 2001, he joined Agere Systems, a spin-off from Lucent Technologies. In April 2007, the LSI Corporation, Allentown, PA, completed its merger with Agere Systems. He was a member of the IEEE 802.3ab Gigabit Ethernet 1000BASE-T standard, and the IEEE Ethernet High-Speed Study Group 10 Gigabit Ethernet. Since 1999, he has led research and development in the area of wireline communication—copper and optical–for telecom and datacom applications. He is currently Director of PHY architecture with the Ethernet Division, LSI Corporation. Dr. Azadet serves on the committees of the IEEE Symposium on Very Large Scale Integration (VLSI) circuits and the VLSI–Technology, System, and Application (TSA) Conference, Taiwan, R.O.C. He has been an associate editor of the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS. He serves as a reviewer for journals such as the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS and the IEEE JOURNAL OF SOLID-STATE CIRCUITS. He was corecipient of the 1998 IEEE JOURNAL OF SOLID-STATE CIRCUITS Best Paper Award for his paper on a color digital CMOS camera.

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Monolithic Integration of a Folded Dipole Antenna With a 24-GHz Receiver in SiGe HBT Technology Erik Öjefors, Member, IEEE, Ertugrul Sönmez, Sébastien Chartier, Student Member, IEEE, Peter Lindberg, Christoph Schick, Member, IEEE, Anders Rydberg, Member, IEEE, and Hermann Schumacher, Member, IEEE

Abstract—The integration of an on-chip folded dipole antenna with a monolithic 24-GHz receiver manufactured in a 0.8- m SiGe HBT process is presented. A high-resistivity silicon substrate (1000 cm) is used for the implemented circuit to improve the efficiency of the integrated antenna. Crosstalk between the antenna and spiral inductors is analyzed and isolation techniques are described. The receiver, including the receive and an optional transmit antenna, requires a chip area of 4.5 mm2 and provides 30-dB conversion gain at 24 GHz with a power consumption of 960 mW.



Index Terms—Dipole antennas, heterojunction bipolar transistors (HBTs), monolithic microwave integrated circuit (MMIC) receivers.

I. INTRODUCTION

T

HE development of monolithic -band RF frontends in Si/SiGe processes, such as the fully integrated receivers reported in [1] and [2], enables the design of low-cost shortrange radar and communication devices for the 24-GHz industrial–scientific–medical (ISM) band. Since all high-frequency components of the receiver, including the low-noise amplifier (LNA), local oscillator (LO), and downconversion mixer, reside on the same chip, no high-frequency interconnects between the building blocks of the receiver are needed. However, the package still has to provide a low-loss high-frequency interconnect to the off-chip antenna. This last off-chip RF interconnect could be removed if the antenna is integrated on the same chip as the rest of the frontend. On-chip antennas have traditionally been considered for III–V monolithic microwave integrated circuit (MMIC) processes where the high-resistivity substrate with a backside ground-plane metallization can be used for microstrip circuits and patch antennas. In the case of Si/SiGe high-frequency

Manuscript received September 29, 2006; revised February 21, 2007. This work was supported in part by the European Commission under the Information Society Technologies ARTEMIS Project. E. Öjefors and A. Rydberg are with the Department of Engineering Sciences, Uppsala University, SE-751 21 Uppsala, Sweden (e-mail: erik.ojefors@ieee. org). E. Sönmez is with microGaN GmbH, 89081 Ulm, Germany. S. Chartier and H. Schumacher are with the Department of Electron Devices and Circuits, University of Ulm, 89081 Ulm, Germany. P. Lindberg is with Laird Technologies AB, SE-184 25 Åkersberga, Sweden. C. Schick is with the Mixed Signal Integrated Circuit Design Group, Ubidyne GmbH, 89081 Ulm, Germany. 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.2007.900315

circuits, lumped passive elements are commonly used instead of transmission line components due to the absence of backside metallization, and high losses in the silicon substrate. The use of lumped passive elements also yields comparatively compact circuit layouts (less than 1.5 mm for the receiver reported in [1]), which, in turn, reduces the cost of the manufactured chips. As a small die size precludes the integration of -band high-directivity antennas, the main requirements for an on-chip antenna are low area consumption and high radiation efficiency in the presence of the silicon substrate. The antenna feed type (balanced or single ended) should preferably match the integrated circuit (IC) topology to avoid the need for baluns. The half-wave dipole antenna offers a balanced feed, and is thus suitable for integration with differentially designed integrated receivers and transmitters. A 10-GHz on-chip dipole connected to a voltage-controlled oscillator (VCO) has been demonstrated in SiGe HBT technology [3], but poor radiation efficiency (10%) was reported due to high substrate losses. Micromachined meander dipole antennas [4] have been suggested as a way of integrating compact high-efficiency antennas on the low-resistivity wafers commonly used in commercial Si/SiGe bipolar and BiCMOS processes. Micromachining does, however, require additional post processing of the manufactured wafer. High-resistivity silicon has been proposed as a substrate for millimeter-wave circuits and it has been shown [5] that low-loss microstrip transmission lines can be implemented on such substrates. Monolithic microstrip antenna arrays have been demonstrated at 94 GHz [6], thus validating the use of high-resistivity silicon in antenna applications. The efficiency of monolithic integrated 77-GHz half-wave dipoles manufactured in a commercial active device silicon process has been improved by the use of an undoped silicon substrate placed adjacent to the low-resistivity active device chip [7]. In this case, the high-resistivity silicon was used to guide the waves to an external lens antenna. As an alternative, high-resistivity silicon wafers can be used directly as substrate material in some commercial Si/SiGe foundry processes to enhance the efficiency of an on-chip antenna. In a previous study [8], monolithic integration of simple shorted dipole with a 24-GHz receiver [1] manufactured on a high-resistivity silicon wafer has been demonstrated. However, the implemented antenna was not impedance matched to the integrated receiver and did not utilize the available on-chip space efficiently. In this paper, we extend the work reported in [8] and provide the results from a full characterization of the 24-GHz receiver where the simple antenna has been replaced with a compact folded dipole, and matched to the input impedance of the

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receiver. Section II presents the characteristics of the selected SiGe HBT process. Section III describes the design of the integrated antenna. Section IV outlines the receiver design. Section V provides details about the utilized on-chip isolation techniques. Section VI then provides the characterization setup and results for the integrated antenna receiver are presented. II. SiGe HBT TECHNOLOGY The 24-GHz receiver and the on-chip antenna were designed for the Atmel SiGe2RF [9] process. It is a 0.8- m Si/SiGe HBT technology with a minimum effective emitter width of 0.5 m. Metal–insulator–metal (MIM) capacitors, four resistor types, and three metallization layers are available. A selectively implanted collector technique increases the transit frequency from 50 to 80 GHz at the cost of a reduced breakdown voltage (2.4 V instead of 4.3 V). The maximum oscillation frequency is 80 GHz. cm) are available High-resistivity silicon substrates (1000 as a process option in addition to the standard (20 cm) low-resistivity wafers, thus facilitating the design of on-chip antennas with reduced substrate losses. In this study, 500- m-thick silicon wafers were used. The surface of the processed silicon substrate is a doped p -type to prevent an unwanted inversion layer formation. Normally, the p layer is connected to circuit ground. The locations where this doping should be omitted are marked by the so-called “channel-stopper” layer, which is a negative mask. This mask can be applied for sensitive passive devices such as spiral inductors and antennas in order to reduce the substrate losses. III. DIPOLE ANTENNA DESIGN AND SIMULATION Integrated antennas offer the possibility of deviation from the standard 50- system impedance in order to allow direct connection of the antenna to the first stage circuits without the need for a matching network. The choice of a suitable antenna impedance for the receiver is governed by the optimum reflecfor minimum noise of the integrated LNA. tion coefficient Noise figure measurements of a standalone version of the receiver LNA suggest a differential optimum antenna impedance . of In previous integration work using this receiver [8], a simple dipole was used, which, due to space constraints, had to be top loaded with lengthening sections perpendicular to the main dipole. However, presence of dielectric materials and conductive bodies close to the dipole radiator tends to lower the radiation resistance, thus causing a significant drop of the antenna impedance at resonance compared to the theoretical free-space value. Shortening the radiator by folding back the ends, as often required due to chip size restrictions, will further decrease the radiation resistance. Printed folded dipole antennas offer an input impedance, which can be adjusted over a wide range using the design formulas derived by Lampe [10]. The input impedance for a dipole of length is given as (1)

Fig. 1. Chip photograph of the designed folded dipole test structure.

where is the impedance of the equivalent dipole, is the is the impedance of the transmission line mode, and ratio of the currents on the driven and parasitic branches of the is obtained radiator. The transmission line mode impedance as the impedance of a shorted stub (formed by half the dipole) (2) is the characteristic impedance and is the propawhere gation constant of the coplanar strip line (CPS), formed by the driven and parasitic radiator. of the two For the special case of equal conductor width , the folded dipole branches and narrow conductor spacing of four at resonance yields a impedance step up ratio goes to infinity) compared to an ordinary half-wave (where dipole with the same printed conductor dimensions. By folding the on-chip dipole, the antenna impedance can thus be increased of the LNA. to a value closer to In Fig. 1, a chip photograph of the designed on chip antenna is shown. The antenna was simulated and optimized using Agilent Technologies’ High Frequency Structure Simulator (HFSS) [11] full 3-D EM simulation package to account for the reduced effective dielectric constant of the finite dielectric substrate created by the dicing of the wafer into individual chips. was limited to 2100 m The length of the straight section as a compromise between chip size and antenna radiation resistance, thus requiring an additional lengthening section m to be added to each of the radiator ends in order to obtain resonance at 24 GHz. These lengthening sections were added along the edge of the chip, perpendicular to the main radiating section of the dipole, in order to minimize current canceling effects in the far field. The sensitivity of the antenna impedance was anand efficiency with respect to the conductor width alyzed by HFSS simulations and found not to be critical. The m was selected to keep the resistive losses width of the aluminum metallization below the dominant substrate losses, while at the same time minimizing the area requirement m, equal to the confor the antenna. A slot width ductor width, was used between the conductors in the radiator. , defined as the distance between The ground clearance the radiator metallization and grounded layers such as the p channel stopper or circuit metallization, is an important design parameter as large conductive areas in the vicinity of the antenna will decrease the radiation resistance of the antenna due to induced currents. Based on simulation results, the p channel stopper doping of the substrate was removed within a distance

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Fig. 3. Block diagram of the integrated receiver, as presented in [8]. The optional transmitter antenna is coupled to the LO through a switchable buffer.

Fig. 2. Simulated antenna impedance of simple (dashed line) and folded dipole (solid line) of the same dimensions with a full 2300 2300 m chip size. Frequency range: 18–26.5 GHz; 1-GHz marker spacing. LNA differential antenna impedance (circles), LNA optimum input reflection coefficient (crosses).

2

m of the dipole in order to reduce the losses caused by this low resistive layer. In the simulation, a substrate thickness of 500 m and a chip size of 2300 2300 m was used, which corresponds to a distance of 100 m between the antenna metallization and the edge of the diced substrate. The antenna is operated above its resonant frequency, as shown by the simulated trace in Fig. 2, to provide an inductive impedance close to a conjugate match of . As referthe LNA at 24 GHz, which also coincides with ence, the simulated impedance for a simple nonfolded dipole of is also shown in this figure. equal dimensions , , and The simulated gain is 1.8 dBi. IV. RECEIVER DESIGN A. Circuit Topology and Passive Elements In contrast to the backside ground-plane metallization available in III–V MMIC technologies, no ground plane is usually present in Si/SiGe-based technologies, which, in conjunction with high substrate losses, makes a microstrip design approach difficult. A coplanar design has been demonstrated at this frequency with good performance [12]. However, a topology based on distributed elements leads to a drastic increase of size compared to a lumped passive element implementation. Therefore, a design based on lumped elements was chosen in this study. To improve the accuracy of the circuit level simulation, the -parameters of the inductors were measured on-wafer on separate test structures with the effect of bond pads removed by the use of on-chip deembedding structures. Due to the lack of a common high-frequency circuit ground plane, it is difficult to define a stable circuit ground. A fully differential design topology has, therefore, been used, which avoids the problem by using 180 phase-shifted signals to provide local virtual ground points at the symmetry axis of the differential circuit. Other advantages of the differential topology

Fig. 4. Simplified schematic circuit diagram of the receiver antenna interface and the first differential cascode amplifier in the three-stage LNA.

are rejection of common-mode noise from supply lines and substrate, increased voltage swing, and suppression of even-order harmonics. B. 24-GHz Receiver Architecture The self contained 24-GHz receiver consists of an LNA, a voltage-controlled LO with an output buffer and a quadrature downconversion mixer, as depicted in Fig. 3. The receiver can either be operated at zero IF or as an image reject receiver, but the IF poly-phase filter for a image rejection was not integrated to avoid increasing the chip size unnecessarily. A separately powered 16 : 1 static frequency divider [13] is connected to the output of the VCO to facilitate phase locking of the LO by an external low-frequency phase-locked loop (PLL). At the LO output, a switchable buffer amplifier is also connected in order to allow the chip to be used as a simple frequency modulated transmitter. The LNA is designed with three stages, each consisting of four transistors in a differential cascode configuration, as shown m emitter size are used in Fig. 4. Transistors with 0.8 m in the cascode, requiring a total dc power of 340 mW for the three stages in the differential configuration. LC networks are used for the inter-stage matching, and the 0.13-nH large spiral provide emitter degeneration. For a single-ended inductors standalone version of the LNA, a total gain of 20 dB has been ob. A mintained with better than 60-dB reverse isolation imum noise figure of 5.8 dB has been measured, however, 6.6 dB

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Fig. 5. Schematic circuit diagram of the 24-GHz VCO with integrated buffer amplifier.

Fig. 7. Schematic of the LC/RC filter network for decoupling of the supply line between different receiver blocks and stages [14].

V

V. ISOLATION TECHNIQUES Successful single chip integration of a complete receiver with an antenna requires that good isolation can be obtained between stages of the different functional blocks, as well as between the circuit and antenna. Poor isolation can lead to instability and self-oscillation of amplifier stages, noise pickup, increased phase noise of the LO, and saturation or dc-offset problems if the LO signal is picked up by the on-chip antenna. A. Circuit Isolation Fig. 6. IQ-downconversion mixer with LC-emitter load and polyphase network. The emitter follower IF output buffers are not included.

is expected at the bias point used in the receiver. For the differential version of the LNA, the noise optimum source impedance is . A differential common base design is used for the VCO, nH, are shown in Fig. 5, where the base inductors used to make the transistors in the oscillator core unstable. Feedback is provided by 0.42-nH large spiral inductors connected to the collectors and the pF output coupling capacitors at the emitters. An integrated cascode buffer is used to amplify the output signal. The oscillator–buffer combination has been tested in a standalone configuration where it yields an output power of 7.5 dBm at a center frequency of 23.9 GHz with a dc power requirement is 200 mW. By changing the bias point of the transistors in the core through the control voltage, the VCO can be tuned over a 2.4-GHz range. The estimated phase noise is 80 dBc at 1-MHz offset. The quadrature downconversion mixer (Fig. 6) consists of two Gilbert switching cores provided with in-phase (I)and quadrature (Q)-channel LO signals from a conventional single-pole RC polyphase network. An input RF transconductance stage is shared by the I and Q mixer cores. A parallel LC circuit between the emitters of the stage and ground is used instead of a conventional transistor current mirror to improve the common mode signal rejection of the stage. Emitter followers are used as buffer amplifiers at the output of the mixers. The buffers together with an RC low-pass filter are designed to limit the I/Q output signal bandwidth to 470 MHz. The conversion gain of the mixer is designed to be 13 dB and the dc power consumption is 300 mW at a supply voltage level of 4 V.

Due to the high integration level, spiral coils are placed in close vicinity. The coupling between these elements has been studied in [14], but due to the weak influence of the evaluated coupling properties on the circuit performance, it can be neglected and the modeling of this effect was not performed. A reduction of the coupling between inductors was obtained by placing a grounded metal ring around these elements, which drastically reduces the crosstalk for a given spacing between these components [15]. The supply voltage lines of the circuit are filtered both at block (such as LNA or VCO) and stage levels, as depicted in Fig. 7. Different stages within a block, such as the core and the buffer connected to in the VCO, use RC filters with the capacitor conground as close as possible to the stage and a resistor rail of the block. nected to the LC low-pass filters are used to isolate the supply of different rail. The inductors are large blocks sharing the same on-chip inductors with an inductance of 1 nH. The block decouare obtained by connecting several capacpling capacitors itors in parallel, yielding a total capacitance in the decoupling network of approximately 30 pF. In order to estimate the isolation provided between the supply node of a single stage, a test voltage line of a block and the structure combining both LC and RC dc-filtering network techniques has been separately manufactured and evaluated [14]. The provided isolation between the two nodes is better than 50 dB between 13–30 GHz. B. Antenna to Circuit Crosstalk Crosstalk with the on-chip antenna will primarily appear as common mode components on the differential signal lines of the due to the compact and symmetrical layout of the stages.

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Fig. 8. Crosstalk simulation setup including LNA tank inductor, optional circuit ground shields, and on-chip antenna.

Fig. 10. Manufactured receiver chip with integrated receive antenna (left) and auxiliary transmit antenna (right).

VI. IMPLEMENTED RECEIVER IC

Fig. 9. Simulated crosstalk (solid line with metallic shields, dashed line without) between the on-chip antenna and spiral inductor in the output of the LNA.

However, differential mode crosstalk can appear on signal lines through spiral inductors unless two inductors with opposite winding directions are used in a balanced configuration. The spiral inductors would also need to be closely spaced to prevent any coupling to the antenna from the differential mode components, which is difficult to achieve in practice due to the comparatively large physical size of the inductors. A critical point for antenna crosstalk is the LNA output matching network. To prevent self-oscillation, larger isolation than the gain of the LNA is needed. The crosstalk between the antenna and a collector load inductor in the output stage of the LNA has been simulated using the method of moments (MoM) package IE3D [16]. An infinite substrate was assumed in the simulation and the complex shaped shielding ground metallization, present in the receiver layout, was replaced by a rectangular shield with an opening for the studied inductor, as shown in Fig. 8. The simulated coupling between inductors in the collector network of the LNA and the antenna is shown in Fig. 9. The use of grounded metallic shields in the vicinity of the inductors reduce the simulated crosstalk from 51 to 72 dB at the antenna resonance frequency.

A chip photograph of the manufactured receiver is shown in Fig. 10. The overall system requires an area of 2100 m , including the receive and auxiliary 2220 transmit antenna. The simulated coupling between the receive and transmit antenna is 6.5 dBi (free-space conditions), which restricts the use of the simple switchable transmitter stage to half duplex communication. The circuit part of the 24-GHz integrated antenna receiver requires only 1480 1150 m of the total area, thus leaving space for integration of other circuit blocks such as a PLL, IF signal processing, or upconversion mixers for a transmit path. The receiver circuit was placed in between the two antennas in a region where the channel stopper layer was not omitted. Both the receive and auxiliary antennas are connected to the circuit using short traces. A separate passive antenna chip, identical to one of the integrated antennas without the receiver circuit, was also manufactured on the same wafer to facilitate characterization. Wafers containing the manufactured receivers and passive antennas were diced before characterization to prevent dielectric loading of the antennas and coupling to nearby circuits on the wafer. VII. RESULTS AND DISCUSSION A. Passive Antenna Characterization The passive dipole antenna test chip has been characterized in an environment equivalent to free space using a modified wafer probe station [17]. For the antenna impedance measurements, the antenna was supported by a low dielectric constant foam material backed by an absorber, thereby preventing impedance shifts due to dielectric loading and reflections from metallic parts. The measured and simulated antenna impedance is shown in Fig. 11 for the separately manufactured passive antenna

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Fig. 11. Measured and simulated antenna impedance: 23–25 GHz. Passive antenna test chip mounted on foam (solid line), HFSS simulation of antenna test chip (dashed line), and HFSS simulation of antenna on receiver chip (dots).

chip, together with the simulated impedance for the full 2300 2300 m large receiver chip. Good agreement between simulated and measured impedance is obtained over the 23–25-GHz frequency range. The de-tuning of the dipole test structure compared the designed antenna for the full size receiver is caused by the smaller die size and, thus, lower effective dielectric constant of the diced substrate. Antenna gain and radiation patterns were measured with the antenna under test (AUT) mounted on a foam dielectric in free space. The RF connection to the antenna was provided by a wafer probe. As the probe positioning equipment blocks part of the relatively omnidirectional radiation pattern, the probe setup had to be covered with absorbers to minimize unwanted reflections. The setup was calibrated for gain measurements by replacing the AUT and wafer probe with a 20-dBi standard gain horn antenna. The measured radiation pattern and gain is shown in Fig. 12 for the passive antenna chip. In the -plane measurement, the probe setup partly shadows the AUT for negative angles. A front-side gain of 2 dBi is obtained in measurements. The low gain compared to the value obtained in the design can be caused by underestimation of the conductance in the p channel stopper doping used for the initial design, as well as decreased substrate resistivity during processing. Simulations of the antenna test structure with a p layer surface resistivity of 250 , also shown in Fig. 12, yield 3-dBi antenna gain in the same direction, which indicates that a larger p layer clearance is needed to fully take advantage of the high-resistivity substrate. The measured 15-dBi -plane cross polarization level is higher than the 45-dBi result obtained in the HFSS simulation. However, the increased cross polarization level can be explained by interference from the wafer probe.

Fig. 12. Measured (co-polarization: solid line; cross polarization: dotted– dashed line) and simulated (co-polarization: dashed line; cross polarization: dots). (a) E - and (b) H -plane radiation pattern and gain of the passive antenna chip.

B. System Characterization For system characterization, the integrated antenna receiver chip was mounted on a metallic carrier, as shown in Fig. 13, to provide a heat sink for the circuit. A thermal compound was used to enhance the heat transfer from the chip to the carrier. To minimize the loading effect of the metal carrier on the integrated antenna, the chip was mounted with the antenna part of the chip sticking out 1 mm from the edge of the carrier. The electrical connections to the chip were provided with wafer probes. Ground–signal–signal–ground (GSSG) probes were used to extract the I- and Q-channel baseband signal, while the dc power was provided with a needle probe. The overall dc consumption is 960 mW at 4-V supply voltage. The LO signal leakage at port of the receiver was characterized in a separate the supply current was supplied through a setup, where the microwave probe and bias-T. Using a spectrum analyzer, an LO leakage power of 51.5 dBm was recorded, thus verifying good line. decoupling of the

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Fig. 13. System noise and conversion gain characterization setup.

Fig. 15. Measured I-channel baseband output spectrum when illuminated with EIRP 5.9-dBm signal at 1.25-m distance. RBW: 1 MHz; analyzer noise floor: 93 dBm.

0

Fig. 14. Measured conversion gain at the I-channel output with illuminated on-chip antenna.

A continuous wave (CW) test signal with a frequency of dB power was generated with a 24.10 GHz and synthesized signal generator and applied to a standard gain horn antenna for illumination of the device-under-test. As dBi at this frequency, an the gain of the horn is effective isotropic radiated power (EIRP) of dB was yielded. The horn antenna was positioned at a distance of 1.25 m from the receiver chip and oriented for maximum received power while the IF output signal was monitored on a spectrum analyzer. The conversion gain characteristics at a constant LO frequency are depicted in Fig. 14. For a fixed LO frequency GHz, a 3-dB bandwidth of 500 MHz and of a maximum conversion gain of 31 dB are obtained. A drop in conversion gain is measured for input signals close to the LO frequency due to the high-pass characteristics of the IF balun used. The receiver input and output 1-dB compression points have not been characterized for the receiver with an integrated antenna; however, for a standalone version, receiver dB and dB has been chip reported [1]. The I/Q channel imbalance was measured at a output frequency of 50 MHz, yielding 1.2-dB amplitude error and a phase

imbalance of 4.3 . DC offsets of 47 and 70 mV have been measured at the I and Q baseband outputs. If the dominant cause of the dc offset is assumed to be self-mixing of the 8-dBm strong LO signal generated on chip, the result indicates better than 46-dB isolation between the LO output signal and antenna. The noise performance of the integrated antenna receiver has been evaluated in the conversion gain setup using the measured power spectrum of the I-channel output, which is shown in Fig. 15. dB is obtained with a A signal-to-noise ratio 1-MHz resolution bandwidth (RBW). The result corresponds to dB/Hz. Cona signal to noise spectral density sidering an equivalent input signal in an isotropic receiver antenna of 56.1 dBm, as calculated above for the setup, and assuming a background noise floor of 173.9 dBm/Hz at an ambient temperature of 21 C, a input signal-to-noise spectral dB/Hz is obtained. A single-sideband density system noise figure for the receiver including the integrated andB. tenna can thus be calculated as If image rejection is obtained through the use of a 90 IF hybrid or processing of the quadrature outputs in a zero-IF application, a double-sideband (DSB) noise figure of 8.7 dB can be obtained. The result is in agreement with the separately measured 6.6-dB noise figure for the LNA and 2-dBi gain for the antenna. Although the measured system noise figure is affected by the measurement setup, as reflections from the heat sink can increase the gain of the antenna and, thus, lower the apparent noise figure in the measurement, the result indicates low pickup of substrate noise by the antenna. VIII. CONCLUSION Monolithic integration of a compact folded dipole antenna on the same chip as a 24-GHz receiver manufactured using highresistivity silicon wafers in a commercial SiGe HBT process has been demonstrated. The implemented radiator requires a 2.1 0.9 mm chip area and provides 2-dBi gain with an impedance level suitable for direct connection of the receiver LNA.

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The system performance of the integrated receiver has been evaluated with a maximum conversion gain of 31 dB and a system noise figure of 8.8 dB obtained for the full system including the on-chip antenna. The noise performance of the receiver is in agreement with the theoretical value calculated from the measured antenna gain and LNA noise figure, thus showing that the performance is not significantly degraded by substrate noise. Isolation techniques, consisting of ground shields around the spiral inductors and RC/LC filtered dc supply lines, have been used to reduce the amount crosstalk. No instability of the receiver is observed despite the close integration of the antenna and the high-gain LNA. The low dc offsets measured at the I/Q signal output indicates that sufficient isolation between the LO and antenna can be obtained on-chip.

ACKNOWLEDGMENT The authors would like to thank Dr. W. Schwerzel, Atmel GmbH, Heilbronn, Germany, for arranging space on wafer runs.

REFERENCES [1] E. Sönmez, S. Chartier, C. Schick, and H. Schumacher, “Fully integrated differential 24 GHz receiver using a 0.8 m SiGe HBT technology,” in Proc. 35th Eur. Microw. Conf., Paris, France, Oct. 2005, pp. 745–748. [2] A. Ghazinour, P. Wennekers, J. Schmidt, Y. Yin, R. Reuter, and J. Teplik, “A fully-monolithic SiGe-BiCMOS transceiver chip for 24 GHz applications,” in Proc. Bipolar/BiCMOS Circuits Technol. Meeting, Toulouse, France, Sep. 2003, pp. 181–184. [3] F. Touati and M. Pons, “On-chip integration of dipole antenna and VCO using standard BiCMOS technology for 10 GHz applications,” in Proc. 29th Eur. Solid-State Circuits Conf., Estoril, Portugal, Sep. 2003, pp. 493–496. [4] E. Öjefors, F. Bouchriha, K. Grenier, A. Rydberg, and R. Plana, “Compact micromachined dipole antenna for 24 GHz differential SiGe integrated circuits,” in Proc. 34th Eur. Microw. Conf., Amsterdam, The Netherlands, Oct. 2004, pp. 1081–1084. [5] A. Rosen, M. Caulton, P. Stabile, A. Gombar, W. Janton, C. P. Wu, J. Corboy, and C. W. Magee, “Silicon as a millimeter-wave monolithically integrated substrate,” RCA Rev., vol. 42, pp. 633–660, Dec. 1981. [6] J. Büchler, E. Kasper, P. Russer, and K. Strohm, “Silicon high-resistivity substrate millimeter-wave technology,” IEEE Trans. Microw. Theory Tech., vol. MTT-34, no. 12, pp. 1516–1521, Dec. 1986. [7] A. Babakhani, X. Guan, A. Komijani, A. Natarajan, and A. Hajimiri, “A 77 GHz 4-element phased array receiver with on-chip dipole antennas in silicon,” in IEEE Int. Solid-State Circuits Conf. Tech. Dig., Feb. 2006, pp. 180–181. [8] E. Öjefors, E. Sönmez, S. Chartier, P. Lindberg, A. Rydberg, and H. Schumacher, “Monolithic integration of an antenna with a 24 GHz image-rejection receiver in SiGe HBT technology,” in Proc. 35th Eur. Microw. Conf., Paris, France, 2005, pp. 677–680. [9] A. Schuppen, J. Berntgen, P. Maier, M. Tortschanoff, W. Kraus, and M. Averweg, “An 80 GHz SiGe production technology,” III-Vs Rev., vol. 14, no. 6, pp. 42–46, 2001. [10] R. Lampe, “Design formulas for an asymmetric coplanar strip folded dipole,” IEEE Trans. Antennas Propag., vol. AP-33, no. 9, pp. 1028–1031, Sep. 1985. [11] High-Frequency Structure Simulator (HFSS). ver. 5.6, Agilent Technol., Palo Alto, CA, 2000. [12] H. Kuhnert, W. Heinrich, W. Schwerzel, and A. Schuppen, “25 GHz MMIC oscillator fabricated using commercial SiGe-HBT process,” Electron. Lett., vol. 36, no. 3, pp. 218–220, Feb. 2000. [13] E. Sönmez, S. Chartier, P. Abele, A. Trasser, and H. Schumacher, “Sensitivity matched static frequency divider using a 0.8 SiGe HBT technology,” in Proc. German Microw. Conf., Ulm, Germany, Apr. 2005, pp. 152–155.

[14] E. Sönmez, S. Chartier, A. Trasser, and H. Schumacher, “Isolation issues in multifunctional Si/SiGe ICs at 24 GHz,” in IEEE MTT-S Int. Microw. Symp. Dig., Long Beach, CA, Jun. 2005, 4 pp. [15] J. H. Mikkelsen, O. K. Jensen, and T. Larsen, “Measurement and modeling of coupling effects of CMOS on-chip co-planar inductors,” in Silicon Monolithic Integrated Circuits in RF Syst. Top. Meeting Dig., Atlanta, GA, Sep. 2004, pp. 37–40. [16] IE3D. ver. 10.1, Zeland Softw., Fremont, CA, 2004. [17] L. Roy, M. Li, S. Labonte, and N. Simons, “Measurement techniques for integrated-circuit slot antennas,” IEEE Trans. Instrum. Meas., vol. 46, no. 4, pp. 1000–1004, Aug. 1997.

Erik Öjefors (S’01–M’06) was born in Uppsala, Sweden, in 1975. He received the M.Sc. degree in engineering physics and Ph.D. degree in microwave technology from Uppsala University, Uppsala, Sweden, in 2000 and 2006, respectively. He is currently with the Department of Engineering Sciences, Uppsala University. His research interests are on-chip integrated antennas, RF microelectromechanical systems (MEMS), and SiGe circuit design.

Ertugrul Sönmez was born in Ulm, Germany, on January 22, 1972. He received the Diplom-Ingenieur degree in electrical engineering from the University of Ulm, Ulm, Germany, in 1998. In 1998, he joined the Department of Electron Devices and Circuits, University of Ulm, as a Member of the Scientific Staff, during which time his main fields of research were compact silicon bipolar transistor modeling and analog RF MMIC design. In 2005, he joined TES Electronic Solutions GmbH, Stuttgart, Germany. Since 2007, he has been a Business Developer with microGaN GmbH, Ulm, Germany, where he is involved with gallium–nitrade-based electronics, sensors, and actuators. He has authored or coauthored over 40 publications and conference contributions. His main activities are ultra-wideband MMIC design.

Sébastien Chartier (S’05) was born in Auchel, France, in 1979. He received the Master’s degree in microelectronic from the University of Lille, Lille, France, in 2003, and is currently working toward the Ph.D. degree at the University of Ulm, Ulm, Germany. In 2004, he joined the Department of Electron Devices and Circuits, University of Ulm, as a Member of the Scientific Staff. His main field of research is the design and testing of millimeter-wave silicon bipolar MMICs, especially for application in automotive radar systems.

Peter Lindberg was born in Uppsala, Sweden, in 1974. He received the M.Sc. degree in engineering physics and Ph.D. degree in microwave technology from Uppsala University, Uppsala, Sweden, in 2000 and 2007, respectively. From February 2000 to July 2002, he was an RF Engineer with Smarteq Wireless AB. In November 2002, he joined the Microwave Technology Group, Department of Signals and Systems, Uppsala University, as a Research Engineer. He is currently an RF Engineer with Laird Technologies AB, Åkersberga, Sweden. His research has concerned the areas of SiGe RF integrated circuit (RFIC) design and terminal antenna development.

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Christoph Schick (S’02–M’06) was born in Böblingen, Germany, in 1975. He received the Dipl.-Ing. degree in electrical engineering from the University of Karlsruhe, Karlsruhe, Germany, in 2001. In 2002, he joined the Department of Electron Devices and Circuits, University of Ulm, Ulm, Germany, as a Member of the Scientific Staff. In 2006, he joined the Mixed Signal Integrated Circuit Design Group, Ubidyne GmbH, Ulm, Germany, where he is currently an RF IC Design Engineer. His main fields of research are analog and mixed-signal silicon bipolar MMICs for fiber-optical communication systems.

Anders Rydberg (M’89) was born in Lund, Sweden, in 1952. He received the M.Sc. degree from the Lund Institute of Technology, Lund, Sweden, in 1976, and the Ph.D. degree from the Chalmers University of Technology, Göteborg, Sweden, in 1988. From 1977 to 1983, he was involved with development and research with the National Defence Research Establishment, ELLEMTEL Development Company, and the Onsala Space Observatory. In 1991, he became a Docent (Associated Professor) with the Chalmers University of Technology. From

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1990 to 1991, he was a Senior Research Engineer with Farran Technology Ltd., Cork, Ireland. In 1992, he was an Associate Professor, and then a Professor in 2001, of applied microwave and millimeter-wave technology with the Department of Engineering Sciences, Uppsala University, Uppsala, Sweden.

Hermann Schumacher (M’93) received the Diplom-Ingenieur and Doktor-Ingenieur degrees from the Aachen University of Technology, Aachen, Germany, in 1982 and 1986, respectively. He then joined Bellcore, Red Bank, NJ, where he was involved with InP-based devices and wideband circuit design. Since 1990, he has been a Professor with the Department of Electron Devices and Circuits, University of Ulm, Ulm, Germany, where he focuses on heterostructure transistors and their circuit applications. In 2001, he founded the Competence Center for Integrated Circuits in Communications, University of Ulm, a public-private partnership dedicated to research and development in RF and wideband opto-electronic ICs. He has authored or coauthored over 150 publications and conference contributions.

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A 4-bit CMOS Phase Shifter Using Distributed Active Switches Dong-Woo Kang, Student Member, IEEE, and Songcheol Hong, Member, IEEE

Abstract—This paper presents a novel 4-bit phase shifter using distributed active switches in 0.18- m RF CMOS technology. The relative phase shift, which varies from 0 to 360 in steps of 22.5 , is achieved with a 3-bit distributed phase shifter and a 180 highpass/low-pass phase shifter. The distributed phase shifter is implemented using distributed active switches that consist of a periodic placement of series inductors and cascode transistors, thereby obtaining linear phase shift versus frequency with a digital control. The design guideline of the distributed phase shifter is presented. The 4-bit phase shifter achieves 3.5 0.5 dB of gain, with an rms phase error of 2.6 at a center frequency of 12.1 GHz. The input and output return losses are less than 15 dB at all conditions. The chip size is 1880 m 915 m including the probing pads. Index Terms—CMOS, distributed phase shifter, phase shifters, phased arrays, satellite communications.

I. INTRODUCTION

M

ICROWAVE AND millimeter-wave phase shifters are essential components in a phased-array system for scanning of the radiated beam in minimal time [1]. Most of the RF phase shifters have been implemented in GaAs technology, but recently Si-based phase shifters have been extensively studied for low-cost and small-size phased-array systems. Several design topologies using standard silicon technology have been demonstrated to realize both digital and analog monolithic microwave integrated circuit (MMIC) phase shifters. A 6-bit p-i-n diode phase shifter was realized in a silicon germanium bipolar technology [2]. Additionally, a 5-bit digital phase shifter using a MOS switch was developed over a 9–15-GHz frequency band [3]. These two digital phase shifters exhibit a high insertion loss and large chip area due to the cascade of several phase bits. In contrast, in analog phase shifters, the differential phase shift is varied in a continuous manner by the capacitance change of a varactor or by the vector sum using active devices. A dual-band phase shifter using two variable gain amplifiers was reported for a smart antenna transceiver [4]. In order to cover the full range of 0 –360 , the phase shifter consists of a cascade of four identical phase shifters with a 90 phase range. Two -band active phase shifters using the vector sum of orthogonal signals were recently reported in 0.18- m CMOS technology [5]; however, using passive couplers and baluns in a bulk

Manuscript received November 21, 2006; revised April 12, 2007. This work was supported in part by the Agency for Defense Development, Korea, through the Radio Detection Research Center, Korea Advanced Institute of Science and Technology. The authors are with the Department of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology, Daejeon 305701, Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2007.900317

silicon substrate, the phase shifters have high insertion losses despite the fact that the variable gain amplifiers provide gains. Analog phase shifters can be implemented using a varactor-loaded transmission line in which the transmission line is synthesized with lumped inductors [6]. They have been demonstrated using microelectromechanical systems (MEMS) varactors or thin-film ferroelectric barium strontium titanate (BST) [7], [8]. An 180 continuous analog phase shifter using varactors that implement a constant-impedance technique was proposed in SiGe [9]. A multiband phase shifter was designed by employing a distributed amplifier between varactor-tuned LC networks for active loss compensation [10]. The low quality factor of varactors and inductors in CMOS results in large insertion losses and loss variations of the conventional distributed phase shifters. In this paper, we present a novel digital phase shifter using distributed active switches. The phase shifter consists of an 180 high-pass/low-pass phase shifter and a 3-bit distributed phase shifter, which utilizes a distributed amplifier technique. The artificial transmission line consists of a ladder network of series inductance and gate capacitance of active switches; in this manner, it becomes a constant- transmission line. The differential phase shift, which is the phase shift between active switches along the gate line, can be obtained by selecting one of the active switches in parallel. A cascode MOSFET is used in a distributed active switch, yielding a significant improvement in gain and gain variation performances. The gain variation can be minimized by adjusting the transconductance of each active switch. All microstrip lines with the first metal ground are simulated using ADS Momentum for planar electromagnetic simulations to take into account the coupling between adjacent microstrip lines. The proposed novel 4-bit phase shifter exhibits low phase and gain variations with a relatively low level of power consumption.

II. ANALYSIS A. Artificial Transmission Lines A transmission line can be modeled as a distributed network circuit consisting of inductors, capacitors, and resistors. In a lossless line, transmission lines have been traditionally modeled with the equivalent circuit shown in Fig. 1. The artificial transmission line exhibits a low-pass filter behavior with an equivagiven by [11] lent line impedance

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(1)

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Fig. 1. Ideal artificial transmission line.

where is the operation frequency and quency given by [12]

is the Bragg fre-

Fig. 2. Proposed circuit topology using distributed active switches.

(2) At a sufficiently low frequency, the characteristic impedance is real and a signal power can be propagated along the line from a matched generator to a matched load. However, as the frequency increases above the Bragg frequency, the line impedance is imaginary, and no power can be delivered to the line. At frequency , waves propagate according to [13] (3) is a complex propagation constant, which is a function where of the frequency. is an attenuation constant and is a phase constant. In the case of a lossless line, the per-section phase shift of the artificial transmission line is equal to the phase constant and can be derived as in [12] (4) At frequencies well below the Bragg frequency, the characteristic impedance and per-section phase shift are

(5) A conventional distributed analog phase shifter has been developed by changing the shunt capacitance [14]. By applying a single bias voltage to either varactors or MEMS bridges, the effective distributed capacitance of the artificial transmission line can be changed, which, in turn, changes the phase velocity, and thus, the associated time delay through the line. However, there is tradeoff between the bandwidth and phase tuning range in the number of T-section networks. Hence, by tailoring the voltage across each capacitor separately, the desired phase shift can be achieved with good impedance-matching performances. B. Digital Distributed Phase Shifter Fig. 2 shows the proposed phase-shifter topology using distributed active switches. The operation of the phase shifter is very similar to that of a traveling-wave amplifier. However, the circuit is associated with only one input artificial transmission line, unlike a conventional traveling-wave amplifier, which consists of input and output lines. The input line consists of a cascaded ladder network with series inductances and shunt capacitances. The shunt capacitors of the gate line are supplied by the

gate capacitance of common source MOSFETs. The capacitance of transistors is related with the per-section phase shift and bandwidth. A low gate capacitance of transistors results in the high Bragg frequency and the low per-section phase shift. It is beneficial to design high-frequency phase shifters as long of transistors is sufficiently high. A cascode design is as the chosen for the individual gain cells of the phase shifter. The cascode arrangement of two MOSFETs provides high gain, high output resistance, and high reverse isolation. In addition, the cascode MOSFET operates as active switch, as the gate bias of the common gate MOSFET can be used as an effective means of switching between VDD and GND. This enables the distributed phase shifter to be controlled digitally. The input signal propagates through the gate line, tapping off some of the input power before being absorbed by a terminating resistor. The input signal sampled by the gate circuits at different phases is transferred to the output through each activated cascode cell. By switching the common gate MOSFETs in succession, the phase shift can be incremented by the steps of the phase constant of the input artificial transmission line. Thereand fore, the smallest phase shift is the unit phase shift of . the largest phase shift is An important parameter in phase shifters is the rms phaseshift error. Most MMIC phase shifters exhibit a phase shift error due to manufacturing process variations and modeling inaccuracies of the circuit elements. For the proposed distributed phase or the series inshifter, the transistor capacitance variation results in the unit phase variation . ductance variation can be obtained in terms of the unit The rms phase error . It can be computed as phase mismatch

(6) In addition, the unit phase mismatch can be a function of the number of stages needed for a required phase shift range. For a given gate capacitance or series inductance variation , which is defined as or , the unit phase mismatch can be expressed by (7) where

is the maximum differential phase shift.

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Fig. 4. (a) Transmission line circuit for the gate of the distributed phase shifter. (b) Simplified small-signal equivalent circuit of a single cascode cell. Fig. 3. Calculated rms phase error versus the number of stages as the gate capacitance or series inductance varies.

Using (6) and (7), the rms phase error can be calculated as

is the effective small-signal transconductance of where each cascode cell. The available gain of each phase state is given by

(8) Fig. 3 shows the calculated rms phase error with the gate capacitance or series inductance variation for a maximum differential phase shift of 180 . The rms phase error is not reduced significantly when more than eight sections are used. Another important parameter of the phase shifter is the state-to-state variation of the insertion loss or gain. In practice, the gate voltage wave traveling throughout the gate artificial line will unequally excite the gates of common source MOSFETs due to the loss present in the gate artificial transmission line. Each successive common source MOSFET receives less signal voltage as the signal travels down the gate line. Hence, the current generators from the drains have different magnitudes. This results in a significant gain variation of the phase shifter. In order to equalize the current generators, it is necessary to change the transconductance of each cascode cell by adjusting the size of the common gate MOSFET. The magnitude of each current generator decreases as the differential phase shift increases, thereby increasing the size of the common gate MOSFET. Fig. 4(a) shows a simplified equivalent circuit for the gate line of the distributed phase shifter. The input voltage wave propaacross gating the gate line produces voltages . The voltage at the th tap of the gate each gate capacitance line is related to the gate line’s segment length and complex propagation constant . If the voltage across the input terminal , then of the distributed phase shifter is

(9) Fig. 4(b) shows a simplified small-signal equivalent circuit of a single cascode unit cell. The current generator of each cascode cell can be expressed as (10)

(11) , in which and are the attenuation where and phase constant of the gate line, respectively. The following general condition for the amplitude equalization of the distributed phase shifter is then obtained:

.. . (12) In general, the transconductance of an MOSFET is directly proportional to the width-to-length ratio. The effective transconductance of a cascode cell can be changed by varying the size of common gate MOSFET. Hence, the width of the common gate MOSFET is given by

.. . (13) Equation (13) indicates that the width of a common gate MOSFET increases as an exponential function of the attenuation constant assuming that other parasitic losses are not considered. This is explained by the fact that the input voltage on the gate line decays exponentially. Process variation can affect the gain of the distributed phase shifter. If the transconductance of each cascode cell is changed at the same ratio by the process variation, there is no change of

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Fig. 5. Schematic of the 4-bit CMOS phase shifter using distributed active switches.

the gain deviation from (11) and (12). The absolute gain value of each phase state can be higher or lower. III. CIRCUIT DESIGN The proposed 4-bit phase shifter consists of a 3-bit distributed phase shifter, a feedback amplifier, and an 180 high-pass/lowpass phase shifter, as shown in Fig. 5. The 3-bit distributed phase shifter, in which eight cascode cells are connected in parallel, ranges from 0 to 157.5 in steps of 22.5 . The basic design of the distributed phase shifter can be carried out easily using (5). For a 50- port impedance and a center frequency of 12 GHz, and were determined as 0.104 pF and 0.26 nH, respectively. A common source MOSFET with a 45- m gatewidth was selected, which corresponds to an input gate capacitance of 0.104 pF. The Bragg frequency can be calculated to be approximately 62.4 GHz. The size of the common gate MOSFET was increased from 19 to 50 m in order to minimize the gain deviations of each phase state. It was designed with different size from 19, 20, 25, 30, 32, 33, 37, and 50 m, respectively. The size was slightly optimized due to other parasitic components and the losses of the connecting lines at the output of the each cascode cell. The 4-bit phase shifter was designed at 12 GHz for a satellite phased-array system. Hence, the output of the distributed phase and shifter is loaded with a high-pass combination of to provide parallel resonance, thereby increasing the gain at the design frequency. The connecting lines of the distributed phase shifter are negligible for the phase performance and bandwidth because the phase performance is mainly dependent on the input artificial transmission line and the output load affects the bandwidth. The connecting lines can have an influence on changing the loss of each state. Fig. 6 shows the simulated operation conditions of a 3-bit distributed phase shifter. Sizing up the common gate MOSFET inof the common source MOSFET, creases the drain voltage thereby increasing the drain current. This results in varying the effective transconductance of the each cascode cell.

Fig. 6. Operation conditions of 3-bit distributed phase shifter.

A two-stage amplifier followed by the distributed phase shifter is used to flatten the gain over a wider bandwidth of frequencies. The second stage is realized as a cascode amplifier forms the feedback, with resistive shunt feedback. Resistor is used for independent biasing of the gate and drain. and is normally large in order to function The capacitance of as a short circuit over the frequency of interest. The resistive feedback provides better stability, gain flatness, and bandwidth. The inter-stage matching network is a high-pass combination and in order to obtain conjugate matching between of and the gate of . the drain of The third stage is a common source amplifier used to enhance the overall gain. The output of the third stage is loaded with a shunt inductor and a series capacitor . This impedance matching network improves the output return loss of the amplifier stage, thereby minimizing the interactions between the output of the third stage and the input of the 180 phase shifter. Fig. 7 shows the simulated gain responses of sub-circuits. The losses of the distributed phase shifter are approximately 4 dB. The two-stage amplifier provides 13–14-dB gain over a frequency range. The overall gain of active circuits varies from

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Fig. 7. Simulated gain responses of active sub-circuits.

Fig. 8. Simulated losses and phase response of 180 phase shifter.

Fig. 9. (a) Cross-sectional view of a microstrip line. (b) Meandered microstrip line pattern. (c) Simulated and measured S -parameter results of the meandered line pattern.

9 to 10 dB for different phase conditions. The simulated input 1-dB compression point of the phase shifter is 8 dBm. The 180 phase shifter switches a T-type high-pass/low-pass phase-shift network using two SPDT MOSFET switches [3]. The gatewidth of each switch is 40 m. The on resistance and off capacitances are approximately 14 and 0.03 pF, respectively. The isolation of the switch is approximately 14 dB at 12 GHz. It is designed to have a characteristic impedance and a series capacitor of 50 . A shunt capacitor are implemented as metal–insulator–metal (MIM) capacitors. In both networks, the inductors are implemented using a microstrip line structure with a first metal ground plane because the relative phase shift of the 180 phase shifter is sensitive to the inductance value of the high- or low-pass filter. Fig. 8 shows the simulated insertion losses and phase difference of the 180 phase shifter. The insertion losses of the high- and low-pass state are better than 5 dB with an amplitude imbalance of within 0.3 dB and a phase error of within 1 . Fig. 9(a) illustrates a typical microstrip configuration using CMOS technology. The microstrip line is realized through the use of the top metal as a signal line and the bottom metal as a ground with a thick SiO layer as a substrate. The top metal

and bottom metal are fabricated with 2- and 0.5- m-thick Al metals, respectively. The metals are separated by an oxide layer approximately 6.5- m thick. In the microstrip structure, most of the electric fields are confined in a silicon–oxide layer due to the ground-plane shielding. Thus, the loss associated with a silicon substrate can be reduced and the coupling effect with the adjacent signal line is minimized. For a compact circuit size, meandered microstrip lines are adopted for the distributed phase shifter and 180 phase bit. Moreover, all microstrip lines including interconnect effects are simulated by a full-wave EM simulator (ADS Momentum). The microstrip line has a simulated quality factor of approximately 5–6 at 12 GHz. Fig. 9(b) shows the meandered microstrip line test pattern that can form the artificial transmission line of the distributed phase shifter. After the momentum condition was verified by the test pattern at the first tape-out, we designed the 4-bit phase shifter at the second tape-out for accurate phase responses. The length of each section can be optimized to be in line with the linear phase response because of different electrical delay according to the different size of the common gate MOSFET in the cascode cell. The linewidth and spacing are 8 and 20 m, respectively. The meandered line was measured using an on-wafer

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Fig. 10. Chip photograph of the fabricated phase shifter.

probing system. Fig. 9(c) shows the simulated and measured results of the standalone meandered microstrip line pattern. There is a good agreement between the simulation and measurement results.

Fig. 11. Measured relative phase shift for all 16 states.

IV. MEASUREMENT RESULTS The proposed 4-bit phase shifter was fabricated using TSMC’s 0.18- m CMOS technology, which provides one poly layer for the gate of the MOSFET and six metal layers for inter-connection. The active device models are based on standard BSIM3 model provided by TSMC. The circuit draws a maximum 14.8-mA dc current from a 1.8-V power supply; thus, the maximum power consumption is 26.6 mW. The 3-bit V with the drain distributed phase shifter is biased at current varying from 5.7 to 6.0 mA. The power consumption of the 3-bit distributed phase shifter itself is 10.8 mW. The to ) of the common gate MOSFET is toggled gate bias ( between 0–1.8 V. In the two-stage amplifier, the bias voltages V and V, resulting in are chosen as dc currents of 5.2 and 3.6 mA for the second and third stages, respectively. The 180 phase shifter using passive MOSFET switches requires two control bias inputs of 0 and 1.8 V, but does not consume any current. Fig. 10 shows a die photograph of the proposed 4-bit phase shifter. The chip size, including the pads, is 1880 m 915 m. The phase shifter was measured using RF probes and a shortopen-line-thru (SOLT) calibration up to the probe tips to measure the -parameter. It was measured with an Agilent 8510C vector network analyzer and a Cascade Microtech probe station. Fig. 11 shows the measured relative phase shifts of 16 phases over 11.6–12.6 GHz. The relative phase shifts can be obtained by the phase difference of each transmission coefficient between the phase and reference states. The phase shift at the designed center frequency of 12.1 GHz is incremented in the steps of 22.5 . A phase shift of 0 –157.5 is achieved with the 3-bit distributed phase shifter and, as expected, increases linearly with the frequency. These time-delay characteristics enable frequency-independent beam steering, which permits the realization of a wideband phased-array system. In the case of an 180 bit, the relative phase shift is constant and flat to within 0.5 over more than 2 GHz of bandwidth. Fig. 12 shows the measured gain responses for the 16 states. The overall gain of the phase shifter is measured to be 3.5 0.5 dB at 12.1 GHz. From 11.6 to 12.6 GHz, the gain varies by less then 0.5 dB.

