PROPOSAL TO THE NATIONAL SCIENCE FOUNDATION Attn: Dr. Lawrence Goldberg for continuation of COMPUTER MODELING OF MICROSTRIP INTERCONNECTS IN MILLIMETER WAVES Submitted by Linda P.B. Katehi, Presidential Young Investigator Associate Professor Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor, MI 48109-2122 Period of Study: May 1, 1990-April 30, 1991 Cost: $25,000 Date of Submission: May 2, 1990.

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A. PROGRESS REPORT 1. NSF Award: ECS-8657951 2. Period Covered By This Report: May 1, 1989-April 30, 1990 3. Title of Proposal: Computer Modeling of Microstrip Interconnects in Millimeter Waves 4. Name of Institution: University of Michigan 5. Principal Investigator: Linda P.B. Katehi 7. List of Manuscripts Submitted or Published Under NSF Sponsorship During This Reporting Period: 1. N.I.Dib and P.B. Katehi, "Modeling of Shielded CPW Dis6ontinuities Using the Space Domain Integral Equation Method (SDIE). Accepted for publication in the Journal of Electromagnetic Waves and Applications. (Appendix C) 2. N.I.Dib, P.B. Katehi, G.E. Ponchak and R.N. Simons, "Coplanar Waveguide Discontinuities for P-I-N Diode Switches and Filter Applications". To be published in the Proceedings of the 1990 IEEE International Microwave Symposium in Dallas, Texas. (Appendix D) 3. N.L. Vandenberg and P.B. Katehi, "Full-Wave Analysis of Aperture Coupled Shielded Microstrip Lines". To be published in the Proceedings of the 1990 IEEE International Microwave Symposium in Dallas, Texas. (Appendix A) 2

4. N.I.Dib, P.B. Katehi, G.E. Ponchak and R.N. Simons, "Theoretical and Experimental Investigation of Coplanar Waveguide Discontinuities". Submitted to IEEE Transactions on Microwave Theory and Techniques. (Appendix E) 8. List of Interim and Technical Reports Generated From This Work: 1. G.M. Rebeiz and P.B. Katehi, "EECS 411: A Microwave Integrated Circuits Course". Radiation Laboratory Report, April 1990. (Appendix B) 3

I. Introduction The research conducted during the third year of this study concentrated on the following four projects; a) High Frequency Characterization of Aperture Coupled Shielded Microstrip Lines b) Monolithic Active Slot Arrays c) Development of a Microwave Integrated Circuits Course d) High Frequency Characterization of Coplanar Waveguide Discontinuities for P-I-N Diode Switches and Filter Applications. The first three tasks were introduced during the third year and will continue for the next two years. The progress in each one of the four projects is summarized below. II. High Frequencv Characterization of Aperture Coupled Shielded Microstrip Lines. Application of monolithic techniques to microwave and millimeter-wave circuits has introduced the need for accurate fullwave analysis to account for the effects of the package and also the interaction between circuit elements which may be closely spaced. These effects cannot be accounted for by other methods such as simple transmission line analysis, especially as frequency increases. High-frequency interconnect approaches are required which can be efficiently incorporated in a design procedure and also can fit well in a monolithic fabrication scheme. Transitions between microstrip lines printed on different layers are very important circuit elements in multilayered configurations. A variety of transitions have been characterized with approximate techniques. As a result, available models are limited to low-frequency applications. Our work in this area started last year and concentrated on the analysis of aperture coupled shielded microstrip lines. This transition, in a variety of forms, finds wide applicability to bots 4

broadband and narrowband systems and can be used as a building block for interconnects, phase shifters, inverters, directional couplers and filters. Our approach uses a three-dimensional fullwave space-domain integral equation method of moments with Galerkin's procedure to account for all possible interactions (see Appendix A). Results from this work will be presented in the 1990 IEEE MTT-S International Symposium in Dallas, Texas, May 5-11. During the upcoming period, we plan to further substantiate the derived model by laboratory measurements. III. Monolithic Active Slot Array. A spatial power combining array of slotted cavities with active elements (see figure 1) has been proposed to overcome the limitations of traditional array design methods for higher frequencies, namely: * High ohmic losses. * Reduced solid state power. The goal of the study is to develop a set of analytical tools to characterize and design the array and to verify the results with available experimental measurements. The effort has been divided into three steps: 1) Analysis of isolated radiating elements. 2) Development of equivalent circuit parameters. 3) Incorporation of mutual coupling analysis into an array design procedure. During this reporting period, the first two steps of the study have been completed. The derived theoretical results have been compared with previously published experimental measurements and have shown excellent agreement. The results are shown in figures 2, 3 and represent the best known model for this class of problems to date. 5

Coupler-fed Array Figure 1: Monolithic Slot Array 6

Normalized Resonant Resistance vs. Slot Offset 1.200 - -— e Shavit's Measurements /..-e..- Shavit's Calculation i 1.000 g-1.000 - pre This Theory /...... Cosine Assumption,/ -i, 0.800, - -) II / 0.600 I, o 0.400,' 0.200/,. 0.000..... -.200 -.150 -.100 -.050 0.000 0.050 0.100 0.150 0.200 0.250 Slot Offset (inches) Figure 2: Normalized Resonant Resistance vs. Slot Offset. Theory and Experiment. 7

Comparison of Resonant Length Calculations 2.100 2.000 1.900 -0 o-,- Shavit's Measurements — W — Shavit's Calculation X — I — This Theory ~A 1.800......................-.-. —-- 1.700 -.200 -.150 -.100 -.050 0.000 0.050 0.100 0.150 0.200 Slot Offset (inches) Figure 3: Resonant Length vs. Slot Offset. Theory and Experiment.

During the next period we expect to further document and evaluate the numerical model by making comparisons with additional measurements which are in progress. Thus, we will assess the role of the numerical/analytical model as a replacement for empirical design methods which involve costly, time-consuming, experimental measurements. IV. Development of a Microwave Integrated Circuits Coaurse. The technological advances in the microwave field should have a strong impact on university teaching and research programs. Nowadays, a good microwave engineer must know passive circuits, active circuits, printed-circuit fabrication, computer-aided analysis, and measurement of microwave networks. As a response to this need, we submitted a proposal to the AT&T Foundation during the 1989 academic year, for the development of a senior/graduate level microwave integrated circuits course. The proposal was approved by the AT&T foundation, and in June 1989, a grant of $50,000 was awarded to the EECS department for the development of EECS-411. This course offers a delicate balance between passive and active microwave networks, without compromising on the basic electromagnetic analysis and understanding of microwave networks. It also includes a modern laboratory, where the students analyze, design, fabricate, and test hybrid microwave integrated circuits. The laboratory includes computer-aided design stations, microwave measurement stations, and a printed-circuit fabrication facility. The goal of course is to prepare the student to design and measure a fairly complicated MIC, such as a low-noise amplifier. The course was offered, for the first time, in the Fall term of 1989. The enrollment was 28 students, with 24 registered and 4 auditing students. The course consisted of 33 lectures and 7 lab 9

sessions intimately tied to the material covered in the class. More information about the course may be found in the attached report (see Appendix B). V. High Frequency Characterization of Coplanar Waveguide Discontinuities for P-I-N Diode Switch and Filter Applications. The coplanar waveguide (CPW) was introduced for the first time in 1969 by C.P.Wen as an appropriate transmission line for nonreciprocal gyromagnetic device applications. Recently, with the push to high frequencies and monolithic technology, CPWs have experienced a growing demand due to their appealing properties. However, the extent of applications of CPW circuits is limited due to the lack of specific circuit components and the unavailability of circuit element models appropriate for computer aided design. Microwave switches are circuit elements widely used in phase shifters and radiometers. During the reporting period we successfully studied various CPW discontinuities as described in Appendix C. Also we analyzed the CPW p-i-n diode switch shown in figure 1 of Appendix D. For the complete characterization of this circuit element, the diode's on and off states were considered separately and their frequency dependent equivalent circuits were derived accurately. The development of the analytical method used for the derivation of the theoretical data and, the experimental setup and the de-embedding technique used for the measurements are described in Appendices D and E. During the next period, the technique will be extended to characterize more complicated discontinuities, such as a CPW airbridge, and to evaluate conductor losses in coplanar waveguide operating in the millimeter wave range. The theoretical data will be tested by comparing to measurements. The experiments will be performed at NASA Louis Research Center, Ohio. 10

B. OUTLINE OF THE TASKS FOR THE THIRD YEAR 1) High Frequency Characterization of Aperture Coupled Shielded Microstrip Lines. This project will continue until completion. All the theoretical results will be compared to experimental data which are to be performed in the facilities of the Radiation Laboratory of the University of Michigan. 2) Monolithic Active Slot Arravy_. The numerical model derived for the isolated radiating elements will be further validated with comparisons to additional measurements and will be used to analyze and design a linear array of slots with specific radiation properties. 3) High Frequency Characterization of Coplanar Waveguide Discontinuities for Diode Switch and Filter Applications., The developed numerical technique will be improved further in order to design CPW circuit elements with specific functions, evaluate their performance at frequencies above 100GHz and compare it to the performance of corresponding microstrip structures. 4) Development of Software for High-Frequency Circuit Desian. A new activity we plan to initiate this coming year is the development of software capable of designing circuits at very high frequencies. This software will be based on the numerical models we have developed during the past five years and will become available to graduate and undergraduate students for their research or other coursework. Specifically, this circuit design package will be use 11

in three courses: EECS411 (Microwave Circuits I), EECS511 (Microwave Circuits II) EECS525 (Solid State Microwave Circuits) and EECS534 (Design and Characterization of Microwave Devices and Monolithic Circuits). 12

APPENDIX A

Full-wave Analysis of Aperture Coupled Shielded Microstrip Lines1 N. L. VandenBerg and P. B. Katehi Radiation Laboratory, The University of Michigan, Ann Arbor, MI Abstract A full-wave space-domain integral equation analysis of aperture coupled shielded microstrip lines is presented based on Pocklington's integrals and the equivalence principle. The derivation of the associated dyadic Green's functions in the form of waveguide LSE and LSM modes is described. The line currents and slot voltage are expanded in terms of subsectional basis functions and the method of moments, together with even and odd mode transmission line analysis, is applied to determine the two-port scattering parameters. A particular case is illustrated which demonstrates the behavior of the coupler as a bandpass interconnect. Introduction Application of monolithic techniques to microwave and millimeter-wave circuits has introduced the need for accurate full-wave analysis to account for the effects of the shielding structure and also the interaction of circuit elements which may be closely spaced. These effects may not be accounted for by other methods such as simple transmission line analysis, especially as frequency increases. Additionally, high-frequency interconnect approaches are required which must be accurately modeled for design and also must fit well in a monolithic fabrication scheme. Transitions from microstrip to slotline have long been recognized as important circuit elements. Numerous investigators have presented approximate analytical techniques to characterize these structures with applications to circuit elements [1] - [4]. A full-wave analysis for this transition was reported in [5] with applications to open structures. 1This paper is a revision of the original which appeared in the 1990 Microwave Theory and Techniques Symposium Digest reflecting changes in the Numerical Results section after the computer program was modified to improve accuracy. L.

In this paper, the analysis of aperture coupled shielded microstrip lines is presented. This structure, in a variety of forms, has a wide range of applications to both broadband and narrowband connections and can be used as a building block for interconnects [6], phase shifters and inverters [7], directional couplers [8] as well as filters. A quasi-static analysis has been provided in [9], however, this may not be sufficient, particularly for higher frequencies where end effects and higher order mode coupling become more significant. A more recent paper [10] presents a transmission line analysis with excellent results, however, similar shortcomings would be expected. Hybrid methods which combine two-dimensional full-wave analysis with transmission line theory, as in [11], should certainly extend the validity of such models, however, may still not account for all discontinuity effects. Our approach uses a three-dimensional full-wave space-domain integral equation method employing the method of moments with Galerkin's procedure to account for all possible interactions. Analysis The basic structure of the coupler to be discussed is as shown in Figure 1. Variations on this geometry include cases where the strips are on the same substrate in a single cavity, addition of multiple layers to the substrates/superstrates, reverse couplers where the lines exit on the same wall, parallel slots and lines, among others, but can all be analyzed using the same approach. The analysis proceeds as follows: the slot is replaced on both sides by an equivalent magnetic current - representing the tangential electric field in the slot - backed by a perfectly conducting wall (all walls will be assumed to be perfect conductors). In this case, the problem is now separated into two independent regions, coupled together by the magnetic current. Using the same current on either side enforces the continuity of the electric field in the slot. The fields in the cavities can now be written in term's of Pocklington integrals as follows: E=-jWJ GeJJ dS'- JGeK K dS' (1) H=JJGmJ.J dS'jweJJGmK. K dS' (2) The subscripts on the Green's functions indicate whether the iulutcluI Us u the electric (e) or magnetic (m) field type for electric (J) or magnetic (K) currents. By using the appropriate rectangular cavity Green's functions, the fields in the cavities satisfy the boundary conditions on the walls. The remaining boundary conditions, zero tangential electric field on the strips and continuous tangential magnetic field in the slot, allow us to write the integral equations which are Jj tr L dS'-JJ CK* KdS" = 0* (3)

f =lot -S ff jslot =s lot 1 dit.'mJ JL dS jW J [CLcmK + EUGmK] KdS |- I mjr * Ju dS' = 0 (4) =.,triptr=:3tripI /seK [[dS"-jwy G| u ds'= o' (5) slot stripu where O* implies that the field is non-zero at gap generator locations. The superscript indicates the evaluation point for the field. The Green's functions are solutions to the dyadic equations: V x V x e - k2eJ = (R- R') (6) V x V x GmJ - k2GmJ = V X [I5(R- r' )] (7) V X V X GmK- k2GmK =- S(RiR- RI) (8) v v XVXGeK-k GeK=VX I6(R- R') (9) Solutions to these equations can be found for geometries of the type illustrated in Figure 2 where the impedance boundary conditions allow for treatment of multilayered substrates and superstrates as may be encountered in monolithic circuits. The Green's function is found for the layer containing the strips or slot by applying transmission line theory to the other layers to obtain the impedance boundary conditions (7) corresponding to each mode. Subsequently, the fields in the other layers can be found from the homogeneous solutions to Equations (6 - 9) and a pair of coupling coefficients for each layer. The solutions are expanded in terms of LSE and LSM modes which facilitates finding the fields in the remaining layers since these modes are decoupled on the boundaries and can be individually matched to like modes in adjacent layers. The coupling coefficients are then readily found by matching the tangential field components at successive layers. The longitudinal current components are now expanded in terms of piecewise sinusoidal functions with a Maxwellian transverse distribution which satisfies the edge conditions. In general, the currents are written in terms of both longitudinal and transverse components however, in this case, it will be assumed that both the slot and the strips are narrow enough so that the longitudinal components of current dominate their behavior and the transverse components can be neglected. The integral equations can now be written as a matrix equation in conventional method of moments fashion. The inverted matrix is then used with even and odd gap generator excitations at the line ends to find the currents on the microstrip lines. From the even and odd currents, the scattering parameters can be found which characterize the coupling behavior. 3