Fig. 12. Measured small-signal gain.

As designed, the distributed phase shifter and the two-stage amplifier provide 9–10-dB gain and the insertion loss of the 180 phase shifter is approximately 5 dB. Hence, the measured gain agrees well with the simulated value. The measured input and output return losses are plotted in Fig. 13(a) and (b), respectively. Both the input and output return losses are better than 15 dB over the band of interest. The output return loss exhibits different characteristics according to the control of the 180 phase shifter. These differences are due to impedance imbalance between the high- and the low-pass path. Due to the isolation of the amplifier, the input return loss is only related to the input artificial transmission line. Fig. 13(c) shows the measured rms phase and rms amplitude error. The rms of the phase error can be calculated from all measured phase shifts using (6). The phase shifter exhibits a maximum rms phase error of 5.5 and a maximum amplitude error of 0.35 dB with 1-GHz bandwidth. The rms phase error is minimized at 2.6 near the designed frequency of 12.1 GHz. The gain variation of the phase shifter can be more reduced by adaptively adjusting the gate bias of the shunt feedback amplifier. Fig. 14(a) shows the measured gain response after a gain calibration. It is found that the

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Fig. 14. Measured results after gain calibration. (a) Small-signal gain. (b) rms phase and rms amplitude errors.

TABLE I COMPARISON OF SILICON PHASE SHIFTERS

Fig. 13. (a) Measured input return loss. (b) Measured output return loss. (c) Measured rms phase and amplitude error.

gain ripple within the same bandwidth is less than 0.25 dB. Hence, a maximum rms amplitude error of 0.19 dB is achieved, as shown in Fig. 14(b). On the other side, the rms phase error slightly changes by approximately 1 through the calibration process. Table I compares the recently reported phase shifters in the silicon process. Switching-type phase shifters [2], [3] using

Continuous phase shift. # Estimated from VGA

passive switches exhibit large insertion loss and chip area because each sub-bit is cascaded for multibit operation. On the other hand, active phase shifters [5], [9], [10] show significant

KANG AND HONG: 4-bit CMOS PHASE SHIFTER USING DISTRIBUTED ACTIVE SWITCHES

state-to-state gain variation with analog control. This work operates with digital control and achieves lower gain deviation and lower power consumption with competitive gain and size compared to other active phase shifters. V. CONCLUSION This paper has presented the development of a 4-bit phase shifter in 0.18- m CMOS technology. The phase shifter consists of a 3-bit distributed phase shifter, a two-stage amplifier, and a 180 high-pass/low-pass phase shifter. The novel 3-bit phase shifter provides a true time-delay phase shift through the use of a distributed active switch that is comprised of the periodic placement of a series inductor and a cascode MOSFET. The distributed approach operates digitally with a small chip area and a low gain variation compared to conventional distributed phase shifters. The two-stage amplifier exhibits a flat gain response over the frequency of interest. The 4-bit phase shifter has a measured gain of 3.5 0.5 dB at 12.1 GHz with a dc power consumption of 26.6 mW. The input and output return losses are better than 15 dB, as the artificial transmission line is on the input port and the high-pass/low-pass network is on the output port. The rms phase error is less than 5.5 over 1 GHz of bandwidth. The low phase deviation makes the phase shifter suitable for modern communication systems that require the phase flatness over an operating frequency band. In addition, the phase shifter can be used for a wideband phased-array system due to the time-delay nature of the distributed phase shifter. REFERENCES [1] S. K. Koul and B. Bhat, Microwave and Millimeter Wave Phase Shifters. Boston, MA: Artech House, 1991, ch. 1. [2] M. Teshiba, R. Van Leeuwen, G. Sakamoto, and T. Cisco, “A SiGe MMIC 6-bit p-i-n diode phase shifter,” IEEE Microw. Wireless Compon. Lett., vol. 12, no. 12, pp. 500–501, Dec. 2002. [3] D.-W. Kang, H. Lee, C.-H. Kim, and S. Hong, “Ku-band MMIC phase shifter using a parallel resonator with 0.18-m CMOS technology,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 294–300, Jan. 2006. [4] D. R. Banbury, N. Fayyaz, S. Safavi-Naeini, and S. Nikneshan, “A CMOS 5.5/2.4 GHz dual-band smart-antenna transceiver with a novel RF dual-band phase shifter for WLAN 802.11a/b/g,” in IEEE RFIC Symp., Forth Worth, TX, Jun. 2004, pp. 157–160. [5] P.-S. Wu, H.-Y. Chang, M.-D. Tai, T.-W. Huang, and H. Wang, “New miniature 15–20-GHz continuous-phase/amplitude control MMICs using 0.18-m CMOS technology,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 10–19, Jan. 2006.

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[6] F. Ellinger, H. Jäckel, and W. Bächtold, “Varactor-loaded transmission line phase shifter at C -band using lumped elements,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 4, pp. 1135–1140, Apr. 2003. [7] N. S. Barker and G. M. Rebeiz, “Distributed MEMS true-time delay phase shifters and wideband switches,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 11, pp. 1881–1890, Nov. 1998. [8] B. Acikel, T. R. Taylor, P. J. Hansen, J. S. Speck, and R. A. York, “A TiO thins films,” new high performance phase shifter using Ba Sr IEEE Trans. Wireless Compon. Lett., vol. 12, no. 7, pp. 237–239, Jul. 2002. [9] T. M. Hancock and G. M. Rebeiz, “A 12-GHz SiGe phase shifter with integrated LNA,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 3, pp. 977–983, Mar. 2005. [10] C. Lu, A.-V. Pham, and D. Livezey, “Development of multiband phase shifters in 180-nm RF CMOS technology with active loss compensation,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 40–45, Jan. 2006. [11] D. M. Pozar, Microwave Engineering. New York: Wiley, 1998, ch. 8. [12] G. M. Rebeiz, RF MEMS Theory, Design, and Technology. New York: Wiley, 2003, ch. 10. [13] T. H. Lee, The Design of CMOS Radio-Frequency Integrated Circuits. Cambridge, U.K.: Cambridge Univ. Press, 2004, ch. 6. [14] A. S. Nagra and R. A. York, “Distributed analog phase shifters with low insertion loss,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 9, pp. 1705–1711, Sep. 1999. Dong-Woo Kang (S’03) received the B.S., M.S., and Ph.D. degrees in electrical engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2001, 2003, and 2007, respectively. His doctoral research concerned the development of Ku-band CMOS phase shifters. His research interests include CMOS phase shifters, beam-steering systems, and miniaturized radar systems.

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, Stanford, CA, and Samsung Microwave Semiconductor, Suwon, Korea. 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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A New Brillouin Dispersion Diagram for 1-D Periodic Printed Structures Paolo Baccarelli, Member, IEEE, Simone Paulotto, Member, IEEE, David R. Jackson, Fellow, IEEE, and Arthur A. Oliner, Life Fellow, IEEE

Abstract—Dispersion and radiation properties for bound and leaky modes supported by 1-D printed periodic structures are investigated. A new type of Brillouin diagram is presented that accounts for different types of physical leakage, namely, leakage into one or more surface waves or also simultaneously into space. This new Brillouin diagram not only provides a physical insight into the dispersive behavior of such periodic structures, but it also provides a simple and convenient way to correctly choose the integration paths that arise from a spectral-domain moment-method analysis. Numerical results illustrate the usefulness of this new Brillouin diagram in explaining the leakage and stopband behavior for these types of periodic structures. Index Terms—Brillouin diagram, leaky waves, periodic structures, surface waves.

I. INTRODUCTION AND BACKGROUND

ECENTLY, 1-D periodic printed structures have received renewed attention because of their widespread use in the design of filters [1]–[5], leaky-wave antennas [6]–[8], and novel metamaterial waveguides and devices [9]–[12]. As is well known, the presence of periodicity may give rise to attractive propagation and radiation features that have been deeply investigated over the last 50 years [13]–[26]. In this paper, we focus our attention on the fundamental properties of bound (nonradiating) and leaky (radiating) modes supported by 1-D open periodic structures printed on a grounded dielectric slab, and we give an explanation of the relevant dispersive behavior by means of a novel Brillouin diagram. 1-D periodic structures are invariant by a translation of length along one infinite (longitudinal) direction (called ). The periodicity allows us to perform a modal analysis by applying [6], [15], [17]. The Floquet’s theorem along one direction longitudinal variation (along ) of the modal fields may be expressed by a product of two terms: a fundamental traveling wave , and a with a complex propagation wavenumber standing wave, which represents the local variations due to the

R

Manuscript received October 5, 2006; revised February 14, 2007. P. Baccarelli and S. Paulotto are with the Department of Electronic Engineering, “La Sapienza” University of Rome, 00184 Rome, Italy (e-mail: [email protected]; [email protected]). D. R. Jackson is with the Department of Electrical and Computer Engineering, University of Houston, Houston, TX 77204-4005 USA (e-mail: [email protected]). A. A. Oliner is with the Department of Electrical Engineering, Polytechnic University, Brooklyn, NY 11201 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.2007.900304

periodicity [6], [15], [17]. The modal field may thus be represented as a superposition of space harmonics, which are traveling waves with complex propagation wavenumbers expressed with , as [6], [15], [17]. The modal field then has the form

(1) It can be observed that each space harmonic propagates along with a different phase constant , but with the same attenuation constant [6], [15], [17]. Furthermore, each space harmonic possesses different longitudinal phase velocities, while the group velocities of the individual space harmonics are identical and equal to the group velocity of the entire mode [6], [15], is assumed and sup[17]. A time–harmonic dependence pressed throughout. As has been known for many years [13], propagating waves, with real propagation wavenumbers, can be supported only in well-specified ranges of frequency called passbands; regions forbidden to propagation, called stopbands, alternate between the passbands. When the transverse (with respect to the longitudinal direction of periodicity) boundaries are not completely reflecting, the 1-D periodic structure is open and a leakage of power may occur [14]–[19]. 1-D open periodic structures support leaky waves that are guided by the structure, which radiate or leak power continuously into the exterior unbounded regions. Two different types of 1-D open periodic structures are commonly found. The first consists of structures surrounded exclusively by free space so that radiation is allowed only into free space. Many well-known structures, such as periodically modulated slow-wave antennas [14], [18], traveling-wave slot or dipole arrays [6], and metal strip gratings on a grounded dielectric slab [6], [16], [20]–[23] belong in the class. For these structures, leakage may occur only into free space [15], [17], [19]. The second type of structure possesses a dielectric substrate, where leakage is allowed both into the dielectric substrate (in the form of a surface wave) and into free space. Printed 1-D periodic structures fall into the latter category, including structures consisting of periodically loaded microstrip [1]–[3], [5], [6], [24], [25], as well as slot or coplanar lines [4]. For these structures, leakage may occur into one or more surface waves of the grounded dielectric slab (background dielectric structure) or simultaneously into space, as with printed nonperiodic structures [27]–[30].

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BACCARELLI et al.: NEW BRILLOUIN DISPERSION DIAGRAM FOR 1-D PERIODIC PRINTED STRUCTURES

(a)

(b) Fig. 1. Reference structures with relevant coordinate system, physical and geometrical parameters. (a) Metal strip grating on a grounded dielectric slab. (b) Periodically loaded microstrip line.

In this paper, the analysis and identification of all the propagation regimes of 1-D periodic printed structures are made. Two examples of 1-D periodic structures are reported. The former, in Fig. 1(a), is a metal strip grating on a grounded dielectric . This slab, which is infinite in the -direction, so that structure has been chosen here as an example from the class of 1-D open periodic structures where leakage may occur only into free space. The structure in Fig. 1(b) is a periodically loaded microstrip line; it is a finite-width structure with a transverse variation of the field along . It is a typical example of a 1-D open periodic printed structure where leakage may occur into above-cutoff surface waves of the grounded dielectric slab, or also simultaneously into free space. The metal strip grating and the periodically loaded microstrip line will be considered in the following discussion. The classification of the guided-wave types on a 1-D periodic structure, like those of Fig. 1(a) and (b), is based on the traveling-wave properties of each space harmonic in the unbounded cross section. Particular attention is devoted here to leakage phenomena both into the substrate and into free space, either in the backward or forward directions. A straightforward representation of all the propagation regimes for these types of structures is commonly given by using the Brillouin dispersion diagram [15], [17]. As is well known, this diagram is a graphical representation for the longitudinal phase constant of the space harmonics as a function of frequency. Using the Brillouin diagram, many properties can be easily displayed for both closed and open periodic structures, e.g., conditions for radiation, the direction of the radiating beams, etc. [15], [17]. The usual Brillouin diagram identifies only two cases: bound modes and modes that leak into space. A novel Brillouin diagram is introduced here to account for the

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different types of physical leakage, thus providing insight into most of the guided and radiative properties of such structures in a very simple fashion. The new Brillouin diagram is a modification of the usual one, in which new regimes are identified, corresponding to the different types of leakage that may exist when a dielectric substrate is present. In this sense, the new diagram is a generalization of the usual one. The new Brillouin diagram is also shown to directly determine the correct path of integration to use in the complex transverse plane for each of the space harmonics when using a spectral-domain method to analyze the modal propagation. The correlation between the value of the phase constant of a guidedmodeonaprintedstructureandthephysicalleakagemechanism of the mode has already been established in [29]. This reference also establishes the correlation between the physical leakage mechanism and the path of integration in a spectral-domain solution of the structure. This discussion is not repeated here. The reader is referred to [29] and references therein for a justification and validation of this correlation. In this paper, this correlation is used to develop the new Brillouin diagram, and to show its relevance in determining (in a simple graphical manner) the physical leakage mechanism of the guided mode and the path of integration necessary in a spectral-domain solution. Some preliminary results on this topic have been presented in [25], where the early idea was discussed that different regions in a Brillouin diagram may be related to different leakage mechanisms. However, this early description did not develop a complete Brillouin diagram that accounts for all of the space harmonics, and this is presented here for the first time. This new Brillouin diagram presented here has allowed us to obtain and display extensive numerical results for a realistic structure, a 1-D periodically loaded microstrip line printed on a grounded dielectric slab, that illustrate the usefulness of the new Brillouin diagram and provide validation for it (as will be explained at the beginning of Section V). Further details on the numerical calculation procedure used to obtain the results may be found in [30]. The main features of the analysis and the discussion presented here apply to the entire class of 1-D open periodic structures printed on multilayer dielectric slabs. This paper is organized as follows. In Section II, propagation and radiation regimes for 1-D periodic structures that allow for radiation only into free space are reviewed by using the conventional Brillouin diagram. In Section III, 1-D periodic structures that allow radiation both into free space and into the dielectric substrate (in the form of a surface-wave field that carries power away from the structure at an angle) are considered, and the new Brillouin diagram is introduced. In Section IV, the correlation between Brillouin regions and integration paths in the complex spectral plane, when using a spectral-domain method to analyze the modal propagation of 1-D periodic printed structures, is described. Numerical results for a specific structure are shown in Section V in order to illustrate the nature of the radiation from the guided mode as the frequency changes and the mode passes through different areas of the new Brillouin diagram. II. BRILLOUIN DIAGRAM FOR LEAKAGE ONLY INTO SPACE Structures that are infinite in the -direction, such as the metal strip grating of Fig. 1(a), allow for leakage (radiation) only into

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free space. The modal electric field may be expressed in the air ) as region (i.e.,

(2) where is the transverse wavenumber in the -direction for the th space harmonic. Such structures may be characterized by using the conventional Brillouin diagram [15], [17]. Here, the basic principles are reviewed so that the extension to the new Brillouin diagram for finite-width structures may be better understood. Modes supported by the metal strip grating of Fig. 1(a) may be characterized by three different regimes of propagation [17], i.e.: 1) a passband regime for which all the space harmonics (if dielectric and have a real propagation wavenumber metallic losses are absent) and the power is purely bound and guided by the periodic structure; 2) a “closed stopband” regime is complex for all the space harmonics with a typwhere ically high attenuation constant; in this case, the mode is still “bound” (no power leaks from the structure), but is purely reactive and is attenuating along the structure; and 3) a leaky-wave regime for which at least one space harmonic radiates into space is complex for all the space harmonics; this is possible and because the metal strip grating is an open periodic structure (unbounded in the -direction). In the bound passband regime (regime 1), all the space har(a slow wave) monics have a phase constant such that being the free-space wavenumber. Hence, each space with harmonic behaves as an evanescent wave along the unbounded -direction with a transverse wavenumber in air . In the bound stopband regime (regime 2), all the space har, although the propagation monics are still slow wavenumber is now complex. In the radiative regime (regime 3), one or more of the space , thus harmonics satisfy the fast-wave condition giving rise to radiation into free space. The radiating harmonics that is almost real in the air have a transverse wavenumber . region for small radiation loss per unit length with , the field of this harmonic will increase exponenIf direction, while for , the field tially in the vertical of this harmonic will decrease exponentially in this direction. This field behavior is common to all physical leaky waves. All three regimes can be effectively described through the Brillouin diagram of Fig. 2 [15], [17]. In the following, we will consider modes with positive group velocity, which implies a power flow in the positive -direction, by displaying in the Brillouin diagram only those branches of the dispersion curve that have a positive slope. The Brillouin diagram reports the dispersive behavior of the versus [13], [15], [17], space harmonics as a plot of where represents the phase constant of any particular space harmonic, and is the period. The black dashed straight lines at 45 in Fig. 2(a) are the curves that divide the Brillouin diagram into a slow wave (bound) region and a fast wave (radia) on the dispersion curve tion) region. If a point ( of the th space harmonic lies within the fast wave region (i.e.,

(a)

(b) Fig. 2. Brillouin diagram showing the radiative and bound regions for a 1-D periodic structure where radiation into only free space is permitted [e.g., the metal strip grating of Fig. 1(a)]. (a) Diagram for one space harmonic. (b) Same as (a), but including all the space harmonics with the bound-mode regions shown (lak and their replicas are plotted using dashed beled as B). The curves is the fast wave region for the nth space harmonic, when the fundalines. mental space harmonic lies within this region.

R

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) for any , this harmonic corresponds to a fast wave radiating into free space [15], [17]. The propagation will then be complex and the total wave is not wavenumber bound [propagation regime (3)]. In Fig. 2(b), the complete Brillouin diagram for a periodic structure is obtained by including replicas of the curves (black dashed lines), spaced apart by multiples of in the wavenumber . The entire Brillouin diagram identifies when a particular space harmonic of the guided mode will be in the fast wave region. By plotting the dispersion behavior (fundamental) harmonic phase constant on of the the Brillouin diagram in Fig. 2(b), we can define the radiation as the set of points , that correspond region to the th harmonic being located in the fast wave region [17] as (3) It can be noticed that, by increasing the frequency , overlapping areas exist between different radiation regions with ); these areas represent the set of points ( dispersion plot for which both the th and th on the harmonics are located in the fast wave region, and thus radiate into free space. When all the space harmonics are simultaneously slow, the mode is completely bound to the interface and no radiation takes

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place; this occurs if all space harmonics lie within the bound[hatched wave triangular regions (B) that exist for and their areas in Fig. 2(b)], delimited by the curves of periodic replicas (black dashed lines) [15], [17]. This will be the lies within any one of the bound-mode triangles. case if Inside the bound areas, two different propagation regimes can be identified: the passband and the closed stopband regimes (propagation regimes (1) and (2), respectively). Closed stopband regions appear as a consequence of contradirectional coupling between space harmonics with group velocities of oppowithin site signs and are characterized by the frequency region of the stopband. (This expression assumes harmonic lies that the harmonics are labeled so that the within the first bound-mode triangle, which is the usual case). [15], [17]. A closed stopband can only occur when The Brillouin diagram also permits a rapid determination of the number of radiating beams and of their pointing angles. For a , the number of beams is equal to the number given value of of space harmonics located in the fast wave region. For the th radiating harmonic, the angle that the radiating beam forms with broadside [see Fig. 1(a)], and the corresponding angle in the Brillouin diagram [see Fig. 2(a)] are related by the following equation [15], [17]:

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Fig. 3. New Brillouin diagram for a 1-D periodic printed structure, shown for a single space harmonic. (The diagram is drawn for a periodically loaded microstrip line structure having a spatial period p = 4 mm and a homogeneous isotropic lossless grounded dielectric slab of thickness h = 0:508 mm, relative permittivity " = 10:2, and relative permeability  = 1.) The different radiation regions for any particular space harmonic are shown, and indicated with ; = k ; and = k are also shown numbers. The curves = k using black solid lines, black dashed–dotted lines, and black dashed lines, respectively. In each region, a label indicates the type of leakage. For example, in region 4 , leakage occurs simultaneously into free space (k symbol) and into the TM and TE surface waves. A horizontal straight solid white line, at the cutoff frequency of the TE mode, separates regions 3 and 4 from regions 3 and 4 , respectively.

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(4) When these two angles are positive, radiation occurs in the forward region of the open periodic structure; negative angles, instead, correspond to radiation in the backward region [15], [17]. Radiation into the forward region corresponds to a forward wave (phase and group velocities are in the same direction), while radiation into the backward region corresponds to a backward wave (phase and group velocities are in opposite directions) [15], [17]. III. NEW BRILLOUIN DIAGRAM: LEAKAGE INTO BOTH SPACE AND SURFACE WAVES A. Diagram for a Single Space Harmonic The periodically loaded microstrip line of Fig. 1(b) has a cross section that is unbounded in the -direction and, hence, radiation into the space above the structure may occur, just as with the metal strip grating. The modal electric field in the air region has the form

(5) where is the transverse wavenumber in the -direction for the th space harmonic. The path of integration in (5) determines the nature of the leakage, and this is discussed further in Section IV. For this type of structure, surface leakage is permitted in addition to space leakage because of the finite width of the metallization and the unbounded grounded substrate in the -direction [7], [25], [27]–[30]. A surface-leaky mode has a complex propagation wavenumber and

is characterized by at least one space harmonic radiating into the above-cutoff surface waves of the dielectric substrate [7], [25], [30]. A space harmonic radiating only into the dominant surface-wave mode of the substrate, and not into space, , where has a phase constant such that is the wavenumber of the dominant surface wave. , the field of this harmonic will increase exponenIf directions, while decaying expotially in the transverse direction. If , the field of nentially in the vertical this harmonic will decrease exponentially in both the transverse directions and the vertical direction. In the periodically loaded microstrip line, when at least one space harmonic radiates into , a simultaneous radiation also occurs into the space, substrate since . The propagation regime is then characterized by both space- and surface-wave leakage. In this case, there is exponential growth of the field in both the trans, and exponential decay verse and vertical directions for in both directions for . The bound regimes of propagation (1) and (2), reported above for the metal strip grating, also exist for modes supported by the periodically loaded microstrip line when all the space harmonics are slow and satisfy the con. dition A new Brillouin diagram is introduced here to describe the dispersive properties of this class of structures (illustrated by the periodically loaded microstrip line), which can address the occurrence of surface leaky regimes. In the following, only the and modes of the grounded dielectric substrate, for the periodically loaded microstrip line shown in Fig. 1(b), are considered, although the discussion could be generalized to account from more than two surface waves if need be. The radiation regions of the new Brillouin diagram are shown in gray in Fig. 3. The black solid and dashed–dotted lines correspond to the phase constants and

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of the and modes supported by the dielectric substrate, respectively, while the black dashed straight lines are the curves. The horizontal white line in Fig. 3 denotes the surface-wave mode, and this line intersects the cutoff of the and curves at either end. For simplicity, Fig. 3 shows the Brillouin diagram for a single space harmonic (whichever one is of interest). Later (in Fig. 4), the diagram will be extended to show results for all space harmonics simultaneously. As in the metal strip grating, if a point on the dispersion curve of the th space harmonic lies within the regions delimited by the black dashed lines (i.e., , regions 3, 3 , 4, and 4 in Fig. 3), this harmonic corresponds to a fast wave radiating into free space. Due to the finite width of the metallization and the unbounded on the grounded substrate along , if a point dispersion curve of the th space harmonic lies between the , black solid lines in Fig. 3 (i.e., encompassing all gray regions in Fig. 3), leakage into the surface wave of the dielectric substrate occurs for this harmonic. surface wave is above cutoff, a point Moreover, when the on the Brillouin diagram (see Fig. 3) that lies between the black dashed–dotted lines and above the horizontal , regions straight solid white line (i.e., 2, 3 , 4 , and 5 in Fig. 3) corresponds to a mode for which surface-wave leakage occurs for the relevant space harmonic. New radiation regions are thus present in the Brillouin diasurface-wave gram of Fig. 3: regions 1 and 6, where only leakage occurs, and regions 2 and 5, where leakage into both and surface waves (but not space) takes place. Moreover, in regions 3, 3 , 4, and 4 , simultaneous radiation into free space and into the above-cutoff surface waves of the dielectric surface wave in regions 3 substrate occurs (i.e., into the and surface waves in regions 3 and 4, and into the and 4 ). Regions 1, 2, 3, and 3 correspond to radiation in the backward region (negative half-space with respect to the -axis), whereas in regions 4, 4 , 5, and 6, radiation in the forward region (positive half-space with respect to the -axis) occurs. B. Diagram for All Space Harmonics A new complete Brillouin diagram that shows results for all space harmonics simultaneously can be obtained by including and periodic replicas of the curves , as shown in Fig. 4. In the following, colors refer to the color version of Fig. 4 available online, while between parentheses the relevant grayscale tonality in the print version is indicated. Two different cases, depending on the value of the of the surfacenormalized cutoff frequency [as in Fig. 4(a)] and wave mode, are considered: [as in Fig. 4(b)]. These values correspond to the horizontal white lines that separate the orange (medium gray) and blue (dark gray) regions. In Fig. 4(a), the new complete Brillouin diagram is shown for a 1-D periodic structure as in Fig. 1(b) with the same parameters ). As a consequence of the presas in Fig. 3 (i.e., ence of the surface-leaky regimes, purely bound regions are not the same triangular regions as in the metal strip grating, but are the hatched portion of them (labeled B) delimited by the curves

surface wave and their periodic replicas (black solid of the lines), as shown in Fig. 4(a). When one of the space harmonics lies within region B, they all must do so and, hence, the mode is completely bound. Two different propagation regimes can be identified here within the bound region B: the passband and the . closed stopband regimes The remaining area of the triangular regions, up to the tips of ) is reported in yellow (light gray) and the triangles (at mode of corresponds to surface-wave leakage into only the the substrate, and not into space. The yellow regions (light gray) may be further divided into three subregions: the left, right, and curves (solid black lines). top regions, as defined by the The left region corresponds to a single space harmonic leaking mode in the forward direction. The right region into the mode in corresponds to a single harmonic leaking into the the backward direction. The top region corresponds to leakage mode in the forward direction from one harmonic into the mode in the backward direction from and leakage into the an adjacent harmonic. The horizontal white line in Fig. 4(a), which denotes the surface-wave mode, divides the orange cutoff of the (medium gray) region from the blue (dark gray) region. The orange (medium gray) region lies outside the triangular regions, surface-wave and extends vertically up to the cutoff of the surmode. Within this region, simultaneous space and face-wave leakage occurs. The white line may be above or mode curves with the below the intersection of the vertical axis, with the former case shown in Fig. 4(a). Within the orange region, there is always one space harmonic that surface wave. Depending leaks into both space and the on the location within the orange region, other harmonics may surface wave. If a also be leaking into space or into the space harmonic is located in the orange region below the curves, then no other harmonics will be leaking. Finally, the surface blue (dark gray) region above the cutoff of the wave corresponds to leakage into space, as well as into both the and surface waves. The blue region may be divided and curves into various subregions, defined by the , though further discussion is and the TEM lines omitted here. Within the blue region, there is always one space harmonic that leaks into space, as well as into both the and surface waves. Depending on the location within the blue region, other harmonics may also be leaking as well, into space and/or some combination of the surface waves. In Fig. 4(b), the new complete Brillouin diagram is shown for a 1-D periodic structure as in Fig. 1(b) with parameters such . The parameters are the same as in Fig. 4(a), that except that the slab is now thicker. In this case, the normalsurface-wave mode (the horiized cutoff frequency of the zontal white line) occurs below the tip of the triangles and a new physical radiation region has to be considered (shown in black), which is discussed below. The white line in this case may intersect the TEM lines either above or below the intersection of the curves [the latter case is illustrated in TEM lines with the Fig. 4(b)]. Purely bound regions are again the hatched portion of the triangular regions (labeled B) delimited by the curves of surface wave and their periodic replicas (black solid the lines).

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Fig. 4. New Brillouin diagram that includes all the space harmonics for 1-D periodic printed structures [e.g., the periodically loaded microstrip line in Fig. 1(b)]. ; = k ; and = k (and their periodic replicas) are reported using black solid lines, black dashed–dotted lines, and black dashed The curves = k lines, respectively. In the following, colors refer to the color version of this figure available online, while between parentheses the relevant grayscale tonality for the print version is indicated. (a) Brillouin diagram for a periodically loaded microstrip line with parameters as in Fig. 3. Hatched areas labeled with B: bound-mode regions. Yellow (light gray) regions: TM surface-wave leakage only. Orange (medium gray) regions: simultaneous space and TM surface-wave leakage. Blue (dark gray) regions: simultaneous space, TM , and TE surface-wave leakage. (b) Same as (a), but for a periodically loaded microstrip line with spatial period p = 4 mm, on a homogeneous isotropic lossless grounded dielectric slab of thickness h = 1:270 mm with relative permittivity " = 10:2 and relative permeability  = 1. Black regions: simultaneous TM and TE surface-wave leakage only.

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The yellow (light gray) regions lie between the and curves (solid black lines and dashed–dotted black lines, respectively) within the triangles, and still correspond to surface-wave mode of the substrate, and not into leakage into only the space. The yellow regions may be divided into three sub regions curves, as already discussed in connection defined by the with Fig. 4(a). curves The new part of the triangular regions above the and up to the tips of the triangles (at ) is reported in black and corresponds to surface-wave leakage into both the and modes of the substrate, but not leakage into space. Within the black region, there is always one space harand modes of the submonic that leaks into both the strate (but not into space). Depending on the location within the black region, other harmonics may also be leaking into one or both of the surface waves. The orange (medium gray) and blue (dark gray) regions in Fig. 4(b) identify the same type of leakage regions as in Fig. 4(a). Hence, in summary, each color in Fig. 4 uniquely identifies a different type of leakage. There are four different leakage types

(colors) altogether. Based on the three categories of radiation surface wave, and surface wave), there are (space, eight different combinations of possible leakage types, but of these, only four (the ones discussed above corresponding to the four different colors) are physical. For example, for structures of the type shown in Fig. 1(b) it is not physical to have a mode surface wave since that leaks into space, but not into the surface wave is larger than the the wavenumber of the wavenumber of free space. Finally, it has to be noticed that by increasing frequency , additional radiation regions may be similarly depicted in this modified Brillouin diagram, where simultaneous radiation into space and other above cutoff higher order surface-wave modes occurs. These are not shown here, however, in order to avoid complicating the figure. IV. BRILLOUIN REGIONS AND INTEGRATION PATHS IN THE COMPLEX SPECTRAL PLANE Propagation regimes on 1-D periodic printed lines are analyzed here by using a rigorous mixed-potential integral-equa-

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tion (MPIE) formulation based on a spectral-domain approach [30]. This formulation has the advantage of providing a clear physical picture of the different modal regimes based on the mathematical location of poles and branch points in the complex spectral-variable plane. The new Brillouin diagram, which gives a clear indication of when the various bound and physical leakage mechanisms occur, is also shown here to directly determine the appropriate path of integration in the complex plane that is needed for any space harmonic in the spectral-domain analysis of such structures. The features of the Brillouin diagram and the discussion regarding the paths of integration may be applied to any modal analysis of 1-D periodic printed structures based on a spectral-domain method [7], [28], [29]. The full-wave numerical approach developed in [30] implements a mixed-potential 1-D periodic Green’s function, corresponding to a complex phase shift between the unit-cell edges, which may be obtained from the corresponding 2-D spectral-domain Green’s function by

Fig. 5. Adapted from [25]. Paths of integration for the nth harmonic in the complex k -plane that correspond to different propagation regimes in a spectraldomain analysis. Legend: paths are reported in gray (solid line on the proper sheet, dashed line on the improper sheet), pole singularities on the proper sheet are represented by a cross, branch points are represented by a dot, and branch cuts by a wavy line. (a) Passband regime. (b) and (c) Closed stopband regime or backward space- and/or surface-leaky regimes. The path in (b) corresponds to  < 0, while (c) corresponds to  > 0. (d) Forward space- and surfaceleaky regime. (e) Forward surface-leaky regime: leakage is into only the TM mode of the substrate. (f) Forward surface-leaky regime: leakage is into both the TM and TE modes of the substrate. When only the TM mode is above cutoff, only one pair of relevant pole singularities (one pair of crosses) is present on the proper sheet in (a)–(e).

(6) where denotes the 1-D periodic Green’s function for any particular potential component of interest, as a function of the and with planar spatial variables [30] [see (5)]. The function is the corresponding 2-D spectral-domain Green’s function for a nonperiodic source, which is known in closed form for a layered structure [30], [31]. A fundamental role of the new Brillouin diagram is its use in determining the appropriate path of integration in the complex plane for (6). In fact, different integration paths have to be chosen in relation to the different regions of the Brillouin diagram if one is interested in obtaining a physical solution. The necessity of associating different integration paths for the th space harmonic to different regimes for the wavenumber of the harmonic, as is well known, depends on the behavior in (6), which has branch of the spectral Green’s function points at and surface-wave pole singuwith and larities at , [7], [28]–[30]. When bound-mode regimes (passband and closed stopband regimes) are considered, the fields are expected to decay away from the structure, and the space harmonics are thus proper (decaying transversely). All the space harmonics are inside regions B of Fig. 4, and the spectral integration is carried out along the entire real axis on the proper sheet of the complex -plane for all terms in the summation of expansion (6), as shown in Fig. 5(a)–(c) [30]. When a leaky regime is considered, at least one of the space harmonics is radiating into space or leaking power into the above cutoff surface waves of the dielectric substrate, and of the mode is the fundamental propagation wavenumber complex. This leakage of power may be both of the forward

and backward type [30]. Here, we are considering leaky waves for with a power flow in the positive -direction so that the forward type of leakage and for the backward type. If the th space harmonic is in the backward surface-leaky regime, i.e., it radiates through the surface waves in the backward direction, the relevant normalized phase constant lies in region 1 of the Brillouin diagram of Fig. 3, i.e., , or in region 2, i.e., . If the th space harmonic is in the backward spaceleaky regime, i.e., it radiates into space and also into the abovecutoff surface waves in the backward direction, the relevant norlies in region 3 or 3 of the Brilmalized phase constant . In both backward louin diagram in Fig. 3, i.e., surface-leaky and space-leaky regimes, this harmonic is proper, with the modal field decreasing transversely along the - and -directions, and the integration is consistently carried out along plane the entire real axis on the proper sheet of the complex [see Fig. 5(b)]. When the th space harmonic is in a forward space-leaky regime, i.e., it radiates into space and into the above-cutoff surface waves in the forward direction, the corresponding normallies in region 4 or 4 of the Brillouin ized phase constant diagram in Fig. 3, i.e., the relevant path of integration crosses the branch cuts in the third and first quadrants, lying partly on the improper region of the -plane [see the dashed line in Fig. 5(d)] [28], [29]. The choice of the integration path is consistent with the improper (increasing transversely) nature of the forward radiation in regions 4 and 4 , where the modal field increases towards infinity along the - and -directions. Finally, if the th space harmonic is in a forward surface-leaky regime, i.e., it radiates only into the surface waves of the dielectric substrate in the forward direction, the relevant normalized phase constant lies in regions 5 or 6 of the Brillouin diagram in Fig. 3:

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Fig. 6. Top view of the unit cell of the periodically loaded microstrip of Fig. 1(b), showing the Delaunay mesh discretization used in the numerical simulation with the approach described in [30]. Relevant physical and geometrical parameters: " = 10:2;  = 1; h = 0:676 mm, and p = 4 mm.

or , respectively. The integration path is now on the proper sheet, but and surface wave poles it detours either around the pole (region 6), and cap(region 5) or only around the tures their residue contributions [28], [29], as shown in Figs. 5(f) and (e), respectively. The modal field is improper horizontally, growing towards infinity along the -direction, but proper vertically in the -direction.

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Fig. 7. Brillouin diagram for the n = 1 and n = 2 space harmonics of the fundamental EH mode of the structure shown in Fig. 6. Legend: n = 1 space harmonic: bound regime (dark gray solid line), backward leaky regime (dark gray dotted line), forward leaky regime (dark gray dashed–dotted line); n = 2 space harmonic: proper complex regime (light gray solid line with curves and their replicas; circles). The black solid lines are the = k the black dashed–dotted lines are the = k curves; the black dashed lines are the = k lines.

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V. NUMERICAL RESULTS In order to show the usefulness and validity of the new Brillouin diagram in the analysis and identification of the propagation and radiation behaviors of 1-D periodic printed structures, numerical results are provided for the structure shown in Fig. 1(b) by using the approach described in [30]. The metallization within the unit cell, for the 1-D periodically loaded microstrip line, consists of a microstrip line with “spurs” on either side, which are dipole elements connected to the line, as shown in detail in Fig. 6. Results will be presented for this typical structure, although the main features of the analysis and the discussion are quite general. We (dominant, quasi-TEM) mode examine here the perturbed of this structure, which becomes the usual microstrip line mode when the periodic load is absent. In Sections V-A–D, results are presented for different frequency regions, where the mode lies within different regions of the new Brillouin diagram. The path of integration in the complex -plane is chosen for each region according to the discussion in Section IV. It is observed from the results (shown in Fig. 7) that a very continuous curve is obtained, which smoothly crosses the boundary of the various regions. This illustrates that the paths of integration are the correct ones to use in order to realize a physical solution at all frequencies, and this reinforces the validity of the new Brillouin diagram. In Fig. 8, results are also shown for the normalized phased constant of the space harmonic, , and for the normalized attenuation constant , as a function of frequency to better understand the details of the different propagation and radiation regimes illustrated in the new Brillouin diagram. In the same figure, as validation of the results obtained with our approach, these results are compared with those obtained with a hybrid Bloch-wave analysis [32] that has been here applied by extracting the scattering parameters of five unit cells of the structure of Fig. 1(b) with Ansoft Designer. An excellent

Fig. 8. Normalized phase constant =k and attenuation constant =k as a function of frequency for the fundamental EH mode of the structure shown in Fig. 6. The frequency range between 10–42 GHz is considered. Legend: Normalized phase constant: proper real (black solid line), proper complex (black dashed–dotted line), improper complex (gray dashed–dotted line). Attenuation constant: proper (black dotted line), improper (gray dotted line). Circles: results obtained with the hybrid method described in [32].

agreement has been obtained for the phase constant in the entire frequency range and for the attenuation constant in the stopband regimes. However, it is worth noting that the hybrid method [32] fails to predict the correct behavior of the attenuation constant when it assumes very low values, as shown in [33]. A. Bound-Mode Regime At low frequencies, the mode is bound, the modal field is confined to the structure, and all the space harmonics, spaced , lie inside the regions B of the Brillouin diagram in by Fig. 4. In Fig. 7, the dark gray solid line corresponds to the space . The black harmonic in its bound region, i.e.,

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dashed line represents the edge of the first triangle, and the black mode of the substrate and solid lines are the curves of the their periodic replicas. As discussed above, two different propagation regimes are possible inside the bound region. In Fig. 7, ; it is shown that a passband regime occurs when . and a closed stopband regime is established for In both cases, the correct integration path in the -plane for the space harmonic is along the real axis and on the proper sheet [see Figs. 5(a) and (b)]. In Fig. 8, solid and dashed–dotted lines show the behavior of when the propagation wavenumber is real and complex, . In respectively, while a dotted line shows the behavior of ), we can observe the bound-mode regime (i.e., that the closed stopband region, from 12 to 14 GHz, is character(up to 0.23), while in the passband ized by high values of regions, below 12 GHz and from 14 to 18.9 GHz, , i.e., the propagation wavenumber is purely real (for all the space harmonics). B. Transition From Bound-Mode Regime to Backward Surface-Leaky and Backward Space-Leaky Regimes By increasing frequency, a bound mode is no longer possible; a transition to a backward leaky regime occurs, and the propagation wavenumber for all space harmonics becomes complex . We can observe in Fig. 7 that the solution for the space harmonic (dark gray dotted line) leaves region B, curve, and lies for a very narrow frecrosses the quency range in region 1 (backward surface-leaky regime), i.e., . As the frequency continues to increase, the mode still remains complex; the solution of the har, monic, reported as a dark gray dotted line up to , and enters region 3 (backward crosses the line . In both cases, the space-leaky regime), i.e., correct integration in (6) is still along the real axis on the proper sheet [see Fig. 5(b)]. In Fig. 8, we can observe that the propagation wavenumber of harmonic becomes complex after 18.9 GHz (black the dashed–dotted line for ) and that, within this complex reassumes low values up to the gion, the attenuation constant second bump around 25 GHz. In Fig. 9, on an enlarged scale, we can see a sharp change in the behavior of the normalized leakage , black dotted curve) at the transition between constant ( space harmonic the real and complex values of the , black solid and dashed–dotted curves). When ( is below the normalized phase constant of the mode (gray solid line), the solution is proper and real . By increasing the frequency, the wavenumber crosses the curve for surface wave and the solution then becomes complex the with , and there is backward leakage into the surface wave. Shortly after becoming complex, there is a well-pronounced maximum in the attenuation constant. At a slightly crosses the gray dashed higher frequency, the curve of line at , and there is now backward leakage into free space, as well as into the surface wave. C. Open Stopband Region (Broadside) When a 1-D open periodic structure radiates at broadside, it produces an open stopband at exactly broadside, causing well-

Fig. 9. Same as in Fig. 8, but plotted on an enlarged scale between 18–23 GHz. In addition, two new curves have been added: The gray solid line is the curve; the gray dashed line is the k k curve.

0

=0

=

known radiation problems [17], [34]. For the structure shown in Fig. 5, for which the Brillouin diagram is presented in Fig. 7, (at the junction of the condition for broadside is the dotted and the dashed–dotted dark gray lines in Fig. 7). In the companion plot in Fig. 8, we can observe the behavior of the as a function of frequency. normalized attenuation constant The open stopband, which occurs in the frequency range between approximately 24–27 GHz in Fig. 8, appears to be similar to the closed stopband, which is located in the frequency range between approximately 12–14 GHz in Fig. 8. Both of these two stopbands arise from the contra-directional coupling between space harmonic and another space harmonic for the which the power flow occurs in the opposite longitudinal direction. We must note from Fig. 7, however, that the closed stop, in region B, within which all band occurs at solutions are purely bound, as opposed to the open stopband, which lies within the radiation region. Due to this radiation, the propagation wavenumber remains complex outside of the central stopband region (as well as inside it). This key feature leads to several characteristic and significant differences between the closed and open stopbands, despite the evident similarities between them. In this particular example, these differences are not pronounced, as they are in other cases, because the attenuation constant appears to be very small (but not zero) outside of this central region. The tiny values for the attenuation constant outside of the central region become evident when looking at Figs. 8, 9, and 10(b). In the open stopband region, a serious degradation of the radiation properties usually occurs. In Fig. 10(a), an enlarged plot of the Brillouin diagram for space harmonic is reported in the neighborhood of the the open stopband. For the same frequency range, Fig. 10(b) and leakage shows the normalized phase constants as a function of frequency. The black lines indicate the proper branch of the complex solution, whereas the gray lines the improper. An almost vertical (horizontal) behavior of around 0 and a maximum of the attenuation constant can be observed. Moreover, the change of determination of the modal solution occurs exactly when

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(a) (a)

(b)

(b)

Fig. 10. (a) Brillouin diagram in the neighborhood of the open stopband for the n 1 space harmonic of the fundamental EH mode of the structure in Fig. 6. Legend: n = 1 space harmonic: backward leaky regime (black solid line), forward leaky regime (gray solid line). (b) Normalized phase constant =k and attenuation constant =k as a function of frequency for the fundamental EH mode of the structure in Fig. 6. The frequency range between 22–29 GHz in the neighborhood of the open stopband is considered. Legend: Normalized phase constant: solid line. Attenuation constant: dotted line. Proper branch: black lines. Improper branch: gray lines.

Fig. 11. (a) Brillouin diagram in the neighborhood of the transition from forward space-leaky to forward surface-leaky regimes for the n = 1 space harmonic of the fundamental EH mode of the structure in Fig. 6. Legend: n = 1 space harmonic: forward space-leaky regime (black solid line), forward surface-leaky (TE and TM ) regime (dark gray solid line), forward surface-leaky (TM ) regime (light gray solid line with squares). (b) Normalized phase constant =k and attenuation constant =k as a function of frequency for the fundamental EH mode of the structure in Fig. 6. The frequency range between 30–42 GHz is considered. Legend: Normalized phase constant: solid line. Attenuation constant: dotted line. Forward space-leaky regime: black lines. Forward surface-leaky (TE and TM ) regime: dark gray lines. Forward surface-leaky (TM ) regime: light gray lines with squares. The black dashed line is the = +k line; the black solid line with circles is the = +k curve; the black dashed–dotted line is the = +k curve.

=0

0

[ at 24.7 GHz, as shown in Fig. 10(b)]. Here, changes sign and enters region 4 of the Brillouin diagram, while the radiated beam moves from the backward into the becomes positive (gray lines forward quadrant. When in Fig. 10), the correct integration path corresponding to the physical continuous solution is that shown in Fig. 5(d). D. Transition From Forward Space-Leaky Regime to Forward Surface-Leaky Regimes By increasing frequency, a further coupling region is obin Fig. 7, characterized by the third served, when

0

0

bump in the attenuation constant between 34–39 GHz (see Fig. 8). Here, the phase constant of the space harmonic crosses line (as shown in Fig. 7), then enters region 5 the (and then 6), resulting in forward leakage into both the and surface waves (and then only into the surface wave), but no longer leakage into free space. (On the scale of Fig. 7, it is not clear that the solution first enters region 5 before entering region 6, but this is made clear in Fig. 11.) The

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integration path corresponding to the physical solution is that shown in Fig. 5(f) [or Fig. 5(e)]. space In Fig. 11(a) and (b), a detailed plot of the harmonic on the Brillouin diagram and the detailed frequency and , respectively, are shown around behaviors of the transition from the forward space-leaky regime to the forward surface-leaky regime. After the coupling occurs, the forward space-leaky solution undergoes a transition to forward surand modes of the dielectric subface leakage into the strate [dark gray lines in Fig. 11(a) and (b), obtained by using the integration path in Fig. 5(f)] Finally, a further transition to mode is shown [light forward surface leakage into only the gray lines with squares in Fig. 11(a) and (b), obtained by using the integration path in Fig. 5(e)]. At higher frequencies, simultaneous radiation from the and space harmonics is easily observable in the new Brillouin diagram in Fig. 7. In fact, it should be noted that, space harmonic leaves before the phase constant of the space regions 4 and 4 , by entering regions 5 and 6, the harmonic starts to give a radiative contribution. It first radiates surface wave of the substrate in the backward diinto the (light rection, as is shown in Fig. 7 when the phase constant gray solid line with circles) enters region 1 of the new Brillouin space harmonic diagram. By increasing frequency, the then reaches regions 2 and 3 of the Brillouin diagram, thus and surgiving rise to backward radiation into the face waves of the dielectric substrate and to backward radiation and surface waves of both into free space and into the the substrate, respectively. The corresponding integration path for this space harmonic in the plane is that shown in Fig. 5(b). VI. CONCLUSION In this study, an investigation into the dispersion and leakage properties of bound and leaky modes supported by 1-D periodic printed structures on grounded dielectric substrates has been performed. Such structures, which have a wide range of applications, including filters, antennas, and novel metamaterial devices, can be divided into two different categories. One is infinite in the transverse direction (perpendicular to the direction of propagation) and is characterized by radiation that is allowed only into free space. The other type is finite in the transverse direction and may also radiate into the surface waves supported by the dielectric substrate. The former type can be described using the conventional Brillouin diagram, whereas for the latter type, a new Brillouin diagram has been introduced to aid in the analysis and physical interpretation of the modal radiation properties. This new Brillouin diagram sheds light on the various radiation regions that are possible, including both surface- and space-wave leaky regimes. In addition to providing this physical insight in a convenient manner, the new Brillouin diagram also directly determines the correct path of integration to use in the transverse complex plane for each of the space harmonics when using a spectral-domain method to analyze the modal propagation. Surface and space leaky waves of both the proper and improper kinds may occur, corresponding to backward and forward radiation, respectively. Numerical results for a specific structure have been obtained and presented on the new

Brillouin diagram, illustrating the use of the diagram in understanding the important radiation characteristics of the structure. REFERENCES [1] V. Radisic, Y. Qian, and T. Itoh, “Novel architectures for high-efficiency amplifiers for wireless applications,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 11, pp. 1901–1909, Nov. 1998. [2] F. R. Yang, K. P. Ma, Y. Qian, and T. Itoh, “A uniplanar compact photonic-bandgap (UC-PBG) structure and its applications for microwave circuits,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 8, pp. 1509–1514, Aug. 1999. [3] C. K. Wu, H. S. Wu, and C.-K. C. Tzuang, “Electric-magnetic-electric (EME) slow-wave microstrip line and bandpass filter of compressed size,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 8, pp. 1996–2004, Aug. 2002. [4] L. Zhu, “Guided-wave characteristics of periodic coplanar waveguides with inductive loading—Unit-length transmission parameters,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 10, pp. 2133–2138, Oct. 2003. [5] S. Sun and L. Zhu, “Guided-wave characteristics of periodically nonuniform coupled microstrip lines—Even and odd modes,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 4, pp. 1221–1227, Apr. 2005. [6] A. A. Oliner, Antenna Engineering Handbook, R. C. Johnson, Ed. New York: McGraw-Hill, 1993, ch. 10: Leaky-Wave Antennas. [7] P. K. Potharazu and D. R. Jackson, “Analysis and design of a leakywave EMC dipole array,” IEEE Trans. Antennas Propag., vol. 40, no. 8, pp. 950–958, Aug. 1992. [8] M. Guglielmi and D. R. Jackson, “Broadside radiation from periodic leaky-wave antennas,” IEEE Trans. Antennas Propag., vol. 41, no. 1, pp. 31–37, Jan. 1993. [9] D. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2059–2074, Nov. 1999. [10] L. Liu, C. Caloz, and T. Itoh, “Dominant mode leaky-wave antenna with backfire-to-endfire scanning capability,” Electron. Lett., vol. 38, pp. 1414–1416, Nov. 2002. [11] A. Grbic and G. V. Eleftheriades, “Experimental verification of backward-wave radiation from a negative refractive index metamaterial,” J. Appl. Phys., vol. 92, pp. 5930–5935, Nov. 2002. [12] C. Caloz, A. Sanada, and T. Itoh, “A novel composite right/left-handed coupled-line directional coupler with arbitrary coupling level and broad bandwidth,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 3, pp. 980–992, Mar. 2004. [13] L. Brillouin, Wave Propagation in Periodic Structures. New York, NY: Dover, 1953. [14] A. A. Oliner and A. Hessel, “Guided waves on sinusoidally modulated reactance surfaces,” IRE Trans. Antennas Propag., vol. AP-7, no. 12, pp. 201–208, Dec. 1959. [15] A. A. Oliner, Radiating periodic structures: Analysis in terms of k versus diagrams. Jun. 4, 1963, short course on microwave field and network techniques. [16] R. A. Sigelmann and A. Ishimaru, “Radiation from periodic structures excited by an aperiodic source,” IEEE Trans. Antennas Propag., vol. AP-13, no. 5, pp. 354–364, May 1965. [17] R. E. Collin and F. J. Zucker, Antenna Theory. New York, NY: McGraw-Hill, 1969, ch. 19 (by A. Hessel) and 20 (by T. Tamir). [18] S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microw. Theory Tech., vol. MTT-23, no. 1, pp. 123–133, Jan. 1975. [19] C. Elachi, “Waves in active and passive periodic structures: A review,” Proc. IEEE, vol. 64, no. 12, pp. 1666–1698, Dec. 1976. [20] S. Majumder, D. R. Jackson, A. A. Oliner, and M. Guglielmi, “The nature of the spectral gap for leaky waves on a periodic strip-grating structure,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 12, pp. 2296–2307, Dec. 1997. [21] H.-Y. D. Yang and D. R. Jackson, “Theory of line-source radiation from a metal-strip grating dielectric-slab structure,” IEEE Trans. Antennas Propag., vol. 48, no. 4, pp. 556–564, Apr. 2000. [22] P. Burghignoli, P. Baccarelli, F. Frezza, A. Galli, P. Lampariello, and A. A. Oliner, “Low-frequency dispersion features of a new complex mode for a periodic strip grating on a grounded dielectric slab,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 12, pp. 2197–2205, Dec. 2001.

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[23] P. Baccarelli, P. Burghignoli, F. Frezza, G. Lovat, A. Galli, P. Lampariello, and S. Paulotto, “Modal properties of surface and leaky waves propagating at arbitrary angles along a metal strip grating on a grounded slab,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 36–46, Jan. 2005. [24] Y.-C. Chen, C.-K. C. Tzuang, T. Itoh, and T. K. Sarkar, “Modal characteristics of planar transmission lines with periodical perturbations: Their behaviors in bound, stopband, and radiation regions,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 47–58, Jan. 2005. [25] P. Baccarelli, S. Paulotto, D. R. Jackson, and A. A. Oliner, “Analysis of printed periodic structures on a grounded substrate: A new Brillouin dispersion diagram,” in IEEE MTT-S Int. Microw. Symp. Dig. , Long Beach, CA, Jun. 12–17, 2005, pp. 1913–1916. [26] R. E. Collin, Field Theory of Guided Waves, 2nd ed. Piscataway, NJ: IEEE Press, 1991, ch. 9. [27] A. A. Oliner, “Leakage from higher modes on microstrip line with application to antennas,” Radio Sci., vol. 22, pp. 907–912, Nov. 1987. [28] J. S. Bagby, C.-H. Lee, D. P. Nyquist, and Y. Yuan, “Identification of propagation regimes on integrated microstrip lines,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 11, pp. 1887–1894, Nov. 1993. [29] F. Mesa, C. Di Nallo, and D. R. Jackson, “The theory of surface-wave and space-wave leaky-mode excitation on microstrip lines,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 2, pp. 207–215, Feb. 1999. [30] P. Baccarelli, C. Di Nallo, S. Paulotto, and D. R. Jackson, “A full-wave numerical approach for modal analysis of 1-D periodic microstrip structures,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 4, pp. 1350–1362, Apr. 2006. [31] L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. Englewood Cliffs, NJ: Prentice-Hall, 1973. [32] S.-G. Mao and M.-Y. Chen, “Propagation characteristics of finite-width conductor-backed coplanar waveguides with periodic electromagnetic bandgap cells,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 11, pp. 2624–2628, Nov. 2002. [33] P. Baccarelli, C. Di Nallo, S. Paulotto, and D. R. Jackson, “Full-wave analysis of 1-D periodic microstrip leaky-wave antennas,” in IEEE AP-S Int. Symp. Dig., Albuquerque, NM, Jul. 9–14, 2007, pp. 4259–4262. [34] T. Kokkinos, C. D. Sarris, and G. V. Eleftheriades, “Periodic FDTD analysis of leaky-wave structures and applications to the analysis of negative-refractive-index leaky-wave antennas,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 4, pp. 1619–1630, Apr. 2006.

Paolo Baccarelli (S’96–M’01) received the Laurea degree in electronic engineering and Ph.D. degree in applied electromagnetics from “La Sapienza” University of Rome, Rome, Italy, in 1996 and 2000, respectively. In 1996, he joined the Department of Electronic Engineering, “La Sapienza” University of Rome, where since 2000 he has been an Associate Researcher. From April 1999 to October 1999, he was a Visiting Scholar with the University of Houston, Houston, TX. His research interests concern analysis and design of planar leaky-wave antennas, numerical methods, periodic structures, and propagation and radiation in metamaterials and anisotropic media.

Simone Paulotto (S’97–M’07) received the Laurea degree (cum laude and honorable mention) in electronic engineering and Ph.D. degree in applied electromagnetics from “La Sapienza” University of Rome, Rome, Italy, in 2002 and 2006, respectively. In 2002, he joined the Electronic Engineering Department, “La Sapienza” University of Rome, where he is currently an Associate Researcher. From November 2004 to April 2005, he was a Visiting Scholar with the University of Houston, Houston, TX. His scientific interests include analysis and design of planar leaky-wave antennas, guidance and radiation phenomena in metamaterial structures, periodic structures, scattering theory, and electromagnetic characterization of materials.

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David R. Jackson (S’83–M’84–SM’95–F’99) was born in St. Louis, MO, on March 28, 1957. He received the B.S.E.E. and M.S.E.E. degrees from the University of Missouri, Columbia, in 1979 and 1981, respectively, and the Ph.D. degree in electrical engineering from the University of California at Los Angeles, in 1985. From 1985 to 1991, he was an Assistant Professor with the Department of Electrical and Computer Engineering, University of Houston, Houston, TX, where from 1991 to 1998, he was an Associate Professor, and since 1998, he has been a Professor. He has also served as an Associate Editor for Radio Science, the International Journal of RF, and Microwave Computer-Aided Engineering. His current research interests include microstrip antennas and circuits, leaky-wave antennas, leakage and radiation effects in microwave integrated circuits, periodic structures, and electromagnetic compatibility (EMC). Dr. Jackson is currently the chair for the International Union of Radio Science (URSI) U.S. Commission B and the chair of the Transnational Committee of the IEEE Antennas and Propagation Society (IEEE AP-S). He is also on the Editorial Board for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. He was the Chapter activities coordinator for the IEEE AP-S, a distinguished lecturer for the IEEE AP-S, an associate editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, and a member of the Administrative Committee (AdCom) for the AP-S. Arthur A. Oliner (M’47–SM’52–F’61–LF’87) was born on March 5, 1921, in Shanghai, China. He received the B.A. degree from Brooklyn College, Brooklyn, NY, in 1941, and the Ph.D. degree from Cornell University, Ithaca, NY, in 1946, both in physics. He joined the Polytechnic Institute of Brooklyn (now Polytechnic University) in 1946, and became Professor in 1957. He then served as Department Head from 1966 to 1974, and was Director of its Microwave Research Institute from 1967 to 1982. He was a Walker-Ames Visiting Professor with the University of Washington in 1964. He has also been a Visiting Professor with the Catholic University, Rio de Janeiro, Brazil, the Tokyo Institute of Technology, Tokyo, Japan, the Central China Institute of Science and Technology, Wuhan, China, and the University of Rome, Rome, Italy. He is a member of the Board of Directors of Merrimac Industries. He has authored over 300 papers, various book chapters, and has coauthored or coedited three books. His research has covered a wide variety of topics in the microwave field, including network representations of microwave structures, guided-wave theory with stress on surface waves and leaky waves, waves in plasmas, periodic structure theory, and phased-array antennas. He has made pioneering and fundamental contributions in several of these areas. His interests have also included waveguides for surface acoustic waves and integrated optics, novel leaky-wave antennas for millimeter waves, and leakage effects in microwave integrated circuits. He has recently contributed to the topics of metamaterials and to enhanced propagation through subwavelength holes. Dr. Oliner is a Fellow of the American Association for the Advancement of Science (AAAS), Washington, DC, and the Institution of Electrical Engineers (IEE), London, U.K. He was a Guggenheim Fellow. He was elected a member of the National Academy of Engineering in 1991. He was the recipient of prizes for two of his papers: the IEEE Microwave Prize in 1967 for his work on strip line discontinuities, and the Institution Premium of the IEE in 1964 for his comprehensive studies of complex wave types guided by interfaces and layers. He was President of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S), its first distinguished lecturer, and a member of the IEEE Publication Board. He is an Honorary Life Member of the IEEE MTT-S (one of only six such persons). In 1982, he was the recipient of its highest recognition, the Microwave Career Award. A special retrospective session was held in his honor at the IEEE MTT-S International Microwave Symposium (IMS) (reported in detail in this TRANSACTIONS, Dec. 1988, pp. 1578–1581). In 1993, he became the first recipient of the Distinguished Educator Award of the IEEE MTT-S. He was also a recipient of the IEEE Centennial and Millennium Medals. He is a past U.S. Chairman of Commissions A and D of the International Union of Radio Science (URSI), a long time member of and active contributor to Commission B, and a former member of the U.S. National Committee of URSI. In 1990, he was the recipient of the URSI van der Pol Gold Medal, which is given triennially, for his contributions to leaky waves. In 2000, the IEEE awarded him a second gold medal, the Heinrich Hertz Medal, which is its highest award in the area of electromagnetic waves. In 2003, “La Sapienza”, University of Rome granted him an Honorary Doctorate, and organized an associated special symposium in his honor.

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A Nonlinear Finite-Element Leaky-Waveguide Solver Peter C. Allilomes and George A. Kyriacou, Senior Member, IEEE

Abstract—A novel hybrid finite-element method for the analysis of leaky-waveguide structures is presented. The possibly radiating structure is enclosed within a fictitious circular contour- in order to truncate the infinite solution domain. The field in the unbounded domain, outside contour- , is expressed as a superposition of transverse electric and magnetic modes. Their radial dependence is expressed in terms of Hankel functions, which satisfy the radiation condition at infinity. The bounded area is discretized using hybrid node/edge elements for an accurate and efficient handling of the electric field vector wave equation. The transparency of the fictitious contour is ensured by enforcing the field continuity conditions according to the principles of a vector Dirichlet-to-Neumann mapping. The whole procedure yields a nonlinear eigenvalue problem for the complex axial propagation constant ( ). The nonlinearity is due to the appearance of within the argument of the Hankel functions. The final nonlinear problem is solved by employing a matrix Regula–Falsi algorithm. Initial guesses for the Regula–Falsi algorithm and a fast estimation of the eigenvalues spectrum are provided by a linear formulation 2 1. The based on a second-order approximation ( 0) proposed method is validated against published numerical and experimental results for both leaky and surface wave modes. Index Terms—Cylindrical harmonics, Dirichlet-to-Neumann mapping, finite elements, leaky waves, nonlinear eigenvalues, open waveguides, surface waves.

I. INTRODUCTION HE MODAL characteristics of waveguiding structures are very important for the essential understanding of wave propagation and particularly the energy leakage from them. The analysis of closed waveguides where there is no leakage—radiation of electromagnetic energy, is well established in electromagnetics. Analytical solutions for almost every canonical cross section can be found in textbooks, e.g., [1]. Besides, for arbitrary cross-sectional closed waveguides, either air filled or loaded with inhomogeneous and/or anisotropic materials, robust numerical techniques have been successfully applied, e.g., [2] and [3]. The study of open waveguides and particularly the analysis of their leaky modes is well developed only for canonical geometries through analytical or approximate techniques. The most common technique employed for this purpose, as early as 1960, is the “transverse network representation” [4]–[6]. Semianalytical techniques like the “method of analytical regularization” are also extensively used for canonical geometries [7], [8], but,

T

Manuscript received January 10, 2007; revised April 4, 2007. This work was supported by the Greek Ministry of Education and by the European Union under Framework Program FP3. The authors are with the Department of Electrical and Computer Engineering, Microwave Laboratory, Democritus University of Thrace, GR-67100 Xanthi, Greece (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.2007.900306

for arbitrary cross-sectional open waveguides, possibly loaded with inhomogeneous and/or anisotropic materials, the employment of full wave numerical methods is inevitable. The application of these methods in the case of the modal analysis of leaky waves is not matured and there are still many difficulties and open issues. In contrary, in source-driven problems like antenna modeling or scattering, they are very well fitted and established, [9], [10]. The major difficulty in the modal analysis of open waveguides and the computation of their propagation constants is the inherent nonlinear nature of the problem. The nonlinearity is introduced through the argument of the modes eigenfunctions since they include the unknown eigenvalue (complex propagation constant). When analytical and/or approximate methods are employed, the nonlinearity is efficiently handled. These techniques end up as a very small system of nonlinear algebraic characteristic equations, solvable within acceptable computational time. In pure numerical methods, this nonlinearity leads to large nonlinear matrix eigenvalue problem. Advanced and computationally costly iterative algorithms, in conjunction with very good initial guesses, are required for the solution of this kind of large dimension nonlinear problems. Numerous attempts to apply full wave numerical techniques for the solution of the above described nonlinear problem can be found in the literature. An effort to apply the method of moments (MoM), for the specific geometry of a laterally shielded top open planar transmission line, is presented in [11]. Even nondirectly cited in [11], their method yields a nonlinear eigenvalue problem in terms of the unknown complex propagation constant of the leaky-wave mode. The completely closed structure provided the starting solution to an iterative scheme handling the nonlinearity. Subsequently, they gradually changed the top wall impedance to the final Marcuvitz radiation impedance through iterations of multiple small steps. Unfortunately, the method is only applicable to restricted type of geometries and it is not general. Many researchers have also tried to calculate dispersion curves by applying more general methods like the finite-element method. Most of these efforts are oriented toward the analysis of optical fibers using full wave 2-D formulations. The leakage in optical fibers is very small and is also known that only the transverse components of the fields are considerably strong. Thus, formulations considering only these transverse components are presented, e.g., [12]. Certain efforts were made to avoid the nonlinearity by the employment of perfect matching layer (PML)-based techniques [13]. In this case, the nonlinearity appears indirectly through the optimization of the PML parameters. This kind of optimization cannot be made fully computerized and mostly depends on the experience of the researcher. Besides that, the major drawback of the PML-based techniques analyzing open waveguides is the occurrence of

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II. FORMULATION The general geometry of Fig. 1 is considered. It is an arbitrary cross-sectional open waveguiding structure loaded with inhomogeneous material. The structure is assumed uniform along the propagation axis- . Time harmonic field dependence is assumed as along with a wave propagation as . Based on these considerations, the classical corresponding simplifiand are introduced into cations Maxwell equations. Note that herein is assumed complex in order to account for leaky-wave modes. Following the well-established technique presented in [2], the electric field vector wave equation yields a system of equations for the transverse and longitudinal components, namely, (1) (2) Introducing the above relations into the electric field vector wave equation, the resulting coupled expressions reads Fig. 1. General waveguide geometry.

(3) nonphysical “Berenger modes” [10]. The same indirect nonlinearity is an inherent property of infinite elements techniques. This is justified by the optimization needed to estimate the unknown decay parameters of the infinite elements [14]. The formulation presented in this study is based on the Dirichlet-to-Neumann mapping technique, first introduced by Keller and Givoli [15]. It is a well established artificial transparent boundary technique able to extend the finite-element method in problems involving semi-infinite unbounded solution domains. According to [15] and [16], the Dirichlet-to-Neumann mapping constitutes a nonlocal transparent boundary condition, which has the inherent property to be “exact.” It was successfully applied in scattering and radiation problems governed by the scalar Helmholtz equation [17], [18]. An extension of the method to the vector wave equation for scattering problems was first presented by Frenni et al. [19]. To the authors’ knowledge, the formulation of a nonlinear full vector eigenvalue problem for leaky waveguides, systematically established on a vector Dirichlet-to-Newmann mapping, is presented herein for the first time. The current method is able to compute dispersion curves for arbitrary shaped open waveguides loaded with inhomogeneous material (Fig. 1). According to [1] and [20], the complex propagation constant of leaky modes provides information on the angle—direction of the radiated beam maximum and its beamwidth in the elevation plane (along the propagation axis). The nonlinearity of the problem is handled by the introduction of a two-step procedure. The approximate linear formulation, developed in our previous studies [21]–[23], is first employed to provide initial guesses for the subsequent iterative solution of the accurate nonlinear formulation derived herein. An improved version of the linear as well as the new nonlinear formulations are presented. In turn, the solution procedure of the nonlinear eigenvalue problem is described. The overall technique is validated through comparisons against numerical and experimental results available in published studies.

(4) where is the free-space wavenumber, is the speed and of light in free space, and are the inhomogeneous relative permittivity and permeability, respectively. The system of (3) and (4) enables the formulation of a full vector finite-element eigenvalue problem for the unknown propagation constants. It worth noting that the coupling of axial and transverse components observed in differential equation (3) and (4) does not pose any difficulty or complication to their finite-element formulation and solution. In contrary, this coupling constitutes a major difficulty whenever their analytical solution is sought. The standard Galerkin procedure is then applied to (3) and (4) yielding the following weak formulation, e.g., [24]:

(5)

(6)

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where is the vector weighting function and denotes the area of region II, which is enclosed by the artificial circular contour- . The contour- is introduced in order to truncate the infinite solution domain according to the Dirichlet-to-Neumann mapping technique. In principle, the finite-element method can be applied at this point for any given solution domain as long as the evaluation of and in (5) and (6) is feasible. In the two contour integrals and coincides with the general, the integration contour of boundary of the solution domain. In the case of closed boundary problems, the solution domain is finite and its boundary coincides with perfect magnetic or electric walls enclosing the waveand are equal guide. In this case, the boundary integrals to zero. Since an open, possibly radiating, waveguiding structure is considered (Fig. 1), the solution domain is essentially extended to infinity. Thus, the electromagnetic field should satisfy the physical requirement of outward power flow. Thus, the solution domain should be appropriately truncated in order to apply the finite-element method. Most of the finite-element method implementations solving open problems consider the introduction of a closed fictitious boundary enclosing the structure to be analyzed. The finite-element method is employed only within the enclosed area. Besides that, this fictitious boundary should be “transparent” to the field solution. Namely, it should ideally not disturb the field distribution by any means. Equivalently, it should not reflect any outgoing wave incident on that in any direction from normal to grazing incidence. Many approaches have been proposed for the realization of this “transparent boundary” and their strengths and drawbacks are already referred to in Section I. The current method falls into a class of implementations where the electromagnetic field outside the fictitious boundary is expanded into a superposition of functions, which are solutions of Maxwell equations fulfilling the physical requirements of outward power flow. Specifically, for surface wave modes, these functions should obey the Sommerfeld radiation condition at infinity, while for leaky-wave modes, the radial wavenumber involved in their argument will be defined according to the corresponding theory, e.g., [25]. In turn, the “transparency” of the fictitious boundary is realized by enforcing on that the continuity conditions between the finite-element method solution in the bounded domain and the field expansion in the unbounded domain. These continuity conditions should be applied following the Dirichlet-to-Neumann mapping mathematical formalism. A. Vector Dirichlet to Neumann Method The Dirichlet-to-Neumann mapping was first introduced by Keller and Givoli [15]. According to Keller and Givoli [15] and Grote [16], the Dirichlet-to-Neumann mapping constitutes a nonlocal transparent boundary condition, which has the inherent property to be “exact.” Contrarily, numerical methods usually employ “local” boundary conditions, which are approximate. Namely, Dirichlet-to-Neumann mapping may yield a highly accurate solution provided that other approximations are not involved in the numerical technique. Boundary conditions derived from Dirichlet-to-Neumann mapping are based on the introduction of an artificial boundary surface of a canonical shape. Thus, the artificial contour- in Fig. 1 is of circular shape. This boundary encloses all the geometrical irregularities, source terms, material anisotropy, and inhomogeneities.

The regular shape of the fictitious boundary and the homogeneity of the surrounding space enables an analytical solution for the radiating field. For the construction of this analytical solution, arbitrary Dirichlet data on the boundary- are assumed. In the case of an electric field vector wave equation, these types of data are values of the tangential electric field. The constructed analytical solution is then used to establish a relation between the field on the boundary and its derivatives. This kind of relation is called a Dirichlet-to-Neumann map. Inside the boundary, the finite-element method is employed, handling all the irregularities. The binding between the finite-element solution and the analytical expressions is carried out into two steps. During the first step, the finite-element method is applied to determine the initially arbitrary Dirichlet data. This is equivalent to the enforcement of the tangential electric field continuity between the two solutions. During the second step, the Dirichlet-to-Neumann map, provided by the analytical solution, is used for the and . This is identical evaluation of the boundary integrals to imposing the tangential magnetic field continuity conditions. In turn, this is also equivalent to imposing the continuity of the tangential components. This come electric field curl as a consequence of the classical duality inherent in Maxwell equations. It can also be deduced from the relations between the magnetic field and the curl of the electric field. The above enables the construction of a Dirichlet-to-Neumann map for the vector wave equation governing the current problem. Below, every step of the vector Dirichlet-to-Neumann mapping is presented in detail. 1) Construction of the Analytical Solution in Region I: Since the medium outside the boundary- is homogeneous (air) and source free, the field can be expressed as a superposition of TM and TE modes. These modes constitute a complete set of vector solutions of the Maxwell equations, e.g., Jackson [26]. Thus, a general solution for the axial components can be considered for , and for . as In order for this to be consistent with the introduced artificial circular contour- , the appropriate cylindrical wave functions obeying the radiation condition at infinity should be adopted; to represent namely, Hankel functions of the second kind outward propagating waves (leaky waves) or McDonald functo represent radially attenuated waves (surface waves). tions The choice of Hankel functions within this study are made since the primary aim is to focus upon leaky waves rather than surface waves. Due to the periodicity in the azimuthal -direction, the corresponding wave function could be either sinusoidal or expowith azimuthal nential. The latter was selected herein as to for convenience. Moreharmonic index over, due to the complexity of the structure it is generally exand modes will be excited. pected that all possible , and can be Thus, the corresponding axial components expressed as a superposition of these modes as (7) (8) where is the radial wavenumber. Following the classical analysis of cylindrical waveguides, the transverse components can be expressed in terms of the axial

ALLILOMES AND KYRIACOU: NONLINEAR FINITE-ELEMENT LEAKY-WAVEGUIDE SOLVER

components. This is done by expanding the Maxwell curl equations in cylindrical coordinates and solving for the desired transverse components, e.g., [27]

(9)

(10)

(11)

(12) Note that the media outside the contour- is air, thus and . The primed Hankel function denote derivatives with respect to their argument . Special attention should be devoted to the argument of the Hankel funcsince this is the cause of the nonlinearity. Explicitly, tion , thus the argument of the Hankel functions de. pends on the unknown eigenvalue 2) Finite-Element Field Formulation in Region II: The next step toward the application of the Dirichlet-to-Neumann mapping technique is the employment of the finite-element method inside the fictitious boundary- (region II). Due to the vector type of the weak formulation stated in (5) and (6), it is necessary to use a mixed-type node/edge triangular element, e.g., [2] and [24]. In these types of elements, the component of the field is discretized by means of parallel to the propagation axis the degrees of freedom assigned on the nodes of the elements. are, respectively, discretized The transverse components exploiting the vector type degrees of freedom assigned on the edges of the element. This discretization strategy of the full vector wave equations in a 2-D solution domain eliminates the spurious solutions problem. The latter was common when only node elements were considered [28]. In this study, the first-order interpolation functions proposed in [29] are used for both the scalar and vector interpolation functions. The fictitious boundary- is approximated by a canonical polygon of more than 100 edges. Further enhancement of the discretization accuracy is expected through the employment of locally curvilinear elements. This is justified by the cylindrical nature of the waves represented by the 2-D Hankel functions. These, in turn, ask for a mathematically circular boundary-

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as the ideal interface to match the modal expansion to the finite-element solution. Any deformation from the ideal circular contour is expected to introduce inaccuracies in the calculated eigenvalues. In view of this, the linear-segment discretization adopted in the current implementation for its compatibility with the edge elements will cause some inaccuracies. Currently this inaccuracy is reduced to an acceptable level by a proper selection of the number of edges comprising the canonical polygon, which approximates the circular contour. The number of edges and its relation to the truncation number of the infinite sum of cylindrical harmonics will be discussed below. This canonical polygon is also discretized by using mixed-type node/edge line elements of the first order. The discretized contour coincide with the boundary edges of the 2-D triangular mesh. Recall that the discretization accuracy of boundary interfaces strictly defines the accuracy of the evaluated eigenvalues (propagation constants). The deleterious effects of staircase approximation of curved surfaces contours whenever a rectangular grid is employed are very well documented in the published literature, e.g., [30]. 3) Binding the Finite-Element Method and the Cylindrical Harmonics Expansion: The electromagnetic field in the two regions I and II (Fig. 1) is expressed above through the cylindrical harmonics and with the aid of the finite-element method, respectively. Binding together these expressions along the fictitious contour- is expected to yield the desired eigenvalue problem for the unknown complex propagation constants. However, this is actually the critical step where the desired “transparency” should be safeguarded. For this purpose, a combination of an electromagnetic point of view and a strict application of the Dirichlet-to-Neumann mapping approach should be followed. Starting from the electromagnetism point of view and since the contour- is an artificial one (not an actual boundary), the continuity of all six electromagnetic field components should be generally enforced along- . Besides that, enforcing only the continuity of the tangential components of both electric and magnetic field ensures a unique solution. The requirement of enforcing the continuity of all four tangential field components along- seems to contradict the uniqueness theorem, e.g., [31], which asks for only two of them, either electric of magnetic. However, it has been well established by Harrington in 1989 [32] that only formulations that satisfy both tangential electric and tangential magnetic field boundary conditions give unique solutions at all frequencies including those corresponding to internal resonances, otherwise the resulting matrices become singular at the frequencies of internal resonances. At frequencies away from these resonances, the information provided by the continuity of the tangential electric and tangential magnetic field becomes redundant according to the classical uniqueness theorem. The same requirement [32] applies in the current case by realizing that the eigenvalue formulation corresponds directly to the propagation constants characteristic equation. The latter is exactly a transverse resonance condition for the 2-D structure under study. To stress this, recall that numerous analytical or semianalytical approaches employs the “transverse resonance” technique for the evaluation of the complex propagation constants, e.g., [25] and [33]. The above approach also safeguards the transparency of the fictitious surface according to [34]. In turn, from the mathematical point of view, the first step of the Dirichlet-to-Neumann mapping theory requires the field solution in the unbounded region I to be based on Dirichlet data

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on the contour- . Namely, the unknown coefficients involved in the corresponding solution given in (7)–(12) should be evaluated from the field values on contour- . These are, in turn, provided by the finite-element method formulation used in the bounded region II. At this point, a question arises concerning which two out of the four tangential components should be enforced to be continuous: simultaneously serving the first Dirichlet-to-Neumann mapping step? Recall also that the binding of the two solutions will be accomplished and , marked in (5) and (6). A through the integrals closer examination reveals that and is involved in integral . is involved Likewise, . The quantities appearing in integrals and in integral should be obtained through the Dirichlet-to-Neumann map according to the theory of Section II-A. By means of the Maxwell curl equation in cylindrical coordinates, the integrals and are rewritten as

tangential magnetic field Dirichlet-to-Neumann map:

to yield the following

(19)

(13) (20) (14)

The continuity of the tangential magnetic field components is then imposed as

Recall that all quantities in (5) and (6) refer to the finite-ele” superscript). ment method representation of the field (“ and , Namely, in (13) and (14), and the media just inside the contour- is also air, and . However, according to the Dirichlet-to-Neumann should be exmapping formalism, the contour integrals pressed in terms of the magnetic field components given in (8) and (10). This is identical to the enforcement of the tangential magnetic field continuity along- . Summarizing, the answer to and the previous question is that the unknown coefficients should be evaluated through the continuity of the tangential along- as electric field components

(21) (22) The integrals (13) and (14) can be derived be exploiting the Dirichlet-to-Neumann map given by the expansions (19) and (20) and using the identities (21) and (22). By retaining in the expressions, these integrals read

(23)

(15) (16) Note that the superscript “expansion” is adopted instead of “ ” ” superscript denotes the finite-elefor emphasis, and the “ ment representation of the field. By exploiting the orthogonality and , these property of the exponential functions coefficients read (17)

(18) Based on (17) and (18), a relation between the Dirichlet data on the boundary and their derivatives can be formed. For obtained in (17) and this purpose, the coefficients (18) are substituted back into (8) and (10), representing the

(24) The consequence of the above procedure is the truncation of the semi-infinite solution domain. Since the evaluation of the boundary integrals (23) and (24) is possible, the application of the finite-element method is, hence, feasible. Moreover, for computational purposes, the infinite summations of cylindrical harmonics must be truncated to some maximum azimuthal index . The inaccuracy caused by this truncation is well documented within publications establishing the exact theoretical foundation of the “Dirichlet-to-Neumann mapping,” e.g., Grote in 1995 [16]. An important criterion is proved should in [35] “the maximum azimuthal index at least be greater than the argument of the Hankel function .” More details on this subject will be provided in Section IV. B. Formulation of the Nonlinear Eigenvalue Problem As was already mentioned in Section II-A.1, the resulting algebraic eigenvalue problem is nonlinear. A matrix eigen-

ALLILOMES AND KYRIACOU: NONLINEAR FINITE-ELEMENT LEAKY-WAVEGUIDE SOLVER

value problem is formulated employing the finite-element method standard matrix assembly procedure plus the additional boundary integrals terms stemming from the Dirichlet-to-Neumann mapping technique. The finite-element method interpolation functions for hybrid edge/node elements are given in [24]. Thus, the field expressions and the corresponding test functions within each triangle and along the discretized contour- are as follows:

interpolation/weighting functions cretized contour- as follows:

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defined along the dis-

elements

(25)

(26) (27)

(28) where is the number of polygon’s edges approximating the and are the unknown normalized cylindrical contour- . electric field values on the vertices and the edges of triangular and are first-order vector and scalar interpolaelements. tion/weighting functions, respectively [24]. Equations (25) and (26) are used to interpolate the triangular elements discretizing region II. Likewise, (27) and (28) interpolate the line elements on contour- . The substitution of the above into the original weak formulation (5) and (6) yields the nonlinear eigenvalue problem of (29) and (30) as follows:

(30) The subscripts and denotes all the possible scalar interpolation/weighting functions and , respectively. Note that, for the derivation of (29) and (30), the following normalization is already adopted [2]:

elements

(31) (32) In this way, the number of terms depending on the unknown eigenvalue are reduced. Since only the terms depending on are recalculated at every iteration, the above normalization reduces the computational cost. The nonlinear eigenvalue problem of (29) and (30) can be written in an abstract form as (33)

(29) where the subscript denotes all the possible vector interpolation/weighting functions defined within a triangle. denotes all the possible vector Likewise,

The longitudinal propagation constant is considered as the is eigenvalue of (33), while the free-space wavenumber handled as a parameter. Recall that the eigenvalues should be compatible with the discipline of the leaky mode propagation theory [25]. As long as the propagation along the positive -axis must be a complex number. Namely, has the form , where both and are positive numwill be referred to as the phase constant bers. The real part will be referred to as the attenuaand the imaginary part defines the modes chartion-leakage constant. The range of and leaky waves acteristics as surface waves for . The radial wavenumber in the unbounded for

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air region is also assumed complex as , where and are the phase and attenuation–leakage constant in the -direction. For surface waves and a lossless waveguide, is purely imaginary , while in the lossy case, there equal to the waveguide is a small inward power flow losses, e.g., [36]. Besides that, leaky waves present an outward for both lossless and lossy waveguides power flow and their particular characteristic is the exponentially increasing ), e.g., [25] amplitude in the radial direction (thus, and [37]. Hence, they are called improper waves since they violate the Sommerfeld radiation condition at infinity. The purpose of the above short review is to acquire the information necessary for the proper selection of the square-root branch defining . This decision bethe radial wavenumber comes obvious by considering the asymptotic approximation of the employed Hankel functions [38]

[40]. A direct result of this feature is its ability to handle a large number of unknowns, exploiting the sparsity of the system.

(34)

(35)

The behavior of (34) is in accordance with the leaky-waves theory briefly described above when the positive sign is chosen for the desired branch, namely, when is located in the first quadrant of its complex plane: . This branch also supports surface waves, but in this case, their radial wavenumber is moved along the and negative semiaxis for when the lossless case or slightly on the left of that the waveguide losses are considered. It is worth noting that all boundary integrals of (29) and (30) are evaluated analytically without the need for any special care concerning singularity problems. This feature is very attractive as it minimizes the computation time needed for their evaluation. In contrary, in the widely used combination of the finite-element method with the boundary integral method, the integrals are singular and usually cannot be evaluated analytically. Moreover, the generated matrix A is sparse in general with a dense part, resulting from the “non-local” Dirichlet-to-Neumann mapping absorbing boundary conditions. This “non-local” nature must be taken into account during the matrix assembly. Practically speaking, this means that along the discretized boundary- , every segment of the polygon interacts with all the others. In contrary, when “local” absorbing boundary conditions are considered, each segment interacts only with its neighboring segments. The advantage of “non-local” boundary conditions is the accuracy enhancement, but the penalty paid is a considerable increase in the computational cost. However, this is not a serious drawback as the method refers to a 2-D solution domain. III. SOLUTION OF THE NONLINEAR EIGENVALUE PROBLEM For the solution of the nonlinear equation (33), a complex number must be determined for a given free space propagarendering the determinant tion constant . Thus, a pair of the generated matrix equal to zero should be sought. Note that is a large, but very sparse matrix. Hence, a method exploiting the sparsity and effectively handling a large number of unknowns should be chosen. For this purpose, a method originally presented in [39] is employed. It is an iterative technique constructed around the implicitly restarted Arnoldi algorithm

A. Mathematical Definition The Regula–Falsi algorithm, which was implemented herein programming language by the authors, is based on in the c the procedure given by Hager [39]. For this paper to be self-sustained, the basic idea of [39] is repeated herein in an abstract , a number must be deterform. Given a matrix function mined such that the matrix becomes singular. This is equivalent to find a number , which makes the determinant of the matrix equal to zero. A linear interpolation between two points and in the complex plane is done using the Lagrange formula. Neglecting the higher order terms, the following linear interpolation formula is derived:

The points and must lie inside the desired part of the eigenvalue spectrum and should be relatively close. Thus, a nonlinear eigenvalue problem of the form

Nonlinear

(36)

can be approximated by the following linear generalized eigenvalue problem:

Linearized

(37)

This approximation is valid only for the part of the eigenvalues spectrum defined between and . The eigenvalues of (37) represent the fractional term arising from the substitution of (35) into (36) as follows: (38) Furthermore, the linearized generalized eigenvalue problem of (37) is sparse and can be quickly solved using software packages like ARPACK . This is similar to the case when nonlinear algebraic equations are solved. The error in the resulting eigenvalue is caused by the linear approximation of the matrix funcin the interval defined by and . However, the tion solution of this system yields a more accurate guess. Thus, by iteratively solving a system linearized around the most recent guess, the accuracy is improved. All the above are summarized in the following algorithm 1 [39]. Alogrithm 1 Regula Falsi, [39] 1: Select , and 2: for until convergence do 3: 4: 5: if update-criterion then 6: end for

ALLILOMES AND KYRIACOU: NONLINEAR FINITE-ELEMENT LEAKY-WAVEGUIDE SOLVER

First an appropriate eigenvalue search interval is selected. For every iteration of the algorithm 1, as it is stated in line 3, the linearized eigenvalue problem of (37) is solved. In the current implementation, the implicitly restarted Arnoldi method provided online (see [40]) is considered and the eigenvalues are computed. In the classical Regula–Falsi method for single algebraic equation (secant method), only one solution of the linearized equation is computed, but, in this case, the linearization is performed on a matrix equation, thus at every iteration, more than one solution–eigenvalue is generated. In turn, from the comof the linearized problem, the plete set of the eigenvalues most appropriate must be chosen. In fact, the eigenvalue with the smallest complex absolute value is the most appropriate. The first evident for this is found in the next step of the algorithm. From step 4 or (38), it is obvious that the smaller the eigenvalue we chose, the closer the to . This means that the algorithm is closer to convergence, as the updated value of is closer to the previous value . It is clearly proven by Hager [39] that when at least one of the -eigenvalues is equal to zero, then is an exact eigenvalue of the nonlinear problem. Thus, choosing the min- at each iteration causes the algorithm to converge to the actual eigenvalue. This analysis justifies the third line of algorithm 1, which connects the solution of the linearized system (37) to that of the actual nonlinear eigenvalue problem. Finally, considering the above perspective, it is obvious that the , something also depicted on algorithm converges when the fourth line of algorithm 1. From the practical application of the algorithm, it was observed that when the eigenvalues computed at step 3 are all close to unity, this means that there are no eigenvalues of the nonlinear system inside the linearization interval. Thus, from the first iteration, it is possible to determine whether the initial guess is good enough. Also, in the current implementation, is updated every ten iterations. Most of the time, this is not necessary since the algorithm converges between the third and sixth iteration, provided that a rather good initial guess is supplied.

B. Approximate Linear Formulation—Initial Guess The most critical part for the application of algorithm 1 is to start with a good initial search interval. The technique employed herein is based on the solution of an approximate linear finiteelement formulation. This linear formulation is based on the assumption that near cutoff , we may approximate the radial propagation constant with the propagation constant of the free space (39) This approximation–simplification is assumed only for the arregument of the Hankel function and its derivatives, while mains as it is elsewhere in the formulation. This type of approximation was always found to yield rather accurate results, even though it seems inconsistent from the first point of view. An extensive study of the above linearized approximation was performed in our previous study [21]. Some doubts about its validity were raised in [41], which were, in turn, answered in [22].

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Summarizing this approach, the resulting cylindrical harmonics expansion when this approximation is adopted can be given as

(40)

(41)

(42)

(43) Following the procedure presented in detail in Section II, but using the approximate expansion of (40)–(43) for the unbounded region, a linear eigenvalue finite-element method formulation is derived. Its eigenvalues are again the longitudinal propagation constants . The resulting eigenvalues serve as appropriate initial guesses for the solution of the accurate nonlinear eigenvalue problem. At this point, it is worth elaborating a little further on the accuracy of the above linear approximation. Firstly, it is expected to . be accurate at cutoff and its neighborhood, namely, However, it was found in our previous study [21], and particularly [22], to perform quite well for normalized eigenvalues up to approximately 0.8. The corresponding worst case deviation from measured results in the case of the leaky metallic waveguide (slotted longitudinally) was approximately . This 2% for the real and 15% for the imaginary parts of could be explained up to a point, from the fact that the above approximation is a second-order one, since it is required that the square of the normalized eigenvalue be much smaller than unity rather than itself. This is usually expected . It is also suspected (or exto relax the restrictions on pected) that a part of the inaccuracy is absorbed into the distribution of the weighting factors involved in the cylindrical harmonics expansion. This expectation is reinforced herein since the numerical results of Section IV clearly shows that the approximate linear formulation is relatively accurate for the whole range of the expected values of the normalized axial propagation constant. C. Solution Procedure In summary, the procedure followed for the mode analysis of a leaky-waveguiding structure is the following. First, the struc-

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ture under study is modeled by means of the approximate linear formulation derived from the expansion presented in (40)–(43). The solution of the approximate formulation provides an initial guess for the nonlinear formulation. At this point, the following combined strategy is followed. Initially all data values derived from the linear formulation are considered as appropriate initial guesses. This data set contains the values of the approximated axial propagation constant and its corresponding frequency. , A problem of lack of good initial guess around which is the limit of surface wave turn-on, may arise due to the failure of the linear approximation in this spectral area. In this case, a “frequency tracing technique” is employed in the algorithm as follows. Assume that the nonlinear problem converges has reached until a certain frequency and then stops before values near unity. The last converged value of the nonlinear formulation is then used as an initial guess for the next frequency instead of the value provided by the linear formulation. This have reached values scheme is applied until the values of near unity or the whole desired frequency range is covered. 1) Eigenvalues of Surface Waves: It is important to note that even though the whole procedure was established for leaky, it was found to perform equally well wave modes . Acand being robust for surface wave modes tually, there is not any mathematical or physical difference in handling either type of modes. The only requirement is that the Hankel functions should be accurate when the imaginary part of their argument becomes large. The adopted subroutines (zbesj.f and zbesy.f [42]) performed quite well and the Hankel functions were accurate even for imaginary arguments. Once again the formulation of (29) and (30) is not valid when . In turn, the previous “frequency tracing” approach is exploited in order to handle surface wave eigenvalues slightly . In this case, the tracking is done in above unity the opposite direction, starting from values much greater than unity and trying to reach values as close to unity as possible. IV. NUMERICAL RESULTS This section is divided in two parts. First, the current method is validated by comparison against published results, and subsequently, novel results derived solely from the current method are presented. Note that for the computation of all the results presented here, the infinite harmonics expansion sum was trun, and the fictitious contour- was approxicated at mated by a canonical polygon of 100 edges. The latter was deestablished cided taking into account the criterion in [35], as well as some experimentation with these two parameters conducted herein. Note that the criterion of [35] considers an ideal mathematical circular contour- , while herein this is approximated by a canonical polygon. In order to compensate for this discrepancy, a number of harmonics a few times more than those predicted by the criterion are used. A. Validation of the Algorithm The first structure analyzed is a laterally shielded top-open microstrip line (Fig. 2). The results of the current method (finiteelement–Dirichlet-to-Neumann method) are compared against those presented in [11], [43], and [44]. The authors of [43] had

Fig. 2. Laterally shielded microstrip line (a = 4:5 mm, D = 1:59 mm, L = 2 mm, " = 2:56).

used a transverse equivalent network technique for the calculation of the complex longitudinal propagation constant. While the authors of [11] and [44] employed a full-wave MoM based either on spectral- and space-domain Green’s functions, respectively, in conjunction with entire domain basis functions In Fig. 3, the normalized phase constant [see Fig. 3(a)] and attenuation constant [see Fig. 3(b)] versus the ratio is presented. is the width of the strip and is the width of the parallel-plate waveguide used to shield the microstrip line. The analysis was carried out at the frequency of 50 GHz, mm, and for a structure (Fig. 2) with dimensions mm, mm, and a dielectric constant . For the phase constant, a very good agreement with the classic spectral-domain MoM can be observed in Fig. 3(a) ratios. For small , the for almost all the different transverse equivalent network method fails, while the present finite-element–Dirichlet-to-Neumann method is in a very good agreement with both the space- and spectral-domain MoM. ratios, the current method is closer Moreover, for large to the results obtained from the classic spectral-domain MoM rather than the space-domain MoM. For the leakage constant [see Fig. 3(b)], we also notice a good agreement of the finite-element–Dirichlet-to-Neumann method with both the spectraland space-domain MoM. The next example is the laterally shielded top-open slot line shown in Fig. 4. Gomez-Tornero et al. [11] have also analyzed this slot line using their space-domain MoM. In Fig. 5, the longitudinal propagation constant is plotted for different lengths of the parallel-plate waveguide used to shield the structure. The frequency is again 50 GHz, the dielectric constant , and its dimensions are mm, mm, mm, and mm. A very good agreement for both the phase [see Fig. 5(a)] and leakage [see Fig. 5(b)] constants is observed in this case. At this point, it is worth mentioning that, in both cases, the approximate linear formulation performed unexpectedly well. In fact, it could be exploited as a fast tool for the design of leaky and surface wave structures. In turn, the nonlinear formulation can serve as a fine tuning (trimming) tool. The last structure considered in the validation procedure is a classical leaky-waveguide antenna (Fig. 6) originally presented by Lampariello and Oliner [45]. The results provided in [45] mm, mm, mm, are for mm, mm, mm, and

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Fig. 3. Normalized longitudinal propagation constant of the microstrip line of Fig. 2 (space-domain MoM: SpD MoM; transverse equivalent network technique: TRE; spectral-domain MoM: SD MoM; finite-element– Dirichlet-to-Neumann method: FEM-DtN). (a) Phase constant. (b) Leakage constant.

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Fig. 5. Normalized longitudinal propagation constant of the slot line of Fig. 4 (space-domain MoM: SpD MoM; finite-element–Dirichlet-to-Neumann method: FEM-DtN). (a) Phase constant. (b) Leakage constant.

Fig. 6. Waveguide-based leaky-wave antenna (a = 23:00 mm, b = 11:95 mm, a = 11:95 mm, c = 15:65 mm, d = 4:55 mm, F = 21:50 mm, F = 15:00 mm).

Fig. 4. Laterally shielded slot line (a = 2:2 mm, D = 1:59 mm, X = 1 mm, X = 2:1 mm, " = 2:56).

mm. In [45], a transverse equivalent network was used for the structure analysis. In the same study, measurements for the longitudinal propagation constant are also provided. These measurements along with the theoretical results of [45] are presented in Fig. 7, where they are compared against both the approximate linear finite-element–Dirichlet-to-Neumann method and the accurate nonlinear formulation of the current method. A good agreement with the measurements can be observed. The numerical results of the current method are closer to the

measurements than those of the transverse equivalent network technique. In order to check the validity of the criterion proposed in [35] for the truncation of the infinite harmonics sum, the convergence versus the number of harmonics is examined for the structure of Figs. 2, 4, and 6. An indicative example for the phase for and leakage constants versus the number of harmonics the case of Fig. 6 at 10 GHz is plotted in Fig. 8. In this case, the criterion of [35] predicts a minimum number of harmonics at which the percent error is 0.0007% and 0.11% for the phase and leakage constants, respectively. Similar investigation at numerous frequencies are carried out for all three structures. The corresponding percent error for a number and are summarized in of harmonics

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Fig. 7. Normalized longitudinal propagation constant of the leaky-wave antenna of Fig. 6 (transverse equivalent network technique: TRE; finite-element– Dirichlet-to-Neumann method: FEM-DtN). (a) Phase constant. (b) Leakage constant.

Table I. The maximum expected criterion value occurs at the or . These are given in leaky mode’s cutoff Tables II. It is obvious that the employed number of harmonics ensure convergence in all the examined cases, as well as in any particular geometry. This renders a software tool independent from the user experience, while the increase in the computational cost is negligible. B. Novel Results and Discussions Section IV-B has clearly validated the proposed method, which may then provide reliable results for the dispersion characteristics of leaky (and surface) wave structures. The main advantage of the proposed finite-element–Dirichlet-to-Neumann method over the above-mentioned spectral- or space-domain MoM techniques is its general application since it may handle open structures with arbitrary cross sections. This feature makes the method capable of analyzing all sort of leaky-wave structures without the need to devise a separate algorithm and write a new computer program for each one of them. Contrarily, the examples presented in [11] and [43]–[45] were analyzed by using methods especially devised for each case. Thus, these methods cannot be considered as a general tool for the analysis and design of arbitrary geometry leaky-wave antennas. It is also well understood that MoM techniques are primarily depended on the particular structure’s Green’s function, which require a heavy analytical and computational effort. Here, novel results will be presented concerning leaky-wave structures with geometries that render their analysis almost impossible with analytical or semianalytical methods as those presented in [11], [44], and [45]. Here, results are obtained

Fig. 8. Convergence of the normalized longitudinal propagation constant versus the number of harmonics for the structure of Fig. 6 (f = 10 GHz,  = 39:3 mm). (a) Phase constant. (b) Leakage constant.

from the nonlinear finite-element–Dirichlet-to-Neumann formulation and always with the aid of an initial guess provided , by the linear formulation. For values of near unity where the formulation exhibits singularities [see (29) and (30)], the “frequency tracing technique” presented in Section III-A is exploited. The first structure studied is the waveguide-based leaky-wave and antenna (Fig. 6) of Section IV-A, but now the flanges form an angle ; namely, is the flare angle with respect to their original orientation parallel to the rectangular waveguide large side. The question to be answered is whether may be exploited as a design degree of freedom to control the leakage rate. Recall that the antenna directivity depends on the leakage rate [20]. Also, when is increased, the aperture takes a form of a horn antenna, which is associated with higher directivity. All the other dimensions of the structure remain the same. Fig. 9 presents both the phase and leakage constants, which are evaluated at frequencies of 8, 9, 10, 11, and 12 GHz and for the angles and . Its very interesting to observe that the phase constant [see Fig. 9(a)] remains almost constant versus for the same frequency. This does not affect the direction means that the flanges angle -plane” of the radiated beam maximum in the “elevation with respect to the broadside- -axis). This phenomenon ( is classically documented, e.g., [20], as . On the other hand, the leakage constant increases until it reaches its and then decreases until . The maximum at maximum is slightly different from expected at .

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TABLE I PERCENT DEVIATION OF PHASE AND ATTENUATION CONSTANTS VERSUS THE NUMBER OF HARMONICS. IS THE ROUNDED MINIMUM NUMBER OF HARMONICS [35]

N

TABLE II MAXIMUM EXPECTED VALUES FOR THE CRITERION

Fig. 9. Normalized longitudinal propagation constant of the leaky-wave antenna of Fig. 6 computed for different flange angles 8. (a) Phase constant. (b) Leakage constant.

This may be due to the asymmetry of the lengths of the flanges. In order to interpret the results of Fig. 9(b), let us review some basic properties. It is already well understood, e.g., [20] and mainly determines the [46], that the leakage constant -plane in Fig. 6) in rather combeamwidth in the elevation ( causes radiation of plicated manner. Summarizing, large

N DEFINED IN [35]

all available energy at a short distance; hence, yields a small radiating aperture corresponding to wide beamwidth or low directivity. This also involves source mismatch, but its main advantage is the short antenna length. In contrary and following the yields high directivity in the elevasame reasoning, small tion plane, avoids source mismatch, but leads to long antenna structures. According to [46], a tapering in the longitudinal direction could be the best solution since it combines the advantage of the two above extremes and additionally lowers the sidelobes. From the above, one may conclude that the leakage rate does not affect the beamwidth in the transverse-azimuthal -plane), which tends to . The horn-type ( structure in the transverse direction was firstly adopted in [46] to with reduce the transverse beamwidth, achieving a flare angle equal to 10 . This corresponds to flanges angle in Fig. 6. A complete characterization of the phenomenon falls outside the scope of this study. The second leaky-wave structure is the laterally shielded slot line of Fig. 4, but now the length of the parallel-plate wavemm and the longitudinal guide shielding is fixed to propagation constant is plotted as a function of frequency. The dispersion characteristics of this structure are also studied in [47] and [48], but the authors therein presented only the lonand no results for the leakage gitudinal phase constant constant are given. Complete dispersion characteristics are evaluated herein and are depicted in Fig. 10. For the phase constant, a good agreement is observed when the current results are compared against those of [47]. At frequencies below 25 GHz, some discrepancies between the results provided by the finite-element–Dirichlet-to-Neumann method and [47] are observed. The finite-element–Dirichlet-to-Neumann method has managed to predict the coupling phenomena between the “channelguide” (labeled “1st mode”) and the desired leaky mode (labeled GHz where both “2nd mode”). The coupling occurs at and of the two modes are equal; namely, their dispersion curves intersect at the same frequency. For more details

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Fig. 10. Normalized longitudinal propagation constant of the leaky-wave antenna of Fig. 4 (space-domain MoM: SpD MoM; finite-element–Dirichlet-toNeumann method: FEM-DtN). (a) Phase constant. (b) Leakage constant. Fig. 12. Normalized longitudinal propagation constant versus frequency for the microstrip line of Fig. 11 with and without the strip (finite-element–Dirichlet-toNeumann method: FEM-DtN). (a) Phase constant. (b) Leakage constant.