Numerical Results A coupler with the geometry of Figure 1 was analyzed using the above techniques. The parameters which can be varied in this design are numerous, consequently, only a few variations will be presented here. In all cases, although not required in general, symmetric geometry is maintained to simplify the even and odd mode analysis. The affect of the slot length is shown in Figure 3. It can be seen that the slot is initially too long for an ideal match at the frequency shown ( 11 GHz ). As the slot is progressively shortened, at a certain length the slot "matches" the two port coupler and with further shortening the match gets progressively worse. Using a transmission line analogy, one can interpret this effect by transforming the impedances at the ends of the slot to the center. These end impedances are nearly short circuits, the difference being due to fringing fields which extend beyond the ends of the slot line, fully accounted for by the full-wave analysis. At the matching length, the resulting transformed reactance at the center, cancels the reactance associated with the junction thus matching the two ports. As the slot becomes very short, the field in the slot is effectively "short circuited", thus coupling is reduced. Thus S21 tends to zero while S11 approaches unity (since the structure is closed and assumed lossless). The phase of Si, referenced to the end of the line then becomes the phase due to the end effect, typically, -10 to -15 degrees. All of these effects would be expected to repeat as the slot length increases in multiples of A, however, for the case studied here, the maximum slot length is limited by the dimensions of the shielding package (.25 x.25 inches) which are chosen to allow only the dominant microstrip mode to propagate. Figure 4 illustrates the same phenomena in the frequency domain. By shortening the length of the slot, we are actually moving the center of the frequency response. The shape of the frequency response does not remain constant with the shift however, because the high end is dominated by a null due to the line stub. This affect appears at the frequency for which the line stub is approximately A/2 in length so that there is a virtual open circuit beneath the slot, in which case there is very little coupling between the line and slot. Figure 5 illustrates this affect by showing the influence of the line stub length (I - see Figure 1)on the frequency response. Variation of these stub lengths has the effect of changing the position of the current maxima (virtual shorts) and minima (virtual opens) on the lines relative to the slot, thus varying the degree of coupling through the slot. Consequently, the peak in the frequency response roughly corresponds to a line stub length which places a current maximum below the slot (a. A/4). In Figure 6 the affect of the line separation parameter (s - Figure 1) on the phase response is shown. Note the increase in phase shift due to the added length of slotline as the separation is increased. Also notice that for no separation, as the line stub lengths approach zero, the phase shift is near 180 degrees (the reference 4

planes are set at the location of the slot). This additional phase shift is due to geometry as has been pointed out in [7] and could be taken advantage of in combination with other slots for phase shifters. The frequency response shown in Figure 7 demonstrates the utility of the structure as a filter. With proper selection of the geometric parameters such as line and slot widths, line separation and substrate heights, the frequency response can be tailored to give the required center frequency, bandwidth, shape, etc. As can be seen, separation of the transitions enhances the bandwidth of the coupler to better than octave performance, similar to that which was demonstrated in [10]. In all of the cases presented here, the width of the slot was the same as the line width (.025 inches on er = 9.7, W/h = 1 substrate) resulting in a microstrip-to-slotline characteristic impedance mismatch of about 2: 1. Matching the slot impedance and modifying the width of the stubs as in [10] to alter the frequency response can also be readily accomplished with this method. Unfortunately, because the slot length for this simple geometry is typically on the order of A/2 to get low loss coupling, it may not represent the best choice for monolithic integrated circuit applications where small size is required. Therefore, future analytical and experimental efforts are planned to study and optimize the performance/size tradeoffs for this class of couplers while taking full advantage of the benefits of full-wave analysis for high frequency applications. For example, it may be possible to introduce combinations of shorter slots of varying lengths to produce a broadband frequency response while maintaining a reasonably small area of chip real estate. Conclusions A set of integral equations for aperture coupled shielded microstrip lines has been introduced based on Pocklington's integrals and the equivalence principle. The associated dyadic Green's functions in the form of waveguide LSE and LSM modes have been derived which allow for a full-wave analysis, accounting for all electromagnetic interactions of a microstrip - slot coupler. By expanding the unknown line currents and slot voltage in terms of subsectional basis functions and applying the method of moments together with even and odd mode transmission line analysis, the two-port scattering coefficients can be determined. The method has been applied to a particular case which demonstrates the behavior of the coupler as a bandpass interconnect. Acknowledgement This work was supported by the National Science Foundation under contract ECS:8657951 and partially by the NASA Center for Space Terahertz Technology.

References [1] S. B. Cohn, "Slot line on a dielectric substrate." IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-17, No. 10, October 1969, pp. 768-778. [2] E. A. Mariani, C. P. Heinzman, J. P. Agrios and S. B. Cohn, "Slot line characteristics." IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-17, No. 12, December 1969, pp. 1091-1096. [3] E. A. Mariani and J. P. Agrios, "Slot-line filters and couplers." IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-18, No. 12, December 1970, pp. 1089-1095. [4] J. B. Knorr, "Slot-line transitions." IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-22, No. 5, May 1974, pp. 548-554. [5] H. Yang and N. G. Alexopoulos, "A dynamic model for microstrip-slotline transition and related structures." IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-36, No. 2, February 1988, pp. 286-293. [6] R. H. Jansen, R. G. Arnold and I. G. Eddison, "A comprehensive CAD approach to the design of MMIC's up to MM-wave frequencies." IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-36, No. 2, February 1988, pp. 208-219. [7] K. G. Gupta, R. Garg and I. J. Bahl, Microstrip Lines and Slotlines, Artech House, Dedham, MA, 1979. [8] T. Tanaka, K. Tsunoda and M Aikawa, "Slot-coupled directional couplers between double-sided substrate microstrip lines and their applications." IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-36, No. 12, December 1988, pp. 1752-1757. [9] S. Yamamoto, T. Azakami and K. Itakura, "Slit-coupled strip transmission lines." IEEE Transactions on Microwave Theory and Techniques, Vol. MTT14, No. 11, November 1966, pp. 542-553. [10] B. Schuppert, "Microstrip/slotline transitions: Modeling and experimental investigation." IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-36, No. 8, August 1988, pp. 1272-1282. [11] H. Ogawa, T. Hirota and M. Aikawa, "New MIC power dividers using coupled microstrip-slot lines: Two-sided MIC power dividers." IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-33, No. 11, November 1985, pp. 1155-1164. 6

Ii.1 Figure 1: Geometry of basic coupler................ ~..... TLU Figure 2: Sample multi-layered structure.

0.00 -3.00 -6.00 -9.00 i -12.00 -15.00 -21.00 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250 Slot Length (inches) Figure 3: Affect of slot length on S21 and S11 magnitudes (f = 11.0 GHz). 0.0 -3.0 ~s', -6.0 - J/ -9.0 -12.0 / / -15.0 - Slot Length: 0.2500 inches C4 -15.0 \.. SlotLength:0.1532 inches -18.0 Slot Length: 0.1048 inches -21.0 4.0 6.0. 8.0 10.0 12.0 14.0 16.0 Frequency (GHz) Figure 4: Coupler frequency response variation with slot length.

0.0 F P' -3.0 _, -'"U, -6.0 i /1 -21.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 Frequency (GHz) 135.00 9035.00 S 8 inches -1.0 0 1 Le Stub 0en. (inche I -— ~-I~ ine~tub- - -, s.O6I25in 135.00,'. Figure 6: Variation of S21 phase with line stub lengths (f = 11.0 GHz). 359.00~9 -180.00..... v 0.00 0 025 0. = 0.075 010.2 ie Stub Length (inches) Figure 6' Variation of S21 phase with line stub lengths ( f 11.0 G-z ).

0.0 -3.0 -6.0 I -9.0'~ -12.0 -1.0 LineStub0.05inches........ Line Stub.- 0.0815 inches -18.0 / Line Stub.-0.03S0 inches A A -21.0 I.....I..... 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 Frequency (GHz) Figure 7: Coupler frequency response variation with line stub length ( s =.0625 inches). 10

APPENDIX B

EECS 411: A Microwave Integrated Circuits Course Supported in part by a donation from the AT&T Foundation Gabriel M. Rebeiz Curtis Ling (Graduate Student) and Pisti (Linda) B. Katehi Radiation Laboratory Electrical Engineering and Computer Science Department University of Michigan, Ann Arbor

Introduction: Microwave technology has been the central nervous system for the AT&T communication network. Most of the world communications are sent across the world via microwave repeater stations, satellite communication links, and fiber-optic lines modulated at microwave frequencies. The microwave field has been experiencing a technological revolution in the last decade, due to advances in solid-state devices, dielectric materials, and microwave-circuits analysis techniques. One can now integrate passive components (such as filters, couplers, matching circuits, and antennas) with active components (such as amplifiers, mixers and detectors) to result in low-cost, low maintenance, low-weight microwave systems. These MICs have been instrumental in the design of microwave systems for satellites and low-cost microwave transceivers for ground-based applications. The technological advances in the microwave field should have a strong impact on university teaching and research programs. Nowadays, a good microwave engineer must know passive circuits, active circuits, printed-circuit fabrication, computer-aided analysis, and measurement of microwave networks. As a response to this need, we submitted a proposal to the AT&T Foundation during the 1989 academic year, for the development of a senior/graduate level microwave integrated circuits course. This course is a modern microwave-circuits course that offers a delicate balance between passive and active microwave networks, without compromising on the basic electromagnetic analysis and understanding of microwave networks. It also includes a modern laboratory, where the students analyze, design, fabricate, and test hybrid microwave integrated circuits. The laboratory includes computer-aided design stations, microwave measurement stations, and a printed-circuit fabrication facility. The goal of the course is to prepare the student to design and measure a fairly complicated MIC, such as as low-noise amplifier. A low-noise amplifier requires input and output matching networks, input and output filters (optional), a transistor and a bias network. Another important aspect of the course is to get the students excited about microwaves, and electromagnetics in general. It is no secret to anybody who has taught undergraduate electromagnetics, that the topic is fairly abstract (mathematical and physically). The payoff is in graduate school, where the students refine their techniques and apply them to complex topics such as antenna theory, scattering and general electromagnetics. This MIC course gives the students a hands-on experience of microwave active an passive circuits and show them how exciting electromagnetics can be!!. It also instills in them a sense of confidence in 2

electromagnetics and the capabilities of modern-day microwave technology, and hopefully prepares them well to tackle the microwave world once they graduate. EECS 411: A microwave integrated-circuits course A- Lectures: The proposal was approved by the AT&T foundation, and in June 1989, a grant of $50,000 was awarded to the EECS department for the development of EECS-41 1. As noted before, EECS 411 is a 3-credit senior/graduate course. The course was offered in the fall term of 1989. The enrollment was 28 students, with 24 registered and 4 auditing students, made up of 12 undergraduates and 16 first-year graduate students. The course consisted of 33 lectures and 7 lab sessions intimately tied to the material covered in class. Homework was assigned nearly every lecture and were due the following lecture. A homework set consisted of one problem, and its main purpose was to make sure that everybody in the class is always upto-date. For the lab assignments, we used an easy-to-learn, easy-to-use CAD program called Puff. The MIC program was written for the IBM-PC by Prof. Rick Compton (Cornell) and Prof. David Rutledge (Caltech). The MIC program allows the student to graphically layout a MIC, analyze his layout, and get a printed mask when he is satisfied with the design. Puff does not have an optimization routine, and therefore the student must think about the physics of the problem. Puff proved to be very useful, and the students learned to use it in the first week of the course. The lecture breakdown was as follows: Review/Transmission-lines 3 (lectures) Scattering parameters 3 Matching networks 3 Signal-flow graphs 2 Couplers(branch-line, ratrace) 2 Coupled-lines and coupled-lines couplers 3 Filter theory (LP, HP, BP) 5 Diodes and microwave detectors 3 Network analyzer theory 2 (Thanksgiving holiday) Noise in microwave networks 2 Microwave amplifier design (low-noise, high gain) 3 Balanced and feedback amplifiers 2 Total: 33 (lectures) 3