Fig. 11. Microstrip line with finite substrate (a = 30 mm, h = 0:762 mm, W = 9 mm, " = 2:55).

on the coupling phenomena refer to [47], and for a more general explanation see [33]. The results provided by [48] refers to . This is the a structure with an infinite lateral shielding reason why there is no “channel-guide” mode present and, thus, no coupling phenomena occurs in the results provided by [48]. Finally, a microstrip line (Fig. 11). but with a finite substrate and with no lateral shielding, is considered. There is only a restricted published literature, e.g., [49], studying leaky waves on finite substrate structures. However, to the authors’ knowledge, there is not any published paper presenting leaky-mode coupling caused by the finite extend of the substrate, which is illustrated below. Most of the published results considers either an infinite substrate [50] or a laterally shielded structure like that of Fig. 3 [51]. Chen et al. [51] computed the longitudinal propagation constant for the microstip line of Fig. 11 where mm, mm, and . They considered infinite lateral shielding and the analysis was based on the integral method presented in [52]. In Fig. 12, the phase and leakage constants are plotted for the higher leaky mode computed by Chen et al. [51]. In contrary, the finite-element–Dirichlet-to-Neumann method predicts a pair of coupled leaky modes. Note that, in this case, the finite-element–Dirichlet-to-Neumann method is applied for

mm without lateral shielding. As a finite substrate a result of the finite substrate, a mode coupling phenomenon analogous to that of the laterally shielded slot line of [47] occurs. This phenomenon is in accordance with the basic requirement for coupling, as stated in [33]; namely, coupling occurs at GHz where both leaky modes have almost the same and simultaneously. It is important to recall that the mode coupling phenomenon occurring in the slot line of Fig. 4 is exactly due to the finite extend of the lateral shielding. This causes the coupling of the “channel-guide” mode with the desired leaky mode. Likewise, the finite extent of the substrate in the microstip line of Fig. 10 yields a coupling of the two leaky modes. Contrarily, when an infinite substrate is considered, only one of these leaky modes occurs according to [50] and, consequently, there is not any coupling effect. Herein the mode labeled “1st mode FEM-DtN” cannot be characterized as a “channel-guide” leaky mode since this is appropriate only for the case when a finite lateral shielding is present, forming in that way a channel waveguide [25]. According to [33], the channel guide modes occurs when the stub length is greater than the half . Thus, these modes build up free-space wavelength due to the field scattering at the open end of the stub. Likewise, in the current case, the nature of this new leaky mode and the effect of coupling between the two modes is expected to be due to the scattering from the finite substrate edges. It is also expected that coupling will occur only when the substrate dimension is

ALLILOMES AND KYRIACOU: NONLINEAR FINITE-ELEMENT LEAKY-WAVEGUIDE SOLVER

greater than . Nevertheless, this requires a particular effort, which is outside the space limitations of this paper. Moreover, the results for the single mode computed in [51] for frequencies between 8–11 GHz (both the phase and leakage constant) initially follow the second mode computed by the finite-element–Dirichlet-to-Neumann method up to approximately 11 GHz, but subsequently, after a small gap at 11.5 GHz, they join the first mode. This extraordinary behavior is compliant with the phenomenon of coupling between leaky modes, as this is presented for the case of a stub loaded rectangular leaky waveguide in [33]. Thus, it is important to note that the finite-element–Dirichlet-to-Neumann method presented herein is capable of handling the modes coupling phenomena by successfully identifying each mode. The last numerical experiment was conducted only for the grounded finite substrate of the microstrip line of Fig. 11. All the dimensions and the dielectric constant are the same, but the microstrip is excluded from the geometry. In this case, there is not any mode coupling since the mode due to the microstrip disappears. Instead of the pair of coupled modes, a single one occurs, labeled “1st Mode (without strip).” As expected from the coupling phenomena, e.g., [33], there is a mode interchange. Namely, the single mode follows the first of the pair up to the coupling point. In turn, there is an interchange and, beyond that, it joins the second coupled mode. V. CONCLUSION A hybrid finite-element and cylindrical harmonics expansion method has been established based on the vector Dirichlet-to-Neumann mapping principles, The latter ensures the transparency of the fictitious circular contour along which the field continuity conditions are imposed. These nonlocal boundary conditions improve the accuracy, but at the expense of a dense part in the final system matrix. Thus, paying a penalty of relative increase in the computation time. However, a comparison in terms of computation time against techniques hybridized with the boundary element method is in favor of the current finite-element–Dirichlet-to-Neumann method. This is justified by the fact that herein all the contour integrals along the separation boundary are analytically evaluated. The presented (finite-element–Dirichlet-to-Neumann) method is also generally applicable while rival MoM techniques require Green’s functions of the particular structure and, thus, their application is restricted. In general, eigenvalue formulation for open-radiating structures lead to nonlinear eigenvalue problems. In the current case, the nonlinearity is caused by the appearance of the unknown propagation constant within the arguments of the cylindrical harmonics. The required initial guesses are provided by a and the second-order linear approximation nonlinear problem is solved employing a matrix Regula–Falsi algorithm. The implemented finite-element–Dirichlet-to-Neumann method is validated against published numerical and experimental results and it is found to be robust, performing accurately in all tested structures. Even though the formulation of the method was aiming at leaky-wave modes, it was found to operate equally well for surface wave modes. This was

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mainly due to the robust subroutines implementing the Hankel functions, which are accurate for complex as well as purely imaginary arguments and, thus, avoiding the need of McDonald functions. The linear approximation given in our previous studies was reinforced herein since it was found to perform unexpectedly well in almost all cases. This makes it a valuable numerical tool providing a very good approximation of the whole spectrum of axial propagation constants at the minimal computational time. Concerning future improvements of the method, the first refers to the unavoidable truncation of the infinite summation of cylindrical harmonics expansion. This can be greatly benefited by an analytical asymptotic evaluation of the infinite tail of the summation, employing large-order approximation of the involved Hankel functions. Second, the accuracy related to the discretization of the fictitious circular contour can be further enhanced by the employment of locally curvilinear elements. A very important enhancement of the general applicability of the method is related to the inclusion of anisotropic materials and periodic boundary conditions. This will enable the analysis of important novel leaky and surface waveguiding structures. REFERENCES [1] C. A. Balanis, Advanced Engineering Electromagnetics. New York: Wiley, 1989. [2] J. F. Lee, D. K. Sun, and Z. J. Cendes, “Full-wave analysis of dielectric waveguides using tangential vector finite elements,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 8, pp. 1262–1271, Aug. 1991. [3] J.-M. Guan and C.-C. Su, “Analysis of metallic waveguides with rectangular boundaries by using the finite-difference method and the simultaneous iteration with the Chebyshev acceleration,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 2, pp. 374–382, Feb. 1995. [4] R. E. Collin, “Analytical solution for a leaky-wave antenna,” IRE Trans. Antennas Propag., vol. AP-10, no. 5, pp. 307–319, Sep. 1962. [5] F. Frezza, M. Gugliemi, and P. Lampariello, “Millimetre-wave leakywave antennas based on slitted asymmetric ridge waveguides,” Proc. Inst. Elect. Eng.—Microw., Antennas, Propag., vol. 141, no. 3, pp. 175–180, Jun. 1994. [6] T. Zhao, D. R. Jackson, J. T. Williams, and A. A. Oliner, “General formulas for 2-D leaky-wave antenna,” IEEE Trans. Antennas Propag., vol. 53, no. 11, pp. 3525–3533, Nov. 2005. [7] L. N. Litinenko, S. L. Prosvimin, and V. P. Shestopalov, “Electromagnetic characteristics of a slotted waveguide,” Radio Eng. Electron. Phys., vol. 19, no. 3, pp. 41–47, 1974. [8] A. I. Nosich and A. Y. Svezhentsev, “Principal and high-order modes of microstrip and slot lines on a cylindrical substrate,” Electromagnetics, vol. 13, no. 1, pp. 85–94, 1993. [9] J. L. Volakis, T. Ozdemir, and J. Gong, “Hybrid finite-element methodologies for antennas and scattering,” IEEE Trans. Antennas Propag., vol. 45, no. 3, pp. 493–507, Mar. 1997. [10] H. Rogier and D. De Zutter, “Berenger and leaky modes in optical fibers terminated with a perfectly matched layer,” J. Lightw. Technol., vol. 20, no. 7, pp. 1141–1148, Jul. 2002. [11] J. L. Gomez-Tornero, F. D. Quesada-Pereira, and A. Alvarez-Melcon, “A full-wave space-domain method for the analysis of leaky-wave modes in multilayered planar open parallel-plate waveguides,” Int. J. RF Microw. Comput.-Aided Eng., vol. 15, no. 1, pp. 128–139, Dec. 2005. [12] H. P. Uranus and H. J. W. M. Hoekstra, “Modelling of microstructured waveguides using a finite-element-based vectorial mode solver with transparent boundary conditions,” OSA Opt. Express, vol. 12, no. 12, pp. 2795–2809, 2004. [13] S. Selleri, L. Vincetti, A. Cucinotta, and M. Zobili, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quantum Electron., vol. 33, no. 4–5, pp. 359–371, Apr. 2001. [14] K. Hayata, M. Eguchi, and M. Koshiba, “Self-consistent finite/infinite element scheme for unbounded guided wave problems,” IEEE Trans. Microw. Theory Tech., vol. 36, no. 3, pp. 591–599, Mar. 1988. [15] J. B. Keller and D. Givoli, “Exact non-reflecting boundary conditions,” J. Comput. Phys., vol. 82, pp. 172–192, May 1989.

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[16] M. J. Grote, “Nonreflecting boundary conditions,” Ph.D. dissertation, Dept. Sci. Comput. Computat. Math., Stanford Univ., Stanford, CA, 1995. [17] D. Givoli, Numerical Methods for Problems in Infinite Domains. Amsterdam, The Netherlands: Elsevier, 1992. [18] M. J. Grote and C. Kirsch, “Dirichlet-to-Neumann boundary conditions for multiple scattering problems,” J. Comput. Phys., vol. 201, no. 2, pp. 630–650, Dec. 2004. [19] A. Freni, C. Mias, and R. L. Ferrari, “Hybrid finite-element analysis of electromagnetic plane wave scattering from axially periodic cylindrical structures,” IEEE Trans. Antennas Propag., vol. 46, no. 12, pp. 1859–1865, Dec. 1998. [20] E. Frezza and P. Lamparieilo, “On the modal spectrum of the channel waveguide,” Int. J. Infrared Millim. Waves, vol. 16, no. 3, pp. 591–599, 1995. [21] P. C. Allilomes, G. Kyriacou, E. Vafiadis, and J. Sahalos, “A FEM analysis of open boundary structures using edge elements and a cylindrical harmonics expansion,” Electromagnetics, vol. 24, no. 1–2, pp. 69–79, 2004. [22] P. C. Allilomes, G. Kyriacou, E. Vafiadis, and J. Sahalos, “Authors’ reply,” Electromagnetics, vol. 24, no. 6, pp. 493–495, 2004. [23] P. C. Allilomes and G. Kyriacou, “A nonlinear eigenvalue hybrid FEM formulation for two dimensional open waveguiding structures,” PIERS Online, vol. 1, no. 5, pp. 620–624, 2005. [24] C. Reddy, M. Deshpande, C. Cockrell, and F. Beck, “Finite elements method for eigenvalue problems in electromagnetics,” NASA, Langley Res. Center, Hampton, VA, Tech. Rep. 3485, Dec. 1994. [25] L. O. Goldstone and A. A. Oliner, “Leaky antennas I: Rectangular waveguides,” IRE Trans. Antennas Propag., vol. 7, no. 4, pp. 307–319, Oct. 1959. [26] J. D. Jackson, Classical Electrodymanics, 3rd ed. New York: Wiley, 1999, p. 431. [27] D. M. Pozar, Microwave Engineering, 2nd ed. New York: Wiley, 1998, pp. 133–. [28] J. Jin, The Finite Element Method in Electromagnetics. New York: Wiley, 1993, pp. 231–232. [29] D. R. Tanner and A. F. Peterson, “Vector expansion function for the numerical solution of Maxwell’s equation,” Microw. Opt. Technol. Lett., vol. 2, no. 2, pp. 331–334, 1989. [30] A. C. Cangellaris and D. B. Wright, “Analysis of the numerical error caused by the stair-stepped approximation of a conducting boundary in FDTD simulations of electromagnetic phenomena,” IEEE Trans. Antennas Propag., vol. 39, no. 12, pp. 1518–1525, Dec. 1991. [31] C. A. Balanis, Advanced Engineering Electromagnetics. New York: Wiley, 1989, p. 314. [32] R. F. Harrington, “Boundary integral formulations for homogeneous material bodies,” J. Electromagn. Waves Applicat., vol. 3, no. 1, pp. 1–15, 1989. [33] H. Shigesawa, M. Tsuji, P. Lampariello, F. Frezza, and A. A. Oliner, “Coupling between different leaky-mode types in stub-loaded leaky waveguides,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 8, pp. 1548–1560, Aug. 1994. [34] K. Zhao, M. Vouvakis, S.-C. Lee, and J.-F. Lee, “Domain decomposition method with DP-FETI technique for solving large finite antenna arrays,” in URSI EMTS, Piza, Italy, May 2004, pp. 840–842. [35] I. Harari and T. Hughes, “Analysis of continuous formulations underlying the computation of time–harmonic acoustics in exterior domains,” Methods Appl. Mech. Eng., vol. 97, pp. 103–124, 1992. [36] R. E. Collin and F. J. Zucker, Antenna Theory: Part 2. New York: McGraw-Hill, 1969, p. 177. [37] R. E. Collin and F. J. Zucker, Antenna Theory: Part 2. New York: McGraw-Hill, 1969, p. 179. [38] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. New York: Dover, 1965, p. 364. [39] P. Hager, “Eigenfrequency analysis: FE-adaptivity and nonlinear eigen-problem algorithm,” Ph.D. dissertation, Dept. Structural Mech., Chalmers Univ. Technol., Göteborg, Sweden, 2001. [40] R. Lehoucq, K. Maschhoff, and D. SorensenC. Yang, ARPACK Homepage. [Online]. Available: http://www.caam.rice.edu/software/ ARPACK/ [41] J.-M. Jin, “Comment on ‘a FEM analysis of open boundary structures using edge elements and a cylindrical harmonics expansion’,” Electromagnetics, vol. 24, no. 6, p. 491, 2004.

[42] D. E. Amos, “A portable package for Bessel functions of a complex argument and nonnegative order,” ACM Trans. Math. Softw. vol. 12, pp. 265–273, Sep. 1986. [Online]. Available: http://www.netlib.org/slatec/ src/ [43] P. Lampariello and A. Oliner, “A novel phase array of printed-circuit leaky-wave line sources,” in Proc. 17th Eur. Microw. Conf., Rome, Italy, 1987, pp. 255–560. [44] P. Baccarelli, P. Burghignoli, C. D. Nallo, F. Frezza, A. Galli, P. Lampariello, and G. Ruggieri, “Full-wave analysis of printed leaky-wave phase arrays,” Int J. RF Microw. Comput.-Aided Eng., vol. 12, pp. 272–285, 2002. [45] P. Lampariello, F. Frezza, H. Shigesawa, M. Tsuji, and A. Oliner, “A versatile leaky-wave antenna based on stub loaded rectangular waveguide: Part III—Comparison with measurements,” IEEE Trans. Antennas Propag., vol. 46, no. 7, pp. 1047–1055, Jul. 1998. [46] T. N. Tringh, R. Mittra, and R. J. Paleta, “Horn image-guide leaky-wave antenna,” IEEE Trans. Microw. Theory Tech., vol. MTT-29, no. 12, pp. 1310–1314, Dec. 1981. [47] J. L. Gomez-Tornero and A. Alvarez-Melcon, “Nonorthogonality relations between complex hybrid modes: An application for the leakywave analysis of laterally shielded top-open planar transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 3, pp. 760–767, Mar. 2004. [48] P. Lampariello, F. Frezza, and A. A. Oliner, “The transition region between bound-wave and leaky-wave ranges for a partially dielectricloaded open guiding structure,” IEEE Trans. Microw. Theory Tech., vol. 38, no. 12, pp. 1831–1836, Dec. 1990. [49] C. D. Nallo, P. Baccarelli, P. Burghignoli, F. Frezza, and A. Galli, “An efficient boundary-integral-equation technique for accurate analysis of leakage and coupling effects in arbitrary transmission lines and waveguides,” in Int. Eur. Electromagn. Compat. Symp., Sorrento, Italy, Sep. 2002, pp. 179–183. [50] J. S. Bagby, C.-H. Lee, D. P. Nyquist, and Y. Yuan, “Identification of propagation regimes on integrated microstrip transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 11, pp. 1887–1894, Nov. 1993. [51] K.-C. Chen, Y. Qian, C.-K. C. Tzuang, and T. Itoh, “A periodic microstrip radial antenna array with a conical beam,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 4, pp. 756–765, Apr. 2003. [52] G.-J. Chou and C.-K. C. Tzuang, “An integrated quasi-planar leaky-wave antenna,” IEEE Trans. Antennas Propag., vol. 44, no. 8, pp. 1078–1085, Aug. 1996. Peter C. Allilomes was born in Thessaloniki, Greece, in December 1977. He received the Diploma degree in electrical and computer engineering (with honors) from the Democritus University of Thrace, Xanthi, Greece, in 2001, and is currently working toward the Ph.D. degree at Democritus University of Thrace. He is currently with the Microwaves Laboratory, Democritus University of Thrace, where he participates in a European Union–Greek Government research program. His research interests are computational electromagnetics, leaky wave antennas, and the finite-element method. Mr. Allilomes is a member of the Technical Chamber of Greece.

George A. Kyriacou (M’90–SM’05) was born in Famagusta, Cyprus, on March 25, 1959. He received the Electrical Engineering diploma and Ph.D. degree (both with honors) from the Democritus University of Thrace, Xanthi, Greece, in 1984 and 1988, respectively. Since January 1990, he has been with the Department of Electrical and Computer Engineering, Democritus University of Thrace, where he is currently an Associate Professor. He has authored over 122 journal and conference papers. His main research interests include microwave engineering, open waveguides and antennas in anisotropic media, applied electromagnetics, and biomedical engineering. Dr. Kyriacou is a member of the Technical Chamber of Greece.

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Effects of Losses on the Current Spectrum of a Printed-Circuit Line Joaquín Bernal, Member, IEEE, Francisco Mesa, Member, IEEE, and David R. Jackson, Fellow, IEEE

Abstract—This paper studies the effects of practical conductor and dielectric losses on the high-frequency current excited on a microstrip line by a gap voltage source. The analysis shows that whereas losses cause an exponential decay in the propagating bound mode (as expected), the continuous-spectrum current is much less influenced by the presence of material losses. As a consequence, the nature of the strip current far away from the source is dramatically affected by the presence of losses, and will be dominated by the continuous spectrum. This results in unusual behavior that is observed for the strip current far away from the source. Index Terms—Conductor losses, continuous spectrum, dielectric losses, leaky mode, microstrip, spurious effects.

I. INTRODUCTION

T

HE EXISTENCE of spurious effects in the current excited on a lossless microstrip line at high frequencies has been well established in the literature [1]–[5]. These spurious effects include significant oscillations and signal attenuation. A physical decomposition of the current [3], [6] shows that the total current on the line due to a source is composed of a boundmode current and a continuous-spectrum current. The boundmode current is the desired quasi-TEM transmission-line current, while the continuous-spectrum current is a radiating type of current that typically increases with frequency. Oscillations and even nulls in the current arise due to interference between the bound-mode and continuous-spectrum currents. The continuous-spectrum current includes the presence of physical leaky modes if any exist at the frequency of interest. It also consists of a “residual-wave” current, which is the leftover part of the continuous-spectrum current that is not representable in terms of the leaky modes [7]. The residual-wave current, in turn, is composed of two types of currents, one called the surface-wave type (associated with the surface-wave modes of the dielectric waveresidual-wave current) and guide that are above cutoff, i.e.,

Manuscript received January 14, 2006; revised March 23, 2007. This work was supported by the Spanish Ministerio de Educación y Ciencia, by the European Union under FEDER funds Project TEC2004-03214, and by the Junta de Andalucía under Project TIC-253. This work was supported in part by the State of Texas under the Advanced Technology Program. J. Bernal is with the Department of Applied Physics 3, E.S. de Ingenieros Industriales, University of Seville, 41092-Seville, Spain (e-mail: jbmendez@us. es). F. Mesa is with the Department of Applied Physics 1, ETS de Ingeniería Informática, University of Seville, 41012-Seville, Spain (e-mail: [email protected]). D. R. Jackson is with the Department of Electrical and Computer Engineering, University of Houston, Houston, TX 77204-4005 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.2007.900335

the other called the free-space type or residual-wave current. For a lossless microstrip line, the continuous-spectrum current shows a decaying behavior when plotted versus distance from residual-wave current decays in amplitude the source. The [7], whereas the residual-wave current decays faster as [5], [8]. In contrast, the bound-mode than this, varying as current propagates without attenuation on a lossless structure. Thus, for the lossless case, it is expected that interference effects decrease as the distance from the source increases. At very large distances from the source, the current stabilizes to that of the bound-mode current. The above conclusions must be carefully revised when a lossy (and, therefore, more realistic) structure is considered. Although loss effects have been studied for several types of structures [9]–[17], few results have been reported that discuss the effects of losses on the high-frequency current excited by a realistic source [18]. It is well known that conductor and/or dielectric losses cause the bound-mode current to decay exponentially with distance from the source. However, the decaying behavior of each component of the continuous-spectrum current is, in general, different and, therefore, the overall behavior of the current at large distances from the source can differ significantly from that expected for a lossless line. In particular, the boundmode current may not be the dominant part of the current at large distances. This leads to unusual behavior in the current on the strip far away from the source, as will be demonstrated. This study will extend considerably the preliminary results reported in [18] on the effect of dielectric and/or conductor losses on the complete spectrum of the current excited on an infinite microstrip line by a gap voltage source. Our analysis employs a mixed-potential integral-equation scheme [19], [20] where the discrete complex image technique [21]–[23] is used to accurately approximate the kernel of the integral equation in the spatial domain, thus improving the efficiency of the computer code [24]. Dielectric losses are modeled by allowing the dielectric permittivity to have complex values, whereas conductor losses are accounted for by means of a conductor surface impedance [25]. The theoretical and numerical results show that losses can change the nature of the current on the line far away from the source significantly in a way that cannot be predicted by a lossless analysis or by simple transmission line theory (which neglects the continuous-spectrum current). In the lossy case, the current that will eventually dominate at extreme disresidual-wave current since this component of tances is the the continuous-spectrum current is the least affected by losses. Results will be reported to validate the approach. Different numerical calculations will be presented to confirm the theoretical predictions on the influence of losses on the total current that is excited on the microstrip. Those calculations will also provide

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Fig. 1. Geometry of an infinite microstrip line fed by a gap voltage source.

further insight into the physical interpretation of the components of the current. II. ANALYSIS The structure under study is shown in Fig. 1; namely, an infinite microstrip line excited by a 1-V gap voltage source of length . The current on the conductor strip of this structure is calculated following the efficient mixed-potential integral-equation scheme reported in [24]. Although detailed information on the implementation of this integral equation is given in [24] and the references therein, the method will be briefly outlined for completeness, and also to point out the nontrivial changes necessary to adapt the method for lossy lines. In the lossless case, the mixed-potential integral equation is obtained by enforcing a zero tangential electric field on the strip (except inside the gap region, where the field must match the impressed one). In this study, losses in the strip conductor have been accounted for by means of the widely used surface-impedance boundary condition. This condition is enforced for the tangential field on the conductor strip, thus yielding a mixed-potential integral equation where the surface impedance term can be easily included. After performing a convenient , Fourier transform with respect to the longitudinal variable the expression for this integral equation can be written as

(1)

(2) where is the component of the spectral Green’s dyadic of the grounded substrate structure that relates the Fourier transform of the tangential electric field with the Fourier transform of the tangential surface current density at the strip interface. The term in (2) denotes the Fourier transform of the impressed delta-gap electric field. Note that (1) and (2) are in a mixed space-spectral form in which the unknown is the Fourier transform (in the -direction) of the surface current on the strip. This is the most convenient form to use when solving for the Fourier transform of the strip current using the mixed-potential integral equation with the

method of images, which uses the space-domain field from an infinite phased line current having a wavenumber . Having the Fourier transform of the strip current is the key step to plane in performing a path deformation in the complex order to decompose the strip current into its constituent parts (discussed momentarily). Physically, (1) and (2) indicate that, on the surface of the strip, the total electric field must equal the surface impedance times the current density on the strip everywhere, except inside the gap region, where the electric field is an impressed field, and is -directed. The total longitudinal is, therefore, equal to outside of the gap, electric field and inside the gap, i.e., (3) where is the unit step function. Inside the gap, the surface impedance term is small compared with the impressed gap field and, therefore, to a good approximation (4) Since the above equation now holds everywhere on the strip surface, it is directly amenable to Fourier transformation in the -direction, resulting in (2). The expression employed for the has been taken from [25, eqs. (20) and surface impedance (21)], which makes it possible to account for the conductor with reasonable accuracy as long as with thickness being the strip width. Also, losses on the ground plane can be readily included in our approach by modeling the lower metallic plane as a semi-infinite dielectric layer with a very high conductivity (frequency-dependent loss tangent). Once the integral equation is posed for the Fourier transform of the strip current, it can be solved by using the method of moments. It is then necessary to expand the unknown functions (the Fourier transform of the components of the surface current density on the strip) in terms of appropriate basis functions. It should be pointed out that although first- and second-type weighted Chebyshev polynomials have shown a very good performance in the lossless case [24], this set of basis functions exhibits a singular behavior for the longitudinal current near the edges of the metallic strip. This singular behavior is not appropriate for representing the actual nonsingular behavior of the longitudinal current for the lossy case. To overcome this, rectangular pulses have been used as basis functions for the longitudinal component of the current, whereas triangle basis functions have been employed to expand the transverse component. The integral equation is finally solved by using Galerkin’s method, giving rise to a linear system of equations for the unknown coefficients of the current. Once this system is solved, the current can be found from the following numerical along the strip inverse Fourier transform: (5) The transform of the strip current in the above equation comes from summing the coefficients of the longitudinal basis functions times the pulsewidth of each basis function in the direction.

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Fig. 2. Complex k -plane showing the original path of integration C and the path deformation.

The path of integration in the -plane is shown in Fig. 2 along with poles and branch points singularities of the integrand in (5) that appear in the complex -plane [5]. If dielectric and/or conductor losses are present, the bound-mode current must have some attenuation and, therefore, its propagation , which wavenumber must be complex: means that the corresponding poles in the -plane are no longer located on the real axis. In Fig. 2, it is assumed for simplicity that surface-wave mode of the grounded substrate is only the . There are then two pairs above cutoff with a wavenumber and one at . The of branch points, one at branch points at are shifted off of the real axis when a lossy dielectric is considered. Conductor losses may also affect the location of these branch points because of the assumed remains losses on the ground plane. The branch points at on the real axis regardless of the possible dielectric or conductor loss. integration path into two It is possible to deform the steepest descent paths of integration (shown in Fig. 2). Inteand grations along the vertical steepest descent paths emanating from the branch points define the residual residual-wave currents, respectively [5]. wave and the Any leaky-mode pole captured during the deformation to the steepest descent paths corresponds to a physical leaky-mode contribution to the continuous-spectrum current, corresponding . The residue contribution from the to the residue path bound-mode pole yields the bound-mode current, which has the form (6) An asymptotic analysis for large [5], [7], [8] reveals that the two residual-wave currents vary as (7) (8) From this analysis, it can be concluded that the bound-mode current is affected by both conductor and dielectric loss since in (6) becomes complex for lossy structures, causing the mode to decay exponentially. Usually simple computer-aided design (CAD) formulas are sufficient to predict the attenuation of the bound-mode current.

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residual-wave current in the lossy case decays both The algebraically and exponentially since the wavenumber of the surface wave is now complex due to loss. However, the residual-wave field is partly in air and partly in the subsurface wave field is). This field is not as strate (just as the well confined to the substrate region as the bound-mode field is. residual-wave curTherefore, the exponential decay of the rent is expected to be typically less than that of the bound-mode current. A very important point of our analysis is that the residualwave current decay rate is not affected by either conductor or residual-wave current will dielectric loss and, therefore, the both for the lossless and lossy cases. decay as The leaky-mode current always decays exponentially due to radiation loss, even in the lossless case. In most cases, the leaky mode attenuation is high enough that practical losses do not significantly change the decay rate of the leaky-mode current. In view of the above discussion, there must be a range of distances where the bound-mode and residual-wave currents become of the same order of magnitude. This is expected to cause strong interference effects (oscillations) and a change in the nature of the current on the line. This will be confirmed by the results presented in Section III. III. RESULTS First of all, the reliability of our approach to characterize lossy structures will be checked. Since material losses come from two sources, i.e., dielectric and conductor losses, our results will be validated for both cases. In the past, the surface impedance technique has provided very good results for the prediction of the attenuation factor of the bound mode for several types of structures [11], [13], [25]. Thus, our results concerning the phase and attenuation constants of the bound mode agree very well with those reported in the literature or provided by CAD formulas (this comparison is not explicitly shown here). In the same way, our results for the attenuation of the bound mode due to dielectric losses have also been checked. However, for a sufficiently high frequency, the total current is no longer dominated by the bound-mode current component because of the increasing relevance of the continuous-spectrum component. It is then necessary to make sure that the surface impedance technique employed to deal with lossy conductor strips is applicable to those high-frequency situations where the total current is not properly accounted for by only the bound-mode component. The above validation has been carried out by comparing our results with those obtained with the commercial electromagnetic solver Ensemble. The calculation of the surface current density excited on the microstrip line by a delta-gap voltage source can be easily carried out in Ensemble. Nevertheless, the presence of termination reflections due to the finiteness of the line required by the simulator makes it difficult to compare Ensemble results with ours. The reflections basically come from the impossibility of matching the continuous-spectrum component of the fields at the end of the line by means of conventional transmission line theory. In an attempt to overcome this problem, the length of the microstrip structure terminated with open ends has been taken as 70- long at 40 GHz. Unfortunately, for practical values of losses, even for this very long

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Fig. 3. Current for a microstrip line (w = h = 1 mm, " = 2:2, tan  = 0:01) versus distance from the source at 40 GHz for a lossless and lossy metallic strip ( = 1:5 2 10 S/m, t = 17 m). Results are compared with those

Fig. 4. Total (TC) and bound-mode (BM) currents for a microstrip line (w = h = 1 mm, " = 2:2) versus distance from the source at 20 GHz for the lossless and lossy cases ( = 4:1 10 S/m, tan  = 0:001).

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provided by Ensemble.

line, reflections from the open-end termination are still significant, causing the current profile on this long line to be rather different from that in our assumed infinite line. Thus, in order to minimize the undesirable reflections for our validation purposes, a high value for the loss tangent of the substrate is imposed to conveniently attenuate the fields at the open ends. It should also be mentioned that Ensemble provides results for the , and our results are for the current . current density Thus, it has been assumed that the longitudinal profile of the current density at the center of the strip is proportional to the , which has required us to find current, namely, the corresponding proportionality constant before comparison. Finally, since Ensemble assumes a small skin depth, the conductivity of the metallic strip has been chosen by enforcing the at the operating skin depth to be a tenth of the strip thickness frequency. After all the previous considerations, Ensemble results for the total current (TC) are compared in Fig. 3 with our computed results for a microstrip line with a lossless/lossy conductor strip printed on a lossy dielectric substrate at 40 GHz (at this high frequency, the continuous-spectrum current component is expected to be significant). This figure shows a good agreement between Ensemble data and our results. Specifically, an excellent match is found when the strip is considered lossy. Both Ensemble and our approach predict the same very noticeable oscillations in the longitudinal profile of the total current due to the interference between the bound-mode and continuous-spectrum current components. In order to discriminate if conductor losses are well accounted for by our approach, a situation with only substrate losses is also shown in Fig. 3. The fact that the total current profile is clearly different for lossless and lossy conductor strips makes apparent that the effect of conductor loss is significant in this structure. Hence, the good agreement found in both cases validates our results both for conductor and dielectric losses. Now that the approach has been validated, several interesting cases will now be studied. In Fig. 4, the total current and the bound-mode current for a low-permittivity structure is shown at 20 GHz. In the lossless case this figure shows that the

Fig. 5. Continuous-spectrum (CS) current versus distance from the source for the microstrip line in Fig. 4 for the lossless and lossy cases.

bound-mode current is dominant at this moderate frequency, although the continuous spectrum is not negligible, giving rise to some interference oscillations. Note that the bound-mode current does not decay with distance from the source, unlike the continuous-spectrum current. For the lossy case, Fig. 4 shows that the bound-mode current decays exponentially due to losses, thus causing the average level of the total current to decay with distance. However, the amplitude of the oscillations does not seem to decrease with distance any faster than in the lossless case. This can be explained taking into account that these oscillations are caused by the presence of a noticeable continuous-spectrum current that does not seem to be significantly affected by the amount of losses considered here. To better illustrate this fact, Fig. 5 plots the continuous-spectrum current excited on the microstrip structure for the lossless and lossy cases in the same range of distances as in Fig. 4. Note that the curves are practically superimposed, which agrees with the aforementioned observation that the amplitude of the oscillations is hardly affected by the presence of losses in Fig. 4. A more detailed study of the behavior of the different components of the continuous-spectrum current with and without

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Fig. 7. Total (TC) and bound-mode (BM) currents for a microstrip line (w = h = 1 mm, " = 2:2) versus distance from the source at 40 GHz for lossless and lossy cases ( = 4:1 10 S/m, tan  = 0:001).

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Fig. 6. Bound-mode (BM) and residual-wave (RW) components of the current versus distance from the source for the microstrip line in Fig. 4 for the lossless and lossy cases.

losses is presented in Fig. 6 for a wider range of longitudinal distance. It can be seen that the bound-mode current clearly exhibits an exponential decay due to loss, which is in agreement with the expected attenuation factor that appears in (6) ( , when material loss is present), as shown residual-wave curin the table accompanying Fig. 6. The rent shows a slightly faster decay for the lossy case than that observed for the lossless one. This is also expected in light of (7) because the presence of dielectric losses will only cause the apwavenumber pearance of a small attenuation factor in the (see the table in Fig. 6; the small value of the attenuation constant makes the lossy and lossless curves almost indistinguishable in the range considered). On the contrary, the pure algebraic residual wave does not seem to be affected by decay for the losses. Again, this is expected looking at (8) since the free-space wavenumber will not change with dielectric and/or conductor losses, and the asymptotic decay is not affected by losses. The results show that the lossless and lossy results are indeed quite close. The study previously presented in Fig. 4 has also been carried out for a microstrip printed on a substrate with a higher dielectric constant (these results are omitted here). Most of the conclusions from the low-permittivity case are valid in the high-permittivity case. In order to analyze the effect of losses at a higher frequency, Fig. 7 shows the behavior of the total current and bound-mode current for a lossless/lossy low-permittivity substrate at 40 GHz. Due to the expected increase of the continuous-spectrum current level at this higher frequency, it can now be observed that oscillations in the total current are increased considerably with respect to those in Fig. 4. A detailed picture of the behavior of the effect of losses on the different components of the total current is presented in Fig. 8, together with the values of the normalized wavenumbers of the bound mode, leaky mode, and modes. It can be observed that a leaky-mode current component now exists, which dominates the continuous-spectrum current

Fig. 8. Bound-mode (BM), leaky-mode (LM), and residual-wave (RW) components of the current versus distance from the source for the microstrip line in Fig. 7 for the lossless and lossy cases at 40 GHz.

at short distances from the source, but clearly decays exponentially at longer distances. The presence of this highly attenuated leaky mode explains the strong oscillations near the source observed in the total current in Fig. 7, but far away from the source, its role is practically negligible. Due to the highly attenuating nature of the improper leaky mode, the wavenumber and amplitude of the leaky mode are not much affected by the considered values of losses. On the other hand, Fig. 8 shows that the bound mode is appreciably affected by losses, as seen by the exponential decay of the lossy bound mode curve and the value of its attenuation constant. With respect to the residual-wave current, this component seems to be only slightly affected by losses close to the source, as seen in the plot of its corresponding lossless/lossy curves and in the small value of its attenuation constant. For larger distances there is a noticeable difference between the lossless and lossy cases, due to the exponential decay of the residual-wave current in the lossy case. The lossless and lossy residual-wave components show an almost

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Fig. 10. Total current (TC) for a microstrip line (w = h = 1 mm, " = 4:0) versus distance from the source at 20 GHz for the lossless and lossy cases (tan  = 0:022,  = 5:8 2 10 S/m). Fig. 9. Bound-mode (BM) and residual-wave (RW) components of the current for a microstrip line (w = h = 1 mm, " = 4:0) versus distance from the source at 20 GHz for the lossless and lossy cases (tan  = 0:022,  = 5:8 10 S/m.

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identical behavior for all distances. In summary, the present case shows that losses seem to affect mainly the bound-mode current, to a lesser degree the residual-wave components, and only to a minor degree the leaky-mode component. Based on the results reported up to now, it can be speculated that the current spectrum of a microstrip line can be modified by losses since the effect of losses on each component of the current is different. In fact, it is possible to find situations where changes in the spectrum are very significant. To illustrate this fact, Fig. 9 plots the bound-mode current along with the components of the continuous-spectrum current for a microstrip line printed on a commercial FR-4 substrate for a wide range of distances from the source at 20 GHz. At this frequency, the continuous-spectrum current is accounted for by the residual-wave current and the residual-wave current. Wavenumbers of residual-wave currents are shown in the bound-mode and the table accompanying Fig. 9. For the lossless case, the boundmode current is clearly the dominant part of the total current at all distances. However, for the lossy case, the bound-mode current is highly attenuated, and the continuous-spectrum curresidual-wave components) become dominant rent ( and at large distances from the source. Moreover, when very large , the distances from the source are considered residual-wave current is the most dominant current component in this lossy case. It can then be concluded that, in the present case, the order of dominance of the components of the total current at large distances from the source is fully inverted from that observed in the lossless case. To investigate the global effect of losses on the total current, Fig. 10 shows the total current for the lossless/lossy structure analyzed in Fig. 9. Although oscillations in the total current near the source exist for the lossless case, they cannot be observed in this figure due to the wide range of the logarithmic vertical scale. For the lossy case, it can be clearly observed that strong

oscillations are now present due to interference effects where the attenuating bound-mode current becomes of the same order of , see magnitude as the residual-wave currents (around Fig. 9). From this point on, the continuous-spectrum current becomes dominant, and the decay of the total current is no longer dominated by the bound-mode exponential attenuation. Oscillations at larger distances from the source are due to interference residual-wave and residual-wave currents. between the For even larger distances from the source (off the scale of this residual-wave plot), the oscillations die out, since only the current would prevail. In the lossless case, oscillations always decrease as distance from the source increases, since the continuous-spectrum current is decaying relative to the bound-mode current. On the other hand, in the lossy case, the relative level of the oscillations may actually increase with distance after the continuous-spectrum current becomes dominant with respect to the bound-mode current, as shown in Fig. 10. At a higher frequency (40 GHz), Fig. 11 shows the behavior of the different current components for the same structure as in Figs. 9 and 10. At this frequency, it can be observed that residual-wave current now exhibits a more noticeable the exponential decay, which is consistent with the significant increase of its attenuation factor reported in the table of Fig. 11 (which, in turn, is consistent with the fact that the fields of the mode are increasingly confined to the substrate as frequency increases). It is interesting to mention that, at 40 GHz, surface-wave mode is simthe attenuation factor of the attenuailar to that of the bound mode (at 20 GHz, the tion factor was an order of magnitude less than the bound-mode one). The residual-wave current still exhibits the same decay rate as in the lossless case. The above change in the behavior residual-wave current is reflected in the form of the of the total current shown in Fig. 12. When compared with Fig. 10, Fig. 12 shows only a slightly stronger decay of the total current for distances not very distant from the source. However, the osat 20 GHz cillations shown for larger distances are not now present because of the negligible relative imporresidual-wave current with respect to the tance of the

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Fig. 13. Bound-mode (BM) and residual-wave (RW) components of the current for a microstrip line (w = h = 1 mm, " = 4:0, tan  = 0) versus distance from the source at 20 GHz for the lossless and lossy strip cases ( = 10 S/m). Fig. 11. Bound-mode (BM) and residual-wave (RW) components of the current for a microstrip line (w = h = 1 mm, " = 4:0) versus distance from the source at 40 GHz for the lossless and lossy cases (tan  = 0:022,  = 5:8 10 S/m).

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Fig. 14. Bound-mode (BM) and residual-wave (RW) components of the current for a microstrip line (w = h = 1 mm, " = 4:0, tan  = 0) versus distance from the source at 20 GHz for the lossless and lossy ground plane (gp) cases ( = 10 S/m). Fig. 12. Total current (TC) for a microstrip line (w = h = 1 mm, " = 4:0) versus distance from the source at 40 GHz for the lossless and lossy cases (tan  = 0:022,  = 5:8 10 S/m).

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residual-wave one. It is interesting to observe that the oscillations in Fig. 12 occur around the distance for which the amresidual-wave currents plitudes of the bound-mode and the coincide. An additional interesting question that has been explored is the sensitivity of the different current components to the ground-plane losses and to the strip losses as separate effects. residual-wave current It should be expected that the is not affected by strip losses, but this component should be attenuated by ground-plane losses as would the associated surface wave propagating in the background dielectric waveguide. To confirm this speculation, Figs. 13 and 14 show the components of the current for the FR4 case previously analyzed in Fig. 9 when either a lossy strip or a lossy ground plane is considered. Results are compared for a lossless case (no dielectric or conductor loss) and cases having only strip or ground plane loss, respectively. An artificially low value

for the metal conductivity S/m has been chosen in order to make the conductor loss effects more apparent (when the actual conductivity is used, it is found that neither the strip, nor the ground-plane loss has a noticeable effect on either of the two residual-wave currents, and the metal loss has only a small effect on the bound mode). In Fig. 13, strip losses are studied for a perfect conductor ground plane. It can be seen that the bound-mode current is strongly attenuated due to the losses on the strip, but the residual-wave currents remain almost unaltered. In Fig. 14, a structure with a lossy ground plane and a perfect conductor strip is studied. In this latter case, the residual wave are both attenuated due to bound mode and losses, while the residual-wave current is barely affected by the losses (and the decay rate is not affected at all). IV. CONCLUSION This paper has studied the effects of dielectric and/or conductor losses on the current excited on a microstrip line by a gap voltage source. The introduction of material losses gives a more realistic and practical orientation to both the analytical and

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numerical results compared to previously published results for lossless structures. Special attention has been paid to the specific effects of each type of loss on the different components of the current, and on their influence on the appearance of spurious effects such as interference oscillations and a change in the decay rate of the current. In general, it has been found that the bound-mode current is more influenced than the corresponding continuous-spectrum current by both substrate and conductor losses. Since the bound-mode current is typically the prominent part of the excited current in lossless structures, this means that the total current can differ significantly in the lossless and lossy cases. With respect to the different components of the continuous-spectrum current, it has been found that the leaky-mode current is hardly affected by dielectric/conductor losses (at least for common cases where the attenuation constant of the physical residual-wave current is found to leaky mode is high). The be appreciably perturbed by substrate losses and ground-plane losses, and less influenced by losses in the conductor strip. Firesidual-wave current seems to be the component nally, the that is the least affected by any type of losses, which is somewhat expected since its associated fields are basically in free space. The above facts have been numerically verified in practical lossy structures, which has also revealed that, at large distances from the source, the relative importance of the different components of the current spectrum is exactly reversed with respect to that reported for lossless structures.

REFERENCES [1] D. Nghiem, J. T. Williams, D. R. Jackson, and A. A. Oliner, “Leakage of the dominant mode on a microstrip with an isotropic substrate: Theory and measurements,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 10, pp. 1710–1715, Oct. 1996. [2] H. Shigesawa, M. Tsuji, and A. A. Oliner, “A simultaneous propagation of bound and leaky dominant modes on printed-circuit lines: A new general effect,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 12, pp. 3007–3019, Dec. 1995. [3] C. Di Nallo, F. Mesa, and D. R. Jackson, “Excitation of leaky modes on multilayer stripline structures,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 8, pp. 1062–1071, Aug. 1998. [4] M. J. Freire, F. Mesa, C. Di Nallo, D. R. Jackson, and A. A. Oliner, “Spurious transmission effects due to the excitation of the bound mode and the continuous spectrum on stripline with an air gap,” IEEE Trans. Microwave Theory Tech., vol. 47, no. 12, pp. 2493–2502, Dec. 1999. [5] F. Mesa, D. R. Jackson, and M. J. Freire, “High-frequency leaky-mode excitation on a microstrip line,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 12, pp. 2206–2215, Dec. 2001. [6] F. Mesa, C. Di Nallo, and D. R. Jackson, “The theory of surface-wave and space-wave leaky-mode excitation on microstrip lines,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 2, pp. 207–215, Feb. 1999. [7] D. R. Jackson, F. Mesa, M. J. Freire, D. P. Nyquist, and C. Di Nallo, “An excitation theory for bound modes, leaky modes, and residualwave currents on stripline structures,” Radio Sci., vol. 35, no. 2, pp. 495–510, Mar.–April 2000. [8] P. Baccarelli, P. Burghignoli, F. Frezza, A. Galli, G. Lovat, and S. Paulotto, “Asymptotic analysis of bound-mode and free-space residualwave currents excited by a delta-gap source on a microstrip line,” Radio Sci., vol. 39, no. 3, p. RS3011, May–June 2004. [9] R. H. Jansen, “High-speed computation of single and coupled microstrip parameters including dispersion, high-order modes, loss and finite strip thickness,” IEEE Trans. Microw. Theory Tech., vol. MTT-26, no. 2, pp. 75–82, Feb. 1978.

[10] D. Mirshekar-Syahkal, “An accurate determination of dielectric loss effect in monolithic microwave integrated circuits including microstrip and coupled microstrip lines,” IEEE Trans. Microw. Theory Tech., vol. MTT-31, no. 11, pp. 950–954, Nov. 1983. [11] T. E. Deventer, P. B. Katehi, and A. C. Cangellaris, “An integral equation method for the evaluation of conductor and dielectric losses in high-frequency interconnects,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 12, pp. 1964–1972, Dec. 1989. [12] W. Heinrich, “Full-wave analysis of conductor losses on MMIC transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 38, no. 10, pp. 1468–1472, Oct. 1990. [13] N. K. Das and D. M. Pozar, “Full-wave spectral domain computation of material, radiation and guided wave losses in infinite multilayered printed transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 1, pp. 54–63, Jan. 1991. [14] W. Heinrich, “Conductor loss in transmission lines in monolithic microwave and millimeter-wave integrated circuits,” Int. J. Microw. Millimeter-Wave Comput.-Aided Eng., vol. 2, no. 3, pp. 155–165, Jul. 1992. [15] D. F. Williams, J. E. Rogers, and C. L. Holloway, “Multiconductor transmission-line characterization: Representations, approximations, and accuracy,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 4, pp. 403–409, Apr. 1999. [16] C. M. Krowne, “Dyadic Green’s function modifications for obtaining attenuation in microstrip transmission layered structures with complex media,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 1, pp. 112–122, Jan. 2002. [17] J. C. Rautio and V. Demir, “Microstrip conductor loss models for electromagnetic analysis,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 915–921, Mar. 2003. [18] J. Bernal, F. Mesa, and D. R. Jackson, “Effects of dielectric and conductor losses on the current spectrum excited by a gap voltage source on a printed circuit line,” in IEEE MTT-S Int. Microw. Symp. Dig., San Francisco, CA, Jun. 2006, pp. 1307–1310. [19] 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. [20] 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. [21] D. G. Fang, J. J. Yang, and G. Y. Delisle, “Discrete image theory for horizontal electric dipoles in a multilayered medium,” Proc. IEEE, vol. 135, no. 10, pp. 297–302, Oct. 1988. [22] 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. [23] 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. [24] J. Bernal, F. Mesa, and D. R. Jackson, “Crosstalk between two microstrip lines excited by a gap voltage source,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 8, pp. 1770–1780, Aug. 2004. [25] J. Aguilera, R. Marques, and M. Horno, “Quasi-TEM surface impedance approaches for the analysis of MIC and MMIC transmission lines including both conductor and substrate losses,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 7, pp. 1553–1558, Jul. 1995.

Joaquín Bernal (M’06) was born in Seville, Spain, in 1971. He received the Licenciado and Doctor degrees in physics from the University of Seville, Seville, Spain, in 1994 and 2000, respectively. In 1995, he joined the Departmento de Electrónica y Electromagnetismo, University of Seville. In 1998, he joined the Departmento de Física Aplicada 3, University of Seville, where he became Associate Professor in 2004. His research interests focus on the analysis of planar structures for integrated microwave circuits and high-speed very large scale integration (VLSI) interconnects.

BERNAL et al.: EFFECTS OF LOSSES ON CURRENT SPECTRUM OF PRINTED-CIRCUIT LINE

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 Associate Professor with the Department of Applied Physics 1, University of Seville. His research interests focus on electromagnetic propagation/radiation in planar lines with general anisotropic materials.

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David R. Jackson (S’83–M’84–SM’95–F’99) was born in St. Louis, MO, on March 28, 1957. He received the B.S.E.E. and M.S.E.E. degrees from the University of Missouri, Columbia, in 1979 and 1981, respectively, and the Ph.D. degree in electrical engineering from the University of California at Los Angeles (UCLA), in 1985. From 1985 to 1991, he was an Assistant Professor with the Department of Electrical and Computer Engineering, University of Houston, Houston, TX. From 1991 to 1998 he was an Associate Professor, and since 1998, he has been a Professor with this same department. He has also served as an Associate Editor for the Journal of Radio Science and the International Journal of RF and Microwave Computer-Aided Engineering. His current research interests include microstrip antennas and circuits, leaky-wave antennas, leakage and radiation effects in microwave integrated circuits, periodic structures, and electromagnetic compatibility (EMC). Dr. Jackson is currently the chair of the Transnational Committee of the IEEE Antennas and Propagation Society (IEEE AP-S), and chair for URSI, U.S. Commission B. He is also on the Editorial Board of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. He was Chapter activities coordinator for the IEEE AP-S, a Distinguished Lecturer for the IEEE AP-S, an associate editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, and a member of the IEEE AP-S Administrative Committee (AdCom).

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Microwave Circuit Design by Means of Direct Decomposition in the Finite-Element Method Valentín de la Rubia and Juan Zapata, Member, IEEE

Abstract—In this paper, a domain decomposition approach for design purposes is proposed. The analysis domain is divided into subdomains according to the arbitrarily shaped parts that should be modified in the synthesis process. A full-wave matrix-valued transfer function describes each decomposition subdomain, namely, an admittance-type matrix. Field continuity between subdomains is directly enforced by an admittance matrix connection. This methodology makes it possible to analyze only those parts of the analysis domain that are supposed to evolve in order to satisfy the design specifications. Furthermore, several modifications in the shape of the components are allowed as a consequence of the easy matrix connection process, where the consideration of different admittance-type matrices or absence of them gives rise to distinct geometric structures. A model order reduction technique is also considered for fast frequency sweeping. Finally, several numerical examples illustrate the capabilities of the proposed procedure, as well as its accuracy. Index Terms—Admittance matrix, design automation, design methodology, domain decomposition methods (DDMs), finiteelement methods (FEMs), piecewise (PCW) linear approximation, reduced-order systems.

I. INTRODUCTION

M

ICROWAVE designs are currently complex enough for requiring full-wave design techniques, demanding computer-aided design (CAD). Most numerical methods are conceived as analysis tools, but current interests encourage us to go beyond simple analysis because a wide range of analyses ought to be carried out when developing a design by means of optimization processes. The finite-element method (FEM) is widely known for its flexibility and reliability, but is still merely thought of as an analysis tool due to its rather time-consuming characteristics, especially when dense meshing and high-frequency resolution over a wide band are demanded. Although a 2-D FEM analysis substantially reduces its computational costs and makes it compatible with CAD [1], there is still intensive research to be carried out in the 3-D case. Over the last three decades, a considerable effort has been made in speeding up the solution time for the FEM, as well as increasing its capabilities. The fine mesh requirement in dielectric rods and conducting insets is mitigated in [2] by the introduction of macro-elements Manuscript received January 3, 2007; revised April 2, 2007. This work was supported by the Ministerio de Educación y Ciencia (MEC), Spain, under a scholarship and by the MEC under Contract TEC2004-00950/TCM. The authors are with the Departamento de Electromagnetismo y Teoría de Circuitos, Universidad Politécnica de Madrid, 28040 Madrid, Spain (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2007.900307

in the stiffness matrix. The inclusion of lumped elements in the finite-element (FE) analysis is also appealing [3]. When the FEM is used to solve open-region scattering and radiation problems, the infinite analysis domain must be truncated to restrict the computational domain. An artificial boundary results and a boundary condition is needed. Different fashions have been proposed to handle this boundary condition. The so-called unimoment method exploits the eigenfunction expansion of the field on the artificial boundary [4], [5]. Boundary integral equations also stand for suitable boundary conditions [6], [7]. A boundary condition preserving the sparse and banded nature of the matrices arisen in the FEM is the absorbing boundary condition [8]–[10] because local boundary conditions relating the fields on the artificial boundary are considered. In addition, interest in the domain decomposition method (DDM) is becoming popular not only for enabling the use of parallel architectures, and thus reducing the CPU time compared with the traditional approach [11], but also for the new possibilities that arise. The general idea is to split the analysis domain into smaller subdomains and solve a sequence of similar subproblems on these new subdomains. In [12], the boundary conditions are adjusted iteratively by ad hoc transmission conditions of Robin type between adjacent subdomains. Further emphasis on the transmission conditions is addressed in [13]. Effective preconditioners for iterative handling of the large sparse system solution come about by considering the polynomial order of the basis functions as the decomposition domain [14]. A noniterative DDM is used in [15] saving memory requirements and computing time in the determination of the Doppler spectrum of a flying target. Likewise, a nonconforming DDM framework is also appealing [16]. There are increasing efforts in extending the FEM capabilities to analyze large problems. The case of large repeating geometries is of special interest in photonic and electromagnetic-bandgap structures. Iterative procedures exploiting geometric repetition in the DDM have been addressed in [17] and [18]. In this paper, we propose a domain decomposition scheme for design purposes. When developing a design by means of an optimization technique, the analysis domain is typically modified, e.g., its shape, the electromagnetic properties of the materials, or their shapes, until the required specifications are satisfied. It seems clear that the global structure ought to be analyzed each time a modification is carried out. However, there is no point in having to solve the whole electromagnetic problem while only a small portion of domain has changed. What we propose is to isolate those parts from the remaining analysis domain in such a way that every modification carried out on these be transparent to the remaining analysis domain, which should only be analyzed once, and thus concentrate on the analysis of those parts that evolve. In other words, we aim to use our computing

0018-9480/$25.00 © 2007 IEEE

DE LA RUBIA AND ZAPATA: MICROWAVE CIRCUIT DESIGN BY MEANS OF DIRECT DECOMPOSITION IN FEM

resources during the optimization process only in those parts of the analysis domain that change. In order to carry out this strategy, a direct decomposition approach in the FEM is considered, and thus, the (arbitrary) subdomains are electromagnetically described by full-wave matrix-valued transfer functions, namely, admittance-type matrices. Once these transfer functions are determined, the response of the overall system is recovered by the appropriate admittance matrix connection. This methodology is compatible with a reduced-order model approach, therefore, a fast frequency sweep (FFS) technique can be carried out in the admittance-type matrices. The general idea of splitting a domain into subdomains is rather old since it has been extensively used in different ways in the past. Our approach draws upon the mathematical literature on DDMs, and mainly upon substructuring methods with regard to the solution of partial differential equations [19]–[21]. However, further exploitation of the matrix-valued transfer function describing the electromagnetics in an arbitrary subdomain is addressed in this study. The theory presented here relies on the formulation adopted in [22]. However, they both work on different purposes. Namely, in this paper, modifications of a given structure are actually addressed by means of a domain decomposition approach, whereas [22] deals with the incorporation of artificial ports inside a single domain, and their use in obtaining the electromagnetics within modifications of strips and slots in a given structure via a single multipurpose admittance matrix. This paper is organized as follows. In Section II, we briefly review the FEM formulation of the boundary value problem, establish the foundations of the domain decomposition procedure, consider a model order reduction, and present the matrix connection scheme in order to restore the response of the original system. Section III deals with numerical examples and is conceived to illustrate the capabilities of the proposed approach, as well as its accuracy. Finally, in Section IV, we comment on the conclusions. II. FORMULATION Applying Galerkin projection onto an admissible function space , the following weak formulation is considered for the boundary value problem arisen in the time–harmonic Maxwell equations: Find

such that

(1) where and are the relative permittivity and permeability is the wavenumber, of the medium, respectively, and . is a source-free bounded domain. on the boundary of is the boundary condition, related to the tangential electric field on , where is the unit , and and are outward normal vector on the boundary the electric and magnetic fields [23]. The FEM is chosen to solve problem (1). In this sense, the solution to this problem requires an approximation of the solu-

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tion space to be defined. Given a tetrahedral mesh of the domain , the following vector functions are defined in the reference tetrahedron: Edge functions Face functions

(2)

with and and being the normalized coordinates for this simplex [24]. As a result, the FE space is made up of vector functions with a continuous tangential component across inter-element faces, the gradient and rotational fields have order 1 and 2, respectively. Thus far, the following linear algebraic system appears [25]: (3) where is the coefficient vector of the magnetic field in is a sparse vector, which takes into account the boundary condiand are sparse symmetric FE matrices, where tion, and the superscript denotes the analysis domain. contains a subset in which a modal spectrum Whenever can be determined, the field may be written as a linear combination of modal fields. As a result of the inclusion of the modal boundary condition in the discretization of problem (1), a matrix relationship between the tangential electric and magnetic fields on the modal boundary , namely, a generalized admittance matrix (GAM), arises (see [25] for all the details). Thus, (4) and are coefficient vectors of the modal tangential electric and magnetic field on . A. Direct Decomposition Typically, when developing a design, only a small part of domain , with reference to shape or electromagnetic properties of the media therein, is modified. However, in view of (3), the overall system ought to be solved each time a modification is carried out, which ends up in a rather time-consuming procedure that is prohibitive in most situations. What we propose is to minimize the impact of parameter modification on the overall numerical computation. In this regard, one may be interested in decomposing the analysis domain into subdomains in which the modal segmentation technique [25] cannot be applied. Thus, let be the part of , which should be modified. Had we known the tangential electric field on the boundary of , the subdomain could have been removed from the computation in , decomposing problem (1) into two separate problems, conand . However this is not our case and, cerning domains instead, our approach defines a suitable basis for the tangential . Hence, electric field on the boundary may be expressed as the tangential electric field on

on

(5)

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with representing the coefficient of the basis function . An admissible function space for the tangential electric field is demanded. Taking advantage of the FEM context, we use piece, which wise (PCW) functions. Given a triangular mesh of of the tetrahedral mesh of indeed will be the restriction on domain , the following vector functions are considered in the reference triangle: (6) and and are the normalized where coordinates for this simplex [22]. It should be stated that tangential continuity across inter-element edges, along with completeness to order 1, are common features in this function space. The electric field in the domain , related to the curl of the magnetic field in , has order 1 as a result of the use of the basis functions (2) in the reference tetrahedron. This is the rational for the function space (6), where the face functions, which have order 2, are not considered. As a result, problem (1) may be decomposed in the following fashion. Without loss of generality, we assume and have disjoint boundaries in our explanation. On the one hand, regarding domain ,

with being the coefficient of the PCW function result, the following integral relationship arises:

[22]. As a

(11) which in matrix form reads (12) stands for the coefficient vector of the tangential magnetic field , and is a sparse matrix with elements on (13)

This nonsingular matrix may be regarded as a change of basis in span , i.e., . denotes the coefficient vector of the tangential magnetic field on with respect to a new basis for the tangential magnetic field. Consequently, the following admittance-type matrix may be considered and describes the electromagnetics in :

(7) (14) where

and are sparse symmetric FE matrices in domain and denote coefficient vectors of the magnetic field in and the tangential electric field on , respectively, and is a sparse matrix with elements (8)

On the other hand, concerning domain

, it follows that

(9) and are the FE matrices in domain where , and represents coefficient vector of the magnetic field in . In addition to having proposed a basis for the tangential elec, we may also consider a basis for tric field on the boundary . In order to the tangential magnetic field obtain a full-wave transfer function describing the electromag, namely, an admittance-type matrix, netics either in or in a mapping between the magnetic field either in or in and the tangential magnetic field on is required in (7) and (9). Let us consider the same basis as already defined in (6), i.e., , then

Analogously, the following admittance-type matrix arises in do: main

(15) To this extent, the original problem has been decomposed into two nonoverlapping subdomains where both admittance-type matrices and denote the electromagnetic behavior therein. However, a link is required in order to restore the overall electromagnetics in . Tangential field continuity across the boundary is what we are talking about. This requirement is achieved through a simple admittance matrix connection. Let us be more precise. The following matrix-valued transfer functions are what we have obtained by means of this formulation:

(16) Tangential field continuity across the boundary is easily apand . As a result, a GAM plied by imposing matrix appears as follows:

(17) on

(10)

which definitely constitutes an alternative way of computing (4).

DE LA RUBIA AND ZAPATA: MICROWAVE CIRCUIT DESIGN BY MEANS OF DIRECT DECOMPOSITION IN FEM

B. Numerical Aspects All admittance-type matrices arising in the above formulation, (4), (14), and (15), are alike in form, viz., (18) Let be the number of degrees of freedom involved in the FE discretization, i.e., the dimension of the stiffness matrix and let be the dimension of the admittance-type matrix , i.e., the total number of modes and PCW functions taken into account. In order to determine the admittance-type matrix at a given frequency, the factorization of the sparse symmetric matrix is carried out taking advantage of its skyline matrix storage. It should be noticed that is an sparse matrix. We have

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PCW functions can properly handle inhomogeneous ports for fast frequency sweeping. Thus, (24) being a specific wavenumber and with diagonal matrix with elements

representing a

for PCW functions, TEM, and spherical modes for TE modes for TM modes

(25)

where

(19) with being the cutoff wavenumber of mode . Equation (23) then reads

Therefore, (19) reads (20) where

is the solution to the sparse triangular system

(26) with

(21) As a result, an sparse triangular system ought to be solved times, a single mode or PCW function considered each time. Concerning efficiency, matrix sparsity is advantageous. Let us be more precise. As suggested in [26], in the th column of whenever through the th column of . Hence, by considering the position of the first nonzero element in each column of , the amount of operations is reduced. This in information can be further used in the determination of (20). Thus,

(27) This symmetric matrix-valued transfer function is symmetric matrix Padé via Lanczos (SyMPVL) compatible [27]. By choosing a wavenumber expansion point and carrying out the factorization of matrix at this wavenumber, instead of (27), we have (28) with

(22) (29) Notice that this matrix is symmetric and, therefore, only of their elements need to be explicitly calculated. It is clear from the above discussions that the number of operations is optimized. C. Padé Approximation via the Lanczos Algorithm As previously stated, all matrix-valued transfer functions arising in the proposed method have the same form. Furthermore, if we made their frequency dependence explicit, we would realize that they behave in a similar way as follows:

where and are square and rectangular matrices, respectively [28]. The SyMPVL algorithm generates a sequence of vectors, known as Lanczos vectors, which span the th block Krylov subspace . By means of these, an th matrix Padé approximant to (28) arises (cf. [29]) as follows: (30) As a result, a reduced-order model, which makes an FFS possible, is allowed. D. Computational Aspects

(23) An analytical frequency variation appears in matrix whenever plane and spherical homogeneous waveguide modes and PCW functions are considered. In this sense, it should be noticed that

The proposed approach permits the computational cost in a , design procedure to be twofold: one static part, as regards which is carried out once and might be time consuming, and a dynamic part, concerning , which ought to be carried out each time a modification in the design parameter is carried out, and

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whose computational impact needs to be minimized. This is a good strategy since numerous changes in the design parameter are normally required. Nevertheless, if we feel that the computational cost of either the static or dynamic part is not low enough, we may proceed with a further direct decomposition on that subdomain, expecting to reduce its computational impact on the overall performance. This situation is of special interest in the dynamic part since it also makes it possible to consider further changes in the objects therein such as shape modification or dielectric texturization. Section III will deal with this in greater depth. In addition, we would like to mention that this strategy allows the numerical simulation of large structures even though their numerical discretization cannot be treated as a whole on a single PC. When a reduced-order model is used for fast frequency sweeping, special care should be taken on the frequency band where the Padé approximant converges on the original response with respect to this formulation because it is through admittance-type matrix reduced-order model connections that the response of the overall system will come about. Therefore, we are not content with some predefined order of the Padé approximation, but we require a specific convergence frequency band. This ensures all admittance-type matrix connections are accurate enough in that frequency band. In this regard, we use the technique proposed in [30], where a measurement of the error between the original response and its reduced-order model is carried out, as well as analyzing the convergence throughout different Padé approximant orders [22]. III. NUMERICAL RESULTS Here, we illustrate the proposed methodology through three different devices in which we wish to carry out several modifications: a dual-mode circular waveguide filter, a hemispherical dielectric resonator antenna, and a cylindrical dielectric resonator filter. The possibilities of this approach will become apparent throughout these examples. All calculations were carried out on a laptop computer with a 1.66-GHz Intel T2300 processor and 1-GB RAM. A. Dual-Mode Circular Waveguide Filter In this example, we consider a four-pole elliptic dual-mode circular cavity filter. Fig. 1 shows the filter configuration, as well as its dimensions. Each circular cavity is connected to corresponding input and output WR75 rectangular waveguides by identical slots. In addition, both cavities are connected by a cross-shaped iris. Finally, each cavity is provided with a horizontal tuning screw and a tilt coupling screw. This filter was designed in [31] by means of a hybrid mode-matching (MM) 2-D FE method to provide a 100-MHz bandwidth centred at 11.8 GHz. Here, we first analyze this structure by means of the segmentation procedure proposed in [28], where the overall domain analysis is divided into subdomains according to availability of modal field description and, therefore, analytical subdomains, or mandatory FEM resolution. This procedure circumvents the cumbersome relative convergence problem appearing in MM methods. Afterwards, we study the influence of both iris arm lengths of the

Fig. 1. Geometry of the dual-mode circular waveguide filter. Cavity length: 43.87 mm, radius: 14 mm, iris thicknesses: 1.5 mm, slot lengths: 10.05 mm, slot widths: 3 mm, arm widths: 2 mm, horizontal arm length: 7.65 mm, vertical arm length: 8.75 mm.

Fig. 2. Side view of the dual-mode circular waveguide filter. Modal segmentation for the analysis of the structure by means of the methodology proposed in [28].

cross-shaped iris on the overall filtering performance by means of the current methodology. Fig. 2 details the modal segmentation carried out in the analysis of the filter. The whole structure is assembled by joining the subdomains highlighted in Fig. 2, which are analyzed by means of the FEM, and corresponding circular waveguide sections. Since both input and output rectangular waveguide transitions to circular waveguide (transitions 1 and 2 in Fig. 2) are identical, it is only necessary to analyze one of these. Since the numerical simulation is different from that carried out in [31], it should be noticed that the tuning and coupling screw depths and has to be readjusted to 3.66 and 3.35 mm, respectively, in order to obtain the four-order elliptic filter response. Fig. 3 compares the results of this approach with those measured in [31]. Good agreement is achieved. Now, we concentrate on how the filter response is affected by the cross-shaped iris when both horizontal and vertical iris arm lengths are changed. Thus, we just take into account the transition between circular waveguides including the coupling iris in Fig. 2. The domain being analyzed is also shown and is dein Fig. 4, where the subdomain splitting tailed. However, subdomain is further split, as Fig. 4(b) shows, since we want to analyze different arm lengths. As a result, the original domain is decomposed into three nonoverlapping

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Fig. 3. Dual-mode circular waveguide filter simulation results are compared with measurements.

subdomains, i.e., and , which, in turn, is further decomposed into 13 subdomains, the numbering of these being detailed in Fig. 4(b). Each subdomain is characterized electromagnetically by its admittance-type matrix, and straightforward admittance matrix connection reinstates the GAM of the overall circuit. It should be noted that whenever an admittance-type matrix of a cross-shaped iris decomposition subdomain is not taken into account in the matrix connection process, the subsequent GAM describes the original structure with a perfect electric conductor replacing that subdomain. Therefore, we use this procedure to determine the response of the circuit for different iris arm lengths. Table I presents the computational cost of the static part in this approach, which concerns the determination of the admittance-type matrices for each subdomain in frequency pointwise analysis and FFS technique, constrained to converging in the band from 11.6 to 12 GHz, as well as the number of PCW functions used. On the other hand, the dynamic part in this example, the admittance matrix connection, takes no longer than 4.8 s/frequency point in all the cross-shaped iris configurations taken into account. It should be mentioned that up to 2 different coupling iris configurations can straightforwardly be analyzed by means of this decomposition. As a consequence of the discrete arm length values analyzed, this last approach just gives an idea about the role played by the cross-shaped iris in the filtering structure. Nevertheless, this strategy may be suitable, e.g., for training purposes in an artificial neural-network-based approximator scheme, where a continuous geometry variation comes about with ease. In order to carry out a full-wave continuous variation of the geometric parameter space, namely, the cross-shaped iris arm lengths, a different decomposition strategy needs to be considered. Fig. 5 proposes a continuous geometric variation compatible domain decomposition, where the whole cross-shaped iris is enclosed by the same subdomain. Thus, the original domain (region C in Fig. 2) is decomposed this time into three nonoverlapping suband . It should be noticed that, on this occadomains sion, the whole subdomain ought to be analyzed whenever a

Fig. 4. Cross-shaped iris. Note that the analysis domain is the dielectric part only since perfect electric conductors are considered. (a) Decomposition of the analysis domain into subdomains. (b) Domain decomposition of the cross-shaped iris in order to allow several iris configurations.

TABLE I STATIC PART COMPUTATIONAL RESULTS IN THE DIRECT DECOMPOSITION OF THE CROSS-SHAPED IRIS IN FIG. 4

modification in the cross-shaped iris dimensions is carried out. Table II details the computational cost of the static part within

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Fig. 5. Domain decomposition of the cross-shaped iris in order to allow a continuous geometric variation on the iris dimensions.

TABLE II STATIC PART COMPUTATIONAL RESULTS IN THE DIRECT DECOMPOSITION OF THE CROSS-SHAPED IRIS IN FIG. 5

TABLE III DYNAMIC PART COMPUTATIONAL RESULTS IN THE DIRECT DECOMPOSITION OF THE CROSS-SHAPED IRIS IN FIG. 5

this strategy, i.e., the CPU time for the calculation of the admitand , constrained to contance-type matrices for domains verging in the band from 11.6 to 12 GHz. On the other hand, the dynamic part within this new approach is increased since whenever a modification in the iris arm lengths is carried out, not only is the admittance matrix connection process to be considered, but also the computation of the admittance-type matrix for domain . Table III shows the computational cost of the dynamic part. These values are the maximum of those obtained within the different iris arm lengths analyses, constrained to converging in the band from 11.6 to 12 GHz. It should be noted that these dynamic computational costs make it possible to use the proposed methodology in a full-wave CAD-oriented strategy. The comparison of the current approach with measurements is shown in Fig. 3. Reasonable agreement is found. Fig. 6 details the filter response for different iris configurations. It can be seen how the four reflection zeros appear. Finally, we mention that each cross-shaped iris analysis carried out with the modal segmentation procedure [28] takes 225 s/frequency point and 305 s with the FFS technique, constrained to converging in the band from 11.6 to 12 GHz. Thus, a substantial saving in computational incremental effort is accomplished by the current formulation. It should be noted that even though this filter can be analyzed by means of 2-D

Fig. 6. Dual-mode filter responses for different iris configurations. All dimensions are in millimeters. (a) L = 8:75 mm. (b) L = 7:65 mm.

Fig. 7. Geometry of the hemispherical dielectric resonator antenna with a concentric conductor and FEM analysis domain.

techniques, the proposed methodology permits the analysis of more complex structures. B. Hemispherical Dielectric Resonator Antenna The radiating structure geometry is detailed in Fig. 7. A hemispherical dielectric resonator of radius with a concentric conductor of radius is excited by a coaxial probe of length and displacement , as Fig. 7 shows. The dielectric resonator has a relative permittivity of 9.8. The 50- coaxial line has 2 and 0.63 mm as outer and inner radii, respectively. This antenna has already been addressed in [32] using the MM method and the method of moments. In order to allow the FEM to analyze this open structure, the computational domain ought to be bounded. Inasmuch as a mode expansion is available on the boundary of the analysis domain, the FEM approach accounts for open structures. Hence, a hemisphere truncates the analysis domain since this antenna is mounted on an infinite ground plane. Thus, a

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TABLE IV COMPUTATIONAL RESULTS IN THE DIRECT DECOMPOSITION OF THE HEMISPHERICAL DIELECTRIC RESONATOR ANTENNA

Fig. 8. Domain decomposition of the hemispherical dielectric antenna in order to modify the probe length and the concentric conductor.

hemispherical port plays the role of radiating port, meanwhile a coaxial port plays the role of exciting port. Fig. 7 also shows this. Once the computational domain has been defined, we aim to study the behavior of this antenna for different concentric conductor radii and probe lengths. In this regard, we consider the and , shown in Fig. 8, in order to cylindrical subdomains enlarge or shorten the probe length , and the hemispherical subor , also shown in Fig. 8, conceived to account domain for different concentric conductor radii and , respectively. Moreover, the inclusion of none of the latter subdomains in the admittance matrix connection process gives rise to a concentric conductor radius , which is the decomposition hemispherical subdomain radius (see Fig. 8). It is clear that the probe length and can be changed by proper consideration of subdomains during the connection process. However, regarding concentric conductor radius modification, we could have proceeded in a similar way as before, but instead we chose to enclose the geometry to be modified under the same decomposition subdomain. , Finally, we mention that the remaining analysis domain is . where Table IV details the number of PCW functions used and the CPU time for the determination of each admittance-type matrix in frequency pointwise computation and FFS technique. It should be noted that convergence in the band from 3.1 to 4.8 GHz is required in the FFS technique. This concerns the computational static part. In regard to the dynamic part, it takes no longer than 520 ms/frequency point to connect the admittance-type matrices. In other words, once all admittance-type matrices are determined, each full-wave analysis takes no longer than 520 ms/frequency point, which is the cost paid for the matrix connection. The results of this approach are compared with those obtained by Leung in [32] in Figs. 9 and 10. Good agreement is found. Coaxial aperture of the feeding line is considered as phase reference plane for the input impedance. We see that the antenna increases its resonance frequency as increases, but its reactance increases as well. Furthermore, increasing when is fixed, increases the input impedance, while a small decrease in the resonant frequency appears, as shown in Fig. 10. Although not displayed, the responses for all combinations between and are possible.

Fig. 9. Influence of the concentric conductor radius c on the input impedance in the hemispherical dielectric resonator antenna. a = 12:5 mm, b = 8 mm, l = 7:5 mm.

Fig. 10. Influence of the probe length l on the input impedance in the hemispherical dielectric resonator antenna. a = 12:5 mm, b = 8 mm, c = 4 mm.

C. Cylindrical Dielectric Resonator Filter In this final example, we consider a coaxial-fed dielectric resonator filter made up of two cylindrical dielectric resonators and , with a concentric cylindrical hole of diameters respectively. Fig. 11 shows the geometry of this filter and details its dimensions and materials. A relative permittivity of 38 has been considered in both dielectric cylinders. This filter has already been addressed in [33] and was designed to perform a bandstop from 5 to 6.8 GHz. On this occasion, we concentrate on how this response is affected by changing the coupling to each resonator, i.e., coaxial probe lengths and , as well as the resonators themselves, modifying

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TABLE V COMPUTATIONAL RESULTS IN THE DIRECT DECOMPOSITION OF THE CYLINDRICAL DIELECTRIC RESONATOR FILTER

Fig. 11. Geometry of the cylindrical dielectric resonator filter. All dimensions are in millimeters.

Fig. 12. Domain decomposition of the dielectric resonator filter in order to consider different probe lengths and modify the shape of the dielectric resonators.

diameters and . The direct decomposition shown in Fig. 12 is carried out in the analysis domain . Concerning the coaxial probes, several cylindrical subdomains are taken into account in order to change the probe lengths, as has already been detailed in the previous example. As regards , a concentric cylindrical the dielectric cylinder subdomain is considered as a decomposition subdomain and we change the size of the dielectric resonator, i.e., diameter , by modifying the material properties of this decomposition subdomain. Fig. 12 details all of this. Thus, subdomains and , concerning and , the coaxial probes, and subdomains with respect to the cylindrical resonators, come about. It should , be mentioned that the remaining analysis domain is the union of the former subdomains, is further where and ), making split by three planes ( further use of the PCW functions, into four subdomains , to decrease the computational cost. Table V shows the number of PCW functions considered and the computational cost for the calculation of each admittancetype matrix in frequency pointwise analysis and FFS technique, constrained to converging in the band from 4 to 8 GHz. On the other hand, it takes no longer than 2.2 s/frequency point to connect the admittance-type matrices. This is the incremental computational effort required in order to obtain the full-wave anal-

Fig. 13. Transmission and return losses of the dielectric resonator filter with l = l = 20 mm and d = d = 4 mm.

ysis for the multiple filter configurations. It should be pointed out that this computational cost is full-wave CAD compatible. Fig. 13 shows the measurement, FEM simulation, and current formulation results for the transmission coefficient and return losses. Reasonable agreement is obtained. Finally, Fig. 14 sets

DE LA RUBIA AND ZAPATA: MICROWAVE CIRCUIT DESIGN BY MEANS OF DIRECT DECOMPOSITION IN FEM

Fig. 14. Transmission in the dielectric resonator filter for different configurations. All dimensions are in millimeters.

out the transmission coefficients for different probe length and dielectric size configurations. It should be mentioned that not all configurations perform a bandstop filter from 5 to 6.8 GHz, as shown in Fig. 14. IV. CONCLUSION A direct decomposition in the FEM for design purposes has been presented. The analysis domain is explicitly divided according to the modifications that we want to consider in a given structure in order to satisfy some preestablished specifications. As a result, only those subdomains that change in the design process are analyzed, meanwhile the analysis of the remaining domain, which may be time consuming, is carried out only once. Each subdomain is electromagnetically described by a full-wave matrix-valued transfer function, an admittance-type matrix, and therefore, a suitable function basis for the boundary field description has to be proposed. Transmission conditions between subdomains are imposed through an admittance matrix connection process, ensuring a direct field continuity. An FFS procedure for the admittance-type matrices has also been considered. In addition, the matrix connection scheme is advantageous enough to allow the modification of conducting objects. Several numerical examples show the possibilities of the proposed methodology and its accuracy. REFERENCES [1] J. M. Gil, J. Monge, J. Rubio, and J. Zapata, “A CAD-oriented method to analyze and design radiating structures based on bodies of revolution by using finite elements and generalized scattering matrix,” IEEE Trans. Antennas Propag., vol. 54, no. 3, pp. 899–907, Mar. 2006. [2] Y. Zhu and A. C. Cangellaris, “Macro-elements for efficient FEM simulation of small geometric features in waveguide components,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 12, pp. 2254–2260, Dec. 2000. [3] M. Feliziani and F. Maradei, “Circuit-oriented FEM: Solution of circuit-field coupled problems by circuit equations,” IEEE Trans. Magn., vol. 38, no. 2, pp. 965–968, Mar. 2002. [4] K. K. Mei, “Unimoment method of solving antenna and scattering problems,” IEEE Trans. Antennas Propag., vol. AP-22, no. 6, pp. 760–766, Nov. 1974. [5] J. M. Jin and N. Lu, “The unimoment method applied to elliptical boundaries,” IEEE Trans. Antennas Propag., vol. 45, no. 3, pp. 564–566, Mar. 1997. [6] S. P. Marin, “Computing scattering amplitudes for arbitrary cylinders under incident plane waves,” IEEE Trans. Antennas Propag., vol. AP-30, no. 6, pp. 1045–1049, Nov. 1982.

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[7] J. M. Jin and J. L. Volakis, “A hybrid finite element method for scattering and radiation by microstrip patch antennas and arrays residing in a cavity,” IEEE Trans. Antennas Propag., vol. 39, no. 11, pp. 1598–1604, Nov. 1991. [8] B. Engquist and A. Majda, “Absorbing boundary conditions for the numerical simulation of waves,” Math. Comput., vol. 31, no. 139, pp. 629–651, Jul. 1977. [9] A. Bayliss, M. Gunzburger, and E. Turkel, “Boundary conditions for the numerical solution of elliptic equations in exterior regions,” SIAM J. Appl. Math., vol. 42, no. 2, pp. 430–451, Apr. 1982. [10] J. P. Webb, “Absorbind boundary conditions for the finite-element analysis of planar devices,” IEEE Trans. Microw. Theory Tech., vol. 38, no. 9, pp. 1328–1332, Sep. 1990. [11] Y. S. Choi-Grogan, K. Eswar, P. Sadayappan, and R. Lee, “Sequential and parallel implementations of the partitioning finite-element method,” IEEE Trans. Antennas Propag., vol. 44, no. 12, pp. 1609–1616, Dec. 1996. [12] J.-D. Benamou and B. Després, “A domain decomposition method for the Helmholtz equation and related optimal control problems,” J. Comput. Phys., vol. 136, pp. 68–82, 1997. [13] B. Stupfel, “A fast-domain decomposition method for the solution of electromagnetic scattering by large objects,” IEEE Trans. Antennas Propag., vol. 44, no. 10, pp. 1375–1385, Oct. 1996. [14] J.-F. Lee and D. K. Sun, “pMUS (p-type multiplicative Schwarz) method with vector finite elements for modeling three-dimensional waveguide discontinuities,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 3, pp. 864–870, Mar. 2004. [15] P. Liu and Y. Q. Jin, “Numerical simulation of the Doppler spectrum of a flying target above dynamic oceanic surface by using the FEM-DDM method,” IEEE Trans. Antennas Propag., vol. 53, no. 2, pp. 825–832, Feb. 2005. [16] C. Bernardi, Y. Maday, and A. T. Patera, “A new non conforming approach to domain decomposition: The mortar element method,” in Nonlinear Partial Differential Equations and Their Applications, H. Brezis and J.-L. Lions, Eds. New York: Pitman, 1994, pp. 13–51. [17] M. N. Vouvakis, Z. Cendes, and J.-F. Lee, “A FEM domain decomposition method for photonic and electromagnetic bandgap structures,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 721–733, Feb. 2006. [18] Y. Li and J. M. Jin, “A vector dual-primal finite element tearing and interconnecting method for solving 3-D large-scale electromagnetic problems,” IEEE Trans. Antennas Propag., vol. 54, no. 10, pp. 3000–3009, Oct. 2006. [19] B. Smith, P. Bjørstad, and W. Gropp, Domain Decomposition, Parallel Multilevel Methods for Elliptic Partial Differential Equations. New York: Cambridge Univ. Press, 1996. [20] A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations. New York: Oxford Univ. Press, 1999. [21] A. Toselli and O. Widlund, Domain Decomposition Methods—Algorithms and Theory. Berlin, Germany: Springer-Verlag, 2005. [22] V. de la Rubia and J. Zapata, “MAM—A multi-purpose admittance matrix for antenna design via the finite element method,” IEEE Trans. Antennas Propag., accepted for publication. [23] J. M. Jin, The Finite Element Method in Electromagnetics, 2nd ed. New York: IEEE Press, 2002. [24] J. P. Webb, “Hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements,” IEEE Trans. Antennas Propag., vol. 47, no. 8, pp. 1244–1253, Aug. 1999. [25] J. Rubio, J. Arroyo, and J. Zapata, “Analysis of passive microwave circuits by using a hybrid 2-D and 3-D finite-element mode-matching method,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 9, pp. 1746–1749, Sep. 1999. [26] I. S. Duff, A. M. Erisman, and J. K. Reid, Direct Methods for Sparse Matrices. New York: Oxford Univ. Press, 1990, ch. 7, pp. 140–143. [27] P. Feldmann and R. W. Freund, “Interconnect-delay computation and signal-integrity verification using the SyMPVL algorithm,” in Proc. Eur. Circuit Soc. Circuit Theory Design Conf., 1997, pp. 408–413. [28] J. Rubio, J. Arroyo, and J. Zapata, “SFELP—An efficient methodology for microwave circuit analysis,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 3, pp. 509–516, Mar. 2001. [29] J. I. Aliaga, D. L. Boley, R. W. Freund, and V. Hernández, “A Lanczostype method for multiple starting vectors,” Math. Comput., vol. 69, pp. 1577–1601, 2000. [30] Z. Bai and Q. Ye, “Error estimation of the Padé approximation of transfer function via the Lanczos process,” Elect. Trans. Numer. Anal., vol. 7, pp. 1–17, 1998. [31] J. R. Montejo-Garai and J. Zapata, “Full- wave design and realization of multicoupled dual-mode circular waveguide filters,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 6, pp. 1290–1297, Jun. 1995.

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[32] K. W. Leung, “Complex resonance and radiation of hemispherical dielectric-resonator antenna with a concentric conductor,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 3, pp. 524–531, Mar. 2001. [33] J. R. Brauer and G. C. Lizalek, “Microwave filter analysis using a new 3-D finite-element modal frequency method,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 5, pp. 810–818, May 1997. Valentín de la Rubia was born in Ciudad Real, Spain, in 1980. He received the Ingeniero de Telecomunicación degree from the Universidad Politécnica de Madrid, Madrid, Spain, in 2003, and is currently working toward the Ph.D. degree at the Departamento de Electromagnetismo y Teoría de Circuitos, Universidad Politécnica de Madrid. His research interests include numerical methods for microwave passive circuit and antenna design.

Juan Zapata (M’93) received the Ing. Telecomunicación and Ph.D. degrees from the Universidad Politécnica de Madrid, Madrid, Spain, in 1970 and 1974, respectively. Since 1970, he has been with the Departamento de Electromagnetismo y Teoría de Circuitos, Universidad Politécnica de Madrid, initially as an Assistant Professor, an Associate Professor in 1975, and then a Professor in 1983. He has been engaged in research on microwave active circuits and interactions of electromagnetic fields with biological tissues. His current research interest includes CAD for microwave passive circuits and antennas and numerical methods in electromagnetism, especially the FEM. Prof. Zapata is a member of the Editorial Board of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES.

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Bandwidth-Compensation Method for Miniaturized Parallel Coupled-Line Filters Seong-Sik Myoung, Student Member, IEEE, Yongshik Lee, Member, IEEE, and Jong-Gwan Yook, Member, IEEE

Abstract—This paper proposes a bandwidth-compensation method for miniaturized filters based on short-ended parallel coupled lines. Capacitive loading of such coupled lines is a relatively simple means of reducing the line lengths. In this study, a method is developed that predicts exactly the degree of reduction in the fractional bandwidth due to miniaturization. Using this method, the fractional bandwidth of the prototype coupled line filter can be adjusted, enabling miniaturized filters to maintain the targeted fractional bandwidth. The proposed bandwidth-compensation method applies for any type of filters with coupled lines realized with various transmission lines, with uniform or nonuniform line lengths. Experimental results are also presented that verify the validity of the method. Index Terms—Bandwidth compensation, capacitor, grounding, hairpin, miniaturization, parallel coupled-line filter.

I. INTRODUCTION

W

ITH THE explosive growth of various modern wireless communication services, the performance of RF and microwave filters is being emphasized more and more. In addition, state-of-the-art communication components and systems are being developed rapidly with an eye toward smaller size without substantial sacrifice in its performance. Due to their simplicity of fabrication, compatibility with various circuit components, and low-cost fabrication, planar transmission line filters have inspired a great deal of interest. However, compared with their lumped-element counterpart, the transmission line filters are not a very attractive solution at low gigahertz bands due to the relatively long wavelengths. As a result, various miniaturization techniques have been developed by many researchers: ladder filters based on periodic structures [1], pseudointerdigital filters [2], hairpin filters consist of meandered resonators [3], combline filters with lumped components [4], and stepped impedance resonator (SIR) technology [5]–[7] are a few examples. Although these miniaturized filters showed excellent results, they require time-consuming full-wave electromagnetic simulations for accurate prediction of the filter performance [1], [6], have limited size reduction [2], [3], or require accurate calculation and control of mutual impedances [4].

Manuscript received February 7, 2007; revised April 14, 2007. This work was supported by the Ministry of Information and Communication, Korea, under the Information Technology Research Center support program supervised by the Institute of Information Technology Advancement IITA-2006-(C1090-0603-0034) and under the Brain Korea 21 (BK21) program, Korea. The authors are with the Department of Electrical and Electronic Engineering, Yonsei University, Seoul 120-749, Korea (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2007.900310

On the other hand, the miniaturization technique for coupled line filters, proposed by Myoung and Yook, is considered as a very straightforward approach [8]. With a small number of capacitors, the technique is applicable to a pair of parallel-coupled lines, each line grounded at one end. Accurate calculation and control of mutual impedances is not necessary. Moreover, popular circuit simulators such as ADS [9] yields very accurate results, thus time-consuming full-wave electromagnetic simulations are not required. In addition to these, the method provides additional advantages such as excellent harmonic suppression and improved skirt characteristics. However, owing to the change of the slope parameters in each stage, the technique suffers from the disadvantage of bandwidth shrinkage. Such a bandwidth reduction, a common characteristic of miniaturized circuits with lumped elements [10], [11], may be a critical problem in today’s microwave filters. In this paper, the process of bandwidth reduction is rigorously analyzed using the group delays and the loaded ’s of resonators, and a novel strategy to compensate for the reduced bandwidth is proposed. Experimental results that verify the validity of the proposed bandwidth-compensation method is also presented. II. MINIATURIZATION OF COUPLED-LINE FILTERS Within a specified range of accuracy, a transmission line can be miniaturized to one having a higher characteristic impedance loaded with shunt capacitors and still yield the same two-port -parameters) as the original parameters (such as - and transmission line [11]. The shunt capacitance values depend on the degree of miniaturization. A similar technique can be applied to miniaturize a parallel coupled line, which can be considered as a superposition of even- and odd-mode equivalent circuits. However, a conventional open-ended parallel coupled line pair requires six capacitors for miniaturization, whereas a shortended parallel coupled line pair requires only two capacitors [8]. A short-ended coupled line shows the same transfer characteristics as an open-ended coupled line, except the phase difference of 180 . Therefore, from a theoretical point-of-view, bandpass filters consisting of short-ended coupled lines are better candidates to develop miniaturized filters. The schematic of a conventional open-ended coupled-line and a miniaturized version using the technique describe above is shown in Fig. 1. This miniaturization technique requires calculation of the even- and odd-mode characteristic impedances, as well as the capacitance at the design frequency. Shown in Fig. 2 are simulated results of a conventional coupled-line filter and a miniaturized version of this filter using ideal coupled lines in ADS [9]. The conventional filter is a prototype third-order 0.5-dB ripple Chebyshev bandpass filter coupled lines, designed to have a center frequency at with

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TABLE I COMPARISON OF MINIATURIZED 0.5-dB RIPPLE CHEBYSHEV BANDPASS FILTERS DESIGN USING =8 COUPLED LINES LOADED WITH CAPACITORS

Fig. 1. (a) Conventional open-ended parallel coupled line. (b) Miniaturized parallel coupled line with capacitors.

Fig. 2. Simulated responses of conventional coupled-line filter and when miniaturization technique in [8] is applied to reduce line length to half. Ideal coupled lines are used for both filters.

5.2 GHz with a 10% fractional bandwidth. Then miniaturization technique in [8] is applied to this filter to reduce the line lengths to half of its original dimension. In terms of the their performances, the two filters are distinct in two aspects. First is the previously mentioned bandwidth shrinkage seen for the miniaturized filter. Second is the increase of the ripples in the passband for the miniaturized filter. This increase of passband ripples stems from the assumption for the design equations in or , where the previous work that and are the normalized even- and odd-mode characis the characteristic teristic impedances and impedance of the coupled line. In reality, such an assumption is rarely satisfied, resulting in increased passband ripples when miniaturized, as seen in Fig. 2. Thus, the even- and odd-mode and ) and the capacitances ( and , see impedance ( [8] for more details) for the miniaturization must be modified to the following generalized expressions: (1) (2) (3) (4) (5) where is the length of the miniaturized coupled line. In accordance with (1)–(5), the equations in [8] also needs to be modified to the following even- and odd-mode characteristic

Fig. 3. Simulated responses of conventional coupled-line filter and when new miniaturization technique is applied. Ideal coupled lines are used for both filters.

impedances, as well as the capacitance values of the th coupled line section of the miniaturized bandpass filter (6) (7) (8)

(9) (10) Summarized in Table I are the calculated impedances and capacitances, using the equations in [8] and the revised equations in this paper, i.e., (6)–(10), for a miniaturized Chebyshev-type bandpass filter with the center frequency of 5.2 GHz. The simulated results for the miniaturized filter designed using the new equations are plotted in Fig. 3, along with those for the conventional version. As can be seen, the passband ripple of the miniaturized filter is exactly 0.5 dB, showing that the modified equations (6)–(10) provide more accurate results in terms of the passband ripple characteristics. However, the bandwidth shrinkage effect still remains as a result of miniaturization. It is worth noting that the bandwidth mentioned in this paper is not the 3-dB bandwidth, but rather it is the bandwidth related to the corner frequency , which is defined by the insertion loss method [12]. Thus, in the case of the 0.5-dB ripple Chebyshev filter, the bandwidth is the frequency range between the

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lowest and the highest frequencies where the insertion loss is 0.5 dB. The bandwidth of the miniaturized filter in Fig. 3 is 4.975–5.42 GHz or 445 MHz according to such a definition. This bandwidth is 0.86 times the targeted 10% fractional bandwidth of 520 MHz. III. BANDWIDTH COMPENSATION METHOD The focus of this study is on compensating the reduced filter bandwidth due to miniaturization. As discussed in Section II, capacitors can be used to miniaturize parallel-coupled lines that can be considered as the superposition of even- and odd-mode lines. Although the even- and odd-mode impedances of a parallel-coupled line do not vary substantially over a relatively wide frequency range, the frequency response of the capacitors varies substantially. Moreover, the asymmetric frequency response of capacitors lead to more coupling at frequencies below the design frequency than at frequencies above the design frequency. Thus, the slope parameters of the coupled-line resonators become different at frequencies other than the design frequency, and the design is no longer valid. of a miniaturized parallel The magnitude and phase of and the coupled line are functions of the electrical length capacitance , shown in Fig. 1(b). Fig. 4 shows simulated of a miniaturized parallel coupled line of various lengths, the even- and odd-mode characteristic impedances of which are 70.61 and 39.24 , respectively. The conventional coupled line in the figure is designed to be at 5.2 GHz. The for miniaturresults show that the magnitude and phase of ized coupled lines with various electrical lengths are identical to at the design frequency. those of a conventional line However, around the center frequency, the variations of both the for miniaturized coupled lines are magnitude and phase of different for different electrical lengths. The symmetry of with respect to the center frequency breaks down, in a complex manner, as the degree of miniaturization increases. However, maintains its linearity around the passband. As the phase of can be seen in Fig. 4(c), the slopes, although different for different electrical lengths, remain constant in the passband. The is [13] relationship between the group delay and phase of

Fig. 4. Simulated S of miniaturized coupled line. (a) Magnitude of (b) Phase of S . (c) Phase of S around passband.

S

.

(11)

Therefore, the fact that the phase of maintains its linearity around the passband indicates that although the group delay changes as the degree of miniaturization changes, it remains constant within the passband. Thus, it is easier to analyze the filter characteristics, including the bandwidth reduction or the group-delay due to miniaturization, with the phase of . characteristics, than with the magnitude of of a resonator is defined by the following The loaded [13], [14]: (12) (13)

Equations (12) and (13) relate with the magnitude and phase characteristics of the resonator transfer function, respec, it is clear that the bandwidth tively. From the definition of of a resonator is determined by the of the resonator. Thus, the bandwidth of a filter consisting of resonators is deter’s of the resonators. is also proportional to mined by the of the resonator. Therefore, the group delay bandwidth reduction is unavoidable when the group delay of a resonator increases due to miniaturization. If the exact relationship between the degree of reduction in the fractional bandwidth and the degree of increase in the group delay due to miniaturization is known, the degree bandwidth reduction can be predicted once the reduced length of coupled lines is chosen. If a bandpass filter is designed with miniaturized coupled lines with identical lengths, the bandwidth reduction trend resembles that of the increase in the group delay . In fact, simu-

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Since all elements of the normalized -parameters in (15) of a coupled line can be exand (16) are purely imaginary, pressed as (17) The phase of

and, hence, the group delay

are (18) (19)

Fig. 5. Extracted bandwidth reduction ratio and group-delay ratio of for miniaturized third-order 0.5-dB ripple Chebyshev bandpass filters.

. It can where be seen that vanishes at the center frequency, the frequency at which corresponds to a length of . Since in this case, (20)

lation results reveal that the degree of reduction in the fractional bandwidth of a filter due to the miniaturization is found to be identical to the degree of increase in the group delay at the center frequency of the coupled line. Shown in Fig. 5 are the fracdue tional bandwidth reduction ratios to miniaturization of coupled lines for miniaturized third-order 0.5-dB ripple Chebyshev bandpass filters of various lengths extracted from simulation results of each. As can be seen, the bandwidth reduction ratio fits exactly to the extracted group . This validates the approach of delay ratio due to miniaanalyzing the bandwidth reduction ratio turization with the group delay ratio

and the group delay of a conventional coupled line expressed as

can be (21)

where

(14)

For a rigorous analysis of the bandwidth reduction, the group delays of the conventional and miniaturized parallel coupled lines are formulated. The phase of a open-ended parallel coupled from that of an short-ended parallel line differs merely by coupled line for all frequency ranges. Hence, only short-ended parallel coupled lines are considered in this paper. The normaland miniaturized ized -parameters of conventional short-ended parallel coupled lines shown in Fig. 1 are shown in , , and (15) and (16) at the bottom of this page, where are the normalized even- and odd-mode characteristic impedances and capacitance of a miniaturized coupled line, which are defined in (1), (2), and (5), respectively.

Since the electrical length of a conventional coupled line is , (21) further simplifies as (22)

In a similar way, the group delay of a miniaturized coupled line can be expressed as

(23)

conventional

(15)

miniaturized

(16)

MYOUNG et al.: BANDWIDTH-COMPENSATION METHOD FOR MINIATURIZED PARALLEL COUPLED-LINE FILTERS

where

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%. Since the have a fractional bandwidth of 10% amount of bandwidth shrinkage after miniaturization is calculated to be 0.86, the resulting fractional bandwidth when miniawill be % %. A similar proturized to cedure can be applied for any fractional bandwidth, e.g., 12%, since the bandwidth reduction ratio depends only on the degree of miniaturization. If each section of a multistage coupled line filter is reduced by of each section would be different amount, the changes in different. This requires the bandwidth reduction ratio and the admittance inverter parameters of each stage must be calculated accordingly with the follow equations:

and

(26) (27)

Using (1)–(5), (23) can be simplified as (28) where (24) Using (22) and (24), the bandwidth reduction ratio of (14) is now expressed as (25) where and are in radians. . Equation (22) is a special form of (24) when It should be noted that the plot of (25) exactly matches the group-delay ratio in Fig. 5. Moreover, it is interesting to notice that although the expressions for group delays in (22) and and , the bandwidth reduction ratio (24) are functions of is a function of the electrical length only. This indicates that the reduction of the filter bandwidth due to miniaturization depends only on the degree of miniaturization and not on any other filter parameters, not to mention the relative or the absolute bandwidth of the filter before miniaturization. Thus, the result in Fig. 5 can be applied to any filter, including Chebyshev, Butterworth, and Gaussian types, that consists of coupled lines that are miniaturized using the method in Section II. With (25), the amount of bandwidth reduction due to miniaturization can be predicted in the prototype design stage prior to miniaturization. This indicates that the bandwidth of the prototype filter can be adjusted so that the targeted fractional bandwidth is maintained after miniaturization. Using (25), the required adjustment in the bandwidth can be calculated exactly. For instance, when the length of each coupled line is reduced to , then the to half the original length, from bandwidth reduction ratio calculated with (25) is 0.86. This implies that the bandwidth of the miniaturized bandpass filter will . If the targeted be 86% of the conventional version fractional bandwidth were 10%, then the resulting fractional %. bandwidth of the miniaturized filter will be 10% If the fractional bandwidth were to be maintained at 10% even after miniaturization, then the prototype filter can be designed to

(29) is the bandwidth-reduction-compensated fractional bandwidth. IV. EXPERIMENTAL VERIFICATION For experimental verification of the bandwidth compensation method, a conventional third-order Chebyshev filter with 0.5-dB coupled lines. passband ripples has been designed with The filter is designed to have a center frequency of 5.2 GHz with a fractional bandwidth of 10%. All the sections of the conventional filter are then miniaturized. Depending on the final resonator length chosen, the corresponding bandwidth reducis calculated. The prototype filter is then retion ratio designed so that the bandwidth can be compensated after miniaturization. The even- and odd-mode characteristic impedances as well as the capacitance values are calculated accordingly. For instance, if the length of the coupled line were chosen to be at 5.2 GHz, the calculated bandwidth reduction ratio using (25) is 0.864 and, thus, the modified fractional bandwidth , required to maintain a fractional band%. width of 10% after miniaturization, is 10% Thus, the prototype filter is designed to have a fractional bandwidth of 11.574%, which is then miniaturized. Table II summarizes the calculated filter parameters for bandwidth-compensated miniaturized filters of various lengths. The simulated results for the filters in Table II are shown in Fig. 6 along with those of the conventional version. Ideal coupled lines are used for all designs without any further optimizations. As shown in this figure, the conventional and miniaturized filters with various lengths reveal nearly identical passband ripples with slightly different skirt characteristics outside the passband. Most of all, the bandwidths of all filters are preserved at the targeted fractional bandwidth of 10%, irrespective of the degree of miniaturization. This validates the proposed bandwidthcompensating method for miniaturized coupled line filters.

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TABLE II FILTER PARAMETERS FOR BANDWIDTH-COMPENSATED MINIATURIZED FILTERS

Fig. 6. Simulated responses of conventional and bandwidth-compensated miniaturized coupled-line filters. Ideal coupled lines are used for all designs.

As shown in Fig. 7, two third-order Chebyshev filters with 0.5-dB passband ripples are fabricated on a Teflon substrate mils) and measured. The first filter is ( a conventional filter with short-ended coupled lines of length at the frequency of 5.2 GHz. The second filter is a hairpintype miniaturized version of the short-ended conventional filter or at 5.2 GHz. with the line lengths reduced to The capacitor for the miniaturization is realized with a lowtransmission line. As seen in Fig. 7(b), the impedance miniaturized filter consists of high-impedance parallel coupled lines and short-length low-impedance transmission lines. The size of the conventional open- and short-ended coupled line filters is 45 12 mm , and the size of the miniaturized hairpintype version is 15 12 mm . The measurement results are shown in Fig. 8. Also shown are the simulation results, obtained from full-wave analysis by IE3D [15]. This figure shows an excellent agreement between the experimental results and the full-wave simulation results. The miniaturized filter based on the proposed method show almost identical passband characteristics, while the second harmonic is effectively suppressed, as mentioned in [8]. The harmonic suppression after miniaturization is similar to that seen in compact SIR filters [5]. Although the miniaturized filter only consists of distributed elements, the harmonic suppression is nearly 30 dB at the second harmonic (around 10.4 GHz). This can be improved by designing each coupled line section to have a different electrical length. It has to be mentioned that the third harmonic is not suppressed as much as the second harmonic.

Fig. 7. Fabricated filters with: (a) conventional ( = 90 ) short-ended (b) miniaturized ( = 45 ) coupled lines in hairpin configuration (c) layout of miniaturized filter of (b) L = 5:02, L = 5:16, L = 1:35, L = 2:52, L = 2:42, W = 0:82, W = 0:86, W = 4:22, W = 4:42, W = 4:32, S = 0:10, and S = 0:57 all in millimeters. (a) Area: 45 12 mm . (b) Area: 15 12 mm .