B- Lab setu:-The laboratory was equipped entirely with Hewlett Packard microwave equipment. Prof. Linda Katehi donated $10,000 from her NSF funds, and the department invested $30,000 in the lab to cover the total start up cost of $90,000. The lab equipment consisted of two microwave stations, each with a computer controlled HP8410 network analyzer, an HP8350B microwave sweeper, an HP437B power meter, and, for the low-noise amplifier lab, an HP8970B noise-figure meter. Each station also had supporting infrastructure such as digital multimeters, oscilloscopes, plotters, calibration standards, precision delay lines, coaxial attenuators and couplers, soldering equipment, etc... All the controlling software was written by Curtis Ling, a graduate student in the EECS department. 4-port microwave substrate holders were designed and machined to allow the students to quickly assemble and disassemble of their microwave integrated circuits. These proved to be extremely useful, and reduced the measurement time considerably. The planar microwave circuits used microstrip line (as opposed to the stripline or the coplanar waveguide) as the transmission-line medium, and the transitions to the coaxial structures (which are necessary for the network analyzer) were easy to machine. The substrate holders had a VSWR better than 1.1 (return loss better than -20dB) for frequencies under 7GHz. The substrates were standard plexiglass PC-board, with a thickness of 1.45mm and a dielectric constant of 4.2. Finally, a library of 15 microwave circuits and systems books was made available to the students in the lab. C- Lab operation: The lab operated as follows. First, a lab assignment was given to the students. The students were generally asked to design, for example, a 5-dB ratrace coupler, a low-pass filter with a cutoff frequency of 4GHz and a rejection of 30dB at 4.4GHz, a microwave amplifier with a noise-figure around 2dB, and gain higher than 9dB at 4GHz, etc... The student used Puff and the class lectures to layout a planar microstrip circuit that satisfies the lab assignment. The masks (software only) were due on the following Monday, and the T.A. combined 12 masks together and sent the circuits for lithography and etching. The circuits returned on Tuesday afternoon, and the students measured their designs on Wednesday and Friday. A student typically spent 1 hour in the lab, about 0.5hr on the network analyzer and 0.5hr discussing the measurement with the T.A. and his classmates. And here comes the interesting part. Almost always, the measurement does not agree precisely with the theory, and the student must explain the reasons for the discrepancies, and, using Puff, modify his theory to match the measured results. In the real world, one would modify the circuit to match the design requirements (this point is made clear to the students), but by modifying the design the student gets a much better understanding of the theory and the second-order corrections that affect the measurements. These include corrections for the capacitance of open-ended stubs due to the fringing field, the short-circuiting of currents in microwave T-junctions, and the variation

f the substrate dielectric constant. The corrections are usually small (typically a few percent in mngth, or the impedance level of transmission lines), and a good student will match theory and xperiment down to the -20dB and even to the -30dB level!!. The lab assignments were: - 50/70 transmission-line - Matching - Branch-line / Rat-race coupler - Chebyshev / Elliptic low-pass filter - Coupled-line Chebyshev band-pass filter - Schottky-diode video detector - GaAs-Mesfet low-noise amplifier the next 7 pages, we present a summary of each experiment, and the comparison between ie measured and predicted results for each experiment. The results are taken directly from tudent reports, and represent a "good" sample of the results expected in the lab experiments. 5

11 50/70O transmission-line lab 1.5-7.5GHz (Student: Brian Kormanyos) The purpose of this first lab is to introduce the students to the hardware and software tools of EECS-41 1. First, they measure a 500 line, and determine the VSWR of the connectors/substrate holder and the typical loss of the microwave substrate. Then, they are given a line with an "unknown" impedance, and by measuring the reflection coefficient vs. frequency, they determine the impedance and the length of the "unknown" microstrip line. For the case shown below, the line was 51mm long with an impedance of 642. -10.00 -12.00 -14.00 _ - -16.00 -18.00 I -20.00ko I -22.00 1 j -24.00 1 -26.00 I -28.00 11 -30.00 1 i 1.50 2.60 3.70 4.80 5.90 7.0 Figure 1: The measured vs. predicted reflection coefficient for a 64CI line with a line length of 51mm. []~~~~~~

2- Matching lab 1.5-8GHz (Student: Mike Delisio) The lab assignment is to design a wideband quarter-wave matching network, and a narrowband stub matching network for a 1002 load. The load is a small hybrid carbon resistor suitable for microstrip circuits. For the results shown below, Mike introduced an attenuator to compensate for the dielectric losses and varied his line lengths and impedances by few percent. This is usually done at every experiment to match theory and experiment. 0.00 quart.lb2 SI I I I I -3.00 -6.00 t -9.00 -12.00 -15.00 -18.00 -21.00 -24.00 -27.00 -30.00 _ _, _ 1.60 2.88 4.16 5.44 6.72 8.00 Figure 2(a): Measured and predicted results for a wideband (2-6GHz) quarter-wave match. Notice the vertical scale of -30dB. The students also match the phase on a Smith chart (right). 0.00 -3.00 -6.00 -9.00 -12.00 -15.00 -18.00 -21.00 -24.00 -27.00 -30.00 1.60 2.88 4.16 5.44 6.72 8.00 Figure 2(b): Measured and predicted results for a narrowband single-stub match at 3GHz. Notice the vertical scale of -40dB. The students also match the phase on a Smith chart (right).

3- Branch-line / Rat-race coupler lab 1.5-6GHz (Students: Yow Peng Lam, Philip Devries) The design here is of a 5dB rat-race and branch-line couplers (some students were asked to design 3dB couplers). This lab is tricky to design, because the student must compensate for 4 T-junctions in the coupler circuit. The results were good, and the shift in the center frequency was a maximum of 300MHz for a 3GHz design. The students either undercompensated (yielding a lower center frequency) or over-compensated (yielding a higher center frequency). After the measurements were done, the students reevaluated their designs, and the modified theory shown below agrees well with the measured results. 0.00 0.00 -3.00 -3.00 -6.00 -6.00 -9.00 -9.00 -12.00 -12.00 -15.00 -15.00 -18.00 - -18.s00 -21.00 / -21.00 -24.00 -24.00 -27.00 -27.00 -30.00 1 -30.00, 2, {[,),, 1.50 2.40 3.30 4.20 5.10 6 1.50 2.40 3.30 4.20 5.10 Figure 3(a): The input reflection coefficient (left), and the coupling value (right) for a -5dB ratrace coupler. 0.00 I I - k ~lect'f,. Cu(c,, t- 0.00.. -3.00 -3.00 -6.00 -6.00' e -9.00 -9.00 -12.00 -12.00 -15.00 -15.00 -18.00 -18.00' -21.00 \ / _ -21.00 -24.00 -24.00.27.00 -27.00 -30.00 _ _ __L__ __ -30.00, 1.50 2.40 3.30 4.20 5.10 6. 1.50 2.40 3.30 4.20 5.10 Figure 3(b): The input reflection coefficient (left), and the coupling value (right) for a -5dB branch-line coupler.

4- Chebyshev / Elliptic Low-pass filter lab 1.5-8GHz (Student: Philip Devries) The students were asked to design a Chebyshev filter with a corner frequency of 4GHz, a ripple of 0.5dB, and an attenuation of 25dB at 6GHz. Another required design was an Elliptic filter with a center frequency of 3GHz and an attenuation of 27dB at 3.5GHz. Philip compensated for the open-end capacitance from the impedance steps, and the T-junctions in the elliptic filter quite well, and his measured responses came close to the predicted responses. Then he modified his compensations by few percent, and the measured and predicted 0.00;21 0'IDGMfe daJd UrckU.& f Tkov - -3.00 0 ~~~~~~~~~~~~~~~~~~~ —900 00 MeCuvCeJ response agreed00 Rexacly!. - -1.00 00) -1.00 -Moo X)-1~0* -2C^ANC - @,CJ QPAZZ i E \^+M18.00 _ 00 66H' a -21.0 X) ]/ -27.00 CX a i -30.00 (W ) 1.50 2.80 4.10 5.40 6.70 8. 1.50 2.80 4.10 5 1.50 2.80 5.40 6.70 8. Figure 4(a): The comparison between the predicted and measured responses of a 7'th order Chebyshev filter, before (left) and after (right) making small second-order corrections. 0.00. U ^ v 13 9, + 2 ^ and c W ~-3.00 I I.S0jnt 2I.79b 40 L3 6.66~1 7.95 J 2.7-3.00 4.06ured6 FigureU es ponsc -6.00 the pd Rapur so' -9.00 0 DV~J_ -12.00 0 7 -15.00 0 -18.00 0 35N~ C -21.00 0 -24.00 o -27.00 o I I I0~ -30.00 1.50 2.79 4.08 5.37 6.66 7.95 1.50 1.50 2.79 4.0e 5.37 6.66 7

5- Coupled-line band-pass filter lab 3-5GHz (Students: Brian Kormanyos, Philip Devries) The assignment is to design a coupled-line band-pass filter centered at 4GHz, with a bandwidth of 300MHz, a ripple of 0.5dB, and 25dB rejection at 3.5 and 4.5GHz. This filter was defined using three 90-deg coupled-line sections. The end-capacitance correction proved to be tricky, and the dielectric losses were quite high. The dielectric losses were modelled by placing 12 series resistors at the current peak in the coupled-line sections. The measured results agreed well with the "modified" predicted response, once the dielectric losses and capacitance effects were taken into account. 0.00 MIwra A TkrIc I 0.00 acsurLed and A4jvs1Pd +LtofC;cCA ReJul+r IB -3.00 T3.#re f c CIATh r-cc300 I -6.00 - -6.900 t /eid tMCCLLAr Ac -9.00 - a;'2.G -9.00' l -12.00 me -, f -12.00 -15.00 -15.00 -24.00 ~f.7 G -24.00 27.00 -/.tadbt, / I27.00 -30.00 / I { I \ 1 -30.00 3.00 3.40 3.80 4.20 4.60 5. 3.00 3.40 3.80 4.20 4.60 0.00 0.00 4.00 -.00 ~~~~~~~~-8.00 L It ~~-8.00 -12i.00 -12.00 -16.00 -16.00 -20.00 20.00 xl -24.00 -20.00 -28.00.2i "'",i'p~9 -28.00 -32.00 l -32. I \\ 200 -36.00 -36.00 40.00 0.00 3.00 3.40'.80 24.60 -.00 32.00 3.0 0.24.65.0 2.90 3.24 3.6 4.12 4.56 Figure 5(a,b): The comparison between the predicted and measured responses of a 3'rd-order losses and open-end capacitance of the coupled-lines. Phil Devries (top) and Brian Kormanyos (bottom)., 10

6- Schottky-diode video detector lab 2-7.5GHz (Student: Walid Ali-Ahmad) The purpose of this experiment is to design a wideband microwave video detector using zerobiased Schottky diodes. The diode is of the small packaged type, and supplied by Metelics Corp. Since the diode has a large RF impedance (typically 2-3KfL), a 100' RF shunting resistor must be placed in parallel with the diode to yield a wideband detector. Walid's detector had a maximum VSWR less than 2 in the 2-7GHz band, and a responsivity of 80V/W at 6.5GHz. In another experiment (not shown here), the resistor was placed in parallel with the diode RF-wise, but not DC wise. This increased the responsivity to 1200V/W. which is very competitive with commercial diode detectors. So, if AT&T needs some video detectors, they can now easily get them for free from UoM!!. 0.00 -2.00 -4.00 -6.00 -8.00 -N 10.00 o -12.00 ~-14.00 -16.00 -18.00 -20.00 1.50 2.90 4.30 5.70 7.10 8.50 45.00 40.00 35.00 30.00 ~~2.0o "-....~ oL..0_ k ~~~~1.5-8GHz detector.~~ 2500 1 0.00 -5.00 00 1.50 2.90 4.30 5.70 7.10 49.53 730.14 1410.76 2091.37 2771.99 detected voltage vs. frequency for a constant input power. The results yield show a wideband 11

7- GaAs-Mesfet low-noise amplifier 1-5GHz (Students: John Galantowicz, Walid Ali-Ahmad) This was the most extensive lab. The students designed the input and output matching circuits for a low-noise amplifier in the 3-4GHz region. Since the transistor was potentially unstable, the students had to check for stability at every frequency, and keep clear of the forbidden regions in the Smith chart. The designs were quite successful, considering the fact that we only know the "typical" scattering parameters of the transistors from the manufacturer's data sheets, and not the "exact" scattering parameters of each individual transistor. They also learned how to use the HP8970B noise-figure meter, and about "noise" in microwave networks. The measured minimum noise-figures (1.5 to 2dB) are competitive with modern microwave receivers, and are very low compared to receivers built in the early 1980's. Predicted and Measured Amplifier Gain Measured and Predicted Amplifier Noise Figure 14 l5!iL' l 1 I' Ill I I 1 1 i,12 — U-Hi I Ii;;L't'%I; I;-Ii-;;I_ - I Measured Amplifier - Amplifier Noise Figure (dB) _ __-_f____- ----.. —-. —-- I —I. — --- Theoretical Amplifie 4 Predicted NoiseFigure 10................................................... 8 -- i. — -- t t- -.' P 0.......... I.. 1g4..f tw de. 8, --— 1 —2 —- IL10.0 I' 5.6.. 4.0 0.4 2.0 2.3 2.6 2.9 3.2 3.5 3.8 4.1 4.4 4.7 50 2.0 2.3 2.6 2.9 3.2 3.5 3.8 4.1 4.4 4.7 0 Figure 7(-_b): The measured and predictedresponse of the amplifier gain (eft) and noise-figure (right) for two different designs, J. Galantowicz (top), and W. Ali-Ahmad (bottom)..12 2.0 2.3 2 + 24. -.8 4_1 4,4 4/ i..... 35 38 41 44 4 Frequency~~~~~~~~~ 4.0z Frqec',,,)

APPENDIX C

Modeling of Shielded CPW Discontinuities Using the Space Domain Integral Equation Method (SDIE) by N.I.Dib and P.B. Katehi Radiation Laboratory The University of Michigan Ann Arbor, MI 48109 Running Head: Modeling of CPW Discontinuities Using the SDIE Method P.B. Katehi 3240 EECS Building The University of Michigan Ann Arbor, MI 48109 USA tel: (313) 747-1796 This paper has been accepted for publication in Journal of Electromagnetic Waves and Applications, JEWA.

Contents 1 Abstract 3 2 Introduction 4 3 Theory 6 3.1 Integral Equation for the Coplanar Waveguide............ 6 3.2 Convergence Properties of Network P,"~'.siieters........... 10 4 Numerical Results 11 4.1 Slotline Short Circuit...................... 11 4.2 Coplanar Waveguide Short Circuit.............. 1:3 4.3 Coplanar Waveguide Open End........... 14 5 Summary and Conclusions 15 6 Appendix A 17 7 Appendix B 21 8 Appendix C 23 9 Acknowledgment 25 2

1 Abstract This paper describes a rigorous numerical approach for characterizing coplanar waveguide discontinuities. This approach develops an integral equation for the unknown electric field distribution on the slot apertures of the coplanar waveguide and then solves it numerically in the space domain using the method of moments (SDIE). As a demonstration, this method is applied to derive frequency dependent equivalent circuits for the case of a shorted slotline and, a shorted and open-end coplanar waveguide. It is shown that a shorted slotline and a shorted coplanar waveguide always behave as inductive loads. However, in the case of an open end coplanar waveguide the even and odd excitations create different effects. Specifically, the even excitation corresponds to an inductive load while the odd excitation to a capacitive one. The results derived by the SDIE method have been extensively studied for numerical convergence and have been compared to other available theoretical or experimental data. The agreement is excellent. 3

2 Introduction The Coplanar Waveguide was introduced for the first time in 1969 by C.P. WVen [1] as an appropriate transmission line for nonreciprocal gyromagnetic microwave device applications due to its ability to generate nearly circularly polarized electromagnetic waves. This new line despite the fact that it exhibited many other properties could not compete with the already established microstrip line in hybrid technology and, as a result, coplanar lines attracted little attention in 1970's and early 80's. Recently, with the push to high frequencies and monolithic technology coplanar waveguides have experienced a growing demand due to their appealing properties [2], [3], [4]. With the increasing use of CPWs in monolithic circuits, the coplanar line technology cannot rely on the approximate semi-empirical techniques which were developed a few years ago. It needs more accurate frequency unlimited models which can result only from rigorous analytical methods. Although the infinite Coplanar Waveguide has been well characterized [4], very little is available in literature about Coplanar Waveguide discontinuities. The first few papers appeared recently and provide a rather limited study of a few simple structures. The characterization of these discontinuities was either experimental [5], [6], [10], [111 or theoretical [8]. [9],[12J. However, much more study is needed in order to understand the electromagnetic behavior of these structures and be able to use them in high frequency applications [14], [15], [16]. As a response to this need, this paper presents a general method which call be implemented easily in order to theoretically characterize coplanar waveguitde 4

discontinuities. This method is based on space domain integral equation approach (SDIE) and provides accuracy, computational efficiency and versatility in terms of the geometries it can solve. As an example, the SDIE method is applied to evaluate discontinuity characteristics such as scattering parameters and equivalent circuits in the cases of a slot-line short circuit, a coplanar waveguide short circuit and a coplanar waveguide open end. The theoretical data are compared to available experimental data for the case of shorted slot [17] and open end coplanar waveguide [5] and theoretical data for the case of shorted CPW [8], and show excellent agreement. Furthermore, the behavior of these circuits on semiconducting substrates is investigated and results are plotted as functions of frequency and other geometrical parameters. The extension of this method to more complicated discontinuities is also discussed. 5

3 Theory 3.1 Integral Equation for the Coplanar Waveguide Figure 1 shows a shielded coplanar waveguide excited at the side wall of the enclosing cavity by either a coaxial feed (odd mode) or a slotline (even mode). In both cases the coaxial line or the slot have been designed to allow only the dominant mode which has the electric field on the transverse plane. In response to this excitation, an electric field distribution E, develops on the two slots of the coplanar waveguide and is responsible for the propagation of electromagnetic power along the line. In the presence of a discontinuity the power which is distributed in the waveguide regions above and below the cavity is reflected back resulting also in a storage of electric and magnetic energies in the vicinity of the discontinuity. The reflected power and the stored energies provide information about the electromagnetic behavior of the discontinuity which can be put into the form of equivalent circuits or scattering parameters. In order to evaluate these parameters the original boundary problem is split into two simpler ones by introducing an equivalent magnetic current M,. This surface magnetic current radiates an electromagnetic field in the two waveguide regions so that the electric field boundary condition on the surface of the slots is satisfied. The remaining boundary condition to be applied is continuity of the tangential magnetic field on the surface of the slot apertures: n x (Hl - H2) = O (1) where H1,2 are the magnetic fields in regions 1,2 respectively and can be ex-'6

pressed in terms of the equivalent magnetic current density M, as shown below: H1,2 = ] ] G2((/) M.(r,)ds'. (2) In equation (2), S is the surface of the slot apertures, kl,2 are the propagation constants in the dielectric regions adjacent to the slot apertures and G. 2 are the dyadic green's functions in the two waveguide regions given in Appendix A. In order to simplify the solution of the above boundary value problem, the waveguide walls are considered infinitesimally thin and perfectly conducting. In addition, the coaxial or slot-line excitation is replaced by two ideal current sources which are out of phase and in phase respectively as shown in figure 2. For best accuracy in the solution these sources are placed at distances Ag/4 from the conducting wall. Under these assumptions equation (1) takes the form n x (Hl- H2) = J. (3) where J, denotes the ideal current sources and is given by: J. = 6(z - A,/4){J16(y - ya) + J6(y - y.2)} (4) with y,1, Y,2 the y-coordinates of the centers of the slots No.1 and No.2 respectively. In equation (4), the constants J1 and J2 denote the amplitude of the current sources and are given by J, = J2 = 1 for slot-line excitation (even) 7(

J1 = -J2 = 1 for coaxial excitation (odd) (6) In view of equation (2), equation (3) takes the form: n x j J [G - ds(21 Mr)ds =is (7) To obtain the unknown magnetic current distribution M,, equation (7) is solved with the application of the method of moments. The aperture of the CPW slots is divided into shells as shown in figure 3 and the unknown current density is expressed to be in the form of a finite double summation: N1, N, N1 N, M;('") = &s, E Vs,i, fi(y')gj(z') + a&' E V,,ij f,(z')gi(y') (8) i=l j=1 i=1 j=1 where fi(y')gj(z') {i = l,Ny;j = 1,N,} is a family of orthogonal rooftop functions. Each of these functions extends over a shell (i,j) with a width 1ei and length ty,. Equation (8) is then substituted into equation (7) and Gallerkin's procedure is applied to minimize the introduced error. In this manner, the original integral equation reduces into the matrix equation of the form: YY V Y) (.- IV:(9) where Vy, V, are the subvectors of unknown coefficients for the y,z components of the magnetic current distribution respectively and I,, I, are the known excitation subvectors. For the ideal current source excitation considered in this approach. the elements of these subvectors are given by: 8

I-j = 0 on every subsection (10) and 1 for even/odd excitation at the position of the first current source 1 for even excitation at the position of the second current source Iy ( ) -1 for odd excitation at the position of the second current source O otherwise The elements of the rectangular admittance submatrices of equation ( 9) are expressed in terms of multiple space integrals of the dyadic green's functions G. 2 and are given in Appendix B. For the solution of equation (7), most of the computation effort is spent on the evaluation of the elements of the admittance submatrices. The complexity of these computations comes from the double summations in the expression for GC,2 which are inherently slowly converging. Details on the numerical evaluation of these summations may be found in [19]. By solving the matrix equation (9), the unknown field distribution on the aperture of the coplanar waveguide slots is evaluated as fitnction of frequency and is used to derive the network parameters for the given discontinuity. Since the method for deriving the network parameters has been described extensively in [18], only a brief summary is given here (Appendix 9

c) 3.2 Convergence Properties of Network Parameters In the expressions for the elements of the admittance matrix as given by (B.1) - (B.4), the summations over m and n are theoretically infinite. For the numerical solution of the integral equation ( 7), these summations are truncated and the number of terms kept depends on the convergence behavior of the matrix. Due to the nature of the problem solved here, the above summations have a convergence behavior similar to the summations of [19] and therefore we follow the conclusions of that paper. Even if the convergence behavior of the elements of the admittance matrix affects the final solution it does not guarantee the convergence of the network parameters. An extensive discussion in [19] proves that the convergence of the various parameters depends on the basis functions, the partitioning of the structure (in this problem the aperture of the slots) and the products of the number of terms in the N summation with the subsection lengths along the longitudinal direction in each shell. These products define the Optimum Sampling Range Criterion. In the present work, the number of terms in the summations were chosen according to the above criterion. As a result, the study of the convergence behavior performed in this paper is limited to a few important parameters. Figure (4) shows the effective length versus samples per guide wavelength for the case of a shorted slotline printed on Alumina substrate with e, = 9.7, h = 0.635mm. The frequency of operation is 14.17 GHz. For the case of a slotline, convergence is achieved when 10

the partitioning of the line exceeds 60 samples per guide wavelength. Figure 5 shows a similar convergence behavior for the case of a coplanar waveguide short printed on a 0.635mm Alumina substrate with e, = 9.7 and excited by an odd mode excitation at f=7.09 GHz. As it can be observed, the convergence in this case is much slower. In all of the discontinuities considered in this paper, the number of subsections per guide wavelength was chosen so that 1 % accuracy is achieved. 4 Numerical Results In this section, numerical results are presented in the form of equivalent circuits for the case of a slot-line short circuit, a CPW short circuit and a CPW open circuit. Where possible, comparisons are made to other theoretical or experimental data for verification purposes. All the considered structures were shielded inside a cavity the dimensions of which were chosen to give cavity resonances above the operating frequency range. The top and bottom walls were placed at a distance H = 5h in order to minimize shielding effects. In addition, the coplanar wavegui(le length was approximately equal to one guide wavelength. In almost all cases the substrate was GaAs with the exception of a few structures on Alumina or Duroid which we characterized in order to compare with available data. 4.1 Slotline Short Circuit As a first example, the problem of a slotline short circuit printed on a GaA-1 substrate (et = 12) with the characteristics shown on Figure 6 was considered. Tle 11

circuit was excited with an ideal current source generator the frequency of which varied between 2GHz and 6GHz. This specific circuit was studied in order to compare theoretical data derived by this method to experimental results provided in [17]. Due to the fact that in a short circuit slotline there is storage of pure magnetic energy around the discontinuity, it is expected that the effect of the short circuit is equivalent to an inductive load. Using the equations presented in Appendix C, an equivalent normalized reactance X = jwL is evaluated and is plotted on Figure 6 as function of the frequency. From this figure it is obvious that the agreement between the theoretical and experimental data is very good for small values of the width-to-substrate thickness ratio. However as the values of W/h become larger than 0.5 the discrepancy observed is as high as 25%. Exactly the same disagreement was observed by R. Jansen [8] whose theoretical data compare very well with the ones derived by the method presented here. This is a good reason to believe that the discrepancy between theory and experiment originates at the experimental method and not in the numerical solution. Figure 7 shows the effect of the W/h ratio on the equivalent reactance of a shorted slotline printed on a 200 GaAs substrate between 30 GHz and 70 GHz. The geometry of the shielding cavity and the location of the slotline is shown on the same figure. As it can be seen, the normalized equivalent inductance is almost insensitive to frequency for slot widths less than one tenth of the substrate thickness. As W approaches five substrate thicknesses the variation with f becomes 12

faster. 4.2 Coplanar Waveguide Short Circuit Another discontinuity studied here, is the coplanar waveguide short circuit shown on Figure 8. The dimensions of the shielding cavity are appropriately chosen to prevent vertical resonances in the cavity. This coplanar waveguide discontinuity is printed on aO.635mm Alumina substrate and has the short circuit at a. distance L1 = from the cavity wall in order to prevent coupling between the discontinuity and the shield. This discontinuity is excited by an even and odd mode separately and the equivalent excess length parameter for both cases is evaluated as described in Appendix C. As with the case of a shorted slotline, a shorted coplanar waveguide behaves as an equivalent inductance. Figure 8 shows the normalized excess length as a function of the substrate thickness h/A0 for the case of odd excitation. On the same figure, the theoretical results derived by the SDIE method are compared to the theoretical data presented in [81 which were derived using a spectral domain cavity resonance method. - The agreement between the two theories is excellent. The effects of the same discontinuity were also studied for the case of a coplanar line printed on' a 2001i GaAs substrate. The excess length is evaluated and plotted for both the odd (Figure 9a) and even excitation (Figure 9b) as a function of tihe width W/h and slot separation s/h for frequencies varying between 20 GHz and 60 GHz. The effect of the normalized separation distance s/h on the magnetic field around the discontinuity for the odd and even case is different. For the odd case 13

due to the fact that the two lines radiate fields which are out of phase, a great deal of cancellation takes place which decreases as s becomes larger. As a result the excess length reduces considerably as s/h becomes smaller than 1 and will tend asymptotically to zero as s -- 0. The sensitivity of the excess length is much less in the case of even excitation where the two slots radiate in phase and magnetic fields around the discontinuity interfere constructively to give an enhanced effect (Figure 9b). The effect of the slot width in both excitations is the same. The larger the W the more the fringing and therefore the more the magnetic energy density stored around the discontinuity which in turn behaves as a larger inductor. For the specific coplanar waveguide considered in this example, the even mode excess length is about twice as large as the odd mode excess length. 4.3 Coplanar Waveguide Open End The last example considered in this paper is a coplanar waveguide open end (see figure 1). At first, for verification purposes, the open circuit equivalent length extension ( as seen at the plane of the open end of the center conductor) of an odd mode channelized coplanar waveguide printed on Duroid was plotted as function of the frequency and it was compared to experimental data published in [7] (see Figure 10). TKe agreement is very satisfactory considering that the experimental values oscillate around the theoretical ones. The effects of the same discontinuity printed on a 500p GaAs substrate in the form of an equivalent normalized reactance were studied for both the even and odd excitation and results are presented on figures (Ila) and (lb). An even mode 14

excitation in a coplanar waveguide corresponds to an electric wall at the center of the gap. In this case, the discontinuity behaves as an equivalent inductance which comes as a combination of two inductances LY dependent on the y-component and L, dependent on the z-component of the of the electric field distribution on the slot aperture (see figure 1la). As a result, Ly, will increase with s and decrease with g while Lt will increase with g and decrease with s. The combination of these two inductances results in an almost linear variation of the total inductance with g for large separations and almost no-variation for small separations (Figure 11a). An odd mode excitation corresponds to a magnetic wall at the center of the gap and, therefore, the discontinuity behaves as an equivalent capacitance which depends on the z-component of the electric field distribution on the slot only. As a result, the equivalent reactance will increase with decreasing separation distance s/h and will increase with increasing gap size. The above behavior is predicted very nicely by the theoretical model derived in this paper and is presented on Figures (11lb). 5 Summary and Conclusions This paper presented a rigorous method for the characterization of shielded coplanar waveguide discontinuities. The approach is based on an integral equation which is solved numerically in the space domain. As a demonstration, the presented method was applied on a slotline short circuit and coplanar waveguide short circuit and open end printed on GaAs substrates. Equivalent circuits for these discontinuities were evaluated numerically over a wide frequency range for various slot widths and separations. 15

The convergence behavior of the derived results and the numerical stability of the method were studied extensively by performing several computational experiments. The theoretical results, whenever possible, were compared to available theoretical or experimental data and they showed excellent agreement. The SDIE method presented in this paper is very accurate, computationaly efficient and can be easily applied to other more complicated structures.