2

2

This is because the short section of low-impedance transmission line section no longer behaves as a capacitor due to the short wavelengths at these frequencies. In addition, the asymmetric response in the passband is due to parasitic coupling between capacitors (low-impedance T-lines). This is not seen in Fig. 6 since circuit simulation cannot take into account such a coupling mechanism. In general, such couplings must be taken into account in the design process, especially when capacitors are closely located. From the measured results, it is obvious that the bandwidths of all filters are nearly identical, showing a high level of agreement with the theoretical results. This experimentally verifies the validity of the proposed bandwidth-compensation method. In this study, without any sacrifice in the filter performance, filter miniaturization has been achieved by the method proposed in Section II. The miniaturization method is applied to a conventional parallel-coupled resonators. The hairpin technique is also applied, resulting in a miniaturized filter that is one-third as wide

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The proposed method has been verified experimentally. A conventional third-order Chebyshev bandpass filter with coupled lines, centered at 5.2 GHz with a 0.5-dB passband ripple and a 10% fractional bandwidth, has been designed and fabricated. Also fabricated was a miniaturized filter with the same filter specifications, the coupled line lengths of which were reduced to half. Using the proposed bandwidth-compensation method, the bandwidth of the prototype of this filter, prior to miniaturization, has been chosen so that the fractional bandwidth is maintained at 10% after miniaturization. The measured result has shown a high level of agreement with the theoretical results, and the passband characteristics of the two filters were essentially the same. The proposed bandwidth-compensation method not only applies for Chebyshev filters with microstrip coupled lines of uniform lengths, but also for any filter type with coupled lines realized with various transmission lines with uniform or nonuniform lengths. Therefore, the proposed method can be a very useful tool when designing miniaturized filters for integration in monolithic microwave integrated circuits and low-temperature co-fired ceramic (LTCC)-based circuits and systems. REFERENCES Fig. 8. Full-wave simulation and measurement results. (a) S . (b) S .

as the conventional version. It is a reasonable expectation that further miniaturization is also possible without any significant degradation in the filter performance.

V. CONCLUSION This paper has proposed a bandwidth-compensation method for miniaturized filters based on short-ended parallel coupled lines. By loading the coupled lines with capacitors, the line length can be reduced in a relatively simple fashion. In addition to miniaturization, other advantages such as harmonic suppression have been achieved without any degradation in the filter performance. However, reduction in the filter bandwidth is inevitable as a result of miniaturization. From the analysis of the relationship between the loaded and the group delay of a miniaturized parallel coupled line, the relationship between the degree of bandwidth reduction and the degree of miniaturization has been obtained in a closed-form formula. Using this, the amount of bandwidth reduction can be calculated exactly, without any simulations. Finally, the fractional bandwidth of the prototype coupled line filter can be adjusted, enabling miniaturized filters to maintain the targeted fractional bandwidth. This method can be applied to any type of filter that consists of coupled lines regardless of the relative or absolute bandwidth since the bandwidth reduction ratio due to miniaturization depends only on the degree of miniaturization and not on other parameters. In real situations, additional bandwidth shrinkage occurs for coupled-line filters due to housing, which must be taken into account [16].

[1] J.-S. Hong and M. J. Lancaster, “Recent advances in microstrip filters for communications and other applications,” IEE Adv. Passive Microw. Compon., pp. 2/1–2/6, May 1997. [2] J.-S. Hong and M. J. Lancaster, “Development of new microstrip pseudo-interdigital bandpass filters,” IEEE Microw. Guided Wave Lett., vol. 5, no. 8, pp. 261–263, Aug. 1995. [3] E. G. Cristal and S. Frankel, “Design of hairpin-line and hybrid hairpinparallel-coupled-line filters,” IEEE MTT-S Int. Microw. Symp. Dig., vol. 71, pp. 12–13, May 1971. [4] S. B. Cohn, “Synthesis of commensurate comb-line bandpass filters with half-length capacitor lines, and comparison to equal-length and lumped-capacitor cases,” IEEE MTT-S Int. Microw. Symp. Dig., vol. 80, pp. 135–137, May 1980. [5] Y. Noguchi and J. Ishii, “New compact bandpass filters using =4 coplanar waveguide resonators and a method for suppressing these spurious responses,” Electron. Commun. Jpn., vol. 77, no. 3, pt. 2, pp. 66–73, 1994. [6] J.-S. Hong and M. J. Lancaster, “Theory and experiment of novel microstrip slow-wave open-loop resonator filters,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 12, pp. 2358–2365, Dec. 1997. [7] M. Makimoto and S. Yamashita, “Bandpass filters using parallel coupled stripline stepped impedance resonators,” IEEE Trans. Microw. Theory Tech., vol. MTT-28, no. 12, pp. 1413–1417, Dec. 1980. [8] S.-S. Myoung and J.-G. Yook, “Miniaturization and harmonic suppression method of parallel coupled-line filters using lumped capacitors and grounding,” Electron. Lett., vol. 41, no. 15, pp. 849–851, Jul. 2005. [9] Advanced Design System 2004A. Agilent Technol., Palo Alto, CA, 2005. [10] I. Bahl and P. Bhartia, Microwave Solid State Circuit Design. Hokoken, NJ: Wiley, 2003. [11] T. Hirota, A. Minakawa, and M. Muraguch, “Reduced-sized branchline and rat-race hybrids for uniplanar MMIC’s,” IEEE Trans. Microw. Theory Tech., vol. 38, no. 3, pp. 270–275, Mar. 1990. [12] G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance-matching Networks, and Coupling Structures. Dedham, MA: Artech House, 1980. [13] J.-S. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications. New York: Wiley, 2001. [14] B. Razavi, RF Microelectronics. Upper Saddle River, NJ: PrenticeHall, 1998. [15] IE3D. Zeland Softw., Fremont, CA, 2002. [16] G. L. Matthaei, J. C. Rautio, and B. A. Willemsen, “Concerning the influence of housing dimensions on the response and design of microstrip filters with parallel-line couplings,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 8, pp. 1361–1368, Aug. 2000.

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Seong-Sik Myoung (S’02) was born in Taean, Korea. He received the B.S. degree in electronics engineering from Soongsil University, Seoul, Korea, in 2002, the M.S. degree in electrical and electronic engineering from Yonsei University, Seoul, Korea, in 2004, and is currently working toward the Ph.D. degree at Yonsei University. He is a Visiting Scholar with the Georgia Institute of Technology, Atlanta. His current research interests include GaAs HBT-based monolithic-microwave/millimeter-wave integrated circuit design, microwave filter design, and communication system design and analysis.

Yongshik Lee (S’00–M’04) was born in Seoul, Korea. He received the B.S. degree from Yonsei University, Seoul, Korea, in 1998, and the M.S. and Ph.D. degrees in electrical engineering from The University of Michigan at Ann Arbor, in 2001 and 2004, respectively. From 2004 to 2005, he was a Research Engineer with EMAG Technologies Inc., Ann Arbor, MI. In September 2005, he joined Yonsei University, Seoul, Korea, as an Assistant Professor. His current research interests include passive and active circuitry for microwave and millimeter-wave applications.

Jong-Gwan Yook (S’89–M’97) was born in Seoul, Korea. He received the B.S. and M.S. degrees in electronics engineering from Yonsei University, Seoul, Korea, in 1987 and 1989, respectively, and the Ph.D. degree from The University of Michigan at Ann Arbor, in 1996. He is currently an Associate Professor with Yonsei University. His main research interests are in the area of theoretical/numerical electromagnetic modeling and characterization of microwave/millimeter-wave circuits and components, very large scale integration (VLSI) and monolithic-microwave integrated-circuit (MMIC) interconnects, RF microelectromechanical systems (MEMS) devices using frequency- and time-domain full-wave analysis methods, and development of numerical techniques for analysis and synthesis of high-speed high-frequency circuits for wireless communication applications with an emphasis on parallel/super computing.

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Miniaturized Dual-Mode Ring Bandpass Filters With Patterned Ground Plane Rui-Jie Mao, Student Member, IEEE, Xiao-Hong Tang, and Fei Xiao

Abstract—Miniaturized dual-mode ring bandpass filters with a patterned ground plane are proposed. By loading the resonator periodically with the butterfly radial slot cells on the ground plane, the size of a ring filter is reduced along with an extended upper stopband. Based on the equivalent circuits, the mode-splitting characteristics and impact of the patterned ground plane on the electrical performances of the filter are investigated. A slotline ring filter with a patterned ground plane is also proposed. The miniaturization is achieved by loading the periphery of the slotline ring with the butterfly radial stubs at the backside. The filter has a higher first spurious response and deeper stopband rejection than a conventional microstrip ring filter. Measured results validate the analysis and theoretical prediction with good agreement. Index Terms—Bandpass filter, dual mode, miniaturization, patterned ground structure. Fig. 1. Microstrip dual-mode ring bandpass filter with patterned ground plane. Geometrical dimensions of the butterfly radial slot cells are given in Fig. 2(a).

I. INTRODUCTION

M

INIATURIZED planar filters at the wireless communication frequency band are a highly active area of research. Among various circuit configurations, the dual-mode ring bandpass filters have been extensively used due to their attractive features such as compact size, high selectivity, and simple design. A ring filter can be considered as a doubly tuned resonant circuit, the degenerate modes of which can be excited by a perpendicular feed structure or by introducing different forms of perturbations. Since the first presentation of the dual-mode ring bandpass filter by Wolff [1], various innovative designs have been proposed [2]–[6]. By applying the meander loop [2], triangular loop [3], and hexagonal loop resonator [4], the size of a filter can be reduced considerably. In [5], a miniaturized ring filter is designed using the butterfly radial stubs as a form of loading along the periphery. A pair of rectangular slots on the ground plane is utilized to split the degenerate modes. A size reduction of 65% is achieved. Recently, a periodic stepped-impedance ring resonator is proposed to design dual-mode bandpass filters with a miniaturized area and desirable upper stopband characteristics [6]. The filter not only has the first spurious response 3.7 times away from the center frequency, but also an area reduction of 60% against a conventional ring filter.

Manuscript received January 16, 2007; revised March 21, 2007. The authors are with the School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan Province 610054, China (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.2007.900337

Fig. 2. (a) Configuration of a butterfly radial slot cell. (b) Its equivalent circuit. (c) Simulated frequency responses of a slot cell as functions of the radial slot radius (r ) where g = 0:4 mm, l = 1:5 mm, and W = 1:5 mm.

In this paper, two miniaturized ring filters with a patterned ground plane are proposed. These designs utilize the loading effects of the patterned ground plane at the backside of the ring

0018-9480/$25.00 © 2007 IEEE

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TABLE I EQUIVALENT-CIRCUIT PARAMETERS AND EXTRACTED SIMULATION RESULTS OF THE BUTTERFLY RADIAL SLOT CELLS

resonators. The proposed filters occupy less area and have improved upper stopband characteristics than a conventional ring filter. II. MICROSTRIP DUAL-MODE RING BANDPASS FILTER WITH PATTERNED GROUND PLANE Fig. 1 shows the geometry of a microstrip ring filter with patterned ground plane. Four butterfly radial slot cells are etched periodically along the periphery of a ring resonator on the ground plane. Since they disturb the current flows on the ground plane, additional propagation paths are introduced. In addition, two open-ended stubs are introduced as the perturbato split the two degenerate tion along the symmetry axis modes. The resonator is fed by a pair of perpendicular 50feed lines and each feed line is coupled with the ring by an interdigital structure. A. Butterfly Radial Slot Cell Modeling and Its Properties The configuration of a butterfly radial slot cell is shown in Fig. 2(a). Two radial slots with the fan angle of 90 are etched on the ground plane of a microstrip line. These slots are connected at one end with a narrow gap [7]. The 20-dB fractional rejection bandwidth (RBW) is used to measure the stopband width of a slot cell, which is given as (1) where and are the upper and lower 20-dB rejection frequencies, respectively, and is the resonance frequency of the transmission characteristics. A slot cell can be modeled by a parallel – – resonance circuit, which is shown in Fig. 2(b). The circuit parameters are extracted from its full-wave simulation results as [8] (2a) (2b) (2c)

where is the 3-dB cutoff frequency, is the reflection is the characteristic impedance of the coefficient at , and microstrip line [9]. The frequency characteristics of a butterfly radial slot cell are . The simulated mainly controlled by the radial slot radius frequency responses of a slot cell using the High Frequency Structure Simulator (HFSS), along with those using the equivalent circuit are plotted as functions of the radius in Fig. 2(c). A

Fig. 3. (a) Even-mode equivalent circuit of the filter in Fig. 1. (b) Its odd-mode equivalent circuit.

CER-10 Teflon substrate with a thickness of 0.635 mm and a relative dielectric constant of 9.5 was used for all simulations. The frequency responses exhibit broad RBWs. The equivalent-circuit parameters and extracted simulation results are summarized in Table I. It is seen that the circuit parameters increase with the increment of the radius, and so does the RBW. However, the resonance frequency decreases with the increment of the radius. B. Filter Equivalent Circuits and Analysis Since the filter is symmetrical with respect to the axis , it is possible to simplify the analysis by the even- and odd-mode networks. When a magnetic wall, i.e., an open circuit, is ap, the even-mode equivalent circuit is obplied along tained, as shown in Fig. 3(a). Similarly, the odd-mode equivalent circuit is obtained by applying an electric wall (short circuit) , which is illustrated in Fig. 3(b). The characteristic along impedance of the ring is denoted by , and its mean circumfer. The slot cells are modeled by the parallel – – ence is resonance circuits, their values are extracted from the frequency response of a single cell by the aforementioned method. The loading period of the slot cells is equal to a quarter of the ring circumference. In addition, the slot cell on the ground plane of the interdigital structure is divided into two series-connected subcircuits to reckon with its influence upon the two propagation paths connecting the ports. These resonance circuits have the param, , and . The susceptance of the open-ended eters of stub perturbation is represented by . The interdigital structure is characterized by a series connected capacitor . Its value is controlled by the number, length, width, and space of the fingers [10]. When the equivalent circuits in Fig. 3 are resonant, their input admittances are zero. The resonance frequencies of the degenerate modes can be obtained accordingly. Given the complexity of the equivalent circuits, it is difficult to derive any analytical expression of the resonance frequencies. The Advanced Design

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Fig. 4. Simulated resonance frequencies of the degenerate modes and calculated coupling coefficient using (3) with different radial slot radius against the perturbation size where C = 1:6342 pF, g = 0:4 mm, l = 1:5 mm, R = 10 mm, W = 1:5 mm, and W = 1 mm. The resonance frequencies are simulated with the structure of r = 5 mm.

System (ADS) [11] was used to simulate the even- and oddmode resonance frequencies. The results are plotted in Fig. 4 when the radial slot radius against the perturbation size increases from 0.75 to 8 mm, the evenis 5 mm. When decreases almost linearly mode resonance frequency from 1324 to 1127 MHz. Nevertheless, the odd-mode resonance remains unaffected. It should be mentioned frequency mm, that, without the perturbation, namely, the ring resonator acts as a single-mode resonator and no mode splitting is observed. The center frequency of the filter is approximated by the algebraic mean of these resonance frequencies. The coupling coefficient between the modes is computed as (3) The calculated coupling coefficient is also shown in Fig. 4 as a function of the radial slot radius. It is observed that the coupling coefficient rises proportionally with the increment of the perturbation size. When no perturbation is introduced, the coupling coefficient is equal to zero, i.e., there is no coupling between the modes. It is interesting to mention that the coupling coefficient decreases slightly with the increment of the radius, which results in a narrower bandwidth. The reason is that the slot cell on the ground plane of the interdigital structure reduces the . The decrement of the latter is proportional actual value of to the etched area. As is well known, a ring filter has inherent transmission zeros when the two propagation paths connecting the ports are out of phase. Since their locations depend upon the perturbation size, tuning seems to be impossible once the passband specifications, especially the bandwidth is fixed. Accordingly, the attenuation characteristics are not discussed in this paper. To observe the impact of the patterned ground plane on the electrical performances of the filter, a series of simulations have been performed using ADS. Geometrical dimensions of these filters are basically identical, minus the radial slot radius. Note the perturbation sizes are slightly adjusted to compensate for the diversity of their mode-splitting properties. As is seen from

Fig. 5. (a) Simulated narrowband frequency responses of the filter in Fig. 1 as functions of the radial slot radius where C = 1:6342 pF, g = 0:4 mm, = 10 mm, W = 1:5 mm, and W = 1 mm. The length l = 1:5 mm, R of the stub perturbations (L ) are 4.1, 3.51, 3.35, and 3.23 mm for the cases of without slot cells and r = 3; 4; and 5 mm, respectively. (b) Simulated wideband frequency responses of these filters as functions of the radial slot radius.

Fig. 5(a), without the slot cells, the filter has a center frequency of 1678 MHz. When increases from 3 to 5 mm with a step of 1 mm, the center frequency decreases from 1414 to 1328, and then to 1250 MHz. A loading factor (LF) is defined to weight the extent of miniaturization produced by the patterned ground plane as (4) where and are the center frequencies of the filter with and without the patterned ground plane, respectively. The larger the LF is, the less the area occupies. The calculated LFs for and mm are 15.7%, 20.9%, and 25.5%, respectively. In addition, the passband width reduces with the increment of the radius, as has been anticipated. From Fig. 5(b), it is seen that the first spurious responses of these filters almost remain at the same position, which means a larger radius will result in a broader upper stopband.

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TABLE II DESIGN PARAMETERS AND GEOMETRICAL DIMENSIONS OF THE MICROSTRIP DUAL-MODE RING BANDPASS FILTER WITH PATTERNED GROUND PLANE

C. Filter Design and Measurement To design a dual-mode filter, it is always desirable to start with the passband specifications, namely, the center frequency and bandwidth. Although a ring filter can be miniaturized by the patterned ground plane, the radial slot radius must be chosen properly to avoid any overlapping of the slot cells. The resonance frequency of the slot cells should be kept near the first spurious response of the filter to be loaded. Once the bandwidth is chosen, the design can be fulfilled by using either a Butterworth or Chebyshev low-pass prototype. The inter-stage coupling coefficient is determined by (5) where FBW is the fractional bandwidth of the filter, and and are the values of the low-pass prototype elements. By tuning the perturbation size, the inter-stage coupling coefficient calculated by (3) can be adjusted to the desired value. is estimated as [12] The value of (6) are the characteristic admittances of the feed where and line and ring resonator, respectively. A two-pole ring filter with patterned ground plane was designed and fabricated on a CER-10 substrate. The mean radius is 10 mm for further comparison. The deof the ring signed center frequency is 1250 MHz and the FBW is 6%. A two-stage Butterworth low-pass prototype was applied to the design. Design parameters and geometrical dimensions of the filter are listed in Table II. It is worth mentioning that the capacitance calculated by (6) can only serve as an estimate for initial design. The reason is that the capacitor has been taken into consideration in the equivalent circuits. Their effects on the bandwidth and center frequency cannot be neglected. In addition, the slot cells on the ground plane of the interdigital structures also influence the capacitance. Hence, it is more suitable to optimize the whole filter using either ADS or HFSS once the preliminary design parameters are given. Fig. 6 shows a photograph of the filter. Its size only amounts , where is the guided wavelength on this to substrate at the center frequency. The measured frequency response, which was obtained using an Agilent E8363B network analyzer, along with the simulated results using ADS and HFSS are illustrated in Fig. 7(a) and (b). Good agreement between

Fig. 6. Fabricated ring filter with patterned ground plane. (a) Top view. (b) Bottom view.

the simulation and measurement is observed. The measured response has a fractional bandwidth of 5.6% at the center frequency of 1252 MHz. The minimum passband insertion loss is 1.73 dB, mainly caused by the conductor and dielectric losses. The in-band return loss is greater than 16 dB. The two transmission zeros are located at 1165 MHz with 42.8-dB attenuation and 1515 MHz with 56.7-dB attenuation, respectively. The sharp rolloff from the passband improves the selectivity of the filter well. Besides, the first spurious response is at approximately 3.5 GHz, which is 2.8 times away from the center frequency. To confirm the effects of circuit miniaturization and upper stopband extension produced by the patterned ground plane, a ring filter without a patterned ground plane is also designed and measured. The filter has a mean radius of 10 mm, which is the same as the filter with patterned ground plane. The interdigital structures of them are also identical. The measured frequency responses of these filters are plotted together in Fig. 8 for comparison. The center frequency of the filter without slot cells is 1660 MHz, which means that an LF of 24.6% is brought by the patterned ground plane. The first spurious response of the filter without slot cells is at 3.2 GHz, which is at approximately twice the center frequency. Consequently, a ring filter can benefit from the patterned ground plane with a reduced size and extended upper stopband. Shown in the inset of Fig. 8 are the passband insertion losses of these filters against the frequency offset from . Clearly, the filter with slot their own passband centers cells has a narrower bandwidth. The minimum insertion loss is 0.56 dB lower than that of the filter without slot cells. The unof the filter with slot cells is estimated loaded quality factor

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Fig. 9. Slotline dual-mode ring bandpass filter with patterned ground plane.

Finally, it is worth mentioning that the ring filter can be further miniaturized with a broader upper stopband by the combination of the periodic stepped-impedance structure and patterned ground plane. Moreover, the cost of substrate suspension has to be paid in a shielded case. The influence of the metallic enclosure on the filter responses has been studied in [14]. Given the low operating frequency range of the proposed filter, shielding seems to be not necessary, as most applications of the defected ground plane structure (DGS) have indicated. The problem of the 3-D size increment can be obviated. III. SLOTLINE DUAL-MODE RING BANDPASS FILTER WITH PATTERNED GROUND PLANE

Fig. 7. (a) Measured and simulated narrowband frequency responses of the ring filter with patterned ground plane. (b) Measured and simulated wideband frequency responses.

The microstrip filter with a patterned ground plane in Section II has exhibited attractive properties of size reduction and upper stopband extension. To offer flexible realization of the desired responses, the slotline ring filter with a patterned ground plane is proposed. The circuit configuration is shown in Fig. 9. A slotline ring forms the basic element of the filter. Four microstrip butterfly radial stubs are loaded along the periphery of the ring periodically at the backside. The shunt capacitive property of these radial stubs is utilized to realize circuit miniaturization. The radial stubs along the symmetry have larger radius than that of other stubs axis for mode splitting. The slotline ring is fed by a pair of perpendicular 50- microstrip feed lines at the backside. Each feed line is connected to an arc coupling arm at the end. By comparing a butterfly radial stub at the backside of a slotline with a radial slot cell on the ground plane of a microstrip line, it is evident that the former is the dual structure of the latter. Consequently, it is logical to classify the radial stub at the backside of a slotline as the “patterned ground plane” for ease of expatiation, even though their frequency characteristics are completely different. A. Filter Equivalent Circuits and Analysis

Fig. 8. Measured frequency responses of the ring filters with and without slot cells.

to be 155, which is higher than slot cells [13].

for the filter without

Fig. 10 shows the even- and odd-mode equivalent circuits of the slotline ring filter with a patterned ground plane. The mean circumstance of the ring is divided into four sections with equal . These sections are separated by four shunt calength of pacitors, which represent the butterfly radial stubs. The capacitances of which mainly depend upon the radial stub radius. To take the effects of the fringing fields and dynamic permittivity

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Fig. 10. (a) Even-mode equivalent circuit of the filter in Fig. 9. (b) Its odd-mode equivalent circuit.

Fig. 11. Simulated resonance frequencies of the degenerate modes and calculated coupling coefficient using (3) with a different value of r against the radius r ), where g = 0:4 mm, l = 1:5 mm, R = 10 mm, difference (d = r W = 1:5 mm, W = 1:5 mm, and  = 60 . The resonance frequencies are simulated with the structure of r = 5 mm.

0

into consideration, the analysis method in [15] is implemented. Similar to the microstrip-slot transition, the arc coupling arm is equivalent to a transformer and two open-ended stubs [16]. The latter are series connected with the feed line at one side. These stubs are utilized to represent the two branches of the arc coupling arm. Since both the capacitances and turn ratio of the transformer vary with frequency, it seems difficult to model the filter in any available circuit simulators. A MATLAB program has been compiled to derive the input admittances of the filter. The transfer and normalized to the characteristic impedance matrices of the ring in Fig. 10(a) and (b) are given by the chain matrix as

Fig. 12. (a) Simulated narrowband frequency responses of the filter in Fig. 9 as functions of the radial stub radius where g = 0:4 mm, l = 1:5 mm, R = 10 mm, W = 1:5 mm, W = 1:5 mm, and  = 60 . The radius difference dr are 0.23, 0.17, and 0.11 mm for the cases of r = 3; 4; and 5 mm, respectively. (b) Simulated wideband frequency responses of these filters as functions of the radial stub radius.

where and are the propagation constants of the microstrip , and are the characteristic imline and slotline, and , pedances of the feed line, arc coupling arm, and slotline ring, respectively. , , and in Fig. 10 The normalized admittances , are expressed as

(9a) (9b)

(7)

(10a) (10b) (8)

where the indices ments.

denote the corresponding matrix ele-

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TABLE III DESIGN PARAMETERS AND GEOMETRICAL DIMENSIONS OF THE SLOTLINE DUAL-MODE RING BANDPASS FILTER WITH PATTERNED GROUND PLANE

Fig. 13. Fabricated slotline ring filter with patterned ground plane. (a) Top view. (b) Bottom view.

The normalized even- and odd-mode input admittances of the filter is written as (11) By letting the input admittances in (11) equal to zero, the mode resonance frequencies can be calculated using numerical methods in MATLAB. The results are shown in Fig. 11. It is seen decreases linearly with the increment of the radius that difference , while remains almost the same value. Also shown in Fig. 11 is the calculated coupling coefusing (3) with a different value of . When ficient against the radius of all four butterfly radial stubs is kept the same, the coupling coefficient is equal to zero. The coupling strength is enhanced by introducing a larger value of . In addition, a smaller coupling coefficient results from a bigger , which corresponds to a filter design with narrower bandwidth. The impact of the radial stub radius on the filter responses has also been studied. Fig. 12 shows the simulated frequency responses of a series of filters using HFSS. Geometrical dimensions of the slotline rings and arc coupling arms are all the same. The radius of the radial stubs varies from 3 to 4, and then to 5 mm. Note the radius of the radial stubs as the perturbation is adjusted due to the different mode-splitting characteristics. It is and mm observed that the center frequencies for are 2311, 1927, and 1631 MHz. Their first spurious responses, however, are 2.13, 2.33, and 2.52 times away from the center frequency, respectively. The filter can clearly be miniaturized with a wider upper stopband by introducing a larger radius. Moreover, the bandwidth of the filter is narrowed when a larger value is applied. It is interesting to mentioning that there is an of additional transmission zero at the upper stopband. This transmission zero is generated by the arc coupling arms when their

Fig. 14. (a) Measured and simulated narrowband frequency responses of the slotline ring filter with patterned ground plane. (b) Measured and simulated wideband frequency responses.

electric lengths are equal to a quarter of the guided wavelength and should be useful for the rejection of the interference in the stopband. B. Filter Design and Measurement Due to the existence of the arc coupling arms, it seems unrealizable to relate the filter specifications directly to the input and output coupling structures. Therefore, it is more appropriate to optimize the whole filter by a full-wave simulator. As is observed in Fig. 11, the inter-stage coupling strength is enhanced by introducing a larger radius difference, which broadens the filter bandwidth. Moreover, the input and output

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coupling strength is controlled by the angle formed by the open ends of the arc coupling arms. The larger the angle is, the stronger the coupling between the ports and resonator results, which corresponds to a wider bandwidth as well. To obtain desired responses, the butterfly radial stubs and arc coupling arms should be designed properly to be consistent with the same bandwidth. A two-pole filter with a center frequency of 1631 MHz and a relative bandwidth of 1.8% is designed and fabricated on a CER-10 substrate. Design parameters and geometrical dimensions of the filter are given in Table III. Note the mean radius of is 10 mm to compare with the responses the ring resonator in Section II-C. Fig. 13 shows a photograph of the filter. Its size , where is the guided waveonly amounts to length of the slotline at the center frequency. The measured frequency responses, along with the simulated results using HFSS, are shown in Fig. 14. Good agreement between them is observed. The measured bandwidth is 2% at the center frequency of 1655 MHz. The frequency shift is due to the fabrication tolerance and simulation precision. The minimum passband insertion loss is 2.49 dB and the in-band return loss is greater than 17 dB. The two inherent transmission zeros near the passband are at 1590 MHz with 39.9-dB attenuation and 1775 MHz with 48.3-dB attenuation, respectively. The transmission zero generated by the arc coupling arms is at 2880 MHz with 64.6-dB attenuation. In addition, the first spurious response is at 4 GHz, which is 2.24 times away from the center frequency. It is worth comparing the responses with those in Fig. 8. Given the same radius of the ring, the filter proposed here has approximately the same center frequency with the microstrip ring filter without a patterned ground plane. The upper stopband width of the slotline ring filter is broadened and the rejection level is improved by the additional transmission zero. Moreover, the stopband rejection level of the filter in Section II can be improved by applying the line-to-ring coupling arms, which bring extra transmission zeros in the stopband as well [6], [17]. IV. CONCLUSION Miniaturized dual-mode ring bandpass filters with a patterned ground plane have been proposed and thoroughly studied in this paper. A ring filter can be miniaturized with an extended upper stopband by the periodically loaded butterfly radial slot cells on the ground plane. An LF of 24.6% has been achieved and the first spurious response is 2.8 times away from the center frequency. The slotline ring filter with a patterned ground plane has also been proposed. The filter has been miniaturized by the periodically loaded butterfly radial stubs along the periphery of the ring at the backside. By introducing the radius difference between the radial stub pairs along and perpendicular to the symmetry axis, the degenerate modes can be excited, and design flexibility is thus enhanced. The first spurious response is 2.24 times away from the center frequency. The upper stopband rejection has been improved by the additional transmission zero. It is believed that the proposed filters will find applications in the compact size and high-performance circuit design.

REFERENCES [1] I. Wolff, “Microstrip bandpass filter using degenerate modes of a microstrip ring resonator,” Electron. Lett., vol. 8, no. 12, pp. 302–303, Jun. 1972. [2] J.-S. Hong and M. J. Lancaster, “Microstrip bandpass filter using degenerate modes of a novel meander loop resonator,” IEEE Microw. Guided Wave Lett., vol. 5, no. 11, pp. 371–372, Nov. 1995. [3] R.-B. Wu and S. Amari, “New triangular microstrip loop resonators for bandpass dual-mode filter application,” in IEEE MTT-S Int. Microw. Symp. Dig., 2005, pp. 941–944. [4] R.-J. Mao and X.-H. Tang, “Novel dual-mode bandpass filters using hexagonal loop resonators,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 9, pp. 3526–3533, Sep. 2006. [5] B. T. Tan, J. J. Yu, S. T. Chew, M.-S. Leong, and B.-L. Ooi, “A miniaturized dual-mode ring bandpass filter with a new perturbation,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 343–348, Jan. 2005. [6] J.-T. Kuo and C.-Y. Tsai, “Periodic stepped-impedance ring resonator (PSIRR) bandpass filter with a miniaturized area and desirable upper stopband characteristics,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 3, pp. 1107–1112, Mar. 2006. [7] B. T. Tan, J. J. Yu, S. J. Koh, and S. T. Chew, “Investigation into broadband PBG using a butterfly-radial slot (BRS),” in IEEE MTT-S Int. Microw. Symp. Dig., 2003, pp. 1107–1110. [8] HFSS. ver. 8.0, Ansoft Softw. Inc., Pittsburgh, PA, 2001. [9] D.-J. Woo and T.-K. Lee, “Suppression of harmonics in Wilkinson power divider using dual-band rejection by asymmetric DGS,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 6, pp. 2139–2144, Jun. 2005. [10] I. Bahl, Lumped Elements for RF and Microwave Circuits. Norwood, MA: Artech House, 2003, pp. 230–233. [11] ADS. ver. 2004A, Agilent Technol., Palo Alto, CA, 2004. [12] A. C. Kundu and I. Awai, “Effect of external circuit susceptance upon dual-mode coupling of a bandpass filter,” IEEE Microw. Guided Wave Lett., vol. 10, no. 11, pp. 457–459, Nov. 2000. [13] K. Chang, Microwave Ring Circuits and Antennas. New York: Wiley, 1996. [14] Z.-W. Du, K. Gong, J. Fu, B.-X. Gao, and Z.-H. Feng, “Influence of a metallic enclosure on the S -parameters of microstrip photonic bandgap structures,” IEEE Trans. Electromagn. Compat., vol. 44, no. 2, pp. 324–328, May 2002. [15] F. Giannini and M. Ruggieri, “Shunt-connected microstrip radial stubs,” IEEE Trans. Microw. Theory Tech., vol. MTT-34, no. 3, pp. 363–366, Mar. 1986. [16] J. B. Knorr, “Slot-line transitions,” IEEE Trans. Microw. Theory Tech., vol. MTT-22, no. 5, pp. 548–554, May 1974. [17] L. Zhu and K. Wu, “A joint field/circuit model of line-to-ring coupling structures and its application to the design of microstrip dual-mode filters and ring resonator circuits,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 10, pp. 1938–1948, Oct. 1999. Rui-Jie Mao (S’06) was born in Sichuan Province, China, in 1978. He received the B.S. degree in physical electronics and M.S. degree in electronic engineering from the University of Electronic Science and Technology of China (UESTC), Chengdu, China, in 2001 and 2005, respectively, and is currently working toward the Ph.D. degree in electronic engineering at UESTC. His research interests include microwave planar filters and millimeter-wave circuits and systems design.

Xiao-Hong Tang was born in Chongqing, China, in 1962. He received the M.S. and Ph.D. degrees in electromagnetism and microwave technology from the University of Electronic Science and Technology of China (UESTC), Chengdu, China, in 1983 and 1990, respectively. In 1990, he joined the School of Electronic Engineering, UESTC, as an Associate Professor, and became a Professor in 1998. He has authored or coauthored over 80 technical papers. His current research interests are microwave and millimeter-wave circuits and systems, microwave integrated circuits, and computational electromagnetism.

MAO et al.: MINIATURIZED DUAL-MODE RING BANDPASS FILTERS WITH PATTERNED GROUND PLANE

Fei Xiao was born in Sichuan Province, China, in 1975. He received the B.S. degree in applied physics from the University of Chongqing, Chongqing, China, in 1997, and the M.S. degree in physical electronics and Ph.D. degree in radio physics from the University of Electronic Science and Technology of China (UESTC), Chengdu, China, in 2002 and 2005, respectively. In 2006, he joined the School of Electronic Engineering, UESTC, as a Teaching Assistant. He has authored or coauthored approximately 20 technical papers. His current research interests are microwave and millimeter-wave devices and computational electromagnetism.

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Design and High Performance of a Micromachined -Band Rectangular Coaxial Cable

K

Michael J. Lancaster, Senior Member, IEEE, Jiafeng Zhou, Maolong Ke, Yi Wang, and Kyle Jiang

Abstract—This paper presents the design and performance of a low-loss rectangular air-filled coaxial cable. A high-precision micromachining technique is used to fabricate the cable. It is assembled by bonding together five layers of gold-coated SU-8 photoresist fabricated using the ultraviolet photolithographic technique. As the cable is air filled, both the dielectric and radiation losses are negligible. The cross coupling is also very weak between the cable and other parts of the circuit in a system. These advantages make the proposed cable a very good candidate for low-cost high-performance miniaturized transmission lines. The cable is designed to work in the frequency range of 14–36 GHz, which covers the whole -band. The size of the cable is only 8.9 mm 8.6 mm 1.5 mm and the measured minimum insertion loss of the as-made cable is approximately 0.6 dB. The return loss is better than 15 dB in the passband. Index Terms— -band, micromachine, rectangular coaxial cable, SU-8, wideband filter.

I. INTRODUCTION

O

NE OF the most fundamental components in a communication system is the transmission line that interconnects various parts in the system. The ideal transmission line should have low loss and low dispersion, be compact in size, and be easily integrated with other components. Quite a few structures have been recently reported to realize the micromachined transmission lines, including the dielectric-filled coaxial line [1], suspended coplanar line [2], and air-filled microstrip [3]. The loss of a transmission line is usually associated with its conductor loss, dielectric loss, and radiation. The conductor loss is proportional to the square root of the frequency, while the dielectric loss is proportional to the frequency. Therefore, the metallic loss becomes less important at higher frequencies compared to the substrate-related losses [4]. Dispersion is also a major problem for some transmission lines in planar structures [5]. The substrate-related losses and dispersion have been

Manuscript received August 25, 2006; revised January 20, 2007. M. J. Lancaster, M. Ke, and Y. Wang are with the Electronic, Electrical and Computing Engineering Department, School of Engineering, The University of Birmingham, Edgbaston, Birmingham B15 2TT, U.K. (e-mail: [email protected]). J. Zhou was with the Electrical and Computing Engineering Department, School of Engineering, Electronic, The University of Birmingham, Edgbaston, Birmingham B15 2TT, U.K. He is now with the Department of Electrical and Electronic Engineering, University of Bristol, Bristol BS8 1UB, U.K. (e-mail: [email protected]). K. Jiang is with the Department of Mechanical Engineering, School of Engineering, The University of Birmingham, Edgbaston, Birmingham B15 2TT, U.K. 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.2007.900339

Fig. 1. Cross section of an ideal air-filled rectangular coaxial cable.

significantly reduced by fabricating the transmission line on a membrane [2], [3], [6]. The high precision of the micromachining technique is useful when fabricating small components for high-frequency applications [7]–[9]. This paper presents the design and high performance of an air-filled self-suspended micromachined rectangular coaxial cable as a transmission line without using a membrane. As the cable is totally air filled, the dielectric loss is negligible. There is also little radiation loss in the coaxial cable and potentially the cross coupling can be very weak between the cable and other circuits in a system. In this case, the highest aspect ratio is 1 : 2.5 as shown in Fig. 1, which is considerably larger than other techniques that use thin-film micromachining techniques [2], [3], [6]. Therefore, the proposed cable can be a good candidate for the ideal low-cost high-performance transmission line. The coaxial cable proposed is assembled by bonding five layers of gold-coated photoresist SU-8. While circular coaxial lines have been used extensively in many applications, rectangular coaxial lines are preferable in this case, as the presence of flat surface structures are particularly suitable for the photolithographic process. It is much easier to fabricate the layered rectangular structure rather than the circular one by the micromachining method proposed. The cable with a square cross section also has lower conductor loss than the rectangular one. The reason why here the cable is rectangular is described in Section II.

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-BAND RECTANGULAR COAXIAL CABLE

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The cable is designed to work in the frequency range of 14–36 GHz, which covers the whole -band. By varying its -band, dimensions, this design can be easily altered for the -band, or even higher frequency band applications. The coaxial cable is an example of the proposed fabrication method. Due to the flexibility in fabricating complex 3-D structures, the same technique can also be adopted to fabricate high-performance filters, antennas, waveguides, and other passive microwave components. The coaxial transmission line, or other components fabricated with such a technique, can be connected to the monolithic microwave integrated circuit (MMIC) using the coax-coplanar transition proposed in this paper or the coax-waveguide and the coax-microstrip transitions [10]. II. DESIGN OF THE RECTANGULAR COAXIAL CABLE A. Characteristic Impedance The cross section of an ideal air-filled rectangular coaxial cable is shown in Fig. 1. The cable has a rectangular inner conductor located symmetrically or asymmetrically inside a rectangular hollow outer conductor. For simplicity, these two conductors are assumed to be concentric in the following discussions. When the thickness of each conductor is equal to its width, the line becomes a square coaxial cable. The characteristic impedance of the cable can be calculated by the expressions given in [11] and [12]. The electromagnetic field in such a square cable resembles that in a circular coaxial cable. Especially for the case of a small inner conductor, the cable can be regarded as a circular coaxial one to calculate the characteristic impedance [13]. If the width is much larger than the thickness, the field resembles that in a shielded strip line. The characteristic impedance can be calculated as in a strip line by the expressions given in [13]. In other cases, the characteristic impedance of a rectangular coaxial can be approximately calculated by [14]

Fig. 2. Configuration of a rectangular coaxial cable with the inner conductor supported by quarter-wavelength stubs. (a) Whole circuit. (b) Middle three layers. The top and bottom layers are rectangular with dimensions of 8900 m 8600 m. The thickness of all five layers is 300 m. The circuit is symmetrical to its geometry center.

2

(1)

where is the permittivity of free space, , , , and and are given by in Fig. 1, and

are given

(2) and (3) The impedance calculated from (1)–(3) is in good agreement with the value obtained from simulation [15]. B. Suspending Stubs As mentioned above, the proposed coaxial cable is air filled so that there is no dielectric loss produced. Ideally, the cable only has the inner and outer conductors, with a characteristic impedance of 50 at any cross section. However, one problem

associated with the air-filled coaxial cable design is that the inner conductor cannot be held up by the dielectric as in the other dielectric material filled coaxial cables. One method to solve this problem is to connect the inner conductor to the outer one with quarter-wavelength stubs at appropriate intervals along the length of the cable. These stubs have characteristic impedances of , , and , respectively, as shown in Fig. 2. Given the impedance the size of these suspending stubs can be calculated in a similar way to that for the coaxial cable detailed above. The impedance and length of the stubs needs to be adjusted to obtain optimal transmission characteristics. This can be achieved by using the concept of wideband filter design as detailed in Section II-C. C. Design of the Rectangular Coaxial Cable As discussed in Section II-B, quarter-wavelength suspending stubs are needed to support the central conductor in the air-filled coaxial cable. Taking these stubs into account, the design of a

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TABLE I DESIGN PARAMETERS FOR THE RECTANGULAR COAXIAL CABLE

coaxial cable can be approached by the design of a wideband bandpass filter, as detailed in [7]. The filter design is begun with a low-pass prototype. A low-pass to bandpass transformation is then applied to obtain the bandpass filter, which is implemented physically as shorted quarter-wavelength stubs (the suspending stubs in Fig. 2) connected by transmission lines (the mainline of the inner conductor in the center, as shown in Fig. 2), which are also quarter-wavelength [13]. For different specifications, a low- or high-pass filter approach can also be used to design the cable. Although the connecting quarter-wavelength stubs limit the bandwidth of the coaxial cable, a fractional bandwidth of over 40% is achievable by using the conventional filter-design techniques [13], [16]. The filter designed here has a center frequency of 25 GHz, three poles with 0.01-dB passband ripple, and a Chebyshev response. The equal-ripple fractional bandwidth is 47.5%, while the 3-dB fractional bandwidth is 88% between 14–36 GHz. The required characteristic impedance and the dimensions of each section of the cable are calculated based on the theory described in [13, Ch. 10] and (1)–(3). Table I contains all the impedances required in the design. The dimensions of the lines and stubs to realize such impedances are shown in Fig. 2. It should be pointed out that the ideal cross section for low conductor loss is square. In this cable, all five layers have the same thickness, which means that the distance from the inner conductors to the top and bottom walls of the cable is fixed. However, as discussed above, the cable has different sections with different impedances. The widths of the inner conductors and the distances from the inner conductors to the sidewalls need to be adjusted to achieve the required impedances. Therefore, the cable designed here is rectangular. The circuit with the calculated dimensions is then simulated and optimized using the 3-D simulator CST Microwave Studio [15]. Optimization is needed mainly because it is necessary to take into account the junction effect where the stubs and the connecting line (the mainline of the inner conductor) meet in order to properly determine the length of the stubs and the connecting line. The full dimensions of the coaxial cable after optimization are given in Fig. 2. The size of the whole device is 8.9 mm 8.6 mm 1.5 mm. It should be noted that the two stubs with impedances of are connected to the ground symmetrically. These two stubs could be replaced by a single stub with impedances of connected to either side of the ground, as is the case of those and . By having two stubs with characteristic impedances two stubs with impedance of instead of a single stub with , not only is the central conductor better susimpedance of pended, but the characteristic impedances of these two stubs are and . In this case, , , and are of also closer to

Fig. 3 Whole package of the rectangular cable. Two alignment holes can be seen on the top. The supporting stubs outside of the main body of the device can be taken off after assembling.

the same value, which makes the design procedure and fabrication process much easier. It should also be noted that the length are slightly different of these two stubs with impedance of from the other two. This is mainly due to the different junction effects as mentioned above. The characteristic impedance of each port of the coaxial cable is 50 . A short section of an air-filled coplanar line is added at each port of the coaxial cable, whose characteristic impedance is also 50 . These coplanar extensions are needed for the measurement of the device using an on-wafer probe station. A similar approach is adopted in [1]. III. CIRCUIT PACKAGING Fig. 2 shows the configuration of the proposed rectangular coaxial cable. It is divided into five separate layers to suit the micromachining process. The inner conductor in the central layer, marked as layer 3 in Fig. 2, is connected to the outer ground on the same level by the perpendicular suspending stubs. Layers 1 and 5 form the top and bottom grounded plates of the enclosed coaxial structure, respectively. Layers 2 and 4 are the layers that separate the inner conductor from the top and bottom grounds. The self-suspended coaxial structure is formed by bonding together these five layers on top of each other. Since layers 2 and 4 have two separate halves, in the fabrication procedure, a pair of temporary links is added at the ends of these two layers, as shown in Fig. 3. These links help hold the two halves in the alignment procedure, and are removed after the five layers are bonded together. Although the two halves in layer 3 are interconnected by the suspending stubs, a pair of small supporting links is also added near the ports on this layer to strengthen the interconnection. The small supporting links on this layer can be either removed

LANCASTER et al.: DESIGN AND HIGH PERFORMANCE OF MICROMACHINED

or kept on the circuit after the packaging. Keeping these two links in the circuit can help balance the grounds of the coplanar line to suppress other parasitic modes at the ports. These two stubs could be especially useful if the cable were designed for higher frequency applications. SU-8 photoresist was chosen to build the coaxial cable. The advantages of utilizing SU-8 photoresist as the construction material are potentially low cost in fabrication, capability of building high aspect ratio structures [17], and suitability for volume production. The fabrication of SU-8 components requires only standard ultraviolet photolithography equipment. Firstly, SU-8 photoresist is uniformly dispersed onto a 4-in Si substrate with a syringe and spun at 600 r/min for 40 s. The resist was then pre-baked at 65 C for 5 min and raised to 95 C for 2 h on a hotplate. A Canon PLA mask aligner was used to expose the resist. Postbake was needed to cross-link the exposed resist and it was also carried out at 65 C and 95 C, respectively, on a hotplate. The resist was then developed with an EC solvent and hard baked to 150 C for 30 min to strengthen the cross-linking bond further. The cross-linked SU-8 pieces were strong and resilient to many known acids and alkalis, and they were then peeled off from the Si substrate with a sodium hydroxide solution. A more detailed description of SU-8 fabrication and how to avoid potential deformation or bending of cross-linked SU-8 pieces during the assembly/bonding process will be published separately. All five layers are made of 300- m-thick SU-8 and coated with 2- m-thick gold on all the surfaces, including all sidewalls, which was achieved through a modified evaporator, which can continuously spin the substrate holder at any given angle. The five pieces are then aligned under an optical microscope and bonded together with a silver loaded epoxy. A photograph of the assembled circuit is shown in Fig. 3.

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Fig. 4. (a) Measured and simulated results of the rectangular coaxial cable. (b) Passband details.

IV. EXPERIMENTAL RESULTS The assembled rectangular coaxial cable is measured using an on-wafer probe station connected to a vector network analyzer. The high-frequency ground–signal–ground coplanar probes are connected to the coplanar line outside of the main body of the coaxial cable. Line–reflect–reflect–match (LRRM) full two-port calibration was carried out before the measurements were performed. The analyzer source power is 10 dBm. The measured response is shown in Fig. 4 and compared with the simulated performance. Loss is not considered in the simulation. The minimum insertion loss measured is 0.6 dB, including the loss of two coaxial-coplanar transitions. It is also estimated by simulation (not shown here) that the gap leakage due to imperfect bonding and the silver epoxy is responsible for approximately 0.1-dB loss. The return loss is better than 15 dB in the passband. The deviation of the measured return loss from the simulated one is mainly due to the slight misalignment of each layer in the package, as can be seen in Fig. 3. While the simulated 3-dB passband is between 14.5–36.5 GHz, the measured one is between 14.0–36.3 GHz. The overall measured performance agrees very well with the simulated one. Two spurious peaks are observed around 47 GHz, which are mainly due to the coupling of two parasitic resonators. One resand , as shown in Fig. 2, the other onator is composed of

Fig. 5. Measured and simulated group delay of the rectangular coaxial cable.

is composed of and . The two stubs connect the resonators to ground and provide the coupling between these resonators. One coplanar filter using a similar coupling structure is reported in [18]. The measured group delay after smoothing is shown in Fig. 5. The ripples of the group delay may be attributed to mismatches along the cable and imperfections due to the inaccuracy in fabrication. The shape of the measured group delay is in good agreement with the simulation. For this 8.6-mm-long coaxial cable, the minimum attenuation is 0.6 dB, approximately 0.069 dB/mm at around 23 GHz. This performance can be improved by a better alignment. A few

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other circuits with similar sizes, using different processing techniques, have achieved comparable performance [6], [19]. Reference [6] reported a membrane-supported low-pass filter having an insertion loss between 0.25–0.9 dB from 10 to 23 GHz, and a bandpass filter having an insertion loss of 1.0 dB from 22 to 32 GHz. In [19], a rectangular coaxial fabricated on a thick oxide Si substrate with copper plating has achieved an attenuation of 0.08 dB/mm up to 40 GHz. Another similar structure is reported in [9], where a bandpass filter fabricated using EFAB technology1 has achieved an insertion loss of 1.17 dB at 30 GHz. Some other interesting structures are reported in [8] and [20]. The fabrication method proposed in this paper is quite different from the above mentioned one. The main advantage of this micromachining process is that only one ultraviolet photolithography step is needed to fabricate the air-filled rectangular cable and the ability to make high aspect ratio coaxial structures. The high performance achieved suggests that this technique may be a very good low-cost solution for the nextgeneration high-frequency wideband communication modules and system. V. CONCLUSION AND FUTURE STUDY The design and high performance of a rectangular coaxial cable have been presented in this paper. A high precision micromachining technique has been used to fabricate the circuit. The rectangular cable has been assembled by bonding together five layers of gold-coated SU-8 photoresist. The minimum insertion loss of the as-made circuit was better than 0.6 dB, and the return loss was better than 15 dB in the passband. In this fabrication, the five pieces have been manually aligned under an optical microscope and bonded together with silver loaded epoxy. In the future, a bonder/aligner will be used for accurate alignment and direct thermal compression bonding. In future research, the same technique can be applied to fabricate a rectangular coaxial cable at a higher frequency band. It is also possible to fabricate other microwave components such as filters, couplers, and antennas with this technique.

[5] D. Mirshekar-Syahkal and J. B. Davies, “Accurate solution of microstrip and coplanar structures for dispersion and for dielectric and conductor losses,” IEEE Trans. Microw. Theory Tech., vol. MTT-27, no. 7, pp. 694–699, Jul. 1979. [6] T. M. Weller, L. P. B. Katehi, and G. M. Rebeiz, “High performance microshield line components,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 3, pp. 534–543, Mar. 1995. [7] I. Llamas-Garro, M. J. Lancaster, and P. S. Hall, “A low loss wideband suspended coaxial transmission line,” Microw. Opt. Technol. Lett., vol. 43, no. 1, pp. 93–95, Jan. 2004. [8] Y. Kim, I. Llamas-Garro, C.-W. Baek, and Y.-K. Kim, “A monolithic surface micromachined half-coaxial transmission line filter,” in 19th IEEE Int. MEMS Conf., Istanbul, Turkey, Jan. 22–26, 2006, pp. 870–873. [9] R. T. Chen, E. R. Brown, and C. A. Bang, “A compact low-loss -band filter using 3-dimensional micromachined integrated coax,” in 17th IEEE Int. MEMS Conf., Maastricht, The Netherlands, Jan. 25–29, 2004, pp. 801–804. [10] M. Pardalopoulou and K. Solbach, “Over-moded operation of waveguide-to-coax transition at 60 GHz,” in Joint 29th Int. Infrared Millim. Waves Con./12th Int. Terahertz Electron. Conf., Sept. 27 –Oct. 1 2004, pp. 475–476. [11] H. E. Green, “The characteristic impedance of square coaxial line (correspondence),” IEEE Trans. Microw. Theory Tech., vol. MTT-11, no. 6, pp. 554–555, Nov. 1963. [12] S. W. Conning, “The characteristic impedance of square coaxial line (correspondence),” IEEE Trans. Microw. Theory Tech., vol. MTT-12, no. 4, pp. 468–468, Jul. 1964. [13] G. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching Networks and Coupling Structure. Norwood, MA: Artech House, 1980. [14] T.-S. Chen, “Determination of the capacitance, inductance, and characteristic impedance of rectangular lines,” IEEE Trans. Microw. Theory Tech., vol. MTT-8, no. 5, pp. 510–519, Sep. 1960. [15] CST Microw. Studio. CST, Wellesley Hills, MA, 2006. [16] J.-S. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications. New York: Wiley, 2001. [17] R. Yang and W. Wang, “A numerical and experimental study on gap compensation and wavelength selection in UV-lithography of ultrahigh aspect ratio SU-8 microstructures,” Sens. Actuators B, Chem., vol. 110, no. 2, pp. 279–288, Oct. 2005. [18] J. Zhou, M. J. Lancaster, and F. Huang, “Coplanar quarter-wavelength quasi-elliptic filters without bond-wire bridges,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 4, pp. 1150–1156, Apr. 2004. [19] I. Jeong, S.-H. Shin, J.-H. Go, J.-S. Lee, C.-M. Nam, D.-W. Kim, and Y.-S. Kwon, “High-performance air-gap transmission lines and inductors for millimeter-wave applications,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2850–2855, Dec. 2002. [20] J. R. Reid and R. T. Webster, “A compact integrated coaxial -band bandpass filter,” in IEEE AP-S Int. Symp., Jun. 20–25, 2004, vol. 1, pp. 990–993.