6 Appendix A As shown in Fig. 12, the original problem is reduced to deriving the Green's dyadic function in both regions directly above and below the slots. The transmission line theory is used to transform the surrounding layers into impedance boundaries. Using the equivalence principle, the problem is divided into four subproblems as shown in Fig. 12. So, we have to solve for the Green's functions in both regions due to magnetic currents in the y and z directions. After this has been accomplished, the continuity of the tangential fields at the interface will be used to arrive at the integral equation. In the analysis, LSE (TE,) and LSM (TM,) modes are used to derive the Green's function. The main steps in the derivation of the Green's function for an infinitesimal z- directed magnetic current inside a cavity (with impedance boundary top side), Fig. 12, will be presented here. The following vector potentials for the LSM and LSE modes are assumed: A~asV F=a,; (Al) Through the manipulation of Maxwell's equations along with 17

H =-V x A (A2) E = — V x F (A3) one can obtain the field components in terms of the magnetic and electric vector potentials which satisfy the homogeneous wave equation. The differential equations for A and F are solved for the unknown field components considering the following boundary conditions (see Fig. 13): E. = En at x = x' (A4) l= HII at x = x' (AS) H' = HII at x = z' (A6) (E)LSE = zLS4 at x = d (A7) Hz''i)LS = LS at x= d (A8) E" - El = 6(2 - z')6(y - y')6z - z') (A9) where 4LS and ZLSM are the LSE and LSM impedances looking up at x = dl which can be computed using transmission line theory. That is, each layer is simply considered a transmission line with a characteristics impedance (ZOLSE or ZLSM) and an eigenvalue ki where 18

ki2 + k2 + k2 = W2,/ (ZLSE)i - and (Z =LSM) k' I Wei The fields which are evaluated through the solution of the wave equation constitute the components of the dyadic Green's function in region (1) due to a z - directed magnetic dipole. In the same manner the dyadic Green's function to a y - directed magnetic dipole in region (1), and the corresponding ones in region (2), can be obtained. The components of the dyadic Green's function for the coplanar waveguide problem are given by:,~<:/:, =2e. I1 + 2Ql = m=O crtO - aLk kkvPI +, k- Q1 m=O nIO x I sin(k,y' cos(k,z') sin(k,y) cos(k,z) (A10) G,(r')(F/) - - 0 2 kVo -, pi- cl ] = or3aLk2 _k2EP1Q sin(k,y') cos(k,z') cos(k,,y) sin(k,z) (Al 1) 00 co 2 It,- Q m,,Oa. kz, -- kI 0 cos(k,y') sin(k,z') sin(k,,y) cos(k,z) (A12) G02e] 1 o.o aL k k2 + cos(kuy') sin(k,z') cos(k,y) sin(k,z) (A 13) where

kp = (k), WL + jk ZLSE tan(k.1di) k\t I,, Z1LZSE +jw1s tankd) (A14) we, + jWel LSM tan(k.1 di) Qi =,, i weizfSM + jk, tan (kd) (A15) en 1 n=O =2 n a 0 (A16) em = 1 m=O = 2 m#O (A17) k2 = W2;zi (A18) and k2 +k2+k2 = k2. In the above expressions G() denotes Hj4) due to an infinitesimal M, at z = 0. The components of G2 are essentially the same as (A10) - (A18) with the following changes: z LSE SE zLSM. zLSM (A19) Cn, Cm ) -en -em kl k2 Il,el 2i, d e2'

7 Appendix B Yv (i, j /mn) = ( (k2P + k 2Q1) k',- e k(kYP2 +kQ2)) (B1) aL k2,-k2, t- k. -kk (kP2 + k-Q2)) Iv(,i)l(zj)hI(yi)Ih(ZY) (B2) Y,.(i,j/m,n) = Y,(i,j/m,n) = L kk. * (k', - k,(P1 - Ql) + k2 -kQ2) 1(zi)I(yj)I2(yi)I (Yj) (B3) 21

where =(Yi) k'I 4 sin(ky,;) k2' k*2 sin(kk*-'-' sin(k + kv)lvy) sin( (kyI-k)lyi) (B4).sin( 4 sin( (B4) 4 4 k' 4 sin(k,z+ ) k2() - k2 sin(k-'2) 2 (k' + kx)lxi (k_ - k_)___ (zi) =- k (sin(k,(zi + - sin(k,(z, - 2)))( I2(zi,) k- (i-(',) + ) -sin(k,(yc, - (B7) and k' = w= /#a (B8)

8 Appendix C In a lossless transmission line, the propagation constant %, is Y9 =jig (C1) where the phase constant /g is given by 2ir P3 -A (C2) where. A is the guide wavelength. This can be calculated as two times the distance between two adjacent current maxima. Another parameter of interest is the effective length extensions Al0c or AlC. AIOC (Adl,) represents the length of an ideal open circuited (short circuited) transmission line which,.as a continuation of the slots, would present the same reflection coefficient, at the plane of the discontinuity, as the open end (short end) discontinuity. They can be computed as Wsc = -dmax Aloc = ~-I - dmin 23

where dmar (dmin) is the distance from the plane of the discontinuity to the closest maximum (minimum). In the case of an open-end CPW with even mode, Ale, can- be defined as (Al).oc) = - dma, since it has shown an inductive reactance at the plane of discontinuity. The normalized inductive reactance for the shorted slot line (or CPW) can be calculated as XL = tan(P A 1.) = wL The normalized capacitance reactance for the open end CPW (odd mode) can be computed as X. cot(A, A 4) =4 C In the same manner, the normalized inductive reactance for the open end CPW (even mode) is given by XL =tan(i3(al.). 2 wL The impedance at any point on the transmission line can be computed from the above teactances. 24

9 Acknowledgment This work was supported by the National Science Foundation under Contract ECS-8657951. The authors wish to thank Mr. George Ponchak and Dr. Rainee Simons at NASA Lewis Research Center for their helpful comments.

References [1] Cheng P. Wen, " Coplanar Waveguide: A Surface Strip Transmission Line Suitable for Nonreciprocal Gyromagnetic Device Applications ", IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-17, No. 12, December 1969, pp.1087-1090. [2] M. Houdart, " Coplanar Lines: Application to Broadband Microwave Integrated Circuits ", Proc. of 6th European Microwave Conference (Rome), 1976, pp. 49-53. [3] R.A. Pucel, " Design Considerations for Monolithic Microwave Circuits ", IEEE Trans. of Microwave Theory and Techniques, Vol.29, June 1981, pp.513534. [4] K.C. Gupta, R. Garg and I.J. Bahl, Microstrip Lines and Slotlines, Dedham, MA: Artech House, 1979. [5] Rainee N, Simons and George E. Ponchak, " Modeling of Some Coplanar Waveguide Discontinuities ", IEEE Trans. on Microwave Theory and Techniques, Vol. 36, December 1988, pp.1796-1803. [6] George E. Ponchak and Rainee N. Simons," Channelized Coplanar Waveguide PIN-Diode Switches", 19th European Microwave Symposium Digest, September 1989, Wembley, London, pp.489-494. 26

[7] R.N. Simons and G.E. Ponchak, " Channelized Coplanar Waveguide: Discontinuities, Junctions and Propagation Characteristics ", Proc. of 1989 IEEE MITT-S International Symposium, Long Beach, Ca, June 1989, pp. 915-918. [81 R.H. Jansen," Hybrid Mode Analysis of End Effects of Planar Microwave and Millimetrewave Transmission Lines ", lEE Proc., Vol. 128, Pt. H, No. 2, April 1981, pp.77-86. [9] C.W Kuo and T. Itoh, " Characterization of the Coplanar Waveguide Step Discontinuity Using the Transverse Resonance Method ", 19th European Microwave Symposium Digest, September 1989, Wembley, London, pp.662-665. [10] N.H. Koster, S. Koblowski, R. Bertenburg, S. Heinen and I. Wolff, " Investigation of Air Bridges Used for MMICs in CPW Technique ", 19th European Microwave Symposium Digest, September 1989, Wembley, London, pp.666671. [11] G. Kibuuka, R. Bertenburg, M. Naghed and I. Wolff, " Coplanar Lumped Elements and their Application in Filters on Ceramic and Gallium Arsenide Substrates", 19th European Microwave Symposium Digest, September 1989, Wembley,; London, pp.656-661. [12] R.W. Jackson, " Mode Conversion at Discontinuities in Finite-Width Conductor-Backed Coplanar Waveguide ", IEEE Trans. on Microwave Theory and Techniques, Vol.37, No. 10, October 1989. 27

[13] R.W. Jackson, " Considerations in the Use of Coplanar Waveguide for Millimeter Wave Integrated Circuits ", IEEE Trans. on Microwave Theory and Techniques, Vol. 34, December 1986, pp. 1450-1456. [14] M. Riaziat et al.," Monolithic MillimeterWave CPW Circuits ", Proc. of the 1989 IEEE MTT-S International Symposium, Long Beach, CA, pp. 525-528. [15] T. Hirota, Y. Tarusawa, and H. Ogawa, " Uniplanar MMIC Hybrids-A Propose of a New MMIC Structure ", IEEE Trans. of Microwavec Theory and Techniques, Vol.35, June 1987, pp.576-581. [16] P.A.R. Holder, " X-Band Microwave Integrated Circuits Using Slotlines and Coplanar Waveguide ", Radio Electronics Engineering, Vol.48, January/February 1978, pp.38-42. [17] J.B. Knorr and J. Saenz, " End Effect in Shorted Slot ", IEEE Trans. on Microwave Theory and Techniques, Vol. 21, No.9, September 1973, pp. 579580. [18] P.B. Katehi and N.G. Alexopoulos, " Frequency-Dependent Characteristics of Microstrip Discontinuities in Millimeter Wave Integrated Circuits ", IEEE Trans. on Microwave Theory -and Techniques, Vol. MTT-33, No. 10, October 1985, pp.1029-1035. [19] L.P. Dunleavy and P.B. Katehi," Generalized Method for Analyzing Shielded Thin Microstrip Discontinuities ", IEEE Trans. on Microwave Theory and Techniques, Vol.36, No.12, December 1988, pp.1758-1766. 28

List of Figures 1 A CPW open circuit discontinuity inside a cavity.......... 31 2 Excitation mechanisms in a Coplanar Waveguide.......... 32 3 Geometry for use in the expansion of the equivalent magnetic current for a CPW open circuit.......... 33 4 Convergence behavior of the normalized equivalent excess length Ae/h for a shorted slotline (h = 0.635mm, h/Ao = 0.03, W/h = 0.2, a/h = W/h + 20, L1 = 5h, e4 = 9.7)................ 34 5 Convergence behavior of the normalized equivalent excess length el/h for a shorted CPW (odd mode) (h = 0.635mm, h/Al = 0.015, W/h = 0.4, s/h = 0.2, a/h = 2(W/h) + s/h + 20, L1 = 5h, e, = 9.7)........................ 35 6 Normalized inductive reactance of a shorted slot line as compared to experimental results from [17] (h = 3.073mm, a/h = (W/h) + 15, LI = 3h, e = 12.0, **** W/h = 0.892 [16], o o o o W/h = 0.562 [16], x x x x W/h-= 0.221 [16], This Paper)....... 36 7 Normalized inductive reactance of a shorted slot line on GaAs substrate. (h = 0.2mm, a/h = (W/h) + 15, L1 = 5h, e, = 13.1).... 37 8 Effective length extension of a shorted CPW (odd mode) as compared to theoretical results from [8] (h = 0.635mm, a/h = 2(W/h) + s/h+20, LI = 5h, e, = 9.7, x x x x Data from [7], s/h = 0.2, - - - s/h = 1.0)..................;3s 29

9 Effective length extension of a shorted CPW on GaAs substrate. (h = 0.2mm, a/h = 2(W/h) + s/h + 20, L1 = 5h, X. = 13.1, a: odd mode b:even mode, -- - s/h = 1.0, s/h = 0.2). 39 10 Effective length extension of an open end channelized CPW (odd mode) as compared to experimental data from [7] (h = 03.175mm, a = 5.08mm, g = 0.381mm, W = 0.254mm, s = 1.14mm, e, = 2.2, 0Cl 0 Experimental data [17],. This paper)........ 40 11 Normalized reactance of an open end CPW on GaAs substrate as a function of the gap width. a: even mode, b: odd mode (h = 0.5mm, a/h = 2(W/h) + s/h + 10, W = 0.25mm, L1 = 5h, f = 20GHz, e, = 13.1)....................... 41 12 The four equivalent subproblems. (All the sides excluding the impedance boundaries are assumed perfectly conducting............. 42 13 The magnetic source was raised to apply the boundary conditions 43 30

Figure 1: A. CPW open circuit discontinuity inside a cavity 3.1

SLOT LINE CPW V- 7/ COAX CPW Figure 2: Excitation mechanisms in a Coplanar Waveguide

, zi.z Figure 3: Geometry for use in the expansion of the equivalent magnetic current for a CPW open circuit.

0.50 = 0.45 OClC) w _e l 0I.4 = 0.30 0.30 O. 20. 40. 80. 80. Number of subsections / uide wovele I ngth Figure 4: Convergence behavior of the normalized equivalent excess length Al/h for a shorted slotline (h = 0.635mm, h/A, = 0.03, W/h = 0.2, a/h = W/h + 20, L1 = 5h, (, = 9.7). 34

0.16 e 0.14._ C I- L L 0.10 0. 50. 100. 150. Number of subsections / guide wavelength Figure 5: Convergence behavior of the normalized equivalent exceu length Al/h for a shorted CPW (odd mode) (h = 0.635mm, h/A, = 0.015, W/h = 0.4, s/h = 0.2, a/h = 2(W/h) + s/h + 20, L = 5h,, =9.7). 35

0. - a 05 *._~ 04 - / W/h=. 892.0.4 ~~~O 0.562 C 0.0 0.221 0.2 0.0I. 2. 3. 4. 5 Frequency, (GHZ) Figure 6: Normalized inductive reactance of a shorted slot line as compared to experimental results from (171 (h = 3.073mm, a/h = (W/h) + 15, LI = 3h, ~, = 12.0, * * * * W/h = 0.892' [16], o o o o W/h = 0.562 [161, x x x x W/h = 0.221 [161, This Paper). 36

0.8' --- o.5w W = W/h=5 C 0,4 Ou 0. 0.3 09 i ~~~~~~0.5 0.2 1 0.Z 0.1 0.0 I I 20. 30. 40. 50. 60. 70. 60. Frequency, (GHz) Figure 7: Normaised inductive reactance of a shorted slot line on GaAs substrate. (h = 0.2mm, a/h = (W/h)+ 15, Li = 5h, c, = 13.1).'37

O. -- w L 0. Wi r s W 0.5 _ _ L 0.4' - W/h=1 0- 0.4 O~~~~~~~~t~~0.4 0.1 2 x x 0.2 ~~~~~~~~0.0 O. _ - x.. X. - X.......'0.2.000.005.010.015.020.025 Substrote thickness to wavelength rotio, h Figure 8: Effective length extension of a shorted CPW (odd mode) as compared to theoretical results from [8] (h = 0.635mm, a/h = 2(W/h)+s/h+20, LI = 5h, c, = 9.7, x x x x Data from (71, --- a/h = 0.2, - - - - s/h = 1.0). 38

k — L --- T St a:odd mode 0.5 0.5 10. 20. 30. 40. 50. 80. 70. Frequency, (GHz) 1,2 W W a \ b:evenmode %.~ %1 *,. N. so. 40o. so.' O. 7O. f requeay. (GHz) Figure 9: Effective length extension of a shorted CPW on GaAs substrate. (h 0.2mm, a/h 2(W/h)+/h+20, L1 = h, = 13.1, a oddmode b: evenmode, -— /h = 1.0, - a/h = 0.2). 39

Iwls Iwl 25 co 20 IO, O 150 0, 00 O 210 0 0 2 4 6 8 10 12 14 16 18 Frequency (GHz) Figure 10: Effective length extension of an open end channelised CPW (odd mode) as compared to experimental data from ([7 (h = 03.175mm, a = 5.08mm, g = 0.381mm, W = 0.254mm, s = 1.14mm, (, = 2.2, 0 0 O 0 Eperimental data [17, -- This paper). 40

2.0 /~~~~ ~/ s/h-5 i. t / a:cven mode 41. g 0.0 I 0.0 o.1 0.2 0.3 0.4 0.5 gop width, (rm);L. W 2 2 aX2= 1. 0. I 0.0.1 O.2 0.3 0.4 0.6 bp widtl (,.m) Figure 1: Normalised reactance of tau open end CPW on GaAs substrate as a function of the gap width. a even mode, b odd mode (h = 0.5mm, a/h = 2(W/h) + /h + 10, W = 0.25mm, L1 = 5h, f = 20GHz, ~, = 13.1).

x=d2 L Y -, a,,, (Impedance boundary sides) LSE LSM Zi or Z d I i d I.: xt.::IIyllhHI:(1) _,_, ~~X X d2 d2 d2 (Impedance boundary sides) LSE LSM Z2 or Z24 igure 12: The four equivalent subproblemn. (All the sides excluding the impedance boundaries re assumed perfectly conducting 42

LSE LSM X ~r Zlor Z ~d II Figyre 13: The magnetic source was raised to apply the boundary conditions

APPENDIX D

COPLANAR WAVEGUIDE DISCONTINUITIES FOR P-I-N DIODE SWITCHES AND FILTER APPLICATIONS N.I. Dib, P.B. Katehi Radiation Lab., University of Michigan, Ann Arbor, MI G.E. Ponchak, R.N. Simons NASA, Lewis Research Center, Cleveland, OH ABSTRACT the method of moments solution. Thus, the ON and OFF states of the switch will be predicted from the same strucA full wave space domain integral equation (SDIE) anal- ture without having to go to different CPW discontinuities ysis of coplanar waveguide (CPW) two port discontinuities for reaization of these states. is presented. An experimental setup to measure the Sparameters of such discontinuities is described. Experimental and theoretical results for CPW realizations of pass-band 2 THEORY and stop-band filters are presented. The S-parameters of such structures are plotted in the frequency range 5-25 GHz. The coplanar line under consideration is shown i shown in Fig.3. The measurements were performed using an open CPW structure. However, in the theoretical analysis the CPW is 1 INTRODUCTION assumed to be inside a rectangular cavity of perfectly conducting walls as opposed to experiments where open strucThe coplanar waveguide (CPW) was introduced for the tures were measured. The cavity dimensions were chosen first time in 1969 by C.P. Wen (1) as an appropriate trans- such that the CPW fundamental mode is not affected by mission line for nonreciprocal gyromagnetic device appli- higher order cavity resonances. cations. Recently, with the push to high frequencies and The original boundary problem is split into two simpler monolithic technology, CPWs have experienced a growing ones by introducing an equivalent magnetic current M, on demand due to their appealing properties (2,3). However, the slot aperture. This surface magnetic current radiates an the extent of applications of CPW circuits is limited due electromagnetic field in the two waveguide regions (above to the unavailability of circuit element models which can and below the slot) so that the continuity of the electric be incorporated into CAD programs. Houdart (2) proposed field on the surface of the slots is satisfied. The remaining several configurations using the coplanar line technique in boundary condition to be applied is the continuity of the order to realize basic types of elements required in MIC's. tangential magnetic field on the surface of the slot apertures Microwave switches are circuit elements widely used in which leads to the following integral equation phase shifters and radiometers. A CPW switchable attenuating medium propagation, SAMP, switch has been demon-' =, (1) strated by Fleming et al (4). This device is useful for GaAs MMIC circuits but it is not easily incorporated into MIC's where 42 are the magnetic dyadic Green's functions in the on passive substrates such as alumina or duroid. P-i-a diodes are good rmicrowave switches since the impedance of two waveguide regions (5) and j. denotes the ideal current the diode can be changed from a very high value to nearly source feeding the CPW, (gap generator model) zero in a short time. Recently, CPW p-i-n diode switches The integral equation (1) is solved using the method were proposed and fabriuc (6). Figl shows an switch were of moments where the unknown magnetic current is expanded in terms of rooftop basis functions. Then, Galerkin's a diode is mounted across the open end of a quarter wave- length stub which is in series with the center strip conductor method is applied to reduce the above equation to a linear of the CPW. The two states of the switch, ON and OFF, can be realized sepat by appropriate CPW disconti- Y Y,, VW (2) nuities which in turn can be used to build pus-banand dy, (2) stop-band filters respectively (2,7,8). / In this paper, the space domain integral equation (SDIE) where Yi(i = y, z; j = y, z) represent blocks of the admethod, presented in (5), is extended to analyze the S- mittance matrix (5), VK is the vector of unknown y and z parameters of the CPW realization of the above switches, magnetic current amplitudes, and Ii is the excitation vector Fig.2. Experimental results are compared to theoretical which is identically zero everywhere except at the position data and a good agreement is achieved. The experimen- of the sources. Finally, the equivalent magnetic current dista setup, measurement technique and the sources of error tribution and consequently the electric field in the slots are are described. As a next step, the study of the switches will obtained by matrix inversion. be completed by including the diode as a lumped element in Away from the discontinuity, the slots fields form st arnd To be published in the 1990 IEEE International Microwave Symposium Digest, Dallas.

ing waves of the fundamental CPW propagating mode. Us- the variation in the gold thickness. The circuit dimensions ing the derived electric field, an ideal transmission line varied by, at most, 1% compared to the designed dimenmethod as described in (9) is applied to determine the scat- sions. The probe placement was repeatable to 5 Am. The tering parameters of the two port discontinuity. probe pressure created the largest uncertainty in the measurements since it changed the contact between the alumina and the duroid substrates. One further consideration is that the spacing between circuits on the substrate was approxThe circuits were fabricated using a liftoff procedure. A imately (S + 2W)/2 which resulted in a finite size ground 2.8 /sm gold layer was electron beam evaporated onto a 25 planes. mil thick polished alumina substrate (e, = 9.9). The alumina substrate was placed on a 125 mil thick 5880 RT/Dur- 5 CONCLUSIONS oid (e, = 2.2) substrate with copper cladding on one side to serve as the bottom ground plane. The space domain integral equation method solved by The RF measurements were performed using HP8510 the method of moments (Galerkin's technique) in conjuncANA and a probe station with design techniques DC - tion with simple transmission line theory was applied to an26 GHz probes. In order to measure the circuit elements, alyze CPW realizations of pass-band and stop-band filters. an LRL calibration (10) was performed to remove the ef- An experimental setup to measure the S-parameters of those fects of the system to the reference plane PP' (Fig.4) using structures has been described and the various sources of erEEsof ANACAT software (11). The calibration standards ror were discussed. The agreement between the theoretical shown in Fig.5 were fabricated on the same substrate as the data and the experimtal measurements was very good, cicuits. thus, the validity of both results is verified. 4 RESULTS AND DISCUSSION 6 ACKNOWLEDGMENT For the soluion of (2), most of the computation effort is The theoretical part of this paper was supported by the spent on the evaluation of the elements of the admittance National Science Foundation under contract ECS-8657951. submatrices. These elements involve double summations in their computation which are inherently slowly convergent. References Details on the numerical evaluation of these summations may be found in (12). Extensive numerical experiments [1] C.P. Wen, "Coplnar Waveguide: A Surface Strip Transmis. were performed on the convergence of the S-parameters with sion Line for Nonreciprocal Gyromanetic Device Applications," respect to the above summations and the number of basis IEEE 7The. os Moiecws Theoy and Techniques, Vol. MTTfunctions used in the expansion of the slot field. Both were 17, pp.1087-1090, Dec.1969. chosen such,that at least 2% accuracy is achieved. (2] M. Houdart, "Coplanar Lines: Application to Broadband MiFig.6 shows the scattering parameters for the short cir- crows nteated circuits," Proc. of 6th European Microwave cuit CPW stub (SC-CPW stub) of length 500S m. It can be Cosnfres (Roes), pp.4953, 1976. seen that the agreement between the theoretical and exper- (3] K.C. Gupta, t. Garge and I.J. Bahl, Miretrip Lines and Slotimental results is very good. The differences can be due to hs**, Dedham, MA: Artech oun, 1979. conduction losses and radiation losses since an open struc- [4] P.L. Fl1ming, T. Smith, H.E. Carbon, and W.A. Cox, "GaAs ture was used in the mesurements. Figl.7 shows the scat- SAMP Device for Ku-Band Switchingl," IEEE TIts. on Mitering parameters for the SC-CPW stub of length 1500 mrm. crouss T&r and T.cniques, Vol. MTT-27, pp.1032-1035, From these results it can be observed that a resonance oc- 1979. cures when the stub length is slightly less than a quarter [51 N.I. Dib and P.B. Katehi, Modeling of Shielded CPW Diswavelength (Aw - 6.14mm at f = 21 GHz). This is due continuiti Uin the Spuec Domain Interl Equation Method to the inductive reactace. which exists at the end of the (SDIE)," nbmitt to JEWA, Spcial Ine on Electromagnetun stub as a result of the ead Iect. A stop-band filter can be S Weodou. realized by cascading several SC-CPW stubs in series. (61 G.E. Ponchack and ILN. Simor, "Chnnelied Coplanar WavegFigs.8 and 9 show the scattering praeters for the open uide PIN-Diode Switches," 1t& 19 opS ta Maicrowce Symposium circuit CPW stub (OC-CPW stub) of length 500 im and Dig"t (Lneu), pp.4994, 198. 1500 #m respectiv It ca be seen from Fig.9 that St [7n R.N. Bata, Desig of Micrastip Spur-Line Band-Stop Fal(St2) has its minimum (maximum) value at f = 19GHz trs" INu. Me Opis enO d Ag tie, Vol.1, No 6. at which A,/4 1.7?mm. This is due to the capacitive pp.20-214, Nov.19?. end effect of the stub. Such structures operate as pasw 81 D.F. Wilhiau and S.E. Schwus, Design and Performance band filters. The agreement between the theoretical and CopinAr Waeguid Band-Pass Filters," IEEE Tran. on i,. experimental results is very good. crouwee hesor and TecAsqu*, Vol. MTr-31, pp.558-566. Jul7 The errors associated with the measured data are nonre- 1963. peatability in probe placement and pressure and circuit ab- [9) P.B. Ktei, A Geralid Mod r the Evaluation of u u ricatioa tolerances. The circuit dimension variations acro Coupling in Microstrip Arrays, IEEE It7m. on Astern*.,,d the 2in. x 2in. alumina substrate were negligible as was Propgioe, Vol. AP-35, pp.12-133, Feb.1987.