Ka

V

ACKNOWLEDGMENT The authors would like to thank Dr. P. Suherman, Electronic, Electrical and Computing Engineering Department, School of Engineering, University of Birmingham, Edgaston, Birmingham, U.K., for her technical support in the measurement. REFERENCES [1] J. A. Bishop, M. M. Hashemi, K. Kiziloglu, L. Larson, N. Dagli, and U. Mishra, “Monolithic coaxial transmission lines for mm-wave ICs,” in Proc. IEEE/Cornell Adv. Concepts Conf., Aug. 5–7, 1991, pp. 252–260. [2] G. Sajin, E. Matei, and M. Dragoman, “Microwave straight edge resonator (SER) on silicon membrane,” in Proc. Int. Circuits Syst. Semiconduct. Conf., Oct. 5–9, 1999, vol. 1, pp. 283–286. [3] C. B. Ashesh, D. Bhattacharya, and R. Garg, “Characterization of V-groove coupled microshield line,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 2, pp. 110–112, Feb. 2005. [4] W. Y. Liu, D. Steenson, and M. B. Steer, “Membrane-supported CPW with mounted active devices,” IEEE Microw. Wireless Compon. Lett., vol. 11, no. 4, pp. 167–169, Apr. 2001. 1EFAB,

Microfabrica, Van Nuys, CA, 1999.

Michael J. Lancaster (M’91–SM’04) received the Physics and Ph.D. degrees from Bath University, Bath, U.K., in 1980 and 1984, respectively. His doctoral research concerned nonlinear underwater acoustics. Upon leaving Bath University, he joined the Surface Acoustic Wave Group, Department of Engineering Science, Oxford University, as a Research Fellow, where his research concerned the design of novel surface acoustic wave (SAW) devices including filters and filter banks. These devices worked in the 10-MHz–1-GHz frequency range. In 1987, he became a Lecturer with the School of Electronic and Electrical Engineering, The University of Birmingham, Edgbaston, Birmingham, U.K. Shortly after, he began the study of the science and applications of high-temperature superconductors (HTSs), involved mainly with microwave frequencies. He currently heads the Emerging Device Technology Research Center, The University of Birmingham. His current research interests include microwave filters and antennas, as well as the high-frequency properties and applications of a number of novel and diverse materials.

LANCASTER et al.: DESIGN AND HIGH PERFORMANCE OF MICROMACHINED

Jiafeng Zhou was born in Jiangsu, China. He received the B.Sc. degree in radio physics from Nanjing University, Nanjing, China, in 1997, and the Ph.D. degree from The University of Birmingham, Edgbaston, Birmingham, U.K., in 2004. His doctoral research concerned high-temperature superconductor microwave filters. Beginning in July 1997, for two and a half years he was with the National Meteorological Satellite Centre of China, Beijing, China, where he was mainly involved with communications of the ground station and Chinese geostationary meteorological satellites. From August 2004 to April 2006, he was a Research Fellow with The University of Bimingham, where his research concerned phased arrays for reflector observing systems. He is currently with the Department of Electronic and Electrical Engineering, University of Bristol, Bristol, U.K. His current research interests include microwave power amplifiers and linearization.

Maolong Ke, photograph and biography not available at time of publication.

Yi Wang was born in Shandong, China, in 1976. He received the B.Sc. degree in physics and M.Sc. degree in condensed matter physics from the University of Science and Technology Beijing, Beijing, China, in 1998 and 2001, respectively, and the Ph.D. degree in electronic and electrical engineering from The University of Birmingham, Birmingham, Edgbaston, U.K., in 2005. His doctoral research concerned microwave superconducting coplanar delay lines. He is currently a Research Fellow with the Electronic, Electrical, and Computer Engineering Depart-

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ment, The University of Birmingham. His main research interests are high-frequency applications of novel materials and structures such as superconductors, ferroelectric materials, left-handed transmission-line structures, and micromachined structures. His most recent research is on microwave and millimeterwave passive devices involving micromachining techniques, which include micromachined rect-coaxial devices, membrane supported filters, and microelectromechanical systems (MEMS)-actuated tunable superconducting resonators. His minor research interests are device and material characterizations.

Kyle Jiang received the Ph.D. degree in mechatronics from King’s College London, London, U.K., in 1993. Since 1999, he has been a Lecturer The University of Birmingham, Edgbaston, Birmingham, U.K. Prior to his current appointment, he was a Research Assistant with King’s College London before joining The University of Birmingham as a Research Fellow in 1997. His current research interest in microdevices includes microwave components, microengines, nanosurface for surface plasmon resonance sensors, microdevices for patch clamping and micro wireless sensors. He is a member of an international journal Editorial Board. He is currently involved with projects funded by the European Commission, DTI, the Engineering and Physical Sciences Research Council (EPSRC), the Royal Society, and the Xian Science Foundation of China. Dr. Jiang has been a member of the Institution of Engineering and Technology (formerly the IEE) and a Chartered Engineer since 1993. He serves on a number of international conference committees.

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A Swept-Frequency Measurement of Complex Permittivity and Complex Permeability of a Columnar Specimen Inserted in a Rectangular Waveguide Atsuhiro Nishikata, Member, IEEE

Abstract—This paper proposes a new method for the swept-frequency measurement of a solid material’s complex permittivity and complex permeability. The proposed method utilizes a columnar specimen inserted in a rectangular waveguide, and the two-port -parameters of the specimen are measured. In the analysis, the -parameters are rigorously formulated in which the waveguide’s fundamental mode, as well as the higher order modes are taken into account. Measurement was performed by using a standard waveguide and three types of materials as specimens. Results were compared with the conventional transmission-line method using the same rectangular waveguide. It is confirmed that the complex permittivity and complex permeability measured by the proposed method agrees very well with those measured by the conventional transmission-line method. Index Terms—Complex permeability, complex permittivity, -parameters, waveguide.

I. INTRODUCTION

A

S MOBILE telecommunication devices or high-speed digital circuits are supported by many types of materials such as printed circuit boards, ferrites, noise suppression composite materials, dielectrics used in filters, etc., there are increasing demands among manufacturers for the determination of the materials’ electromagnetic properties in microwave and millimeterwave frequencies. For the swept-frequency measurement of complex permittivity and complex permeability of materials in microwave frequencies, transmission-line-based methods has been developed over the years. They include basic or specific measurement techniques [1]–[3], [5]–[8], calculation algorithms for material parameters [1]–[6], [8], or tensor permittivity and permeability measurement [8]. When restricted to permittivity measurement, determination of the unknown layer in a multilayer specimen [9] and a fast calculation algorithm [10] have been investigated. The transmission-line method typically uses a coaxial line or rectangular waveguide to determine the -parameters of a section of the transmission line filled with the material under test. Today we can easily perform two-port -parameter measurement by using a vector network analyzer. Relative com-

Manuscript received September 7, 2006; revised April 5, 2007. The author is with the Center for Research and Development of Educational Technology, Tokyo Institute of Technology, Tokyo 152-8552 Japan (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.2007.900340

Fig. 1. Specimen loading of proposed method and conventional method. (a) Proposed method. (b) Conventional method.

and relative complex permeability are plex permittivity then separately determined from measured -parameters by a well-known algorithm [3], [4]. The transmission-line method basically assumes that the fields in the waveguide is single moded. It is based on the assumption that a section of waveguide is completely filled with the material, which is linear, uniform, and isotropic. However, the actual situation is more or less imperfect, which can cause the higher order mode excitation. In the standard rectangular mode can propagate when the waveguides, only the frequency is within the range of (1) is the cutoff frequency of the mode, is the where velocity of light, and is the longer side length of the inner dimension of waveguide’s cross section. If the waveguide is uniformly filled with lossless material having and , the above inequality is replaced with (2) Therefore, the single-mode conditions for the empty section and the specimen-filled section are not consistent if the dielectric specimen has greater than 4. Thus, high-permittivity material measurement tends to have an erroneous result due to the influence of higher order modes in the specimen because the source of the higher order mode more or less exists. In this paper, a new waveguide method is proposed in which a columnar specimen is inserted between two wider walls of a rectangular waveguide, as depicted in Fig. 1(a). The conventional waveguide method requires the specimen to have a precise rectangular shape with six flat faces and to precisely fit with the cross-sectional dimension of the waveguide, as depicted in Fig. 1(b). Compared to the conventional waveguide method, the proposed method’s specimen has a simpler columnar shape, which may easily be machined with the lathe.

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only the fundamental mode survives and the scattered field is expressed with the scattered wave amplitudes and . The analysis aims for the formulation of -parameters, which relate and with and by (3) where and comes from the symmetry of the analysis model with respect to the plane. B. Decomposition of Total Field

Fig. 2. Model for the analysis.

In the analysis, the -parameters of a columnar specimen placed in a rectangular waveguide are rigorously formulated without ignoring the generation of higher order evanescent modes around the specimen. Related previous studies include the variational solution to the equivalent circuit of a dielectric post in the waveguide [11], mode-matching analysis for a double-layered circular cylinder [12], method of moment analysis for a dielectric post having an arbitrary-shaped cross section including the column [13], and liquid materials’ permittivity measurement by utilizing a dielectric container tube piercing through a pair of holes made on the rectangular waveguide’s wider walls [14], [15]. Those are all restricted to the analysis or measurement of nonmagnetic materials. Another investigation for permittivity and permeability measurement by a partially filled waveguide can be found [16] in which the rectangular specimen partially occupies the rectangular waveguide along the longitudinal axis and the axis parallel to the -field, which is a different configuration from the current investigation. II. FORMULATION A. Model for the Analysis The model for the analysis is shown in Fig. 2. Cartesian coor, as well as the cylindrical coordinates dinates are introduced. The longer side length of the waveguide’s internal cross section is , and the shorter side length is . A and height is placed columnar material of diameter at the center of the waveguide. The material is assumed to be linear, uniform, and isotropic, whose relative complex permittivity is , and the relative complex permeability is . mode varying as with Incident waves of the angular frequency are assumed from both the left- and rightand , respectively. hand sides with different amplitudes The reference plane is set at the plane, which includes the center axis of the columnar specimen. The scattered field modes and many higher order is expressed as a sum of evanescent modes. At some distance apart from the specimen,

The analysis model is uniform in the -direction and the problem is reduced to a 2-D problem. The electric field has a component only, and the magnetic field has both and components. Thus, the fields have , , and . The total electric field in the waveguide, excluding the material region, is decomposed into (4) is the incident field, is the field directly scatwhere is the sum of scattered by the columnar material, and tered fields coming from the infinite number of images of the columnar material, which reflect in the parallel mirrors of wave. guide walls at C. Incident Field The incident field is the sum of two oppositely traveling waves of the mode and is written as (5) , , is the where , is equal wavenumber in free space, which is equal to to , where is the cutoff wavelength of the mode, and is the wavelength in free space, which is equal . is essentially equivalent with a superposition of to four plane waves. For simplicity, two excitation modes are considered. One is , and the other is the the symmetrical excitation . In the former antisymmetrical excitation case, the scattered amplitudes and are both equal to , and in the latter case, and . The analysis for symmetrical excitation will first be given. The incident field for the symmetrical excitation is written as (6) This can be rewritten in terms of cylindrical waves (see Appendix I) as

(7)

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where

where as (see Appendix III)

for the columnar material is derived

(8) where is equal to 1 for and is equal to 2 for nonzero integer , is the angle between the propagation direction of plane, four plane waves and the transversal plane such as and is given by

(16) Therefore, we obtain the relation

(9)

D. Determination of Total Field Considering the symmetry, the scattered field from a single columnar material can be expanded in terms of outgoing cylindrical waves as (10)

(17) To solve it numerically, an assumption that overrides (16) is introduced as follows: even number

(18)

A matrix equation of finite order is then obtained as follows:

The scattered field coming from all of the images of columnar material is then

.. .

.. .

.. .

..

.. .

.

(11) If we expand the image field in the form

.. .

.. .

(19)

(12) the coefficients

can be expressed as (13)

where Appendix II)

is derived to be (see

The above matrix equation can be solved numerically to ob. Others are assumed to tain be null from (17) and (18). Once they have been obtained, the total electric field can be calculated as a sum of (6), (10), and (11). E.

-Parameters for Symmetric Excitation

For symmetric excitation, , which is equal to . The columnar material emanating the outgoing cylindrical waves are equivalent with a superposition of monopole and multipole sources, which excites the waveguide’s fundamental mode. As a result (see Appendix IV), we have (20) (14)

where

The infinite series in (14) converges slowly. The use of Aitken’s process is found to be effective for the acceleration of convergence. The applied field for the single columnar material is , while the scattered field from the single columnar material is . The columnar material responds to the applied field and yield the outgoing cylindrical wave for each quantum number . Mathematically, it is expressed by

For antisymmetric excitation, it can be formulated in a similar manner. The incident wave (6) is replaced with

(15)

(22)

(21) If the right-hand side of (20) converges as is expected to approach the rigorous one. F.

increases, the value

-Parameters for Antisymmetric Excitation

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Its cylindrical wave expansion is

where

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is the Jacobian matrix given by

(23)

(30)

(24) can be determined by Therefore, small quantities solving the following linear system of equations:

The matrix equation (19) is replaced with (25), shown at the is an odd number given by . bottom of this page, where In this case, the wave amplitude is written as

(31) (26) If is close to singular, the iterative calculation based on the above equation shall fail. Therefore, the Jacobian, which is the determinant of , should not be too close to zero, in order for the above equations to be successfully solved.

where (27)

H. Uncertainty Analysis

Thus, the -parameters of a columnar material, i.e., and , can be obtained by adding or subtracting the results of (20) and (26). On the other hand, (20) and (26) themselves have physical meaning as the one-port reflection coefficients in special boundary conditions: the former with a perfect magnetic con, the latter with a perfect ductor (PMC) wall placed at electric conductor (PEC) wall at the same position. G. Determination of

The main sources of uncertainties for the proposed method are expected to be the -parameters’ uncertainties and specimen volume’s uncertainty, as well as the air gaps between the specimen and waveguide walls. The influence of the latter two sources may be physically analogous with the case of the conventional transmission-line method, while the influence of -parameters’ uncertainties is more complicated and is considered here. The propagation of uncertainties of -parameters can be investigated by a relationship similar to (31) as follows:

and

The -parameters of a columnar material are now calculable must as functions of and . In practice, however, and be inversely determined from the measured -parameters. Let the right-hand side of (20) and (26) be denoted as and , respectively. They are complex functions of two complex variables and . The system of nonlinear equations are then to be fulfilled by unknown and

The preceding multiplying sides, we have

(32) represents its deviation from true value. By from the left and taking the 2-norm of both

(28) (33) A numerical iterative technique, such as Newton’s method, with two complex variables can be used to find the solution. If the set of relative complex permittivity and relative complex permeand is expressed ability is already close to the solution as , then

where

is a natural norm of a matrix

, and (34)

Therefore, (29)

.. .

.. .

.. .

..

(35)

.

.. .

.. .

.. .

(25)

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Thus, the uncertainties of -parameters propagate to the unceror tainties of and , via a coefficient, which is equal to less. As for the uncertainty of , the following formula can be derived from (32): (36) If and are statistically independent and each probability density of them is isotropic on the Gaussian plane having , we can derive their influidentical standard uncertainty as ence on

Fig. 3. Snapshot of E -field strength in 2-D FDTD simulation. The FDTD model is sinusoidally excited from the left, and the TE mode amplitudes of scattered waves are extracted. In this example, the specimen (indicated in online version via a yellow circle) has a displacement of approximately 1 mm in the positive y -direction. Other parameters are " = 5, d = 4:5 mm, a = 22:86 mm, and f = 10 GHz.

(37) where represents the contribution of -parameters’ uncertainty to the ’s uncertainty. Therefore, the sensitivity coefficient, which relates the -parameter’s uncertainty and the relative uncertainty of , is expressed as

They can be inversely solved with respect to formulas in calculating order are

and

. The

(44)

(38)

(45) In a same way, the sensitivity coefficient for the relative uncertainty of is given by

(46) (47)

(39)

I. Formulas for Conventional Transmission-Line Method Assume that the waveguide is sectionally filled with a material having thickness , as depicted in Fig. 1(b). The material . and of occupies the waveguide at the material can then be derived from the measured -parameters. Here, the formulas for the conventional transmission-line method [3] will be reviewed for the case of the rectangular waveguide. It will be with a minor modification to have a form compatible with the proposed method’s formulation in which both of the reference planes are set at the center of the specimen that plane. is the If the fields in the waveguide is assumed to be single moded mode, the -parameters with their reference with the are given by planes at (40)

(41) where (42) (43)

The solution has ambiguity due to the logarithmic function, which is a multivalued function. Concretely, (46) involves an arbitrary integer which must be determined by some means. For specimens thicker than a half guided-wavelength in the material-filled waveguide, should be increased. If multiple specimens with different thicknesses are available, one can easily identify the correct solution, which commonly appears within different thickness specimens. If the real part of encounters a discontinuity while increasing the frequency, one must add 1 to at the frequency so as to cancel the discontinuity of ’s real part. III. NUMERICAL INVESTIGATION In the following numerical investigation, a standard rectangular waveguide (IEC R-100, 8.2–12.4 GHz) is assumed, whose longer and shorter side lengths of the inner cross-sectional dimm and mm, respectively. mension are The RF source frequency is assumed to be 10 GHz. A. Comparison With Finite-Difference Time-Domain (FDTD) Simulation To check the validity of the current analysis, 2-D FDTD simulation is performed and compared with analytical result in terms of -parameters. The FDTD model whose cell size is mm and time step is ps is sinusoidally exmode amplitudes of cited from the left-hand side, and the scattered waves are extracted. Fig. 3 shows a top-view snapshot of the -field strength. In this example, the specimen having

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Fig. 5. Variation of 0 and 0 for various " .

Fig. 4. Comparison of current analysis with FDTD simulation. S (= S ) and S (= S ) are plotted versus: (a) d or (b) " . The current analysis result given by solid or dashed lines agrees well with FDTD simulation result given by circle or cross marks. a = 22:86 mm, f = 10 GHz. (a) S , S versus d. " = 2. (b) S , S versus " . d = 10 mm.

and mm (indicated in online version via a yellow circle), has a displacement of approximately 1 mm in the positive -direction. One can observe the loss of field symmetry with respect to waveguide axis by the displacement. Fig. 4 shows the comparison of -parameters of current analysis with FDTD simulation. Here, no displacement of the specand are plotted versus imen is assumed. or in (a) or (b), respectively. The current analysis result given by solid or dashed lines agrees very well with the FDTD simulation result given by a circle or cross marks. B. Basic Characteristics of

Fig. 6. Variation of 0 and 0 for various  .

and

Physically, corresponds to the reflection coefficient when plane. The specthe PMC wall is virtually placed at the imen is then placed at the position where the electric field is maximum and the magnetic field is minimum. To the contrary, corresponds to the reflection coefficient when the PEC wall is virtually placed and the magnetic field there is maximum and the electric field is minimum. Therefore, it is expected that mainly depends on , which interacts with the electric field, while mainly depends on , which interacts with the magnetic field. . Fig. 5 shows calculated and for various and are shown as dots on the complex plane. and are discretely moved as and . The specimen is assumed to be of nonmagnetic . The diameter of the specimen material, i.e., is 6 mm. The frequency is 10 GHz. If the specimen is absent, and , the reflection coeffiwhich corresponds to and . From this figure, cients have the value of is indeed larger than that of , as expected. the variation of Fig. 6, on the other hand, shows the calculated and for . is moved discretely and is fixed various

to

. The range of variation of is the same as that of in the former figure. Other conditions are the same. From is relatively larger than that of , as Fig. 6, the variation of expected. Compared to the variation of in Fig. 5, the variation in Fig. 6 is smaller. It may be due to the direction of the of magnetic field applied to the specimen. The applied magnetic field is perpendicular to the axis of the columnar specimen, and the antimagnetic field disturbs the polarization of the material than in the case where the field is applied longitudinally to the axis of the specimen. C. Influence of -Parameters’ Measurement Uncertainties and do not vary sufficiently or independently with If and , the measurement uncertainty of and respect to shall increase. It is mathematically expressed by (35), which implies that the measurement error of the -parameters is multo contribute to the uncertainty of tiplied, at most, by and . It means that from the measurement’s point of view, the ’s value is the better. Fig. 7 shows the value of smaller as functions of , assuming . Each curve has a different as described. Other assumptions are the same as those

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Fig. 7. Natural matrix norm kK k for various " .

Fig. 8. Natural matrix norm kK k for various  .

in Fig. 5. It is found that for lossless dielectrics, is relatively small, while it increases for dielectrics with higher losses is not more than 20 or higher permittivities. Still, however, in the range of calculation. in a similar way, but is varied and is Fig. 8 shows . Results similar to those for the former figure are fixed to also obtained from this figure. Fig. 9 is restricted to the dielectric measurement. It shows the sensitivity coefficient, which shall be multiplied with the -parameter’s uncertainty to yield the relative uncertainty of permittivity. The specimen is assumed to be low-loss dielectrics. One can see that depending on the value of , the specimen has the optimum diameter relative to the wider side length of the waveguide . For example, for the measurement of material or , the optimum ratio is found to be having around 0.35 or 0.15, respectively. D. Influence of Specimen Displacement In the current method, the specimen is assumed to be centered in the -direction. Actually, however, the positioning error of the specimen shall more or less exist. If there is a displacement of the specimen in the -direction, denoted as , the

Fig. 9. Sensitivity coefficient, which relates the uncertainty of S -parameter measurement to the relative uncertainty of " as functions of relative diameter of columnar specimen. f = 10 GHz and a = 22:86 mm are assumed.

-parameters shall deviate. Since and are even functions of , the deviation of the measured and , as a consequence of specimen displacement, shall be expressed by polyno. If is small enough mials having the terms of so that the -term is dominant, the deviation shall be divided is halved. Therefore, it is expected to be suffiby 4 when ciently small for small . To evaluate this quantitatively, the FDTD simulation with specimen displacement is utilized, and and are inversely calcuthe consequential deviations of lated. Fig. 10 shows the result, assuming the specimen parameand , ters ( , , mm ) being in Fig. 10(a)–(c), respectively. These diameters are close to the optimum values taken from Fig. 9. The figure shows that the influence of displacement as much as 1 mm (which is a noticeable displacement, as shown in Fig. 3 is less than 0.0034 of ), 0.036 ( of ), and 0.23 ( of ) in permittivity deviation. Though the de( viation tends to be large for larger , it is still considered to be very small. IV. EXPERIMENT A. Experimental Setup A standard waveguide for -band (8.2–12.4 GHz) is utilized. Fig. 11 shows the waveguide set for the experiment. A pair of coaxial-to-waveguide converters are connected to a vector network analyzer via test port cables. Each coaxial-to-waveguide converter is joined to a waveguide section so as to attenuate higher order evanescent modes, which shall exist in the vicinity of probes of the coaxial-to-waveguide converters. The flange surface of each waveguide section is defined as the reference plane of -parameters. B. Procedure of -Parameter Measurement After thru-reflect-line (TRL) calibration [17] by which the reference planes are set at the flange surfaces of test ports, twoport -parameters of specimen holder (a section of waveguide) are measured without or with the specimen loaded, as depicted in Fig. 12(a) and (b), respectively. The measured data include the phase shift and attenuation due to the propagation in the

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Fig. 11. Experimental setup.

Fig. 12. Measurement of specimen holder without or with the specimen. (a) Unloaded waveguide. (b) MUT-loaded waveguide.

• Let the measured -parameters of the specimen-loaded . The reflection -parameter waveguide section be and transmission -parameter of the specimen itself is obtained by two types of averaging calculation as follows:

(49)

Fig. 10. Deviation of measured " and  due to the specimen displacement in y -direction. a = 22:86 mm, f = 10 GHz. (a) " = 2, d = 8 mm. (b) " = 5, d = 4:5 mm. (c) " = 10, d = 3:5 mm.

where the geometric mean is utilized in order to eliminate the possible positioning error of the specimen toward the -direction. The same procedure is also used to measure the -parameters of specimens of rectangular shape. Once the -parameters of specimens are obtained, they are used to determine and of the material by the proposed method or by the conventional transmission-line method. C. Measured Results

waveguide section. To eliminate the propagation factor and to obtain the -parameters of the columnar specimen solely, the following procedure is used. • Let the measured -parameters of the unloaded waveguide . The transmission coefficient of the section be waveguide is obtained as (48)

Three types of materials are prepared. For each material, both the columnar specimen and the rectangular specimen are manufactured, as listed in Table I. The columnar specimen is measured by the proposed method, while the rectangular specimen is measured by the conventional transmission-line method for comparison, both by using a similar rectangular waveguide. Each measurement is done on 201 frequency points spanning 8.2–12.4 GHz.

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TABLE I SPECIMENS MANUFACTURED FOR THE MEASUREMENT

PTFE is a typical low-loss dielectric material. TONEREPOXY is a mixture of the laser printer’s black toner and epoxy resin. Since the black toner particle is dielectric-coated carbon black, the composite material is expected to be a nonmagnetic dielectric material with some dielectric loss. FERRITE-EPOXY is a mixture of ferrite powder and epoxy resin. It is expected may be different to be a lossy magnetic material whose may be greater than zero. from unity and as functions of Fig. 13 shows the PTFE’s , , , and frequency in each flame from top to bottom. The columnar specimen’s result measured by the proposed method is shown by solid line curves, while the rectangular specimen’s result measured by the conventional transmission-line method is shown by dotted line curves. Very good agreement is observed between the proposed method and transmission-line method. The lower two flames show that is almost equal to unity, which is appropriate since PTFE is a nonmagnetic material. The measured results, regardless of the measurement method, show a small fluctuation, which is periodic with respect to frequency. This may be due to the interference of residual reflected waves at connector junctions or coaxial-to-waveguide converters not being completely eliminated by TRL calibration. and Fig. 14 shows TONER-EPOXY’s measured by the proposed method and conventional transmission-line method. They are in very good agreement. Measured ’s are almost frequency independent having small losses. The dielectric loss may be due to the conduction current in the carbon particles. Measured ’s by two methods are both close to unity. They are considered to be appropriate since TONER-EPOXY is made from nonmagnetic materials. The third material is FERRITE-EPOXY (ferrite composite). Fig. 15 shows measured results of FERRITE-EPOXY obtained by the proposed method and conventional transmission-line method. Results are almost the same between two measurement methods. In this case, is almost constant with frequency and is dispersive with frequency and dielectric loss is low, while is less than unity, magnetically lossy. It is observed that which is popularly seen for ferrite composite materials. Throughout the results including low-loss dielectric material, lossy dielectric material, and lossy magnetic material, the differences between the proposed method and transmission-line method are seen to be no more than 2% for components that are not close to zero. Thus, the validity of the proposed method is shown. If two methods are compared in terms of the width of periodic fluctuation, these figures seem to suggest that the proposed

Fig. 13. PTFE’s " , " ,  , and  (from top to bottom) measured by proposed method with columnar specimen and by conventional transmission-line method with rectangular specimen.

method is more precise for permittivity, but less precise for permeability than that of the conventional method, regardless of

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Fig. 14. TONER-EPOXY’s " , " ,  , and  (from top to bottom) measured by proposed method with columnar specimen and by conventional transmissionline method with rectangular specimen.

Fig. 15. FERRITE-EPOXY’s " , " ,  , and  (from top to bottom) measured by proposed method with columnar specimen and by conventional transmission-line method with rectangular specimen.

what type of material. However, since the specimens used here are not of optimized dimensions, a fair comparison between the

two methods’ accuracy is difficult. Still it worth pointing out that the proposed method places the specimen at the center of

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the waveguide where the traveling wave’s electric field is maximum. This could cause an advantage in the permittivity measurement by the proposed method. To the contrary, the aforementioned antimagnetic field shall weaken the lateral magnetic polarization of the specimen, especially in the higher permeability case, which could cause a disadvantage in the permeability measurement. V. CONCLUSION A new method for the swept-frequency measurement of material’s complex permittivity and complex permeability has been proposed. The proposed method utilizes a columnar specimen inserted in a rectangular waveguide. The -parameters of the specimen are rigorously formulated in which the waveguide’s fundamental mode, as well as the higher order modes are taken into account. Since the structure essentially generates multimodal fields in the waveguide, it is free from the single-moded assumption that the conventional transmission-line method is based upon. Moreover, the columnar shape is considered to be advantageous as a specimen compared to the rectangular shape since the columnar shape is easily manufactured by the lathe. Measurement was performed in 8.2–12.4 GHz by using an IEC R-100 standard waveguide. Three types of materials including low-loss dielectric material, lossy dielectric material, and lossy magnetic material were measured successfully. It was confirmed that the complex permittivity and complex permeability measured by the proposed method agreed very well with those measured by the conventional transmission-line method. APPENDIX I DERIVATION OF A plane wave whose wavenumber vector -plane can be expanded as

is within the

Fig. 16. Image of material as a source and an observation point.

By letting reduced to

, the sum of four cosine functions is

(53) This leads to have (7) and (8). APPENDIX II DERIVATION OF Consider the wave coming from an image of material whose center is apart from the origin, as depicted in Fig. 16. Bessel’s summation theorem states that the cylindrical wave emanated from the image source is expressed as

(50) where is the angle between the direction of the observation point and the direction that the plane wave is coming from. Since the incident field (5) can be rewritten as a superposition of four plane waves,

By substituting

and

(54) , we have

(55) (51) If we make the linear combination of cosine and sine functions to yield the exponential function, we have

which can be expanded as

(56)

(52)

(57)

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If we reconstruct the cosine function, we have

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It is worth extending the formula to the case of coaxially stratified material for future use. Assume that the material consists of layers. The material constants of the layers are and from inside to outside. Cylindrical surfaces bound these layers, and is the outer at surface of the outermost layer. The extension only needs to add and a recurrence formula, which relates the admittances at adjacent interfaces of and , respectively, by (58)

The wave reflected times between two narrow walls reverses . the sign for odd , resulting in having the multiplier There are two image sources that experience -time reflection. Their positions are at and . Thus, the sum of the two image source’s contribution yields

(64) where

(65)

(66) (67)

(59) Multiplying with and summing over positive integer yield . Therefore, is derived as

will (68) (69) and (70)

(60) which yields (14). APPENDIX III DERIVATION OF Assume a columnar material having relative complex permittivity , relative complex permeability , and radius . Its center axis coincides with the -axis. Consider an elementary wave out of the expansion of the external field, and the material scatters the elementary wave to produce outgoing a cylindrical as wave with scattering coefficient (61) Let us introduce the normalized wave admittance on the cylindefined by drical surface

in (16) is then replaced with The normalized admittance for the stratified case. APPENDIX IV DERIVATION OF Consider a -directed surface electric current distribution on the cylindrical surface integer

(71)

where represents the Dirac’s delta function. In free space, the above current is to produce the outgoing cylindrical wave of the unit coefficient such as (72) The internal field is then

(62) (73) By calculating from and substituting them into the above equation, the first equation of (16) is obtained. While, the electric field inside the material is expressed as (63) By calculating from the above and substituting them into (62), the second equation of (16) is obtained.

By equating the discontinuity of the tangential magnetic field with the surface current density, we have

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(74) Therefore,

wave of unit amplitude (72) in free space. It means that is identical with as follows:

is determined to be

(83)

(75) Next, consider a -directed line current of unit strength in the waveguide

By the help of integer (84)

(76) The waveguide’s fundamental modes, as well as higher modes order evanescent modes are excited. Only are excited and the -directed electric field is [18]

(77) (78) The current distribution (71) is equivalent with

integer

(79)

Therefore, the electric field produced by the current distribution in the waveguide is calculated by (80)

From the symmetry of the current distribution, only survives in the summation. For (port 1 side), the electric field is expressed by the modal expansion as (81) where

(82) , represents the mode’s amplitude excited For by the source (71), which emanates the outgoing cylindrical

one can execute the integration of (83) to obtain (21) and (27). ACKNOWLEDGMENT The author would like to thank Prof. K. Ono, Ehime University, Matsuyama, Ehime, Japan, for his suggestion about the analytical solution to the structure discussed in this paper. The author also thanks Dr. T. Aoyagi, Tokyo Institute of Technology, Tokyo, Japan, for the discussion on the mathematical treatment of the problem. REFERENCES [1] A. R. Von Hippel, Dielectric Materials and Applications. Cambridge, MA: MIT Press, 1954. [2] M. Ookouchi and T. Makimoto, Maikuroha Sokutei (in Japanese). Tokyo, Japan: Ohm-sha, 1959. [3] A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of materials by time-domain techniques,” IEEE Trans. Instrum. Meas., vol. IM-19, no. 4, pp. 377–382, Nov. 1970. [4] 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. [5] H. B. Sequeira, “Extracting  and " of solids from one-port phasor network analyzer measurements,” IEEE Trans. Instrum. Meas., vol. 39, no. 4, pp. 621–627, Aug. 1990. [6] K. E. Dudeck and L. J. Buckley, “Dielectric material measurement of thin samples at millimeter wavelengths,” IEEE Trans. Instrum. Meas., vol. 41, no. 5, pp. 723–725, Oct. 1992. [7] R. Luebbers, “Effects of waveguide wall grooves used to hold samples for measurement of permittivity and permeability,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 11, pp. 1959–1964, Nov. 1993. [8] G. Mazé-Merceur and P. Naud, “Microwave characterization of magnetic materials under uniaxial and biaxial mechanical compressive stress,” IEEE Trans. Instrum. Meas., vol. 50, no. 3, pp. 742–748, Jun. 2001. [9] M. J. Havrilla and D. P. Nyquist, “Electromagnetic characterization of layered materials via direct and deembed methods,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 1, pp. 158–163, Jan. 2006. [10] L. P. Ligthart, “Fast computational technique for accurate permittivity determination using transmission line methods,” IEEE Trans. Microw. Theory Tech., vol. MTT-31, no. 3, pp. 249–163, Mar. 1983. [11] N. Marcuvitz, Waveguide Handbook. New York: McGraw-Hill, 1951, pp. 266–267. [12] E. D. Nielsen, “Scattering by a cylindrical post of complex permittivity in a waveguide,” IEEE Trans. Microw. Theory Tech., vol. MTT-17, no. 3, pp. 148–153, Mar. 1969. [13] K. Sarabandi, “A technique for dielectric measurement of cylindrical objects in a rectangular waveguide,” IEEE Trans. Instrum. Meas., vol. 43, no. 6, pp. 793–798, Dec. 1994. [14] Y. Kuriyama, N. Ueda, A. Nishikata, K. Fukunaga, S. Watanabe, and Y. Yamanaka, “Liquid material’s complex permittivity measurement using a rectangular waveguide and a dielectric tube at 800 and 900 MHz band,” in Proc. Int. Electromagn. Compat. Symp., Sendai, Japan, 2004, pp. 645–648.

NISHIKATA: SWEPT-FREQUENCY MEASUREMENT OF COMPLEX PERMITTIVITY AND COMPLEX PERMEABILITY OF COLUMNAR SPECIMEN

[15] N. Ueda, Y. Kuriyama, A. Nishikata, K. Fukunaga, S. Watanabe, and Y. Yamanaka, “Liquid material’s complex permittivity measurement using seamless dielectric tube and rectangular waveguide,” in Proc. Asia–Pacific Radio Sci. Conf., Qingdao, China, 2004, p. 537. [16] J. M. Jarem, J. B. Johnson, and W. S. Albritton, “Measuring the per-band using a partially mittivity and permeability of a sample at filled waveguide,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 12, pp. 2654–2667, Dec. 1995. [17] G. F. Engen and C. A. Hoer, “Thru-reflect-line: An improved techniques for calibrating the dual six-port automatic network analyzer,” IEEE Trans. Microw. Theory Tech., vol. MTT-27, no. 12, pp. 987–993, Dec. 1979. [18] R. E. Collin, Field Theory of Guided Waves, 2nd ed. New York: Wiley, 1991, pp. 78–86.

Ka

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Atsuhiro Nishikata (M’01) was born in Tokyo, Japan, on January 30, 1961. He received the B.Sc. degree in physics and M.Eng. and Dr.Eng. degrees in electrical and electronic engineering from the Tokyo Institute of Technology, Tokyo, Japan, in 1984, 1986, and 1989, respectively. From 1989 to 1993, he was with the Communications Research Laboratory, Tokyo, Japan. Since 1993, he has been with the Tokyo Institute of Technology. In 1995, he became an Associate Professor with the Center for Research and Development of Educational Technology, Tokyo Institute of Technology. Since 2004, he has also been a Short-Term Researcher with the National Institute of Information and Communications Technology, Tokyo, Japan. His main interests are in electromagnetic compatibility and learning environments. Dr. Nishikata is a member of the Institute of Electrical, Information and Communication Engineers (IEICE), Japan, and the Japan Society for Educational Technology (JSET).

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Phase and Amplitude Noise Analysis in Microwave Oscillators Using Nodal Harmonic Balance Sergio Sancho, Member, IEEE, Almudena Suárez, Senior Member, IEEE, and Franco Ramirez, Member, IEEE

Abstract—In this paper, a nodal harmonic balance (HB) formulation is presented for the phase and amplitude noise analysis of free-running oscillators. The implications of using different constraints in the resolution of the perturbed-oscillator equations are studied. The obtained formulation allows the prediction of the possible spectrum resonances without ill conditioning at low frequency offset from the carrier. The noise spectrum is meaningfully expressed in terms of the eigenvalues of a newly defined matrix, obtained from the linearization of the nodal HB system about the steady-state solution. The cases of real or complex-conjugate dominant eigenvalues are distinguished. The developed phase-noise formulation is extended to a system of two coupled oscillators. The phase and amplitude noise analyses have been applied to a push–push oscillator at 18 GHz, a bipolar oscillator at 1 GHz, and a coupled system of two field-effect transistor oscillators at 6 GHz. Index Terms—Microwave oscillator, nonlinear computer-aided design (CAD), phase noise.

I. INTRODUCTION HE theoretical phase-noise spectrum of an oscillator at the has, as predicted by Leeson’s model [1], a frequency characteristic pattern versus the frequency offset from the cardependence, close to the carrier, rier , evolving from a to the flat noise floor. However, additional resonances, at frequencies different from , may give rise to an irregular noise spectrum with local maxima [2]–[5]. In the frequency domain, these resonances can be predicted with the conversion-matrix approach [6]–[8]. However, the accurate determination of the close-to-carrier noise using this technique is demanding [7], [9], [10]. On the other hand, the time-domain formulation in [2] and [3], based on the introduction of a stochastic time in the perturbed-oscillator equations, enables the prediction of the noise-spectrum resonances, together with an accurate determination of the close-in noise. Due to the autonomy of the free-running oscillator, the obtained system has one degree of freedom, thus requiring an additional constraint for its practical solution. Depending on the selected constraint, different definitions for the phase and amplitude processes are obtained [2], [3]. As shown in [3], there is a particular constraint that provides uncoupled equations in the amplitude and phase processes. For these

T

Manuscript received January 10, 2007; revised April 16, 2007. This work was supported by the Spanish Ministry of Education and Science under Project TEC2005-08377-C03-01/TCM and under the Ramon y Cajal Program. The authors are with the Communications Engineering Department, University of Cantabria, Santander 39005, Spain (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.2007.900213

uncoupled time-domain equations, the additional-circuit resonances give rise to an increase of the amplitude process spectrum without affecting the phase process [3]. In [11], a formulation was presented, enabling the frequencydomain determination of the phase-noise sensitivity functions defined in [3], [12], and [13]. Piecewise harmonic balance (HB) was used for this derivation. As shown in [11], for moderate frequency offset from the carrier, the phase-noise formulation [11] gives similar results to the carrier-modulation approach [14]. However, this approach cannot predict additional resonances that can take place in the oscillator circuit [2]–[5], as it is based on the first-order Taylor series expansion of the HB equations about the free-running solution at , both in terms of the perturbations in the independent voltage/current variables and the oscillation frequency. The aim of this study is to develop a general noise-analysis formulation, based on nodal HB [15], capable of accurately predicting the phase and amplitude noise spectrum of the oscillator circuit, including the effect of any possible additional resonances. The circuit distributed elements are described by means of a modified convolution term, which allows an explicit analysis of the influence of these distributed elements on the oscillator noise. The equations in the amplitude and phase processes will be derived from the nodal HB formulation of the perturbed-oscillator system. A detailed study of the implications of using different constraints in the resolution of the perturbed-oscillator equations will also be presented. As will be shown, the phase processes obtained from different constraints agree when the time derivative of the amplitude process is neglected. It will be seen that, in a manner similar to [3], [12], and [13], when uncoupling the nodal HB equations in the amplitude and phase processes, the additional resonances only affect the amplitude spectrum. Under other conditions, the resulting processes will be coupled and the additional resonances will have an influence on the phase process, which justifies why these resonances are commonly obtained in the measurements. In the time-domain analysis, the noisy-oscillator spectrum is related to the Floquet multipliers of the periodic oscillation [3], [12]. Here it is shown that, in the frequency domain, it can be related to the eigenvalues of a newly defined matrix, obtained from the linearization of the nodal HB system about the steadystate solution. Here, these eigenvalues will be called the solution poles since, as will be shown, they agree with the roots of the Laplace transform of the linearized system impulse response. The power spectral density (PSD) of the perturbed solution due to the phase and amplitude processes will then be expressed in terms of the system poles. In the case of uncoupled phase and amplitude processes, the phase process is determined from the eigenvector associated to

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SANCHO et al.: PHASE AND AMPLITUDE NOISE ANALYSIS IN MICROWAVE OSCILLATORS USING NODAL HB

the pole , responsible for the singularity of the Jacobian matrix of the oscillator equation system. The amplitude process will be obtained from the rest of poles and their associated eigenvectors. Constraints not leading to uncoupled phase and amplitude processes will also be considered, deriving the corresponding noise spectra from the solution poles and conveniently defined projectors. Complex-conjugate poles near the imaginary axis will give rise to resonances in the noise spectrum. On the other hand, a negative real pole close to zero may be due to a high quality factor or to operation near singular conditions. While the former situation gives rise to a reduction of the oscillator phase noise, the latter provides a significant increase of this noise. This different behavior will be analytically justified. The noise formulation in terms of the solution poles will be illustrated by means of its application to a push–push oscillator and a bipolar-based Colpitts oscillator with different kinds of dominant poles. The formulation will be extended to the noise analysis of a system of coupled oscillators. The variation of the phase-noise spectral density along the operation band will be examined and analytically related to the dominant poles. This paper is organized as follows. In Section II, the nodal HB formulation for the phase and amplitude processes is presented. In Section III, the cases of uncoupled and coupled processes are analyzed, deriving the corresponding phase and amplitude noise spectra in terms of the eigenvalues of the steady-state solution. In Section IV, practical examples of the effect of the dominant poles are presented in a push–push oscillator and a bipolar oscillator, respectively. In Section V, the formulation is extended to the phase-noise analysis of a system of two coupled field-effect transistor (FET)-based oscillators. II. PHASE AND AMPLITUDE NOISE ANALYSIS BASED ON A NODAL HB FORMULATION A. Decomposition of the Nodal HB Formulation Into Phase and Amplitude Perturbations

contains all the white and colored noise where the vector sources. These noise sources are assumed of small amplitude and narrow band PSD [11] about each harmonic component , where is the oscillation frequency. Thus, the vector can be expressed in the following manner: (3) where and is the number of considered harmonic are the vectors of terms. The harmonic components narrowband stochastic processes. As proposed in [2] and [3], the solution in the presence of noise sources can be decomposed into phase and amplitude perturbations. The harmonic equation of this decomposition takes the form (4) contains the harmonics of the unperturbed steadywhere state solution . The variable is the common phase deviation for all the circuit variables affected by the harmonic then constitutes order at the different harmonic terms. the phase process of the oscillator, which provides the instantaneous frequency perturbation (5) The instantaneous frequency perturbation is modu. Since these sources are assumed lated by the noise sources to be narrowband around the harmonic components, the phase process can be considered as slow varying (see Appendix A). contains the perturbations of the The complex vector different harmonic terms and constitutes the so-called amplitude process. It gives rise to an additional phase deviation that will depend on the particular variable and harmonic term. Due to the low power of the noise sources, and assuming that the probability of noise “spikes” is low, the amplitude process will fulfill the condition

The modified nodal equation describing the dynamics of a microwave oscillator in the presence of noise sources is

(1) is the -dimension vector of state variables comwhere conposed by the node voltages and inductor currents, tains the sums of resistive currents (that enter each node) and are the linear and nonlinear charges and loop voltages, are the bias sources, and is the fluxes of the circuit, modified nodal impulse matrix of the distributed elements. The represent the first-order system rematrices sponse to the white and colored noise sources, respectively. For compactness of the formulation, the terms of (1) involving the noise sources will be included in a unique term (2)

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(6) The phase and amplitude perturbation variables and can be seen as a collection of random variables indexed versus the time variable . These variables are then stochastic processes, whose variation is ruled by a differential equation with noise perturbations. Following the notation used in [2], [3], [12], and [13], these variables will be called the “phase and amplitude processes.” B. Time-Frequency Formulation of the Perturbed System Next, the phase and amplitude processes of (4) are grouped in a single vector of time-varying harmonics. This allows the use of a time-frequency formulation of the modified nodal equation (1) (see Appendix B)

(7)

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where is the vector containing the time-varying harmonics of the state variables and the rest of vectors contain the timevarying harmonic components of the corresponding variables equations in (1). The nonlinear system (7) contains unknowns, given by the components of the vector in of time-varying harmonics. is the Toeplitz matrix correOn the other hand, sponding to the time-domain matrix . This matrix, con, is used in (7) to taining the time-varying harmonics of perform the frequency-domain convolution corresponding to the of time-domain product of (2). The harmonic contribution the distributed elements can be expressed in the frequency domain using the convolution theorem

(8) is The bandwidth of the frequency domain components limited to the maximum offset frequency considered in the analysis. As this value is relatively low compared to the oscillator fundamental frequency, the transfer function can be approached by a first-order Taylor-series expansion. The subindex stands for the derivative with respect to the frequency. As shown in the envelope-transient formulation of [16] in the case of broader band, it is possible to consider a higher order in the Taylor-series expansion of the transfer functions. Using (8), the convolution term of (7) is given by

of harmonics of the Equation (10) relates the vector and the amplitude instantaneous frequency perturbation to the time-varying harmonics of the noise process sources. Note that the steady-state terms do not appear in the above equation due to their inherent cancellation. On the other hand, the higher order terms in the different increments have been neglected. In order to express (10) in a compact manner, the following generalized Jacobian function will be defined: (11) which, due to the use of the nodal HB formulation, shows an associated explicit dependence on the transfer functions to the distributed elements. When particularized to the steadystate regime, function (11) becomes

(12) which agrees with the Jacobian matrix of the nodal HB equations. This constant matrix will be used to obtain a compact equation of the system (10). The frequency derivative of the Jacobian function (11) at the steady-state solution will also be necessary, which is given by (13) Using (12) and (13), the perturbation system (10) can be expressed as

(9) The explicit introduction of (9) into (7) will allow a more direct evaluation of the effect of the distributed elements on the oscillator noise. Equation (7) will now be used to obtain an explicit system on the amplitude and phase processes. For this purpose, the in (7) are expressed in terms of the time-varying harmonics amplitude and phase processes (see Appendix B). A linearization of (7) can be carried out in the amplitude process, which is considered of small amplitude. Introducing the Fourier expan(see Appendices A and B) into (7) and taking into sion of account the orthogonality of the Fourier series, the following system is obtained:

(10) being a diagonal matrix used to mulwith tiply each component of the vector of spectral components by , and being the Toeplitz matrix of the vector , which contains the harmonics of . and represent the freOn the other hand, quency-domain Jacobian matrices of and , evaluated at the steady-state solution .