(101 P.R. Pantoja, M.J. Howes, J.R. Richardson, and R.D. Pollard, "Improved Calibration and Measurement of the Scattering Parameters of Microwave Integrated Circuits," IEEE Truns. on Microwave Theory and Techniques, Vol. MTT-37, pp.1675-1680, Nov. 1989. (11] EEsof Inc., -Anacat Users Manual. [121 L.P. Dunleavy and P.B. Katehi, Generalized Method for Ana- Lciration lyzing Shielded Thin Microstrip Discontinuities," IEEE Trans. on I lstend 1 Microwave Theory and Techniques, Vol. MTT-36, pp.1758-1766, Dec. 1988. ClrIu I Circuit reference Coax cWl Coax reerence P / r reerence........,... I I - - _ _ _- _ ~:.:.:.::.:.::-:::::::.::.:.:........ Figure 1: Schematic diagram of CPW p-i-n diode switched serietub Figure 4: Basic concept of de-embeding. switch. CPW probes Fiu /(er5u:-Wstand-tars GrLuRa ) Cciba strio on pi Da' r 3 A,, i In L [ I -t............. Figue 5: CPW *tum fot &RL calibr&tioa: (a) Delay,r'-,; Fisure 3: A coplnar waveguide inside ~ cavity. L,, (b) Delay of length LJ, (C) Refection of arbitrary marn&,tu:

o 01 SII THEORETICAL.+ US- a CKS1. IX.i. - THEORETICAL Q 000 000 QOQO 1 — I 3' 1 r 1 1' 7 I W2 -2. 000 13. 00' F1180~8Q! ~ 0 t-o0s aX. 00r Figure 6: S-parameters for SCCPW stub with L=500 pm, (Di = Figure 8: S-parameters for OC-CPW stub with L=500;pm, (Di = 25mil, D2 = 125mil, e,e = 9.9,te -- 2.2). 25mil, D2 = 125mi,,eI = 9.9,,2 =- 2.2). Oa KSal. 011S-t THEORETICA a0 On cosIl - THEOIITICAL 41 00 1 — I -~- 17 1 ~~ ~Lq( ~II Q~dO r I r i r a-1 000'a "' i 00! Figure T: S-paramters for SC-CPW stub with L=1500 pm, (Other Figure 9: S-paraeter for OC-CPW stub with L=1500,,,R dimensions are the same as in Fig.6). dimensions am the ame as in Fig.8).

APPENDIX E

COPLANAR WAVEGUIDE DISCONTINUITIES FOR P-I-N DIODE SWITCHES AND FILTER APPLICATIONS N.I. Dib, P.B. Katehi Radiation Lab., University of Michigan, Ann Arbor, MI G.E. Ponchak, R.N. Simons NASA, Lewis Research Center, Cleveland, OH ABSTRACT A full wave analysis of coplanar waveguide two port discontinuities using the space domain integral equation method (SDIE) is presented. The experimental setup to measure the S-parameters of these discontinuities is also described. Experimental and theoretical values of the S-parameters for coplanar waveguide (CPW) pass-band and stop-band filters are plotted in the frequency range 5-25 GHz. In addition, a lumped element equivalent circuit is proposed to model these filters. The values of the capacitances and inductances, in this equivalent circuit, are plotted as a function of the stub length. This paper has been submitted to IEEE Transactions on Microwave Theory and Techniques (symposium issue), March 1990.

1 INTRODUCTION The coplanar waveguide (CPW) was introduced for the first time in 1969 bv C.P. Wen [1] as an appropriate transmission line for nonreciprocal gyromagnetic device applications. Recently, with the push to high frequencies and monolithic technology. CPWs have experienced a growing demand due to their appealing properties [2.:3]. However, the extent of applications of CPW circuits is limited due to the lack of specific circuit components and the unavailability of circuit element models appropriate for computer aided design. Microwave switches are circuit elements widely used in phase shifters and radiometers. A CPW switchable attenuating medium propagation, SAMP, switch has been demonstrated by Fleming et al [4]. This device is useful for GaAs MMIC circuits but it is not compatible with MIC's printed on passive substrates such as alumina otduroid. P-i-n diodes are good microwave switches since the impedance of the diode can change from a very high value to nearly zero in a short time. Recently, CPW p-i-n diode switches were successfully designed by Ponchak and Simons [5] with insertion loss of 1.0 dB, isolation of 19dB at 9 GHz and with switch times of few nanosecondk. Fig.la shows a switch where a diode is mounted across the open end of a cljuar.eX wavelength stub printed on the center strip conductor of the CPW. The two stato, of the switch, ON and OFF, can be realized separately by appropriate CP\W 4di 2

tinuities shown in Fig.l(b) and l(c). These discontinuities can also be used to build pass-band and stop-band filters [2, 6, 7]. In section 2, the space domain integral equation (SDIE) method, presented in [S], is extended to analyze the S-parameters of the CPW realizations of the above switches. The experimental setup and the measurement technique are described in section 3. Theoretical and experimental results for the scattering parameters are presented in section 4 and are in very good agreement. Furthermore, a lumped element equivalent circuit is proposed to model these pass-band and stop-band filters from 5 GHz to their first resonance. Values of the capacitances and inductances are presented as a function of the stub length. 3

2 THEORY 2.1 Derivation of the Integral Equation Figure 2 shows a shielded coplanar waveguide. This structure can be excited in the odd mode (coplanar waveguide mode) or in the even mode (coupled slot mode). The cavity dimensions were chosen such that the CPW fundamental mode is not affected by higher order cavity resonances. Since the theoretical analysis has been presented in [8], a brief summary of the SDIE method will be given here. The original boundary problem is split into two simpler ones by introducing an equivalent magnetic current M, on the slot aperture. This surface magnetic current radiates an electromagnetic field in the two waveguide regions (above and below the slots) so that the continuity of the electric field on the surface of the slots is satisfied. The remaining boundary condition to be applied is the continuity of the tangential magnetic field on the surface of the slot aperture, n x (H1 -H2)Jo = where J. denotes an ideal current source used to model the excitation mechanism. I,(l are the magnetic fields in the regions above and below the slot aperture respectl f'a and can be expressed in terms of the equivalent magnetic current density Ml, as ia\ 4

below:,2 = s Gl,2(r/) 2Ma(r)ds'. (2) where Gt,2 is the magnetic dyadic green's function in the two waveguide regions above and below the slot aperture. The expressions for the components of this dyadic function are given in the Appendix. In view of equation (2), equation (1) takes the form: n x |[GI - G2] M,(r)dsr = J. (3) where iA is the vector normal to the dielecteric interface pointing towards region 1. To obtain the unknown magnetic current distribution M,, equation (3) is solved by applying the method of moments. First, the unknown current density is expressed as a finite double summation: N, No Ny Ns Mi(f') = a E E Vyijfi(y')gj(z') + a E E Vz, fj(z')g,(y') (4) i=l j=l i=l j=1 where f.(y') gj(z') {i =, N,;j = 1, N5) is a family of orthogonal rooftop functions and V/,oj, V/;,, are the unknown coefficients for the y and z components of the magnet ic current density. Equation (4) is then substituted into equation (3) and Gallerkitl'u procedure is applied to minimize the introduced error. In this manner, the omiir:i integral equation reduces into the matrix equation of the form: 5

where Vy, Vz are the subvectors of unknown coefficients for the y,z components of the magnetic current distribution respectively and Iy,,I, are the known excitation subvectors. Expressions for the elements of the rectangular admittance submatrices are given in [8]. For the solution of equation (5), most of the computational effort is spent on the evaluation of the elements of the admittance submatrices. The complexity of these computations comes from the double summations in the expression for G12 which are inherently slowly converging. Details on the numerical evaluation of these summations may be found in [9]. By solving the above matrix equation, the unknown field distribution on the aperture of the coplanar waveguide slots is evaluated. Then, the ideal transmission line method, described in [10], is used to determine the scattering parameters of the two port discontinuity. The theoretical analysis is general so that both the CPW mode and the coupled slot mode may be considered. However, for the applications discussed here only the CPW mode is of interest, because it tends to concentrate the fields around the slot apertures as desired. 6

2.2 Convergence Properties of Scattering Parameters In the expressions for the Green's functions given in the Appendix, the summations over m and n are theoretically infinite. However, for the numerical solution of the matrix equation, these summations are truncated to MSTOP and NSTOP which are chosen so that covergence of.the S-parameters is assured. Fig.3 shows the convergence behaviour of the scattering coefficients with respect to each one of the above parameters. It can be seen that the S-parameters are more sensitive to the choice of NSTOP than MSTOP. All of the theoretical results presented herein were obtained using NSTOP = 700 and MSTOP = 300 with a cavity length of twice the guide wavelength. Another critical parameter in the convergence of the results is the number of subsections considered on the slot apertures. Since the slots are assumed to be fairly thin. only a longitudinal magnetic current in the slot apertures away from the discontinuity was assumed. Furthermore, it has been found that the transverse current in the main line around the discontinuity has a negligible effect on the current distributior0 and on the scattering parameters. Extensive numerical experiments have shown that the mesh which provided the best convergence has a subsection length Az (for t Il current in the main line) equal to g/2 approximately. This mesh is shown in Fig I The program was written in a general form such that different subsection lenlt!, 7

may be chosen for the y and z components of the magnetic current. The CPU time depends mainly on the number of basis functions. All of the symmetries of the physical structure were taken into consideration in order to improve the computational efficiency. 8

3 EXPERIMENT The circuits were fabricated using gold electroplating to build up the ground planes and center strip conductor, and photoresist mask to define the slots. The plated gold thickness was 2.8 Jrm placed onto a polished alumina substrate (Er = 9.9). The alumina substrate was placed on a 125 mil thick 5880 RT/Duroid (,- = 2.2) substrate with copper cladding on one side to serve as the bottom ground plane. The RF measurements were performed on an HP 8510 ANA. A probe station with Design Techniques DC- 26.5 GHz probes was used for providing the RF connections to the circuits. A two tier calibration was used. The first tier calibration was an open, short, load 3.5 mm coaxial calibration to the probe head/coax reference plane shown in Fig.5. The second tier was an LRL calibration which provided a calibrated system to the circuit reference planes. The LRL calibration standards shown in Fig.6 were fabricated on the substrate along with the circuits to eliminate errors caused by fabrication tolerances. 9

4 RESULTS AND DISCUSSIONS 4.1 Scattering Parameters Fig.7 shows the scattering parameters for the short circuit CPW stub (SC-CPWV stub) of length 1500 pm. It can be seen that the agreement between the theoretical and experimental results is very good. The differences are due to radiation and conductor losses since an open structure was used in the measurements and the theoretical model did not account for ohmic losses. From these results it can be observed that a resonance occures at f _ 21 GHz. Fig.8 shows a comparison between theoretical and experimantal values of the resonant frequency as a function of the stub length. In addition, superimposed are values of the resonant frequencies for the above CPW stub operating under ideal conditions: no discontinuity effects and zero electromagnetic interactions. This comparison indicates that at lower resonant frequencies (larger stub lengths) the specific stub geometry introduces negligible discontinuity effects and is almost free of parasitic electromagnetic interactions. As a result, the resonant frequencies computed by t tle theoretical method described in this paper agree very well with the ones predict e! for the ideal stub. However, as the frequency increases, the parasitic effects bet,rie1 stronger reducing the resonant length appreciably. From the characteristic beliavn t: 10

of the above stub, it can be concluded that a stop-band filter can be realized by cascading several of these stubs in series. Fig.9 shows the scattering parameters for the open circuit CPW stub (OC-CPW stub) of length 1500 #.m. The resonance is not as sharp as that observed in SCCPW stub. Such structures operate as pass-band filters. The agreement between the theoretical and experimental results is very good. Other examples have been reported in [11] which verify the validity of both the experimental and theoretical results. In the theoretical analysis the CPW stubs were assumed to be inside a cavity while for the derivation of experimental data these structures were measured in opentenvironment. However, a very good agreement between both results was obtained. The loss factor of the measured stub discontinuities has been investigated and has shown a maximum value of -10dB at the stub's resonant frequency. This indicates that CPW stub discontinuities radiate much less as compared to corresponding microstrip stubs which may radiate as much as -3, -4 dB of the incident power [12]. 4.2 Equivalent Models Accurate equivalent circuit models valid over a wide frequency range are required for CAD programs. Two equivalent circuits for the short circuit terminated stub have been presented in the literature. The first model proposed by Houdart [2] was a series inductor, fig.lOb. This equivalent circuit cannot predict the resonant nature of the 11

stub as L approaches A,/4 nor the asymmetry of the discontinuity which is indicated by the difference in the phase of S11 and S22 (Fig.7). Therefore, this model is valid only in the limit as the stub length approaches zero. The model used by Ponchak and Simons [5] shown in Fig.lOc can be used to accurately predict the resonant nature of the stub but it cannot predict the asymmetry of the structure. To accurately model the short circuit terminated stub over the frequency range from 5 GHz to the first stop-band resonance, the model shown in Fig.lOd was developed. Touchstone was used to determine the lumped element values from the measured data with L varying from 250 to 2500 gm. The element values are shown in Fig.ll. It should be noted that the fringing capacitances C1l and Cf2 and the inductors L1 and L3 converge as L approaches zero, as required, since a symmetric model is expected for a simple notch in the center conductor of the CPW. The open circuit terminated stub was modeled by Houdart [2] as a series capacitor as. shown in Fig.12b. This model is also too simple to predict the resonant nature of the stub or its asymmetry and is valid only for stubs with very small lengths Williamns [7] expanded this model to a capacitive II-network and selected a reference plane which removed the element asymmetry. This improved model is difficult to incorporate into CAD programs since the reference planes are not at the plane of t!t. discontinuity as shown in Fig.12. The model by Ponchak and Simons [5] slhow l 12

Fig.12c again cannot predict the element asymmetry. The model shown in Fig.12d was matched to measured data by Touchstone for stub lengths from 250 to 2500 msm. It is valid from 5 GHz to the first pass-band resonance. The element values are shown in Fig.13. In the limit as L approaches zero, the inductances reduce to zero resulting in a capacitive II-network which is expected for a series gap. 4.3 Discussions The errors associated with the measured data are nonrepeatability in probe placement and pressure, and circuit fabrication tolerances. The circuit dimension variations across the 2 in. x 2 in. alumina substrate were negligible as was the variation in the gold thickness. The circuit dimensions varied by, at most, 1%o compared to the designed dimensions. The probe placement was repeatable to 5 ym. The probe pressure created the largest uncertainty in the measurements since it changed the contact between the alumina and the duroid substrates. One further consideration is that the spacing between circuits on the substrate was approximately (S + 2W)/2 which resulted in a finite size ground planes. The equivalent circuit models are valid for the limited amount of discontinuities characterized. Further characterization is required to extend the frequency range of the models beyond the first resonance. In addition, further measurements will permit a refinement of the models in the limits as L approaches zero when the discontinuit i1 13

become a series gap and a notch in the center conductor for the open and short terminated stubs respectively. 14

5 CONCLUSIONS A space domain integral equation method solved by the method of moments (Galerkin's technique) in conjunction with simple transmission line theory was applied to analyze CPW realizations of pass-band and stop-band filters. An experimental setup to measure the S-parameters of those structures has been described and various sources of error were discussed. The agreement between the theoretical results and the experimental data was very good, thus, the validity of both results is verified. CPW discontinuities have shown lower radiation losses than those encountered in microstrip form. Lumped element equivalent ciruits were proposed to model the above filters and the values of the capacitances and inductances were plotted for different stub lengths. 15

6 APPENDIX The components of the dyadic Green's function for the coplanar waveguide problem in the dielectric layer directly above the slot aperture are given by: MSTOP NSTOP 2en 1 ~G(ly~)(f) If_ ae k'-k 2[kyPl + kZQ1] m —0 n=O xi 1 sin(kYy') cos(k,z') sin(ky) cos(k,z) (Al) MSTOP NSTOP 2 kyk = Zoe Z 2k - m-0 n —O xi I sin(k,y') cos(kzz') cos(kyy) sin(k.z) (A2) MSTOP NSTOP 2 kykz Gly(r/r) Ifo aek -L [P - Q1] n= n=O n=0 cos(k,,y') sin(kz') sin(kyy) cos(kzz) (A3) MSTOP NSTOP 2em 1 G(lz)(f If) = mOk ek- zPl +kkQl] mIO nO0 x k cos(ky1y') sin(kz') cos(kyy) sin(k,z) (A4) where |= ( k \wI + j 1jk, ZLSE tan(k,, dl) w, I) k,,Z ZLSE +jwps tan(k,, dl) w(w \ k:1 + jW IZLSM tan(k/L dl) = k J ILSM + j k, tan(k,, dl) 16

en =1 n=O 2 n~O em = 1 m=O 2 m O k1 = E2ulet ky = mzr/a kz = nrle and k., + ky + k2 = k2, where a and e are the width and length of the cavity respectively. In the above expressions GQ) denotes H(1) due to an infinitesimal.i at x = O. ZLSE and ZLSM are the LSE and LSM impedances looking up at the upper interface of the layer directly above the slot apertures (with thickness dl) which can be computed using transmission line theory [12]. This makes it possible to include multi-dielectric layers above and below the CPW. The components of G2 (i.e. in the dielectric layer directly below the slot aperture) are essentially the same as (Al) - (A4) with the following changes: 17

ZLSE zLSE ZLSM, ZLSM 1 2 en7 em p -en,-em kl k2 /l, El' 12, 62' where 7LSE and ZLSM are the LSE and LSM impedances looking down at the lower interface of the layer directly below the slot apertures (with thickness d2). 7 ACKNOWLEDGMENT The theoretical part of this paper was supported by the National Science Foundation under contract ECS-8657951. 18

References [1] C.P. Wen, "Coplanar Waveguide: A Surface Strip Transmission Line for Nonreciprocal Gyromagnetic Device Applications," IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-17, pp.1087-1090, Dec. 1969. [2] M. Houdart, "Coplanar Lines: Application to Broadband Microwave Integrated circuits," Proc. of 6th European Microwave Conference (Rome), pp.49-53. 1976. [3] K.C. Gupta, R. Garge and I.J. Bahl, Microstrip Lines and Slotlines, Dedham. MA: Artech House, 1979. [4] P.L. Fleming, T. Smith, H.E. Carlson, and W.A. Cox, "GaAs SAMP Device for Ku-Band Switching," IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-27, pp.1032-1035, Dec. 1979. [5] G.E. Ponchak and R.N. Simons, UChannelized Coplanar Waveguide PIN-Diode Switches," Proc. of 19th European Microwave Conference (London), pp.489-4!)4. 1989. [6] R.N. Bates, "Design of Microstrip Spur-Line Band-Stop Filters," lEE Joltr l,', Microwaves, Optics and Acoustics, Vol.l, No.6, pp.209-214, Nov. 1977. 19

[7] D.F. Williams and S.E. Schwarz, "Design and Performance of Coplanar WVaveguide Band-Pass Filters," IEEE Trans. on Microwave Theory and Techniques. Vol. MTT-31, pp.558-566, July 1983. [8] N.I. Dib and P.B. Katehi, "Modeling of Shielded CPW Discontinuities U-sing the Space Domain Integral Equation Method (SDIE)," submitted to JEWA, Special Issue on Electromagnetism and Semiconductors. [9] L.P. Dunleavy and P.B. Katehi, "Generalized Method for Analyzing Shielded Thin Microstrip Discontinuities," IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-36, pp.1758-1766, Dec. 1988. [10] P.B. Katehi, "A Generalized Method for the Evaluation of Mutual Coupling in Microstrip Arrays," IEEE Trans. on Antennas and Propagation, Vol. AP-35.3 pp.125-133, Feb. 1987. [11] N.I. Dib, P.B. Katehi, G.E. Ponchak, and R.N. Simons, "Coplanar Waveguide Discontinuities for P-i-n Diode Switches and Filter Applications," in 1990 [EEE MTT-S International Microwave Symposium Digest, May 1990. [12] W.P. Harokopus and P.B. Katehi, "Characterization of Microstrip Discont irl i t it'> on Multilayer Dielectric Substrates Including Radiation Losses," IEEE Tram,.,, Microwave Theory and Techniques, Vol. MTT-37, pp.2058-2065, Dec. 1!'')' 20

List of Figures 1 (a) Schematic diagram of CPW p-i-n diode switched series-stub switch. (b) CPW stop-band filter. (c) CPW pass-band filter......................... 23 2 A coplanar waveguide inside a cavity................... 24 3 Convergence behaviour of the S-parameters for open circuit CPW stub. (Dimensions are as in Fig.9 with L=500 prm and f=16 GHz)..... 25 4 Rooftop functions used to represent the magnetic current in the slots. 26 5 Basic concept of de-embeding.................... 27 6 CPW standards for LRL calibration: (a) Zero length thru line, (b) Delay of length Ld where 0 < Ld < A/2, (c) Open circuit reflection standards......................... 28 7 S-parameters for SC-CPW stub with L=1500 am, ( D1 = 25mil, D2 = 125 mil, re, = 9.9,,2 = 2.2 )............ 29 8 Resonant frequency of SC-CPW stub of different lengths, ( D1 = 21.5 mil, D2 = 125 mil, Erl = 9.9,,2 = 2.2 ) Other dimensions are as in Fig.10...................30 9 S-parameters for OC-CPW stub with L=1500 pm, ( D1 = 25mil, D2 = 125 mil, e1 = 9.9, e,2 = 2.2 )............:31 21

10 Equivalent circuits for SC-CPW stub................... 32 11 Parameters of the lumped element equivalent circuit in Fig.lOd, ( D1 = 21.5 mil, D2 = 125mil,, = 9.9,,r2 = 2.2 )........... 33 12 Equivalent circuits for OC-CPW stub.................. 34 13 Parameters of the lumped element equivalent circuit in Fig.12d, ( D1 = 21.5 mil, D2 = 125 mil, EI, = 9.9, er2 = 2.2 )..........:35 22

~;:;;;;;;;.''::::1:::;.:;:;.-..-..:..-.-.-.-. —. I I................. -- - - - -i' ii - -'' -'i:i:' -': - - - -.~~~~ ~ ~... -..'.-.'.'.'. ".'. -".'.':. "..: ".' -'.;..................... p'.p,...........P.................. ~..................[.....................................'i ~~~~~~~~:.'-.!......-..........-... ~ -' —-'-' ~~ ~ p, p' ~~~~,, ~ Figur e'...... o..e.-...witch. (b) -P stop-band f il ter. (c).P.- ft......................................................... ~r ~...................................................................................................................................................................................................................................................~....................r............................................................................. ~ ~ r,...................................................................... r..........r........................................................... ~ r - - - - - - - - - - - - - - - - - - - -........................................................... iiiiiiiiii.............................................b...................................................................................................(C......................::.............. ~ ~ ~ ~.....................................~ ~ ~.............................~~ ~ ~ ~~ ~ ~ ~............................................................................ d.; ~~~ ~~ ~~~~P e ~~~ ~ a~

Co W S W - -lo TrrITUTUTNI * * JWWWW vWW DI me %u~~eFigure 2: A coplanar waveguide inside a cavity uSI gr 1 /I hihhhi ihiA os143jOPS;P.;S1;pO L L1q'4.;; 2"~"''''" *:'**":$: *'~~ ~~Fgr 2 opar eraveud inid:cviy "" —''""';""'' ~ "~~....~24:

-2.6 ~S11 -2.5 _~ 3.~ [~i NSTOP=500J -3.2 -3.4 -3. l. I ~, * I _ _ ~ 200W. 3. 400. N0.. 700 MSTOP -2.6 D~ -3.*~luMSTOP=5o00o -3.2. -3.4 S12 3W. US. 4W. MI. 6. 7M. 6. NSTOP Figure 3: Convergence behaviour of the S-parameters for open circuit CPW stub. (Dimensions are as in Fig.9 with L=500 pm and f=16 GHz) 25

Yr,Sinusiod wal lrooftop functions - rz / %/ (" /1 Magnetic current < Figure 4: Rooftop functions used to represent the magnetic current in the slots. 26

Calibration standard aI Circuit reference I Coax CPW planes CPW Coax reference ~ ~ i referen plrne plane a a1 a1 CPW I discontinuity a2 2 2 II "C# bb1 | b b1 b2 b b2 Figure 5: Basic concept of de-embeding. 27

CPW probes Circuit ref. plane (a) p p p' P P CPW open circuits Figure 6: CPW standards for LRL calibration: (a) Zero length thru line, (b) Delay of length L where 0 < Ld < A/2, (c) Open circuit reflection standards.

( Z'Z = t'"'6'6 = I"-'l!tu SZ = ma't!tu S = Ta )'turl 009=1 rl ql! qnis Md"OdS iOJ ulamrud..S:L amns!l 00 I? 00 sl; 0'" U, 0't~T000 01 1V -13O03H 1 -71tm 9 WSJt + I0TS) 3 006 *Lz 00.S 000 00 000 llo 000'O,1 i i IX i i l i 1V313j.03HI [IS ] + [TS J9 o 11otI. AOSf=S i SZ= | —I SL

30. Theory x x x x Experiment \ -— ~~~~ Ag/4 — r- 25. I\ 20. x *x 15. x 10. 1.00 1.50 2.00 2.50 3.00 Stub length, L (rrn) Figure 8: Resonant frequency of SC-CPW stub of different lengths, ( D1 = 21.5 mil, D2 = 125 mil, rtl = 9.9, rc2 = 2.2 ) Other dimensions are as in Fig.10 30

'( Z'6 Z'' = Ca'1!W ='6 = = = )'w0 0oSI=1 qm!dA qnms Md)O JOJ oj iaueamd-S:6 aln"!l 000 D0 1'31.LIUOIHL 9131 Y31 131 a 00m z1e-Ulu looT 000ooo 009W St_-tIIIII-?~ —-~-~~-~ll~lllflll~ ~ II II.... I M 000111 I I I I I000__ 1V31132103HI'- l~ ) + IRSt BO t00'9- I I II II 9"11 gZZi=MI TlsL

*qnjs MdO3-S ZOJ "!nono lu*A!nbal:01 ain3!t (p) 3- L;O d d SO ddo O (o) dc d,d,d d d

0.80 _L2 0.60 L3 0.40 C - 0.20 0.30 250. 500. 750. 1000. 1250. 1500. 1750. 2000. 2250. 2500. Stub length, L (microns) 0.20 "'0 10 Cfl 0.00 250. 500. 750. 1000. 1250. 1500. 1750. 2000. 2250. 2500. Stub length, L (microns) Figure 11: Parameters of the lumped element equivalent circuit in Fig.lOd, ( D1 = 21.5 mil, D2 = 125 mil, c.1 = 9.9, er2 = 2.2 ). 33

*qnls MdoerO JOj sj!nmno juapA!nba:ZT azn!tj do,s Zed Z~~I o id o.... —.o — i (D) do / d idi.d,d c, l l d d

0.70 L3 0.60 0. 50 0.40 L - 0.30 0.20L1.210 - 0.00 250. 500. 750. 1000. 1250. 1500. 1750. 2000. 2250. 2500. Stub length, L (microns).2500 -.2000 0f2,3.1500 o.1000' Cfl.0500 ~0000 250. 500. 750. 1000. 1250. 1500. 1750. 2000. 2250. 2500, Stub length, L (microns) Figure 13: Parameters of the lumped element equivalent circuit in Fig.12d, ( D1 = 21.5 mil, D2 = 125 mil, c~t = 9.9,,r2 = 2.2 ). 3 9015 02527 7842 35