(14) . In the following, the equations with in the phase and amplitude processes will be derived from the above compact system. C. Equations for the Amplitude and Phase Processes The amplitude and phase processes will be expressed in terms of the oscillator-solution poles. These poles are the roots of the Laplace transform of the linearized system impulse response (see Appendix C). As will be derived below, these poles can be obtained from the eigenvalues of a newly defined matrix, obtained from the linearization of the nodal HB system about the steady-state solution. The solution poles agree with the roots of the characteristic determinant (15) In the case of a circuit containing lumped elements only, this characteristic determinant takes the form (16) In the more general case of a circuit containing lumped and distributed elements, the first-order Taylor-series expansion (9)

SANCHO et al.: PHASE AND AMPLITUDE NOISE ANALYSIS IN MICROWAVE OSCILLATORS USING NODAL HB

of the distributed-element transfer functions, in terms of the perturbation frequency, must be used, which leads to an equation of the same form (16). From the inspection of (11), it is seen associated to that the characteristic matrix the nodal HB formulation is linear in the parameter , which reduces the pole calculation to a simple eigenvalue problem. Indeed, considering (16), the system poles clearly agree with the . Taking into account eigenvalues of the matrix the Toeplitz structure of the Jacobian matrices (12) and (13), the that can be expressed system poles [17] are given by a set as (17) where is the number of considered harmonic terms. Associand that ated to each pole , there are a pair of vectors fulfill the following equations: (a) (b) (18) where the symbol indicates conjugate transpose. Each set of and can be grouped in a unique vectors matrix as

(19) and have Toeplitz strucwhere the so-defined matrices ture. From (18) and expanding the Jacobian function as in (16), the following algebraic condition can be derived [18]: (20) is the identity matrix and is the Kronecker delta. where Due to the autonomy of the oscillator, the Jacobian matrix defined in (12) is singular and, therefore, the system has, at . The associated right eigenvector is least, one pole . This vector corresponds to the direction for which the oscillator solution is invariant. Thus, par, it is possible to write ticularizing (18) to the pole (21) , asIn turn, the left eigenvector of the Jacobian matrix , will be denoted as , fulfilling sociated to (22) is normalized by the module The module of the vector through the condition (20). The vectors will of be used to obtain different projections of phase and amplitude perturbations. The left multiplication of the system (14) by the matrix provides the following equation for the harmonics of the instantaneous frequency perturbation

(23)

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is mainly due to As already stated, the phase deviation the dc harmonic of the frequency perturbation. From (23), the equation corresponding to this harmonic is (24) Next, a closed-form equation will also be derived for the am. Following a procedure similar to that plitude process of the time-domain analysis [3], the following operator is defined to eliminate the frequency perturbation from system (14) (25) The left multiplication of system (14) by this projector leads to the following differential equation in the amplitude perturbation:

(26) Due to its dependence on the time derivatives , the above equation for the amplitude noise is not quasi-stationary, which, as will be shown, will allow the prediction of any possible additional resonances in the noise spectrum. equations in The system (14) contains unknowns. The unbalance is due to the in the set of presence of the common phase process unknowns. For the practical resolution of the subsystems (23) and (26) in the frequency domain, it is necessary to introduce additional equations, one for each harmonic of . Here, the constraint will be imposed in terms of the vector only. Therefore, as gathered from (24), the phase processes resulting from different constraints will only be the same in the case of a negligible amplitude of the time derivative . When this term is not neglected, each arbitrarily chosen constraint will give rise to a different value of the phase and . In Section III, the conamplitude processes tribution of the phase and amplitude processes to the oscillator phase and amplitude noise spectra will be analyzed in detail. III. DERIVATION OF THE OSCILLATOR PHASE AND AMPLITUDE NOISE SPECTRA The phase and amplitude processes determine the phase and . amplitude noise observed in a measured circuit variable , it Using the phase and amplitude processes will be possible to separate the actual real-valued amplitude perand phase perturbation of each harturbation monic term and express the signal as [19]–[21]

(27) and are the amplitude and phase of each where steady-state harmonic , respectively. Equation (27) shows that the so-defined phase and amplitude noise depend, in general, on the selected variable and harmonic component. These phase and amplitude noise components can be obtained in terms of the phase and by equating (27) and (4). amplitude processes

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All the possible definitions of and the common phase will give rise to the same amplitude and perturbation and . It can be easily phase-noise components derived that, while the amplitude noise is determined depends by the amplitude process, the phase noise both on the phase and amplitude processes. The phase noise and amplitude noise at a given node and harmonic component are then coupled processes. While for is common each particular constraint the phase process will depend on the chosen variable to the entire circuit, comprises and harmonic component. This is because the contributions of both the common phase and the . Since the phasor corresponding phasor is different at the different harmonics and nodes, so will be the total phase noise. This is why the measured phase noise provides slightly different results when measured at different nodes of the same circuit. As an additional comment, the conversion-matrix approach obtains the complex sidebands due to the noise perturbation, which are then transformed to the total amplitude and phase-noise spectra [6]. Therefore, the simulated phase-noise spectrum will be slightly different at the different circuit nodes. The carrier-modulation approach provides the obtained when the time variation common phase noise of the amplitude perturbations is neglected. The different constraints to be used in the resolution of (23) and (26) will give rise to a different distribution of the total phase-noise component between the common phase process and the component of the amplitude process. In the following, two different resolutions of the perturbed system (23)–(26) will be considered. In the first case, the used constraint leads to uncoupled phase and amplitude processes. In the second case, the oscillator phase and amplitude remain coupled in the balanced equations. A. Uncoupled Phase and Amplitude Processes The first considered constraint will lead to uncoupled amplitude and phase-noise processes, thus providing a good insight into the nature and characteristic of the common phase deviation of the oscillator circuit. The decoupling of the amplitude from and phase processes requires the elimination of (23), which is achieved by imposing (28) As gathered from (23), this is the only constraint that gives not affected by the amplitude rise to a phase process process. Note that the condition (28) provides a set of equations, one for each harmonic component. In particular, the central term of the matrix (28) provides the following relationship:

lies in the hyperplane orthogonal to the vector , belongs to this provided the vector of initial conditions . plane. The amplitude perturbation will then be denoted By condition (20), the hyperplane corresponds to the sub. It is important to note space spanned by the vectors is not orthogonal in general, the that since the basis is not orthogonal to the vector amplitude perturbation . , there will be no conWith this particular choice for version of amplitude noise into phase noise. Indeed, taking (28) into account, (23) simplifies to an uncoupled equation for the harmonics of the frequency perturbation (31) Equation (31) can be particularized to obtain the dc harmonic of the frequency perturbation, which mainly determines the phase process

(32) where and are the vectors of time-varying harmonics of the white and colored noise sources and and are the Toeplitz matrices corresponding to the first-order system response to these noise sources. The above equation allows a frequency-domain determination of the phase deviation obtained by Kaertner in [3]. Equation (32) shows that the phase perturbation grows unboundedly when the average of the right-hand term is different from zero. This is in correspondence with the time-domain analysis [3], [12] and with the carrier modulation approach [14] in the frequency domain. can be Now, the equation for the amplitude process obtained from (26) making use of the operator , defined in (25). This operator projects the noise perturbation onto the hylies perplane , in which the amplitude perturbation as a result of the constraint (28). Applying the constraint (28) to the amplitude process in (26), the dynamics of the amplitude will then be ruled by process (33) The next step will be the derivation of the noisy oscillator spectrum from the presented decomposition into phase and amplitude perturbations. The PSD at the frequency is obtained from (34) with the different matrices being given by

(29) where

. Integrating the preceding equation, (35) (30)

is not affected by the time inNote that the constant vector tegration. Condition (30) assures that the harmonic perturbation

where the operator represents the Fourier transform, and represents ensemble averaging. For this derivation, white and uncorrelated colored noise sources will be considered.

SANCHO et al.: PHASE AND AMPLITUDE NOISE ANALYSIS IN MICROWAVE OSCILLATORS USING NODAL HB

In the following, analytical equations will be obtained for the terms introduced in (35) considering the definition of the phase and amplitude processes given earlier in this section. These analytical equations will be based on the presented nodal HB formulation. 1) Phase Spectrum: The starting point for the derivation of the oscillator spectrum due to phase noise will be the nonlinear differential equation for the phase perturbation, obtained in the time-domain analysis [3], [13]. In these studies, the phase perturbation is given by

(36) where the vectors and are periodic with period . The instantaneous frequency perturbation calculated through (36) agrees with the one defined in (5), whose dc component is provided by (32). The oscillator spectrum due to the phase process is written as follows [3], [13]: (37) where the phase-noise spectrum at each harmonic lowing Lorentzian shape:

has the fol-

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and are the vectors of time-varying where harmonics of the white and colored noise sources. Comparing and (41) with the derived equation (32), the coefficients can be obtained by doing

(42) where is the left eigenvector associated to the pole , which can be calculated from (22). Therefore, from (42), it and is possible to easily obtain the sensitivity coefficients that determine the oscillator spectrum (38) due to the phase process. 2) Amplitude Spectrum: Here, a novel approach for the amplitude spectrum in the frequency domain will be presented in terms of the system poles. As stated in Section II-C, when using the nodal HB formulation, the calculation of these poles is reduced to a simple eigenvalue problem. By using the equations provided here, the possible resonances in the amplitude spectrum will then be easily characterized. Considering the same noise sources as in Section III-A.1, the oscillator spectrum due to amplitude noise is determined from system (33). The autocorrelation function of the amplitude can be directly obtained from this equation by process using the property [22] (43)

(38) with being the PSD of the th colored noise source. The and are calculated phase-noise sensitivity coefficients as

is the impulse response matrix of the linear time where invariant (LTI) system associated to (33) and the operator denotes convolution. The Laplace transform of the impulse response is obtained from the system (33) as (44) with (45)

(39)

can be factorized using the vectors (18), The matrix taking into account the property (20) [18]

where is the autocorrelation matrix of the white noise sources. Here it is taken into account that the above integrals can be di, rectly obtained from the vector of harmonics of the vector , and the dc component of , given by , i.e., given by

(46)

(40) The nodal HB formulation presented in Section II can be used to obtain the noise sensitivity coefficients and . Expressing the time domain equation (36) in terms of the harmonic components , , the dc component of the instantaneous frequency perturbation (5) is given by

(41)

where the terms corresponding to are cancelled by the projector due to the property (20). This leads to a time-doin terms of the main equation of the impulse response system poles (47) As gathered from (47), the correlation time of the amplitude process will be determined by the absolute value of the of the dominant poles , as . real part The phase process changes slowly with respect to so

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it can be considered invariant in the range of for which the Fourier integral is calculated in (35). This leads to the following approach for the amplitude spectrum:

(48) where the amplitude noise sensitivity matrices are given by

and

(49) To obtain the amplitude spectrum at the output node, the corresponding voltage variable should be selected in (48). The amplitude spectrum (48) is determined by the sensitivity matrices and . As can be seen in (45), these matrices contain the frequency-dependent Jacobian matrix. This frequency dependence will allow the prediction of the possible resonances. Indeed, the factorization (46) provides the following equation of in terms of the system poles: the matrix

to the imaginary axis, the spectrum will be flat up to high fre. The preceding theory quency offset from each harmonic on the pole influence on the amplitude-noise spectrum is illustrated in Section IV by means of its application to a push–push oscillator and a bipolar oscillator. As stated in (32), the phase process does not depend on the amplitude process and, therefore, the possible resonances of the amplitude spectrum will not affect the phase spectrum. On the in (33) and (46) is to other hand, the aim of the projector so that the cancel the component in the direction of . Thus, amplitude process is not affected by the pole this projector prevents the amplitude perturbation from growing unbounded as the frequency offset tends to zero. This also avoids any accuracy degradation of (48) near the carrier. Note that, prior to this work, the frequency-domain prediction of resonances at large frequency offset from the carrier was only possible with the conversion-matrix approach [6]–[8]. However, the linearization used in this approach tends to a singularity condition at small frequency offset from the carrier [10] and is highly dependent on the accuracy in the calculation of the steady-state oscillation [7]. In contrast, the method proposed here does not suffer from any ill-conditioning problem at small frequency offset. This is because the solution autonomy has been taken into account when deriving the perturbed-system equations and the singularity has been removed when using the additional constraint (28). B. Coupled Phase and Amplitude Processes. Phase and Amplitude Noise

(50) with the scalar functions

being given by (51)

Equations (48)–(50), in terms of the solution poles, provide great insight into the amplitude-noise behavior. From the inspection of (50) and (51), the amplitude spectrum can be deterof system poles and their associmined by the subset ated residue matrices . The poles with small magnitude of the real part give rise to resonances in the amplitude spec, trum. For a pair of complex conjugate poles and the magnitude of the corresponding functions undergoes a resonance at the frequency of the poles due to the small magnitude of . Considering all the values of in the summation (50), it is seen that this resonance will take from each carrier . As gathplace at a frequency offset ered from (51), the resonance will be narrower and higher for a smaller magnitude of . For a frequency offset from each car, the noise contribution of will rier decrease as 40 dB/dec. Note that if the magnitude of is relatively high, there is no resonance and the frequency offset from which the amplitude noise starts to drop is given by . The noise floor will then be determined by the poles . This frequency may be outside with the largest value of the offset-frequency range with interest to the designer. If the frequency of the dominant poles is high or no poles exist close

As stated in Section II-C, expressing the perturbed solution in terms of the variables and leads to one degree of freedom in the perturbation system (14). Thus, the use of an additional constraint is required for the practical resolution of this system. From the inspection of (23), it is clear that the constraint (28) is the only one that provides uncoupled phase and amplitude processes. When using any other constraint, the phase process will then be affected by the amplitude process. In some time-domain studies [2], [23], the condition has been used, with . This vector is clearly tangent to the limit cycle in the phase space so the mentioned condition implies that the amplitude perturbation is, by definition, orthogonal to the cycle. In the frequency domain, this condition leads to the following linear system of equations: (52) contains the harmonics of the vector and is where the Toeplitz matrix associated to this vector. The constraint (52) to the hyperplane restricts the amplitude perturbation orthogonal to and, thus, will be denoted here as . By condition (20), this hyperplane agrees with the subspace gener. ated by the vectors The constraint (52) will now be used in the resolution of the perturbation system (14), containing one degree of freedom. The procedure will be similar to the one followed in Section III-A. In this case, the amplitude term is not cancelled in the harmonic equation (23), and the equation for the lowest

SANCHO et al.: PHASE AND AMPLITUDE NOISE ANALYSIS IN MICROWAVE OSCILLATORS USING NODAL HB

harmonic component of the frequency perturbation takes the form (53) As gathered from the above equation, in this case, there is conversion of amplitude noise into phase noise. This amplitude conversion is determined by the vector , defined in (29). Due to this, resonances generated in the amplitude process will be transferred to the phase noise. Note that when the time variation of the amplitude perturbations is neglected, the equation for the phase process is the same for the two different conis orstraints (28) and (52). Since by definition (52) thogonal to , the more parallel is to , the smaller the influence of the amplitude term. to , the Taking advantage of the orthogonality of equation for the amplitude deviation is obtained from system (14) through the application of the projector . For condition (52), this projector takes the form (54)

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these resonances do not affect the phase noise. In absence of resonances, the phase-noise spectrum obtained when applying the constraints (28) and (52) is very similar, due to the small value of the amplitude perturbation. In the following, tools for the prediction of the existence and location of these resonances will be provided. Note that, for high frequency offset , the component is highly attenuated. The amplitude conversion terms will then dominate (57). As stated in Section III-A.2, in the case there are no poles located close to the imaginary axis, the amplitude noise will be flat versus the offset frequency. Thus, in the case of coupled amplitude and phase processes, the phase-noise spectrum will also become flat versus the offset frequency, in agreement with the predictions of Leeson’s formula and the results of the conversion-matrix approach [7], [14]. 2) Amplitude Spectrum: In order to obtain the amplitude , the LTI system associated to (56) is noise spectrum analyzed following the same procedure as in Section III-A.2. Due to the explicit dependence on the projector in the LTI system associated to (56), the matrix must be modified in the following way:

with (55) The left multiplication of system (14) by this projector provides the following equation in the amplitude perturbation:

(59)

(56)

with and with being the set of system poles. It must be noted that the terms corresponding are cancelled by the projector defined in (54). The to same formal equations (48) and (49) then provide the ampliis the same tude-noise spectrum. Although the set of poles for both (50) and (59), the associated matrices of residues are formally different and their values can slightly vary depending on the used constraint (28) or (52). Equation (59) shows that when using constraint (52), the amplitude-noise spectrum exhibits the same resonances as in (50). These resonances can be easily predicted by obtaining the system poles through (16). Due to the amplitude to phase-noise conversion in (57), these resonances will also be observed on the phase-noise spectrum, altering the Lorentzian-shaped characteristic.

must be combined The above equation in terms of with (53) in order to obtain the phase and amplitude noise spectrum. In comparison with the results of Section III-A, a different distribution of the spectrum into phase and amplitude components will be obtained with the same total power of the oscillator spectrum. 1) Phase Spectrum: From (53), it is seen that the phase-noise contribution to the oscillator PSD takes the following form: (57) The component of (57) corresponds to the Lorentzianshaped spectrum due to phase noise, obtained from the first term of (53). This term agrees with the spectrum (37) and (38) obtained by using the previous constraint (28). The terms and correspond, respectively, to the direct amplitude conversion into phase noise and a correlation term between amplitude and phase perturbations. In order to analyze the influence of the amplitude process on the phase-noise PSD, the equation for the direct amplitude conversion contribution is obtained from (53) (58) Equation (58) shows that when using the constraint (52), the will possible resonances in the amplitude noise PSD be projected to the phase-noise PSD through the vector . Note that, as shown in Section III-A.1, when using the constraint (28),

IV. ANALYSIS OF THE INFLUENCE OF THE SYSTEM POLES ON THE NOISE SPECTRUM As already discussed, resonance frequencies are likely to be observed in the spectrum of solutions having poles with small real part , which is equivalent to a small stability margin. Note that this does not imply an actual unstabilization in the parameter ranges of the circuit operation. In the following, the cases of complex-conjugate dominant poles and a real dominant pole will be treated. For notation convenience, the phase-noise PSD when obtained using the constraint will be denoted as (28), and as when obtained using the constraint (52). A. Complex-Conjugate Dominant Poles The push–push oscillator analyzed in [24] will be considered. The circuit is constituted by two oscillators at 9 GHz con-

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TABLE I FIRST FOUR POLES WITH HIGHEST RESIDUE ASSOCIATED TO THE LINEARIZED SYSTEM ABOUT THE STEADY-STATE SOLUTION FOR V = 3 :5 V

Fig. 1. Schematic of the bipolar-based push–push oscillator with 18-GHz output frequency. The circuit has been implemented on a plastic substrate CuClad 2.17 (" = 2:17 and h = 0:254 mm) using specific microwave drilling tools instead of chemical processes, in order to match the physical dimensions of the lines. The tuning network uses an M/A (ML46450) varactor diode in series with a microstrip line (equivalent to L = 0:6 nH) and a resistor R = 1:5 .

nected by a microstrip coupler, oscillating out of phase (see Fig. 1). The output signal at the double frequency of 18 GHz is obtained with a Wilkinson combiner. Each sub-oscillator contains an NPN bipolar transistor BFP405 Infineon, which has GHz. For the simulations, a transition frequency of the Gummel–Poon model has been used, in combination with several noise sources [25]. The flicker noise is modeled with a voltage source in series with the internal base terminal. The Hz. The shot spectral density of this source is 9 10 noise is modeled with a current source of spectral density with being the electron charge, which is connected between the base and emitter terminals. Thermal noise generators also have been added to all of the resistive elements. The circuit is intended to operate as a voltage-controlled oscillator, containing, for this purpose, a varactor diode in each sub-oscillator, biased . with a voltage . The circuit is analyzed versus the varactor bias voltage is determined with HB The steady-state solution using an additional technique [26], [27] to avoid the default HB convergence to the unstable dc solution that coexists with the oscillation. In order to illustrate the influence of the system poles in the output signal spectrum, the matrix summations (50) and (59) have been particularized to the fundamental frequency of the output voltage variable, obtaining the following scalar function:

(60) have been neglected due to their where the terms for small influence on the above function for the considered freagree with the term quency range. The scalar coefficients of the residue matrices corresponding to the first harmonic of the output voltage variable. Although all poles 2 to contribute

to the amplitude noise, in order to illustrate the spectrum resonance, only some of the poles will be considered. The bias V was initially selected. For this voltage voltage value, the solution poles have been obtained. As was stated in Section II-C, these poles agree with the eigenvalues of the ma. Once the poles are obtained, the associtrix and are calculated ated eigenvectors using (18) and normalized by condition (20). The residues are then determined from the obtained eigenvectors. The first four poles with highest residue are shown in Table I. No poles are particularly close to the imaginary axis for this bias value. The two poles with highest residue in (60) are real: s and s . The next two s and s . poles are Fig. 2(a) presents an evaluation of the individual contributions in the summation (60), where the of the terms magnitude of each component has been traced versus the offset . For this calculation, constraint (52) has been frequency used, although considering constraint (28) leads to similar results. For this bias voltage, the real poles do not give rise to resonance, and the level of each component depends both on the pole magnitude and the associated residue. The respective contributions approximately drop 20 dB per decade from a frequency offset agreeing with the magnitude of the real pole. The phase noise has been calculated using both the constraint ] and the constraint (52) [denoted ]. (28) [denoted has also been deterThe amplitude noise spectrum mined in order to evaluate the effect of the amplitude to phase. The results are shown in Fig. 3. As noise conversion in V, the amplitude noise spectrum can be seen, for is flat versus the frequency offset, and the phase and amplitude noise spectra obtained with both definitions are nearly the same. The phase-noise PSD exhibits a transition from 30 dB/dec near the carrier to a 20-dB/dec slope, corresponding to the oscillator response to the flicker and white noise sources. V, the real As the bias voltage decreases from poles , merge and split into two complex conjugate poles. V, these complex poles take For the bias voltage s . Note the value that they are remarkably closer to the axis than the formerly V. Although their assoreal poles obtained for and do not noticeably ciate residues change, the poles vary under this bias-voltage reduction. In Fig. 2(b), the magnihas been traced versus tude of each component the offset frequency - . The components corresponding to the pair of complex-conjugate poles with small real-part magnitude give rise to a well-distinguished resonance at the frequency . From this frequency value, the contribution

SANCHO et al.: PHASE AND AMPLITUDE NOISE ANALYSIS IN MICROWAVE OSCILLATORS USING NODAL HB

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Fig. 3. Phase and amplitude noise of the push–push oscillator. Simulations = 3:5 V, the using both the constraint (28) and (52) are presented. For V = 2:7 V, the amplinoise spectra are nearly identical in both cases. For V tude noise undergoes an amplification at the frequency offset f = 2 MHz. The spectrum S ( ) does not suffer any significant change, whereas S ( ) undergoes an amplification due to the amplitude to phase conversion. Measurements are superimposed for V = 2:7 V.

the other port of the mixer. The phase detector then converts the phase fluctuations of the input signal into their voltage equivalent for measurement. Simplifying, the output of the delay-line frequency discriminator may be approximated as (61)

Fig. 2. Evaluation of the most relevant individual contributions to the summa= 3:5 V. No poles are particularly close to the imaginary tion (60). (a) V axis. The two poles with highest residue are real:  = 2 5 10 s and  = 2 1 10 s . (b) V = 2:7 V. The complex-conjugate poles  = 2 0:01 10 j 2p2 10 s are close to the imaginary axis.

0 1 1 1 1

0 1 1

6

1

starts to drop 40 dB per decade. As in the case of real poles, the relevance of the contribution to the summation depends on the pole magnitude and the associated residue. and and The corresponding phase-noise spectra the amplitude noise have also been determined and represented in Fig. 3. As can be seen, the amplitude noise exMHz. hibits a local maximum at the frequency offset is very When using the constraint (28), the phase noise similar to the one obtained for V. On the other hand, underwhen the constraint (52) is used, the phase noise MHz due to the amplitude to goes amplification about phase-noise conversion. The noise analysis results have been experimentally validated. For the phase-noise measurement, a delay line plus discriminator system has been used. In this implementation, the oscillator signal is split in two channels. One channel is applied directly to one port of a double-balanced mixer, which will operate as a phase detector. The other channel is delayed through some delay element, here implemented as a low-loss coaxial cable and a variable phase shifter, and then input to

(V/rad) is the sensitivity of the phase detector, which where is dependent of the amplitude of the input signals and mixer is the phase difference introduced by the delay loss, and of the network. The phase difference between the two channels can also be approximated as (62) The measurement setup has been calibrated by evaluating the response of the system [delay line network, mixer, low-pass filter, and low-noise amplifier (LNA)] to a single FM tone with a known sideband/carrier ratio. Once the quadrature is established, the spectral density of the voltage fluctuations are measured on the baseband analyzer. For the amplitude-noise measurement, the amplitude-noise fluctuations are converted to baseband fluctuations with a diode detector. Once more, a calibration is needed before the oscillator under test is connected to the setup. The reference level is established by using an AM signal with a known level of sideband/carrier ratio. After this, and applying some corrections for the spectrum analyzer effects, the amplitude noise measurement is straightforward. measured at the output node The phase noise (Fig. 1) will contain contributions from both the phase and amplitude processes. Therefore, as explained in Section III-B, the resonances will be observed both in the measured phase and amplitude noise. This can be seen in Fig. 3, showing the results of applying the phase and amplitude noise characterization

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is no significant variation of the amplitude noise versus the frequency offset, and the phase noise obtained through both definitions is nearly identical. When the resistance value is increased , relatively close to the bifurcation, amplitude to . noise amplification is obtained about the frequency offset When using the constraint (52), conversion from amplitude to . phase noise is observed in the phase-noise PSD C. Real Poles

Fig. 4. Schematic of the Colpitts oscillator. L = 8 nH, R 1:5 pF, C = 10 pF, C = 100 pF, V = 5 V, and V

= 300 , = 05 V.

C

=

The small amplitude value of the dominant real pole may be due to two different causes: a very high value of the oscillator quality factor or the circuit operation near a singular point. The latter case is typically obtained at turning points (or infinite slope) of the oscillator solution curve versus a parameter. At the turning point, a real pole crosses the imaginary axis through zero. This can be associated to hysteresis, as in the case of a voltage-controlled oscillator, or to loss of synchronization, e.g., in the case of two coupled oscillators. To derive the turning-point condition, the steady-state solution curve of system (14) versus a parameter in the absence of noise perturbations will be considered. For a small increment , the corresponding variation of the steady-state solution can be approached as follows: (63) where the vector represents the Jacobian of the HB system with respect to the circuit parameter . The left multithen provides the plication of (63) by the projection vector frequency variation (64)

Fig. 5. Phase and amplitude noise of the Colpitts oscillator. For the simulations, both constraints (28) and (52) have been used. For R = 20 , the phase-noise spectrum resulting from both definitions is nearly identical. For R = 30 , the amplitude noise undergoes amplification about the frequency offset f =2. The spectrum S ( ) undergoes an amplification due to the amplitude to phase conversion, while S ( ) does not exhibit any significant change. Measurements for R = 30 are superimposed.

to the push–push oscillator at V. Resonances are obtained in both the amplitude and phase spectra. B. Complex-Conjugate Dominant Poles at As an example of complex-conjugate dominant poles at , a bipolar-based Colpitts oscillator at GHz will be analyzed [28] (see Fig. 4). The same bipolar transistor as in the previous example has been used. When increasing the resistance , this oscillator undergoes a frequency division by two at the bifurcation value , this corresponding to a pair of crossing complex conjugate poles at the divided frequency the imaginary axis [29]. Fig. 5 shows the oscillator noise spectrum for two different . For the value , there values of the resistance

As the turning point is approached, the slope tends to infinity [27]. From (64), this means that the module tends to infinity at the bifurcation. From the definitions (42) of the and , it is derived that, as the real pole coefficients approaches the origin of the complex plane, the phase noise significantly increases. This amplification, associated to the large value of , can be equally predicted with the phase processes that result from constraints (28) and (52). This phenomenon will be observed in Section V in the case of a system of coupled oscillators. Note that the real pole with near-zero value will also give rise to an amplitude-noise increase, as it contributes to one of the matrix terms of summation (50). Due to the associated small value of the denominator magnitude, it will be a dominant contribution to the amplitude-noise spectrum. In the case of a high value of the oscillator quality factor, the has very small absolute value, but is not in danger of pole crossing the imaginary axis of the complex plane. The behavior is similar to that of stability center [30]. The high quality factor is associated to high frequency sensitivity, which in the nodal formulation (14), corresponds to a high value of the norm of . For convenience, the central term of the matrix the matrix (20), obtained for , will be considered here. This term takes the value (65)

SANCHO et al.: PHASE AND AMPLITUDE NOISE ANALYSIS IN MICROWAVE OSCILLATORS USING NODAL HB

From the above relationship, as the norm of the matrix increases, which means a higher quality factor, the module of must decrease since, in general, neither , nor the vector are eigenvectors of [18]. From (42), a higher quality factor then gives rise to a smaller values of the noise-sensiand , thus improving the phase-noise tivity coefficients spectrum. V. PHASE-NOISE CALCULATION IN A SYSTEM OF TWO COUPLED OSCILLATORS Here, the nodal HB formulation for the phase-noise analysis will be extended to a system of two coupled oscillators. As will be shown, the formulation is easily extended to the general case oscillators. For illustration, the equations will be applied of to two coupled FET-based oscillators at 5 GHz. A. Phase-Noise Formulation

different sub-vectors of equal dimension being the considered number of harmonics

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with

(70) In the following, formulation (66)–(70) will be applied to the phase-noise analysis of a system of two identical oscillators. The possible coupling networks will be divided into two classes: without and with a path to ground. Networks containing transmission lines belong to the second class. 1) Coupling Network With no Path to Ground: A coupling network with no path to ground is, in fact, a passive branch of total admittance , connecting the output of the two oscillators [31]. This coupling network can be described with an admittance matrix of the form (71)

A periodic oscillation at the frequency in a coupled system of two oscillator elements will be assumed. The state vector contains the harmonic components of the state variables of the two individual oscillators (66) refers to each of the two oscilwhere the superindex lators. Considering the division (66) of the state vector, the Jacobian matrix (12) of the HB system, evaluated at the synchronized solution, takes the form

and are the vectors containing the harmonics of where is the vector containing the network-terminal voltages, and the harmonics of the current from the coupling network entering the oscillator . As will be shown in the following, the symmetrical structure of the admittance matrix (71) allows the analytical determination of the phase noise of the in-phase solution in terms of the phase noise of the two individual oscillators, assumed identical. Due to the structure of the matrix relationship (71), the coupling matrices in (67) must fulfill the symmetry condition

(72) (67) where and are the Jacobian matrices (12) associated to each of the two oscillators and the frequency-dependent marepresent the contribution of the coupling nettrices rework. Note that only the admittance elements of and to the lating the harmonics of the injected currents output-node voltages of the two oscillators are different from zero. The uncoupled and coupled terms of the Jacobian function (67) can be separated into two different matrices as (68) where the matrix contains the terms . Due to the symmetry of the coupled system, the Toeplitz matrices of the defined in (2) will system response to the noise sources take the following uncoupled structure: (69) and corresponds to the oscillator where each matrix and the zeros denotes zero matrices. Next, the projection vectors and of the coupled oscillator system will be determined. For convenience, the vectors and will be divided into two

For the in-phase solution, the two state vectors satisfy . The node voltages are then equal , and by (71), . In spite of the zero value of these steady-state currents, there will be noise current flowing through the coupling network. This is because the circuit is necessarily unbalanced with respect to the noise perturbations. Thus, the system is actually coupled with respect to the noise perturbations. By introducing the equalities (72) into the HB equation of the and coupled-oscillator system, it is found that with and being the vectors of harmonics of the free-running solution and time derivative of each individual oscillator, respectively. On the other and of the in-phase sohand, the Jacobian matrices lution are equal and so are their frequency derivatives with being the frequency derivative of each individual Jacobian matrix, evaluated at the free-running solution. Next, the projection vector , determining the phase-noise sensitivity coefficients, will be obtained from all the preceding equalities. Using the property (20) and (67), the following normalization condition is derived for : (73) Note that the frequency derivative of the matrix term is cancelled in (73) due to the equalities (72). Using the property

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(20) again, the normalization condition for each free-running oscillator is given by (74) where is the projection vector corresponding to each individual free-running oscillator. Comparing (73) and (74), the following equation is then obtained for the vector :

insight into the problem, the line dispersion is neglected. The in (68) can then be expressed in terms coupling matrix . The Jacobian of of the delay as the coupled system (67) is periodic in the variable , while the can be expressed using the defrequency derivative matrix composition (68) as

(75)

(78)

For the derivation of the phase-noise spectrum, the correlation matrices of the noise sources of both oscillators will be assumed identical. Inserting (75) into (42), and taking the uncoupled structure of the noise-response matrices (69) into account, the following equations are then derived, after some algebraic manipulation, for the phase-noise sensitivity coefficients:

being the electrical length of the coupling line. The with second term of (78) indicates that the norm of the matrix follows an oscillatory pattern of growing amplitude versus the is periodic in . Now, time delay since the matrix can be written as a function of the time the projection vector delay , using (18) and (20) as follows: (a)

(76) (b) where and are, respectively, the sensitivity coefficients of the coupled oscillator system, associated to the white and colored noise sources included in each oscillator . The phase-noise spectrum of the system of coupled oscillators at each harmonic component is then obtained by combining (37) and (38) with (76)

(77) is the phase-noise PSD of the individual free-runwhere ning oscillator. Extending this analysis to coupled oscillators is straightforward. The resulting phase-noise reduction of the dB, which is in agreement with in-phase solution is [32]. 2) Coupling Network With Path to Ground: In the case of an in-phase solution obtained with a coupling network having a path to ground, the load seen by each oscillator is different from the one seen in free-running conditions. Therefore, the curentering the coupling network are different from rents zero and the symmetry conditions (71) and (72) are not fulfilled in general. The projection vector of the coupled system then has to be numerically obtained from the Jacobian matrix (67) following (22). Provided the coupling network has a high will have input impedance, the elements of the matrices will only slightly differ from small amplitude and the vector (75). The phase-noise reduction for the in-phase solution will or in the case of coupled osthen approach cillators. In other cases, the phase-noise improvement with re, spect to the free-running solution could be higher than provided the network parameters are suitably customized. To particularize the study, a coupling network containing a transmission line will be considered, as this is a usual situation in coupled-oscillator systems. The phase-noise dependence with the transmission-line length will be analyzed by means of the presented formulation. To get manageable equations and better

(79)

is periodic in the variwhere the vector able . Due to the aforementioned dependence of the norm of the maon the variable , in order to fulfill (79) (b), the norm trix will exhibit an oscillatory variation versus the delay with minima proportional to . This result is in correspondence with the phase-noise analysis in [33]. From the definiand , the tions (42) of the noise-sensitivity coefficients . phase-noise spectrum must follow the same pattern as The validity of this derivation will be verified in the following practical example. B. Application to a Coupled-System of Two FET-Based Oscillators The noise-analysis technique has been applied to a coupled system of two FET-based oscillators at 5 GHz (see Fig. 6). The coupling network consists of a transmission line with characterand electrical length 360 , loaded, istic impedance . Each osat each end, with a high value resistance cillator contains an NE-3210S01 NEC HJ-FET transistor. The considered noise sources are a voltage source in series with the V Hz , accounting internal gate terminal for the Flicker noise, and a current source of spectral density , in parallel with the input Schottky diode, accounting for the shot noise. Thermal noise generators have also been added to all the resistive elements. Two different analyses will be performed. In the first one, the phase noise of the coupled system is evaluated versus the phase shift between the two oscillators, which is varied with one of the tuning varactors. In the second analysis, the system phase of the noise is evaluated versus the electrical length coupling transmission line. In Fig. 7, the phase shift between the oscillator elements, obtained with HB simulations, is represented versus the tuning . The synchronization band is delimited, at each voltage end, by a turning-point bifurcation. At each of these two turning points, the coupled system desynchronizes due to an

SANCHO et al.: PHASE AND AMPLITUDE NOISE ANALYSIS IN MICROWAVE OSCILLATORS USING NODAL HB

Fig. 6. Schematic of the coupled system of two FET-based oscillators. In each oscillator, the transistor is an NE3210S01 NEC HJ-FET and the tuning network uses a MACOM MA46H070 varactor diode connected to the drain of the active device. The two oscillators are connected through the source terminal.

Fig. 7. Variation of the phase shift between the two FET-based oscillator elements obtained versus the tuning voltage V . The synchronization band is delimited by the two turning points T , T , at which loss of synchronization takes place.

excessive deviation of the individual free-running frequencies. Thus, synchronized operation is obtained in the range V V . At the turning-point bifurcations, a real pole crosses the imaginary axis through the origin, thus, in (due to the circuit autonomy) there is a second addition , which, as explained in Section IV-B provides zero pole an additional amplification of the phase noise. This is shown in Fig. 8, where the phase-noise difference with respect to free-running conditions is represented versus the tuning voltage at the constant frequency offset of 100 kHz. The phase noise increases as the edges of the synchronization band are approached (see Fig. 8). It tends to infinite at these band edges due to the inherent limitations of the linearized analysis. Measurement points are superimposed. As can be seen, a maximum V corresponding improvement of 3 dB is obtained at to the in-phase solution. The technique presented in Section V-A.2 allows an efficient analysis of the phase noise versus the coupling-network elein (68) and (78). ments, modeled with a separate term

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Fig. 8. Phase-noise improvement of the coupled-oscillator system with respect to free-running operation. The phase-noise spectral density at 100-kHz offset is traced versus the phase shift between the oscillator elements. Measurements are superimposed.

Fig. 9. Phase-noise improvement of the coupled-oscillator system with respect to free-running operation. The phase-noise spectral density at 100-kHz offset is traced versus the electrical length of the coupling transmission line.

Here, the effect of the coupling-line length on the system phase noise has been studied. The improvement in the spectral density with respect to free-running conditions, at the offset frequency of 100 kHz, has been represented in Fig. 9 versus the electrical length . For near zero , the effect of the transmission line vanishes and the phase-noise reduction tends to , as expected from the fact that there is no longer a path to ground in the increases, the phase noise follows an coupling network. As oscillatory pattern with decreasing minima, in agreement with the results of Section V-A. A complementary stability analysis, based on [33], has shown that the in-phase solution keeps stable values. Thus, the phase-noise imin the considered range of provement can be maximized through a suitable choice of the electrical length. VI. CONCLUSION In this paper, a nodal HB formulation for the oscillator-noise analysis has been presented. The formulation explicitly relates

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 7, JULY 2007

the resonances in the noise spectrum to the system poles, which has allowed a rigorous analysis of the influence of the stability margin on the amplitude and phase noise. A detailed study of the implications of using different constraints in the resolution of the perturbed-oscillator equations has been presented. The obtained phase-noise spectra agree when neglecting the time variation of the amplitude perturbations. In the case of coupled equations in the amplitude and noise perturbations, the additional resonances have an influence on the phase noise, unlike what happens in the case of uncoupled equations. Thus, the usual measurement of these resonances in practice. The obtained equations have been extended to the case of two coupled oscillators, investigating the effect of the coupling network on the phase-noise reduction with respect to the free-running regime. The developed formulation has been successfully applied to a bipolar-based push–push oscillator at 18 GHz, a bipolar-based oscillator at 1 GHz and a coupled system of two FET-based oscillators at 5 GHz.

can be considered narrowThe harmonic components is slow varying, which band provided the phase process has been justified in Appendix A. Equation (83) allows the use of a time-frequency formulation of the modified nodal (1)

(84) where is the vector containing the time-varying harmonics of the state variables and the rest of vectors contain the timevarying harmonic components of the corresponding variables in (1). APPENDIX C IMPULSE RESPONSE MATRIX AND SYSTEM POLES If no decomposition into phase and amplitude perturbations is considered, the perturbed solution is expressed as

APPENDIX A HARMONIC REPRESENTATION OF THE INSTANTANEOUS FREQUENCY PERTURBATION The instantaneous frequency perturbation is related to the . Its time dependence is due to phase process as the noise sources, which, as already indicated, are narrowband can be about the different harmonic components. Thus, expanded in a Fourier series with time-varying harmonics as (80) where the harmonic components can be considered as slowly varying. The phase perturbation will be obtained through time integration of (80) as follows: (81) Taking into account the slow varying characteristic of the har, the phase perturbation will be mainly due to monics

(85) being the harmonic perturbation. Performing a simwith ilar analysis as in Section II-B, the linearized system about the oscillator steady-state solution is given by (86) Equation (86) represents a multiinput–multioutput (MIMO) LTI system whose impulse response matrix in the Laplace domain is given by (87) where the Jacobian function has been considered linear in the parameter . This assumption is justified in are called the system poles Section II-C. The roots of associated to the steady-state solution . From the inspection of (87), the system poles agree with the roots of the determinant (88) which is called the characteristic determinant.

(82) REFERENCES APPENDIX B DERIVATION OF THE TIME-FREQUENCY FORMULATION OF THE MODIFIED NODAL EQUATION An envelope representation of the state variables will be obtained by grouping the terms in (4) as follows:

(83)

[1] D. B. Leeson, “A simple model of feedback oscillator noise spectrum,” Proc. IEEE, vol. 54, no. 2, pp. 329–330, Feb. 1966. [2] F. X. Kaertner, “Determination of the correlation spectrum of oscillators with low noise,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 1, pp. 90–101, Jan. 1989. [3] F. X. Kaertner, “Analysis of white and noise in oscillators,” Int. J. Circuit Theory Applicat., vol. 18, pp. 485–519, 1990. [4] K. Taihyun and E. H. Abed, “Closed-loop monitoring systems for detecting incipient instability,” in Proc. 37th IEEE Decision Control Conf., 1998, pp. 3033–3039. [5] S. V. Hoeye, A. Suárez, and S. Sancho, “Analysis of noise effects on the nonlinear dynamics of synchronized oscillators,” IEEE Microw. Wireless Compon. Lett., vol. 11, no. 9, pp. 376–378, Sep. 2001. [6] J. M. Paillot, J. C. Nallatamby, M. Hessane, R. Quéré, M. Prigent, and J. Rousset, “A general program for steady state, stability, and FM noise analysis of microwave oscillators,” in IEEE MTT-S Int. Microw. Symp. Dig., 1990, pp. 1287–1290.

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[7] P. Bolcato, J. Nallatamby, R. Larcheveque, M. Prigent, and J. Obregon, “A unified approach of PM noise calculation in large RF multitone autonomous circuits,” in IEEE MTT-S Int. Microw. Symp. Dig., 2000, pp. 417–420. [8] S. A. Maas, Nonlinear Microwave Circuits. Norword, MA: Artech House, 1988. [9] E. Ngoya, J. Rousset, and D. Argollo, “Rigorous RF and microwave oscillator phase noise calculation by envelope transient technique,” in IEEE MTT-S Int. Microw. Symp. Dig., 2000, pp. 91–94. [10] S. A. Maas, Noise in Linear and Nonlinear Circuits. Boston, MA: Artech House, 2005. [11] A. Suárez, S. Sancho, S. V. Hoeye, and J. Portilla, “Analytical comparison between time and frequency-domain techniques for phase-noise analysis,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 10, pp. 2353–2361, Oct. 2002. [12] A. Demir, A. Mehrotra, and J. Roychowdhury, “Phase noise in oscillators: A unifying theory and numerical methods for characterization,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 47, no. 5, pp. 655–674, May 2000. [13] A. Demir, “Phase noise in oscillators: DAEs and colored noise sources,” in IEEE/ACM Int. Comput.-Aided Design Conf., 1998, pp. 170–177. [14] V. Rizzoli, F. Mastri, and D. Masotti, “General noise analysis of nonlinear microwave circuits by the piecewise harmonic-balance technique,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 5, pp. 807–819, May 1994. [15] A. Demir, D. Long, and J. Roychowdhury, “Computing phase noise eigenfunctions directly from harmonic balance/shooting matrices,” in Proc. IEEE Int. VLSI Design Conf., 2001, pp. 283–288. [16] V. Rizzoli, A. Neri, and F. Mastri, “A modulation-oriented piecewise harmonic-balance technique suitable for transient analysis and digitally modulated signals,” in 26th Eur. Microw. Conf., 1996, pp. 546–550. [17] F. Bonani and M. Gilli, “Analysis of stability and bifurcations of limit cycles in Chua’s circuit through the harmonic-balance approach,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 46, no. 8, pp. 881–890, Aug. 1999. [18] J. H. Wilkinson, The Algebraic Eigenvalue Problem. New York: Oxford Univ. Press, 1965. [19] A. B. Carlson, Communication Systems. New York: McGraw-Hill, 1986. [20] K. Kurokawa, “Some basic characteristics of broadband negative resistance oscillators,” Bell Syst. Tech. J., vol. 48, pp. 1937–1955, Jul.–Aug. 1969. [21] E. Mehrshahi and F. Farzaneh, “An analytical approach in calculation of noise spectrum in microwave oscillators based on harmonic balance,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 5, pp. 822–831, May 2000. [22] A. Papoulis, Probability, Random Variables, and Stochastic Processes. New York: McGraw-Hill, 1991. [23] A. Hajimiri and T. H. Lee, “A general theory of phase noise in electrical oscillators,” IEEE J. Solid-State Circuits, vol. 33, no. 2, pp. 179–194, Feb. 1998. [24] S. Sancho, F. Ramírez, and A. Suárez, “General stabilization techniques for microwave oscillators,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 12, pp. 868–870, Dec. 2005. [25] P. R. Gray, P. J. Hurst, S. H. Lewis, and R. G. Meyer, Analysis and Design of Analog Integrated Circuits. New York: Wiley, 2001. [26] A. Suárez, J. Morales, and R. Quéré, “Synchronization analysis of autonomous microwave circuits using new global stability analysis tools,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 5, pp. 494–504, May 1998. [27] A. Suárez and R. Quéré, Stability Analysis of Nonlinear Microwave Circuits. Boston, MA: Artech House, 2003. [28] E. E. E. Hegazi, A. A. Abidi, and J. Rael, Designer’s Guide to HighPurity Oscillators. New York: Springer-Verlag, 2004. [29] G. M. Maggio, O. D. Feo, and M. P. Kennedy, “Nonlinear analysis of the Colpitts oscillator and applications to design,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 46, no. 9, pp. 1118–1130, Sep. 1999.

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[30] CADENCE, Virtuoso SpectreRF Simulation Option Theory. ver. Product ver. 5.0, Cadence Des. Syst., San Jose, CA, 2004. [31] J. J. Lynch and R. A. York, “Synchronization of oscillators coupled through narrowband networks,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 2, pp. 237–249, Feb. 2001. [32] H. C. Chang, X. Cao, U. Mishra, and R. York, “Phase noise in coupled oscillator arrays,” in IEEE MTT-S Int. Microw. Symp. Dig., 1997, pp. 1061–1064. [33] A. Suárez and F. Ramírez, “Analysis of stabilization circuits for phasenoise reduction in microwave oscillators,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 9, pp. 2743–2751, Sep. 2005.

Sergio Sancho (M’04) was born in Santurce, Spain, in 1973. He received the Physics degree from Basque Country University, Basque Country, Spain, in 1997, and the Ph.D. degree in electronic engineering from the University of Cantabria, Santander, Spain, in 2002. In 1998, he joined the Communications Engineering Department, University of Cantabria. His research interests include the nonlinear analysis of microwave circuits and frequency synthesizers, investigation of chaotic regimes, and phase-noise analysis.

Almudena Suárez (M’96–SM’01) was born in Santander, Spain. She received the Electronic Physics degree and Ph.D. degree from the University of Cantabria, Santander, Spain, in 1987 and 1992, respectively, and the Ph.D. degree in electronics from the University of Limoges, Limoges, France, in 1993. She is currently a Full Professor with the Communications Engineering Department, University of Cantabria. She has been the leading researcher in several Spanish research and development projects and has taken part in a number of Spanish and European projects in collaboration with industries. She has been Technical Referee of the Spanish Evaluation Agency for research proposals. Her research interest include the nonlinear design of microwave circuits, especially the stability and phase-noise analysis and investigation of chaotic regimes. She coauthored Stability Analysis of Microwave Circuits (Artech House, 2003). She has authored or coauthored over 40 papers in international journals, 36 of which appeared in IEEE journals. Dr. Suárez is a member of the Technical Committee of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium (IMS). She has given invited talks in different conferences and institutions in Germany, France, and the U.S. She is an IEEE Distinguished Microwave Lecturer (2006–2008).

Franco Ramírez (S’03–A’04–M’05) was born in Potosí, Bolivia. He received the Electronic Systems Engineering degree from the Antonio José de Sucre Military School of Engineering, La Paz, Bolivia, in 2001, and the Ph.D. degree in communications engineering from the University of Cantabria, Santander, Spain, in 2005. In 2001, he joined the Communications Engineering Department, University of Cantabria. Since April 2006, he has been a Post-Doctoral Visiting Researcher with the Electricity and Electronics Department, University of the Basque Country, Basque Country, Spain. His research interests include the development of nonlinear techniques for the analysis and design of autonomous microwave circuits.

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J. Grimm A. Griol D. R. Grischowsky E. Grossman Y. Guan S. Guenneau T. Guerrero M. Guglielmi J. L. Guiraud S. E. Gunnarsson L. Guo Y. Guo A. Gupta C. Gupta K. C. Gupta M. Gupta B. Gustavsen W. Gwarek A. Görür M. Hafizi J. Haala J. Hacker S. Hadjiloucas S. H. Hagh S. Hagness D. Haigh A. Hajimiri A. Halappa D. Halchin D. Ham K. Hanamoto T. Hancock A. Hanke E. Hankui L. Hanlen Z. Hao A. R. Harish L. Harle M. Harris O. Hartin H. Hashemi K. Hashimoto O. Hashimoto J. Haslett G. Hau R. Haupt J. Hayashi L. Hayden T. Heath J. Heaton S. Heckmann W. Heinrich G. Heiter J. Helszajn R. Henderson H. Hernandez K. Herrick J. Hesler J. S. Hesthaven K. Hettak P. Heydari R. Hicks M. Hieda A. Higgins T. Hiratsuka T. Hirayama J. Hirokawa W. Hoefer J. P. Hof K. Hoffmann R. Hoffmann M. Hoft A. Holden C. Holloway E. Holzman J. S. Hong S. Hong W. Hong K. Honjo K. Horiguchi Y. Horii T. S. Horng J. Horton M. Hotta J. Hoversten H. M. Hsu H. T. Hsu J. P. Hsu C. W. Hsue R. Hu Z. Hualiang C. W. Huang F. Huang G. W. Huang K. Huang T. W. Huang A. Hung C. M. Hung J. J. Hung I. Hunter Y. A. Hussein B. Huyart H. Y. Hwang J. C. Hwang R. B. Hwang M. Hélier G. Iannaccone Y. Iida P. Ikonen K. Ikossi K. Inagaki A. Inoue M. Isaksson O. Ishida M. Ishiguro T. Ishikawa T. Ishizaki R. Islam Y. Isota K. Ito M. Ito N. Itoh T. Itoh Y. Itoh F. Ivanek T. Ivanov M. Iwamoto

Digital Object Identifier 10.1109/TMTT.2007.903660

Y. Iyama D. Jablonski R. Jackson A. Jacob M. Jacob D. Jaeger N. A. Jaeger I. Jalaly V. Jamnejad M. Janezic M. Jankovic R. A. Jaoude J. Jargon B. Jarry P. Jarry J. B. Jarvis A. Jastrzebski A. S. Jazi A. Jelenski S. K. Jeng S. Jeon H. T. Jeong Y. H. Jeong E. Jerby A. Jerng T. Jerse P. Jia X. Jiang J. M. Jin Z. Jin J. Joe J. Joubert M. Jungwirth P. Kabos W. Kainz T. Kaiser T. Kamei Y. Kamimura H. Kamitsuna H. Kanai S. Kanamaluru H. Kanaya K. Kanaya P. Kangaslahtii V. S. Kaper N. Karmakar T. Kashiwa K. Katoh R. Kaul T. Kawai K. Kawakami A. Kawalec S. Kawasaki H. Kayano H. Kazemi M. Kazimierczuk S. Kee L. Kempel P. Kenington A. Khalil A. Khanifar A. Khanna F. Kharabi S. Kiaei J. F. Kiang B. Kim B. S. Kim H. Kim I. Kim J. H. Kim J. P. Kim M. Kim W. Kim N. Kinayman P. Kinget S. Kirchoefer A. Kirilenko V. Kisel M. Kishihara A. Kishk T. Kitamura T. Kitazawa J. N. Kitchen M. J. Kitlinski K. Kiziloglu B. Kleveland D. M. Klymyshyn L. Knockaert R. Knoechel K. Kobayashi Y. Kogami T. Kolding N. Kolias J. Komiak G. Kompa A. Konczykowska H. Kondoh Y. Konishi B. Kopp B. Kormanyos K. Kornegay M. Koshiba J. Kosinski T. Kosmanis S. Koul I. I. Kovacs S. Koziel A. B. Kozyrev N. Kriplani K. Krishnamurthy V. Krishnamurthy C. Krowne V. Krozer J. Krupka W. Kruppa D. Kryger H. Ku H. Kubo A. Kucharski C. Kudsia W. Kuhn T. Kuki A. Kumar M. Kumar C. Kuo J. T. Kuo P. Kuo

H. Kurebayashi K. Kuroda N. Kuster M. Kuzuhara Y. Kwon G. Kyriacou M. K. Kärkkäinen F. Ladouceur K. Lakin P. Lampariello M. Lancaster U. Langmann G. Lapin J. Larson L. Larson J. Laskar C. L. Lau A. Lauer D. Lautru P. Lavrador G. Lazzi C. H. Lee J. F. Lee R. Lee S. Lee S. Y. Lee T. Lee T. C. Lee Y. Lee Y. H. Lee D. Leenaerts Z. Lei G. Leizerovich Y. C. Leong S. Leppaevuori G. Leuzzi Y. Leviatan B. Levitas R. Levy G. I. Lewis H. B. Li H. J. Li L. W. Li X. Li Y. Li H. X. Lian C. K. Liao S. S. Liao D. Y. Lie L. Ligthart E. Limiti C. Lin F. Lin H. H. Lin J. Lin K. Y. Lin T. H. Lin Y. S. Lin E. Lind L. Lind D. Linkhart P. Linnér A. Lipparini D. Lippens A. S. Liu J. Liu L. Liu P. K. Liu Q. H. Liu S. I. Liu T. Liu T. P. Liu I. Lo J. LoVetri S. Long N. Lopez M. Lourdiane G. Lovat D. Lovelace Z. N. Low H. C. Lu K. Lu L. H. Lu S. S. Lu V. Lubecke S. Lucyszyn N. Luhmann A. Lukanen M. Lukic A. D. Lustrac J. F. Luy G. Lyons J. G. Ma Z. Ma S. Maas G. Macchiarella J. Machac M. Madihian K. Maezawa G. Magerl S. Mahmoud F. Maiwald A. H. Majedi M. Makimoto J. Malherbe V. Manasson T. Maniwa R. Mansour D. Manstretta M. H. Mao S. G. Mao A. Margomenos R. Marques G. Martin E. Martinez K. Maruhashi J. E. Marzo D. Masotti G. D. Massa D. Masse A. Materka B. Matinpour A. Matsushima S. Matsuzawa G. Matthaei J. Mayock J. Mazierska

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